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Fundamental experimental study of the operation of gunn effect diodes in resonant circuits Gergis, Isdoris Sobhi 1968

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FUNDAMENTAL EXPERIMENTAL STUDY OP THE OPERATION OF GUNN EFFECT DIODES IN RESONANT CIRCUITS by ISORIS SOBHI GERGIS . B.Sc,  U n i v e r s i t y o f A i n Shams, 1964 C a i r o , U.A.R.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OP  MASTER OF APPLIED SCIENCE  i n the Department o f Electrical  Engineering  We accept t h i s t h e s i s as conforming required  standard  Research S u p e r v i s o r s Members o f Committee  Head, of Department Members of the Department of E l e c t r i c a l  Engineering  THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1968  t o the  In presenting this thesis  in p a r t i a l fulfilment of the requirements  for an advanced degree at the University of B r i t i s h Columbia, I agree  that the Library shall make it freely. available  Study.  thesis  I further agree that permission for extensive  copying of  this  for scholarly purposes may be granted by the Head of my  Department or by h.ils representatives.  or pub 1ication of this thesis  Department of  &IZc-t^Cjd  The University of B r i t i s h Columbia Vancouver 8 , Canada A,A  f  .  ar  ;  M?  It is understood that  copying  for financial gain shall not be allowed  without my written permission.  pate  for reference and  ABSTRACT T h i s work i s concerned with a fundamental e x p e r i m e n t a l study of the o p e r a t i o n of Gunn e f f e c t d i o d e s i n microwave c a v i t i e s . Diodes w i t h d i f f e r e n t parameters were f a b r i c a t e d s i n g l e GaAs c r y s t a l s .  The d i o d e s were t e s t e d i n r e s i s t i v e  from circuits  f o r measuring some of t h e i r c h a r a c t e r i s t i c s and f o r a s s e s s i n g t h e i r performance.  The e f f e c t of the ohmic c o n t a c t s on the  of the d i o d e s was demonstrated.  performance  D e t e r m i n a t i o n of the c a r r i e r  con-  c e n t r a t i o n and m o b i l i t y f o r the GaAs c r y s t a l was made u s i n g the H a l l e f f e c t and c o n d u c t i v i t y measurements. Microwave c o a x i a l c a v i t i e s were designed and b u i l t f o r the o p e r a t i o n of the d i o d e s .  The c a v i t i e s had wide t u n i n g range  and  p r o v i d e d v a r i a b l e impedance at the diode by changing the c o u p l i n g t o the output l i n e . The r e s u l t s o b t a i n e d showed that the upper l i m i t of the f r e q u e n c y of o s c i l l a t i o n v a r i e d from about 1.8 12 product of the diode v a r i e d from 0.4  t o 3x10-  f  t o 2.5 f as the n l o o . -2  cm  .  At f r e q u e n c i e s  s m a l l e r than the t r a n s i t t i m e . f r e q u e n c y the o p e r a t i o n was  i n two  modes; one was the d e l a y e d domain mode a t l o w e r v a l u e s of b i a s . o t h e r was  The  observed at h i g h e r b i a s and was c o n s i d e r e d t o be a m u l t i p l e  domain mode i n which the domain reaches the anode b e f o r e the v o l t a g e swings below t h r e s h o l d , and a new b e f o r e a new  cycle begins.  domain forms and then g e t s quenched  For the delayed domain mode, a t h e o r y  has been developed which was i n f a i r agreement w i t h the e x p e r i m e n t a l results.  I t was found t h a t the range of b i a s i n which the diode  c o u l d o s c i l l a t e c o h e r e n t l y decreased as both the frequency and the  n l p r o d u c t were  decreased.  The r e s u l t s o b t a i n e d i n t h e LSA mode showed t h a t t h e lower l i m i t  of o s c i l l a t i o n  d e c r e a s e d , but i n c r e a s e d  d e c r e a s e d as t h e c a r r i e r  when t h e b i a s  iii  increased.  concentrati  TABLE.OF CONTENTS  _  P  LIST OF SYMBOLS  a  g  e  iv  LIST OF ILLUSTRATIONS  viii  LIST OF TABLES  i  x  ACKNOWLEDGEMENT  xiii  1.  1  INTRODUCTION 1.1  The Gunn E f f e c t  1.2  Gunn Diode f o r Microwave G e n e r a t i o n  4  The Scope of the P r e s e n t Work  7  •1.3 2.  1  THE FABRICATION TECHNOLOGY OF GUNN DIODES  8  2.1  8  2.2  3.  '.  Diode Parameters  ..'  2.1.1  Choice of the M a t e r i a l and the Diode Thickness •  8  2.1.2  The Diode Area  9  F a b r i c a t i o n Process  . ... of S l i c e s  10  2.2.1  Preparation  10  2.2.2  C l e a n i n g of S l i c e s  11  2.2.3  Chemical E t c h i n g  12  2.2.4  Ohmic C o n t a c t s  I  2.2.5  C u t t i n g of the Diodes  16  2.2.6  Diode P a c k a g i n g  16  2  THE MICROWAVE CAVITIES  20  3.1  Introduction  20  3.2  Requirements of the C a v i t y  21  3.3  Design of the C a v i t i e s  21  3.3-1  Cavity Configuration  22  3.3-2  Cavity Construction  22  3.3.3  C a l c u l a t i o n of the Impedance  25  iv -  Page 4.  PRELIMINARY TESTING OF GUNN DIODES 4.1  4.2  5.  .  30  Measurement of the I-V C h a r a c t e r i s t i c s  30  4.1.1  The C i r c u i t Arrangement  31  4.1.2  Observations  31  O s c i l l a t i o n of the Diodes i n R e s i s t i v e C i r c u i t s . . .  34  4.2.1  37  Observations  . 4.3 H a l l E f f e c t and Conductivity Measurements  37  4.4 Determination of the Minimum n l Product  40  4-5  40  Ohmic Contacts....  GUNN DIODES IN MICROWAVE CAVITIES  45  5.1  Theory of Operation of Gunn E f f e c t Diode i n a Resonant C i r c u i t  45 45  5.1.1  Quenched Domain Mode f > f  45  5.1.2  Delayed Domain Mode  5.2  Q  47  Experimental I n v e s t i g a t i o n of the Operating of Gunn E f f e c t Diodes i n Resonant C i r c u i t s  49  5.2.1  Experimental Set-up  50  5.2.2  Experimental Surveying of the Operation of Gunn Diodes at D i f f e r e n t Frequencies ....  52  5.2.3  Power Versus Frequency  54  5.2.4 5.2.5  The E f f e c t of Bias Observation of the Waveform and Harmonic Distortion  57 61 61  5.2.6  T h e o r e t i c a l I n t e r p r e t a t i o n of the R e s u l t s . .  60  5.2.7  The Diode F a i l u r e  65  APPENDIX I  69  Theory of the D i f f e r e n t i a l Negative Conductivity i n Two-Valley Semiconductors  69  1.1  The Band Structure  69  1.2  The Transfer Mechanism  70  v  Page APPENDIX I I A Theory of Gunn Domain Dynamics  74-  II. 1  S t e a d i l y P r o p a g a t i n g Domain  74 .  II. 2  C h a r a c t e r i s t i c s of the Domain  77  APPENDIX I I I  79  Theory of t h e L i m i t e d Space Charge A c c u m u l a t i o n Mode...  79  III. l  Conditions f o r O s c i l l a t i o n  79  III. 2  Power and E f f i c i e n c y  81  III. 3  On t h e Minimum Value of ^  83  REFERENCES  -..  vi  . 85  LIST OP SYMBOLS A  diode area.  B  magnetic f l u x density.  C  shunt capacitance to the diode.  c  propogation v e l o c i t y of the high f i e l d domain.  D  d i f f u s i o n constance.  E E omax  electric f i e l d intensity. maximum e l e c t r i c f i e l d i n t e n s i t y i n s i d e the domain,  E  e l e c t r i c f i e l d i n t e n s i t y outside the domain, and a l s o  P  Q  J  . ac e l e c t r i c f i e l d i n t e n s i t y amplitude, e  e l e c t r o n charge  f  frequency of o s c i l l a t i o n  f-^jf^ffj  resonant frequencies  f  t r a n s i t time frequency  f(e)  density of populated quatum s t a t e s per unit e l e c t r o n energy,  G-  diode  G ,Gn p  growth and decay f a c t o r s of space charge accumulation. )  H  magnetic f i e l d  L ,L  growth and decay exponents of space charge accumulation.  I Is  t o t a l current. s a t u r a t i o n current i n "junction,  I,, th',1v  diode currents at threshold and a f t e r threshold,  J  current density.  k  Boltzmann's constant.  n  k  Q  L,l  conductance.  intensity.  J  free space propogation constant. ' diode thickness.  vii  L p ,Ls  primary and secondary inductances i n the equivalent c i r c u i t of the c a v i t y .  1  equivalent length of the c a v i t y .  M  mutual inductance i n the equivalent c i r c u i t of the cavity.  m  free e l e c t r o n mass.  mf.m*  e f f e c t i v e masses of electrons i n the lower and the upper valleys.  N(e)  density of quantum states per u n i t e l e c t r o n energy.  ' B' o ,  N  N  - D ,  N  n s  c a r r i e r s concentration i n the bulk. c a r r i e r concentration at ."junction, 0  1  c a r r i e r s concentration-diode  thickness  product.  ]°in, P ^. input and output powers. ou  P ^  ac power.  R  H a l l constant.  R ,R  diode low f i e l d r e s i s t a n c e .  T  temperature.  Q  T,T T  Q  G  electrons temperature. l a t t i c e temperature,  t  time.  V  voltage.  v~b'V"th'V"o b i a s , threshold, and ac voltages across the diode. v,v  average d r i f t v e l o c i t y of electrons.  V  average d r i f t v e l o c i t y outside domain,  Q  w  thickness of the H a l l sample,  x  distance  Y  admittance  viii  impedance. c h a r a c t e r i s t i c impedance. r a t i o between d e n s i t y  of s t a t e s i n the two v a l l i e s .  a constant. average e l e c t r o n s dielectric  energy,  constant,  a constant, angle. carriers mobility. e l e c t r o n s ' m o b i l i t y i n the l o w e r and upper v a l l i e s . c o n d u c t i v i t y , and a l s o charge  density.  time. energy r e l a x a t i o n time c o n s t a n t of hot e l e c t r o n s , energy d i f f e r e n c e between the c o n d u c t i o n band edge Fermi l e v e l , barrier electron  height, affinity,  angular frequency.  ix  LIST OF ILLUSTRATIONS  Page  F i g . 1.1  v-E C h a r a c t e r i s t i c s of Bulk GaAs (McCumber and Chynoweth(9))  F i g . 2.1  The P o l i s h i n g Holder  11  F i g . 2.2  The Diode Package  17  F i g . 2.3  Assembled Diode Package  19  F i g . 2.4  Gunn Diode Mounted on the Diode Package Brass Holder  '  •  F i g . 3-1  Schematic Diagram of the Microwave C a v i t i e s . . .  F i g . 3.2  E f f e c t of C  3  19 23  on the Antimode Resonance-  Frequencies-^  24  F i g . 3.3  The Construction of Cavity No. 1  26  F i g . 3.4  Equivalent C i r c u i t of Cavity No. 1  27  F i g . 3.5  The Coupling Loop  28  Fig. 4.1  C i r c u i t Arrangement f o r Measuring I-V Characteristics I-V C h a r a c t e r i s t i c s of Diodes a) E-112, b) K-103. I-V C h a r a c t e r i s t i c s of Diode E-98 a)One Bias Direction,b) Reversed, and Diode E-112, c) One Bias D i r e c t i o n , d) Reversed Schematic Diagram of the C i r c u i t used i n Operating the Diode i n a R e s i s t i v e C i r c u i t . . . .  F i g . 4.2 F i g . 4.3 Fig. 4.4 Fig. 4.5  •  32 33 33 34  Spectrum of the Current i n Diodes Mounted i n R e s i s t i v e C i r c u i t s a) Diode K-101, b) Diode K-107  36  Fig. 4.6  Spectrum of the Output from Cavity Mounted Diodes a) Diode K-101, b) Diode K-107  36  F i g . 4.7  V a r i a t i o n of the C a r r i e r Concentration and M o b i l i t y with Temperature.'  39  Fig. 4.8  Energy Level Diagram f o r Direct Contact between Metal and n-Type Semiconductor  42.  F i g . 4-9  Contact with Inversion Layer  42  x  Page P i g . 5-1  Time Dependent I-V C h a r a c t e r i s t i c s  48  P i g . 5.2  O p e r a t i o n i n t h e Delayed Domain Mode...  48  F i g . 5.3  E x p e r i m e n t a l Set-up f o r T e s t i n g Diodes i n .51  Resonant C a v i t y F i g . 5.4  Frequency Ranges of O s c i l l a t i o n  52  F i g . 5.5  Output v e r s u s Frequency  55  F i g . 5-6  E f f i c i e n c y v e r s u s Frequency  56  F i g . 5.7  Output v e r s u s B i a s  58  F i g . 5.8  Lower L i m i t of Frequency of O s c i l l a t i o n i n The LSA Mode v e r s u s Bias...Waveforms of t h e Output of Diode K-107 a t D i f f e r e n t F r e q u e n c i e s a) 1GHz,. b) 1.6HGz, c) 2.6 GHz  Fig. 5.9  59 60  'Fig. 5.10 D i s t o r t e d Waveform of Output of Diode E-112...  60  F i g . 5.11 Maximum Power E f f i c i e n c y . (CopelandUV))  62  versus-Frequency  F i g . 5.12 V a r i a t i o n of Output w i t h B i a s a t D i f f e r e n t  F i g . 1.1  F r e q u e n c i e s and a t D i f f e r e n t Values of n l  64  Band S t r u c t u r e of GaAs  70  F i g . 1.2  v-E C h a r a c t e r i s t i c s of GaAs (McCumber and Chynoweth) F i g . I I . 1 The E l e c t r i c F i e l d and C a r r i e r C o n c e n t r a t i o n I n s i d e the Domain  76  F i g . I I . 2 The E q u a l Area Rule and the Dynamic Characteristics  77  Fig. I I I . l Fig. III.2  72  The O p e r a t i n g C h a r a c t e r i s t i c s c f C a r r i e r s i n LSA Mode Diodes  80  Maximum E f f i c i e n c y v e r s u s B i a s ( B o t t and Hislum(l4))  84  xi  LIST OF TABLES  •  Table 2.1  S p e c i f i c a t i o n s of the GaAs Materials .  Page 10  Table 4.1  Gunn Diodes and Their C h a r a c t e r i s t i c s  ' 35  xii  ACKNOWLEDGEMENT • The author would l i k e to express h i s a p p r e c i a t i o n t o his supervisors, Dr. M.M.Z. Kharadly and Dr. L. Young, f o r many h e l p f u l suggestions and guidance during the course of t h i s work. G r a t e f u l acknowledgement i s given to the National Research Council f o r the assistance received through Operating Grants A-3344, and A-3392, and a Major Equipment Grant(E 1192) awarded to h i s supervisors, and t o the Defence Research Board of Canada ( f o r Contract ECRDC T79).  •  The author also wishes t o thank the U n i v e r s i t y of B r i t i s h Columbia f o r the Graduate Fellowships awarded during 1966-67, and 1967-68. The author would l i k e t o thank Mr. A. Torrens and Dr. M. S. Tyagi f o r t h e i r help and f o r many valuable d i s c u s s i o n s . The author wants to thank Messrs. C. Chubb, S. Huff, A. Mackenzie, J . Stuber, and E. Voth f o r t h e i r t e c h n i c a l help. The author would l i k e to acknowledge the help of Messrs. R. Olsen, H. Orton, A. Shankowski, K. Suryanaryanan, and L. Wedman f o r proofreading the t h e s i s . F i n a l l y , the author wants to thank Miss A. Hopkins f o r typing the t h e s i s . .  1  1. 1.1  INTRODUCTION  The Gunn E f f e c t A new bulk effeet.phenomenon was discovered by Gunn  i n 1963, while he was working on micrwave noise emission from bulk GaAs biased at high f i e l d i n t e n s i t y .  He found that a r e l -  a t i v e l y short sample would o s c i l l a t e coherently at microwave frequencies i f the b i as was i n c r e a sed beyond a c e r t a i n value corresponding to a f i e l d i n t e n s i t y of about 3KV/cm.  The  frequency  of o s c i l l a t i o n was very near to the r e c i p r o c a l of the t r a n s i t time of electrons through the sample. (2) An explanation of the phenomenon was given by Kroemer i n 1965,  when he l i n k e d t h i s discovery with the t h e o r e t i c a l pred-  i c t i o n s of R i d l e y and Watkins (1961) , and of H i l s u m ^ ^  (1962)  on  the p o s s i b i l i t y of bulk d i f f e r e n t i a l negative c o n d u c t i v i t y i n manyv a l l e y semiconductors  due to the t r a n s f e r of electrons at high  f i e l d i n t e n s i t i e s from a lower energy v a l l e y of high c a r r i e r mobi l i t y to a higher energy v a l l e y of low c a r r i e r m o b i l i t y . The phenomenon of current o s c i l l a t i o n was t h e o r e t i c a l l y (5)  predicted by R i d l e y  f o r materials e x h i b i t i n g voltage c o n t r o l l e d  d i f f e r e n t i a l negative c o n d u c t i v i t y . He found that spontaneous i n s t a b i l i t y i n a diode biased i n the negative c o n d u c t i v i t y region w i l l give r i s e to the formation of a high f i e l d domain.  This  domain nucleates at some s i t e of doping i r r e g u l a r i t y , grows and consumes more voltage u n t i l the rest of the diode becomes biased under threshold.  At the same time the domain d r i f t s with the car-  r i e r stream towards the anode where i t eventually disappears, and the mechanism repeats i t s e l f p e r i o d i c a l l y .  The current i n the  2 external c i r c u i t f l u c t u a t e s with a frequency equal to the r e c i p r o c a l of the t r a n s i t time taken by the domain to traverse the i n t e r electrode spacing. Experimental v e r i f i c a t i o n of Kroemer's theory was c a r r i e d out by several w o r k e r s ^ ' ^ ' ^ , among them Gunn^^, who v e r i f i e d the existence of high f i e l d domains t r a v e r s i n g the diode from the cathode to the anode, using a c a p a c i t i v e probe and sampling technique. Kroemer also proposed a lower l i m i t to the product of the length and c a r r i e r concentration " n l " below which no Gunn o s c i l l a t i o n would occur i f the sample was mounted i n a r e s i s t i v e c i r c u i t . He based h i s p r o p o s i t i o n on the assumption that the domain formation would be i n h i b i t e d i f the diode length was smaller than that of the domain.  This c r i t e r i o n was confirmed by other a u t h o r s ^ ' . Another v e r i f i c a t i o n of the existence of bulk d i f f e r e n -  t i a l negative r e s i s t a n c e was demonstrated by the discovery of bulk microwave a m p l i f i c a t i o n by Thim e t - a l . ^ ^ . 12-2 with an n l product l e s s than 10  cm  They found that diodes  , when mounted at the end of  a c o a x i a l l i n e , would act as a r e f l e c t i o n type a m p l i f i e r with a u n i form gain at low frequencies and a peak near the frequency corresponding to the t r a n s i t time of electrons through the sample. Theoretical i n v e s t i g a t i o n s of both current o s c i l l a t i o n and microwave a m p l i f i c a t i o n — a s s u m i n g the mechanism responsible to be that proposed by R i d l e y , Watkins, and Hilsum—were out by many workers.  carried  One of the most important treatments of these  phenomena was by McCumber and C h y n o w e t h ^ . They used a quasi-thermodynamic approach, i n that they considered the e l e c t r o n d i s t r i b u t i o n among the two v a l l e y s of the conduction band to obey a non-degenerate Fermi d i s t r i b u t i o n .  The temperature of the electron was  3 assumed to e x i s t and to be r e l a t e d t o the e l e c t r i c f i e l d through the energy transport equation,  It <*>  I  3X  T  T  2  T  + Ev  1.1  where e i s the average e l e c t r o n energy, E i s the e l e c t r i c f i e l d , v i s the c a r r i e r d r i f t v e l o c i t y , T i s the temperature of the e l e c trons which i s d i f f e r e n t from the l a t t i c e temperature T , and Trp i s the thermal r e l a x a t i o n time constant of the electrons cooled by the l a t t i c e .  In the steady state case equation 1.1 reduces to , T -T Ev = 2 2 ZT o  1.2  The r e l a t i o n between the average d r i f t v e l o c i t y of electrons and the e l e c t r i c f i e l d under the steady state could be found by s o l v i n g equation 1.2 with other equations r e l a t i n g the e l e c t r o n d i s t r i b u t i o n among the two v a l l e y s with t h e i r temperature.  (For more  d e t a i l s see Appendix I ) . The s o l u t i o n f o r the v-E c h a r a c t e r i s t i c of bulk GaAs was found to have a region of d i f f e r e n t i a l negative c o n d u c t i v i t y at e l e c t r i c f i e l d s higher than 3-2 KV/cm. ( F i g . l . l ' XW'cm/s  1  1  20 30 ELECTRIC FIELD  F i g . 1.1  1  1  40 50 (KV/cm)  v-E C h a r a c t e r i s t i c s of B u l k GaAs (McCumber and Chynoweth^9).  4 One can always formulate the problem of domain dynamics using the complete energy transport equation, with other equations of c a r r i e r t r a n s p o r t , Poisson's E q u a t i o n e t c .  However, apart  from the d i f f i c u l t computational work needed, one can get b e t t e r i n s i g h t i n t o the problem when s i m p l i f y i n g i t by n e g l e c t i n g the energy transport term and the c a r r i e r s p e c i f i c heat.  This m o d i f i -  c a t i o n does not introduce much e r r o r f o r c a r r i e r concentration 18 smaller than 10  -3 cm.  case the temperature  , and frequencies l e s s than 100 GHz.  In t h i s  of the c a r r i e r s and consequently' the d r i f t  v e l o c i t y and the d i f f u s i o n constant are instantaneous functions of the e l e c t r i c f i e l d .  In t h i s case we w i l l be d e a l i n g merely  with bulk d i f f e r e n t i a l negative c o n d u c t i v i t y . This kind of m o d i f i c a t i o n to the McCumber and Chynoweth treatment shows that no s e r ious errors are introduced f o r frequencies up to 100 1.2  GHz.  Gunn Diodes f o r Microwave Generation One of the most important p o t e n t i a l uses of the Gunn  e f f e c t i s the generation of microwave power.  Gunn e f f e c t  diodes would have the advantage of l i g h t weight, and small s i z e , and would require only a small power supply to operate.  Moreover, they  would have the ruggedness and r e l i a b i l i t y of s o l i d state devices. Bulk negative c o n d u c t i v i t y diodes e x h i b i t e l e c t r i c a l c h a r a c t e r i s t i c s s i g n i f i c a n t l y d i f f e r e n t from other negative r e s i s tance two terminal devices such as tunnel diodes, i n which the a c t i v e region occurs at a junction.  Such devices can be made to  o s c i l l a t e i n resonant c i r c u i t s , the frequency of o s c i l l a t i o n being determined mainly by the c i r c u i t parameters. be suppressed by c i r c u i t l o a d i n g .  Also, o s c i l l a t i o n can  On the other hand, a Gunn e f f e c t  diode, whose n l product i s higher than the c r i t i c a l value given by  5 Kroeiner,  o s c i l l a t e s spontaneously i n a low impedance c i r c u i t a t a  f r e q u e n c y equal t o the r e c i p r o c a l  of the t r a n s i t time f r e q u e n c y .  The o s c i l l a t i o n cannot be damped by c i r c u i t l o a d i n g .  In higher  impedance c i r c u i t s , such as-would be used f o r power g e n e r a t i o n , t u n a b i l i t y w i t h o u t much power v a r i a t i o n  c o u l d be a c h i e v e d over  about 60% of the f r e e r u n n i n g f r e q u e n c y .  For diodes w i t h n l  product s m a l l e r than the c r i t i c a l v a l u e , the diode w i l l not o s c i l l a t e i f mounted i n a r e s i s t i v e c i r c u i t , a l t h o u g h i t may a m p l i f y . I t can o s c i l l a t e i f mounted i n a resonant c i r c u i t w i t h s u f f i c i e n t feedback t o m a i n t a i n o s c i l l a t i o n .  The t u n a b i l i t y i n t h i s  w i l l be l i m i t e d around a f r e q u e n c y n e a r l y equal t o the  case  reciprocal  of the' t r a n s i t time of the e l e c t r o n s through the d i o d e . In the Gunn domain mode, t u n a b i l i t y i s a c h i e v e d through two mechanisms.  One i s the e x t i n c t i o n  the a n o d e T h i s  of the domain b e f o r e r e a c h i n g  i s f o r the case of f r e q u e n c i e s h i g h e r than the  t r a n s i t time f r e q u e n c y .  The second mechanism i s the d e l a y i n g of the  f o r m a t i o n of the domain f o r a p a r t of the r . f . c y c l e , f r e q u e n c i e s l o w e r than the t r a n s i t time f r e q u e n c y .  i n the case of Both t h e  extinc-  t i o n and d e l a y i n g a r e caused by the drop of the v o l t a g e under t h r e s h o l d d u r i n g a p a r t of the r . f . c y c l e .  A h i g h v a l u e of the impedance  at the diode t e r m i n a l s i s needed t o a c h i e v e a l a r g e v o l t a g e swing. As the f r e q u e n c y i n c r e a s e s the e x t i n c t i o n p a r t of the r . f . c y c l e  p e r i o d becomes a  larger  and c o n s e q u e n t l y the e f f i c i e n c y drops as the  f r e q u e n c y goes f a r from the t r a n s i t time f r e q u e n c y , u n t i l we r e a c h a f r e q u e n c y where no power c o u l d be o b t a i n e d from the diode operat i n g on t h i s mode.  The l o w e r l i m i t of the frequency of o s c i l l a t i o n  i s o b v i o u s l y h a l f the t r a n s i t time frequency as the domain i s del a y e d f o r h a l f of the r . f . c y c l e threshold^ ). 1 5  and the b i a s i s j u s t over the  6  A bulk d i f f e r e n t i a l negative c o n d u c t i v i t y could be u t i l i z e d more e f f i c i e n t l y i n generating r.f.power and with more f l e x i b i l i t y i f negative c o n d u c t i v i t y could be maintained w h i l s t avoiding the formation of high f i e l d domains.  This was r e a l i z e d  i n what i s known as the l i m i t e d space charge accumulation mode (LSA mode) which was discovered t h e o r e t i c a l l y and v e r i f i e d experimentally by Copeland^^'" "^^ . 1  In order to achieve t h i s mode, the  diode should operate at a frequency high enough so that no appreci a b l e space charge accumulation can grow.  At the same time, the  e x t i n c t i o n of the space charge accumulation i n each cycle by the voltage f a l l i n g below the threshold, i s necessary f o r no space charge to b u i l d up through successive cycles.  Some of the advan-  tages of t h i s mode are: i)  The e f f i c i e n c y of operation can approach the i d e a l value to be obtained from a p a r t i c u l a r type of bulk d i f f e r e n t i a l nega t i v e r e s i s t a n c e , since most of the c a r r i e r s can operate (23) under the optimal condition of dc to ac conversion  ii)  .  More f l e x i b i l i t y i n tuning, since the frequency i s mainly determined by the c i r c u i t .  iii)  The l i m i t a t i o n on the thickness i s much l e s s r e s t r i c t i v e as with operation i n the Gunn mode, since no t r a n s i t time cond i t i o n i s imposed.  However, the inhomogeneity of the m a t e r i a l  and the l i a b i l i t y of avalanche c a r r i e r m u l t i p l i c a t i o n occurring imposes an upper l i m i t to the thickness. Inhomogeneity  will  reduce the e f f i c i e n c y a s the f i e l d i n s i d e the diode w i l l be non-uniform and l a r g e r space charge accumulation w i l l occur. Avalanche c a r r i e r m u l t i p l i c a t i o n might occur i n the high f i e l d  7  domain  during the t r a n s i t i o n from the Gunn domain mode  to LSA mode on the onset of the o s c i l l a t i o n .  The peak e l e c -  t r i c f i e l d i n the domain increases almost l i n e a r l y with the thickness. iv)  The frequency-impedance  product of LSA diodes i s t y p i c a l l y  much l a r g e r than that of t r a n s i t time devices such as Gunn (28) or IMPATT diodes  . A device with a l a r g e r power-impedance-  can be matched to the c i r c u i t more e a s i l y .  Thus, more power  at higher frequencies can be obtained. I. 3  Scope of the Present Work This work c o n s i s t s of two parts.  One i s concerned with  the f a b r i c a t i o n technique of Gunn diodes from s i n g l e c r y s t a l GaAs material.  The second i s a fundamental, study of the operation of  Gunn e f f e c t diodes i n microwave c a v i t i e s i n the 1 GHz range. In Chapter 2 the f a b r i c a t i o n technique i s presented.  Chapter 3 i s  concerned with the design and construction of the microwave cavities.  Chapter 4 deals with preliminary t e s t i n g of diodes i n r e s -  i s t i v e c i r c u i t s and measurements on the GaAs m a t e r i a l .  Chapter 5  i s concerned with the theory of operation and a fundamental experimental study of the operation of the diode i n microwave c a v i t i e s . . The conclusions drawn from the work are given i n Chapter 6.  A brief  presentation of the theory of the negative d i f f e r e n t i a l c o n d u c t i v i t y i n many v a l l e y s semiconductors i s given i n Appendix I. In Appendix I I , a theory of Gunn domain dynamics i s given, and i n Appendix I I I , a theory of LSA mode i s presented.  8 2.  THE FABRICATION TECHNOLOGY OF GUNN DIODES The development of the technique of f a b r i c a t i n g diodes  with c o n t r o l l a b l e c h a r a c t e r i s t i c s was one of the main targets i n t h i s work.  Since diodes might be destroyed during the t e s t s , i t  was necessary to obtain a high degree of r e p r o d u c i b i l i t y .  Apart  from e f f e c t s due to inhomogeneity i n the bulk m a t e r i a l , the technique used provided the desired diode c h a r a c t e r i s t i c s with reasonable tolerance. Diodes from the same s l i c e had r e s i s t a n c e w i t h i n 20% of the desired value. 2.1 Diode Parameters I t was intended to b u i l d diodes with an n l product i n 12 -2 the range 0.5-5.0x10 cm . This range i s the most s u i t a b l e f o r the generation of microwave power when operating i n the tunable (17) domain mode The choice of the other diode parameters—namely  thick-  ness and area—was governed by several considerations, some based on p r a c t i c a l convenience; others n e c e s s i t a t e d by the c a p a b i l i t i e s of the devices used f o r t e s t i n g the diodes. 2.1.1 Choice of the M a t e r i a l and the Diode Thickness The diode thickness was chosen to be about 100 |i f o r the f o l l o w i n g reasons:. 1.  I t was possible to reduce the s l i c e to t h i s thickness by mecha n i c a l abrasion, and to handle i t during the f a b r i c a t i o n processes without much d i f f i c u l t y .  2.  The free running frequency f o r t h i s thickness was 1 GHz which  9 was s u i t a b l e f o r operation i n a c o a x i a l c a v i t y with reasonable size. The choice of the m a t e r i a l was made according to the c a r r i e r concentration and the highest c a r r i e r m o b i l i t y . of GaAs were purchased.*  Two types  The s p e c i f i e d r e s i s t i v i t y , c a r r i e r con-  c e n t r a t i o n and m o b i l i t y of these two materials are tabulated i n Table 2.1. The f i r s t type seemed t o be reasonably homogeneous'; the estimated r e s i s t i v i t y varied i n the range 3-0 - 5.0 ohm.cm. The second type was l e s s homogeneous, and the estimated r e s i s t i v i t y v a r i e d from 3 - 30 ohm.cm.  However, i t was possible to f i n d some  s l i c e s w i t h s u f f i c i e n t homogeneity to b u i l d several diodes with about the same c h a r a c t e r i s t i c s . 2.1.2  The Diode. Area The area of the diode was chosen under the f o l l o w i n g  considerations: 1.  The impedance of the diode could be matched e a s i l y to the coax i a l c a v i t y whose l i n e c h a r a c t e r i s t i c impedance was about 50 ohms  with loaded Q-factor greater than 20. This gave an r . f .  r e s i s t a n c e of 1000 ohms  corresponding to a low f i e l d r e s i s -  tance greater than 50 ohms 2.  For a bias of 100 v o l t s , a minimum  diode r e s i s t a n c e of 33 ohm  was required.  I t was d i f f i c u l t to cut and handle diodes with areas smaller than 150 x 150 p . 2  *  ).  The power handling capacity of the pulse a m p l i f i e r used was 100 v o l t s at 3 amperes.  3.  (see Copeland  Monsanto Chemical Co., U.S.A.  10 Diodes with areas from 200 x 200  2  to 400 x 400 u  2  These had resistances ranging from 30 - 200  w ere f a b r i c a t e d . ohms  LI  f o r m a t e r i a l E and from 30 - 1000 ohms  f o r m a t e r i a l K.  Direction of cut  R e s i,s t i^v i t y - M o b i l i t y C a r r i e r ohm.cm. OT . volt/seccorcenttatiai  , Ingot Number  J  H  2  m  cmr3xI0 4 1  E  G - 1299  K  G - 1300(170*/1"79) Table 2 . 1  2.2  •3.6  2  .  3  6550 7300  111  3-0 .0.42.2  S p e c i f i c a t i o n s ofthe GaAs M a t e r i a l s  HI •  F a b r i c a t i o n Process The GaAs materials were supplied i n the form of wafers of  0.4 mm. t h i c k .  S l i c e s with areas s u f f i c i e n t to produce 10-20  diodes were used.  The s l i c e was f i r s t reduced to the desired t h i c k -  ness by mechanical abrasion, then polished mechanically, followed by chemical etching t o remove surface damage. . Ohmic contacts were e i t h e r applied to the whole s l i c e ,  or the s l i c e was cut i n t o square  dices and contacts then applied to each i n d i v i d u a l d i c e .  The diodes  were mounted i n diode packages. 2.2.1  Preparation of S l i c e s An aluminum holder with a f l a t depression,(Figure 2.1)  was used to hold the s l i c e f o r mechanical abrasion.  The depression  was surrounded by a number of concentric graded r i n g s to monitor the s l i c e thickness during abrasion.  The s l i c e was cemented to  the holder with wax, and polymerized coldmourit p l a s t i c was used to hold i t . f i r m l y i n place. The s l i c e was abraded to reduce i t s thickness using  11 graded emery papers.  The thickness  at t h i s 'stage was 50 u l a r g e r  than the desired thickness. P o l i s h i n g was then done using a p o l i s h i n g wheel with 1 \x alumina powder and water.  The polished  surface, when examined under the microscope, was found quite smooth and void of scratches.  The p l a s t i c and the wax were d i s -  solved i n acetone and t r i c h l o r o e t h y l e n e , r e s p e c t i v e l y . was then turned over and the other side was polished.  The s l i c e The t h i c k -  ness a t t h i s stage was about 10 \x greater than the desired  thick-  ness t o make allowance f o r etching and contact d i f f u s i o n .  PLASTIC  Ga As  GRADED  RINGS  P i g . 2.1 The P o l i s h i n g Holder 2.2.2  Cleaning of the S l i c e A f t e r p o l i s h i n g the specimen, i t was cleaned i n acetone  and t r i c h l o r o e t h y l e n e , d r i e d , washed c a r e f u l l y i n a detergent s o l u t i o n , and f i n a l l y rinsed i n d i s t i l l e d water and dried. The specimen was cleaned again i n boiled acetone, methanol, and dried i n a i r .  12 2.2.3  Chemical Etching The specimen was chemically etched to remove the surface •  damage caused by mechanical e f f e c t s at the contacts.  abrasion, which might cause undesirable  Several etchants were t r i e d , some of them  g i v i n g s a t i s f a c t o r y r e s u l t s , others g i v i n g rough surfaces.  A  b r i e f account of the etches i s given below: a)  Sulphuric-Peroxide Etch This etch has been used by Northern E l e c t r i c Co.  The composition of the etch was by volume 8:1:1 H S0 (96/o), H 0 (30>), and 2  4  2  2  (Canada).  of  H0 2  The specimen was etched f o r one to two minutes at 80°C.  This etch  gave s a t i s f a c t o r y r e s u l t s and was used i n the majority of specimen preparation. b)  Bromine i n Methanol Concentrated  both sides.  solutions  (20-30$) gave rough surfaces on  A d i l u t e d s o l u t i o n (2-5$) was used f o r l i g h t etching  just before evaporating contacts to remove surface c)  3HN0 (70$) + HF(48$) + 5  H0  oxidation.  2  This was used to etch high r e s i s t i v i t y GaAs and gave moderate r e s u l t s when used f o r a short time (20-30 s e c ) . A f t e r etching, the specimen was washed c a r e f u l l y i n d i s t i l l e d water and 2.2.4  dried.  Ohmic Contacts One of the important f a c t o r s governing the q u a l i t y of a  Gunn diode i s the q u a l i t y of the e l e c t r i c a l contact to the semi-  13  conductor.  These contacts should be of low r e s i s t a n c e , non-  i n j e c t i n g , uniform and reproducible. Several types of contacts have been t r i e d on 3 types of GaAs m a t e r i a l having r e s i s t i v i t i e s of 3«6, 23, and 500 ohm.cm. Some were u n s a t i s f a c t o r y and e r r a t i c i n t h e i r behaviour, some were found e x c e p t i o n a l l y good.  An account of these contacts and  a d e t a i l e d d e s c r i p t i o n of the gold-germanium evaporated  contacts  i s given below: a)  Tin contacts Tin spheres of 15 to 20 mils diameter  ( p u r i t y of 69)  were used f o r ohmic contacts by a l l o y i n g them to the clean surface of GaAs.  The a l l o y i n g was done i n  atmosphere.  The spheres were  f i r s t s l i g h t l y f l a t t e n e d , then cleaned i n acetone, etched i n HC1 to remove surface o x i d a t i o n , washed i n water and d r i e d .  A GaAs  dice was l a i d on the f l a t bottom of a graphite j i g , and a t i n sphere was placed on the d i c e .  The j i g was placed on a tantalum s t r i p  heater i n s i d e a small vacuum chamber.  The temperature  was measured by a .thermocouple welded to i t . pumped down to about 50 uHg.  of the heater  The chamber was  gas was then admitted and a steady  stream of the gas was kept f l o w i n g during a l l o y i n g . done at 300 - 450°C f o r a time of 30 - 120 sec.  A l l o y i n g was  The sample was  then turned over and another t i n sphere was placed on the other side and the a l l o y i n g process was  repeated.  Due to poor wetting of t i n on GaAs surface, the contact area was not c o n t r o l l a b l e and the contacts were not always symmetrical.  To improve wetting a f l u x of methyl a l c o h o l , ammonium  Chloride, and water was used.  14 b)  Tin-Nickel^ ^ 1 0  N i c k e l e l e c t r o l e s s p l a t i n g was  used on clean GaAs sur-  face, but the r e s u l t i n g contact was mechanically weak and the n i c k e l separated i n t o f l a k e s during a l l o y i n g . o  To improve the  a 300 A t h i c k evaporated t i n l a y e r was e l e c t r o l e s s process.  contacts,  applied before applying  the  The r e s u l t i n g contacts were then a l l o y e d at  250 - 300°C f o r 30 - 40 sec.  This type of contact was  t r i e d once  s u c c e s s f u l l y on two types of GaAs - the r e s i s t i v i t i e s of which were ' 3.6  and 500 ohm.cm.  In another attempt with the f i r s t  material,  i t gave non-ohmic c h a r a c t e r i s t i c s , presumably because of contamination before the a p p l i c a t i o n of the contacts.  surface  The  elect-  r o l e s s s o l u t i o n used had the f o l l o w i n g composition.(^O) N i c k e l Chloride - N i C l - 6 H 0 2  30 g / l  2  Sodium Hypophosphite - NaH P0 -H 0 2  2  2  •  10 g / l  Ammonium C i t r a t e - (NH^)^HCgH^O^  65 g / l  Ammonium Chloride - NH^Dl  50 g / l  Ammonium Hydroxide - NH^OH was  added u n t i l the s o l u -  t i o n turned from green to blue.  The s o l u t i o n was  kept  at 98°C during the deposition which took about 5 minutes. (21) c)  Gold-Germanium Evaporated Contact This type of contact was  diodes i n t h i s work.  used i n the majority of the  Evaporated gold and germanium were used i n the  e u t e c t i c proportion with a small amount of n i c k e l . helped i n improving wetting during a l l o y i n g .  The n i c k e l  I t s f u n c t i o n was  to  cover the evaporated e u t e c t i c and prevent i t from c o l l a p s i n g i n t o i s l a n d s due to the surface tension.  The f o l l o w i n g are the steps  15 followed to prepare the e u t e c t i c and the contacting process: i)  E u t e c t i c preparation N-type germanium was cleaned i n acetone, etched, washed  i n water and d r i e d .  Gold was cleaned i n acetone and d r i e d .  An  amount of 880 mg. of gold and 120 mg. of germanium were put together i n a quartz c r u c i b l e around which a s p i r a l tungsten heater was wound. The c r u c i b l e was mounted i n a vacuum system which was pumped down to 3 x 10 ^mm.Hg. The c r u c i b l e was heated u n t i l the charge melted, and then allowed to cool i n vacuum.  The r e s u l t i n g e u t e c t i c was  broken i n t o small pieces of about 100 mg. each, using a clean s t e e l cutter. ii)  Evaporation Precautions were always taken to avoid contamination of  the surface of the specimen during or a f t e r evaporation. A tungsten boat was used to evaporate the e u t e c t i c .  The boat.was mounted  i n s i d e the vacuum system and the system was pumped down to about 5 x 10 ^mm.Hg., and the boat was heat a few minutes.  cleaned at white heat f o r  A charge of about 120 mg. of the e u t e c t i c and about  7 mg. of n i c k e l ( p u r i t y 99$) was placed i n the boat.  The GaAs  specimen was mounted at about 3 inches above the source between a b e r y l l i u m copper mask and a mask holder.  The system was then  pumped to about 2 x 10 ^mm.Hg. A l i q u i d nitrogen trap was used t o improve the vacuum. The boat was heated u n t i l the contents of the charge melted together, then heated to white heat u n t i l the m a t e r i a l i n the boat evaporated completely. The system was allowed to cool down and the sample was turned over and another evaporation was c a r r i e d  16 out.  The r e s u l t i n g surface of the contact had a shiny golden  colour. iii)  Alloying A l l o y i n g was done i n the small vacuum chamber used prev-  i o u s l y f o r a l l o y i n g t i n contacts.  The gaseous atmosphere used  during a l l o y i n g was e i t h e r pure  or a mixture of Rv, and  HC1.  The s l i c e was put i n a graphite j i g which was mounted on a tantalum s t r i p heater.  The temperature of the heater was r a i s e d to  about 450°C f o r 30 - 40 sec.  The surface of the contacts  was  observed to turn from golden to a somewhat d u l l yellow colour. 2.2.5  C u t t i n g of the Diode S l i c e s on which ohmic contacts had been applied were  cut i n t o squares of the desired s i z e .  The c u t t i n g was made e i t h e r  by s c r i b i n g and c l e a v i n g , or by using a tungsten wire saw.  The  surface of the s l i c e was scribed i n t o a mesh using a sharp-pointed carborundum s c r i b e r , then cleaved i n t o squares by immersing i n an u l t r a s o n i c wapher.  However, the r e s u l t i n g cleaved sides were not  always smooth or even.  The otner method used f o r c u t t i n g was  by  using a wire saw loaded with 10 p, s i l i c o n carbide powder and o i l . The s l i c e was cemented by wax on a piece of aluminium to hold i t during c u t t i n g . 2.2.6  Diode Packaging A diode package s u i t a b l e f o r mounting on the inner con-  ductor of a c o a x i a l c a v i t y was designed.  I t consists of two  brass  studs threaded to screw to the ends of a short polystyrene tube as shown i n Figure  2.2.  17  F i g . 2.2  The Diode Package  The diode was soldered to one of these studs.  The other  contact was e i t h e r a soldered s i l v e r wire or a t h i n spring s t r i p of b e r y l l i u m copper pressing on the upper contact (see F i g . 2.3). The s o l d e r i n g of diodes with evaporated Au-Ge contacts was done i n a way s i m i l a r to that used f o r a l l o y e d t i n spheres.  The  s o l d e r i n g metal was an a l l o y of t i n (60$) and lead (40$) which melted at about 180°C.  The temperature was r a i s e d just enough  f o r s o l d e r i n g so that the e u t e c t i c contact would not melt with the (32) solder, and t i n , which has a tendency to punch through GaAs would not get i n touch w i t h the bulk m a t e r i a l of the diode.  , A  commercial f l o u r i d e f l u x dissolved i n e t h y l a l c o h o l was used to f a c i l i t a t e the s o l d e r i n g process.  A f t e r s o l d e r i n g two spheres of  the s o l d e r i n g metals to both contacts of the diode, the diode was placed on the top of the brass stud.  Soldering was then performed  18 i n IL, gas i n the vacuum chamber.  Figure 2.4 shows a soldered diode  on the top of the brass stud. The r e s u l t i n g packaged diodes were mechanically rugged and easy to assemble.  Fig.. 2.4  G-unn Diode Mounted on the Diode Package Brass Header  20 3.  THE MICROWAVE CAVITIES  This chapter describes the requirements and the d e s i g n d e t a i l s of the microwave c a v i t i e s that were used i n t h i s work. 3.1  Introduction There are 3 f a c t o r s that determine the operation of a  given Gunn diode as a harmonic o s c i l l a t o r .  They are frequency,  b i a s , and the r . f . impedance at the diode terminals.  At a c e r t a i n  bias and frequency, coherent o s c i l l a t i o n i s p o s s i b l e only f o r impedances i n a c e r t a i n range.  An optimum impedance i s needed f o r max-  imum e f f i c i e n c y of d-c to a-c conversion. (17) Copeland  has shown t h e o r e t i c a l l y that the optimum  impedance of Gunn diode operating on tunable domain mode i s equivalent to a negative r e s i s t a n c e of 30-50 times the low f i e l d r e s i s tance, shunted with a c a p a c i t i v e susceptance 3 times l a r g e r than (22) the negative conductance.  However, recent i n v e s t i g a t i o n s  for  determining the v-E c h a r a c t e r i s t i c s of bulk GaAs have shown that the value of the negative c o n d u c t i v i t y i s about twice that obtained (9)  from the McCumber and Chynoweth  ' model which was used by Copeland.  Thus a smaller value of the equivalent r . f . r e s i s t a n c e should be expected. Two c o a x i a l c a v i t i e s have been designed and b u i l t f o r the operation of Gunn diodes as microwave generators.  The f i r s t c a v i t y  had a l a r g e outside diameter, with a magnetic coupling loop to couple the c a v i t y to the output l i n e . about 500^ to 1900 MHz.  This c a v i t y was tunable from  The second c a v i t y had a smaller outside  diameter and had c a p a c i t i v e probe coupling. about 300 to 1800 MHz.  I t was tunable from  3.2  Requirements of the C a v i t i e s The requirements imposed on the design of these c a v i t i e s  were: i)  T u n a b i l i t y over a. wide range of frequencies with r e l a t i v e mechanical ease.  This was to help survey the.frequency domain  of p o s s i b l e coherent o s c i l l a t i o n .  This requirement was  achieved by using a c o a x i a l c a v i t y working on the fundamental TEM -j wave mode.  The diode was mounted near the end of the  c a v i t y at which the impedance was maximum and corresponding to a p a r a l l e l resonant c i r c u i t . ii)  A v a i l a b i l i t y of v a r i a b l e i n t e r a c t i o n between the diode and the c a v i t y , i . e . , having the impedance at t h e diode terminals v a r i a b l e i n a wide range.  A l s o , a good degree of r e j e c t i o n  of undesired modes was required. i i i ) The loaded Q-factor of the c a v i t y should be high enough f o r low harmonic d i s t o r t i o n and f o r e f f i c i e n t operation.  A value  of 10 - 30 would be reasonable. iv)  High unloaded Q-factor was d e s i r a b l e , although not very c r i t ical.  A usual f i g u r e of the unloaded Q-factor of c o a x i a l cav-  i t i e s i n the 1 GHz range would be 200 - 2000. duce n e g l i g i b l e  This would i n t r o -  losses i f the unloaded Q-factor was of the  order mentioned before. 3.3  Design of the C a v i t i e s The design of the f i r s t c a v i t y w i l l be given i n d e t a i l s  w h i l e the minor d i f f e r e n c e s between the f i r s t and the second cavi t i e s w i l l be discussed l a t e r .  22 3.3-1  Cavity Configuration The operating mode was the fundamental TEM, {-wave length.  Higher order modes other than the TEM's had no a x i a l symmetry, which means that t h e i r i n t e r a c t i o n with the diode was v i r t u a l l y zero.  The diode was mounted along the inner conductor near the  end of the c a v i t y as shown schematically i n Figure 3.1 (a) and (b) . In c a v i t y No. 1 the p o s i t i o n of the diode could be changed i n s i d e the c a v i t y by moving the inner conductor.  This gave more f l e x i b -  i l i t y and a l s o helped i n suppressing the 3/4-wave rno^e. The impedance seen by the diode was that of T-wave length shorted l i n e shunted by a capacitance. i)  This capacitance Cp would comprise:  The p a r a s i t i c capacitance between the two ends of the inner conductor between which the diode was mounted, and a l s o that between the inner conductor and the end p l a t e .  i i ) The package capacitance. Since the diode c h a r a c t e r i s t i c was severely nonlinear, i t was d e s i r a b l e to have the higher order TEM mode unharmonically r e l a t e d to reduce the harmonic content i n the output.  This would  be p o s s i b l e i f the l i n e was loaded by a constant capacitance, thus destroying t h i s harmonic r e l a t i o n s h i p as i l l u s t r a t e d in.Figure  3.2  (a) and (b). The capacitance Cp was estimated to be about 0.3 p.f. which gave a s h i f t i n the resonant frequency of the T-wave mode of about 8$, and a s h i f t of about 30$ i n the 3/4-wave mode. 3.3-2  Cavity Construction The length of the f i r s t c a v i t y was chosen to be 12  corresponding to a minimum resonant frequency of 550 MHZ.  cm.  The  inner diameter of the outer conductor and the diameter of the  23  CAVITY No.1  OUTPUT  CAVITY No.2  P i g . 3.1 Schematic Diagram of the Microwave C a v i t i e s inner conductor were 3-6 and 1.3 cms., r e s p e c t i v e l y , g i v i n g a l i n e c h a r a c t e r i s t i c impedance of 63 ohms. S l i d i n g r . f . contacts were constructed bronze d i s c s 10 m i l s , t h i c k .  from phosphor  These discs were s l o t t e d and bent  around t h e i r peripheries to form s p r i n g f i n g e r s .  One of the two  shorts of the c a v i t y was f i x e d , and consisted of two phosphor bronze discs with spring f i n g e r s contacting the inner conductor. These two discs were f i x e d to a t h i c k brass disc and i s o l a t e d from it  b y t h i n polystyrene  sheets to form an r.f.. by-pass capacitance.  24  F i g . 3.2  E f f e c t of Cp on the Antimode Resonance Frequencies  The bias was applied between the inner and the outer conductors through t h i s short.  The other short was movable and had two d i s c s  with spring f i n g e r s contacting both the inner and the outer conductors.  The mechanical d r i v e to t h i s short was designed i n such  a way that the inner conductor could e i t h e r be moved with t h i s short as one part or i t could be moved r e l a t i v e to i t .  25 The coupling loop was located near the f i x e d short at which the current was maximum.  The loop was connected between the  inner and the outer conductors of an N-type output connector  so  that i t would be p o s s i b l e to use s e v e r a l coupling loops with d i f ferent s i z e s .  The d e t a i l e d construction of the f i r s t c a v i t y i s  shown i n Figure  3.3-  The second c a v i t y had outside and i n s i d e diameters  of  2.1 and 1.0 cm. r e s p e c t i v e l y , and a l i n e c h a r a c t e r i s t i c impedance of 50 ohms.  The diode was mounted between the f i x e d r . f . short and  the inner conductor.  The coupling was achieved by using a capac-  i t i v e probe whose penetration i n t o the c a v i t y could be v a r i e d . The l o c a t i o n of the probe was near the f i x e d short next to the diode at which the e l e c t r i c f i e l d i s maximum as shown i n Figure 3-1-b. 3.3.3  C a l c u l a t i o n of the Impedance The c a v i t y was designed to operate with diodes of equiv-  alent r . f . r e s i s t a n c e s i n the range of 1000 - 8000 ohms. the loop s i z e and the use  Changing  a double stub tuner on the output l i n e  provided a way i n which i t was p o s s i b l e to change the impedance at the diode terminals.  The f o l l o w i n g c a l c u l a t i o n s ware h e l p f u l i n  checking the value of the impedance at some intermediate frequencies with a proposed loop s i z e .  The c a v i t y c h a r a c t e r i s t i c impedance Z. ,  w o u l d have been changed i f w i t h i n p r a c t i c a l loop s i z e the impedance was f a r from the desired value.  A s • w i l l be i l l u s t r a t e d by an  example, i t was found that the c a v i t y used was quite s u i t a b l e f o r the operation of diodes with low f i e l d r e s i s t a n c e s . i n the range 100 - 500 ohms. To estimate the impedance seen by the diode, we w i l l use an equivalent c i r c u i t that represents the c a v i t y as a 4-terminal  DIODE PACKAGE  F i g . 3.3  RF BY PASS  CAPACITOR  The C o n s t r u c t i o n of C a v i t y No. 1.  27  Pig.  3.4  Equivalent C i r c u i t of Cavity No. 1  ' •  Cp i s the shunt capacitance to the diode as explained before. The coupling between the c a v i t y and the output l i n e i s represented by a transformer whose mutual inductance i s defined as  L  _ 1_ voltage induced i n the loon "~ jto current flowing i n the inner conductor (27) i s the s e l f inductance of the loop, and i s given by L  s  =  ud ~~ 2  (In 8d - 2) r  where d i s the diameter of a s e m i c i r c u l a r loop with the same area, r i s the radius of the loop wire.  Lp i s neglected, since in.the  case the l o o p i s open, the impedance at the end of the c a v i t y w i l l 0  be v i r t u a l l y zero.  A c t u a l l y , the e f f e c t i v e length of the c a v i t y i s  modified by the presence of the loop.  However, the estimated  value  of the r e s i s t i v e part of the impedance at the diode terminals i s not s e n s i t i v e to t h i s length.  On the other hand, i f the r e a c t i v e  part i s required to be estimated, a more accurate method should be used. Using the equivalent c i r c u i t of the f i r s t c a v i t y , the  28  IT  i 8 v////////////y//, ,K 9mm  8mm  It mm  I P i g . 3.5  The Coupling Loop  f o l l o w i n g r e s u l t s were obtained at 1 GHz, f o r a coupling loop with dimensions shown i n Pigure 3»5. M  2r  1.66 x 10 ^ Henry o  where A i s the area of the loop, and r  Q  i s the mean radius of the  f l u x l i n k i n g the loop (see Pigure 3-5.) Cp was taken to be 0.6 p.f. The impedance at the diode terminals i s now  Cp i n p a r a l -  l e l with an admittance equal to Y  _! o Z  Z  Z  o s + +  j  V  where Z  = (oM) /(Ro + jwL ) = 1.16 - j l . l , s ^ t = tan k 1 , k i s the free space propagation constant, and Z i s the o eq o ° l i n e c h a r a c t e r i s t i c impedance of the c a v i t y and equal to 63 ohms. As we have Z ^ Z , G  Q  29 Y  _ -  _1 jZ t o  ,  _J2_  + z  2 o  _  1_ jZ t  +  , 1-16 3660  D  . 1.1 3660  At resonance the r a c t i v e part i n Y i s equal to the susceptance wCp, while the r e s i s t i v e part i s equal to 3400 ohms.  A load r e s -  istance of 3400 ohms i s quite s u i t a b l e f o r diodes with low f i e l d r e s i s t a n c e i n the range of 120-600 ohms.  30 4.  PRELIMINARY TESTING OF GUNN DIODES  Two experiments were performed on each diode i n r e s i s t i v e circuits.  The f i r s t was to'measure the I-V c h a r a c t e r i s t i c s using  a s a w t o o t h wave form.  The second was to test the diode f o r f r e e -  running o s c i l l a t i o n under pulsed operation, and to inspect the current through the diode using a spectrum analyzer. These experiments served i n assessing the q u a l i t y of the diodes and i n p r e d i c t i n g t h e i r performance as microwave o s c i l l a t o r s . The important r o l e played by the contacts on the performance of the diodes was  demonstrated.  To perform these experiments, the diode package was mounted along the inner conductor of a short c o a x i a l s e c t i o n .  A  d i s c r e s i s t o r to monitor the current was f i t t e d between the inner and the outer conductors, at about 1 cm. from the diode. end of the coaxial' s e c t i o n was terminated by a 50 ohm  The other  shunt c o a x i a l  pass-through termination to which the bias was applied. Determination of the c a r r i e r concentration and m o b i l i t y of the GaAs c r y s t a l was c a r r i e d out f o r temperatures between -50°C and 80°C by using H a l l e f f e c t and c o n d u c t i v i t y measurements. 4.1  Measurement of the I-V C h a r a c t e r i s t i c s The threshold voltages and the low f i e l d resistances of  the diodes were determined from the measurement of the I-V characteristics.  The c o n d u c t i v i t y and the c a r r i e r concentration of the  bulk m a t e r i a l were then estimated using the two equations o-  =  ^  j  and  f5 =  \x n e  Where R i s the low f i e l d r e s i s t a n c e , A i s the diode area, 1 i s the  31  diode length, and u i s the low f i e l d m o b i l i t y taken to be equal to the manufacturer s p e c i f i e d value. The measurement of the threshold voltage helped i n assessing the q u a l i t y of the contacts.  High r e s i s t a n c e contacts  caused an increase i n the value of the measured threshold voltage as a r e s u l t of the voltage drop at the contacts.  The percentage  current drop a f t e r threshold gave i n d i c a t i o n of the order of the n l product, since i t would be l a r g e f o r l a r g e values of n l products as explained i n Appendix I I . 4.1.1  The C i r c u i t Arrangement The c i r c u i t used i s shown i n Figure 4-1;  s a w t o o t h generator driven by a pulse generator. erator was amperes.  i t comprised a The  s awto o th gen-  capable of d e l i v e r i n g a voltage up to 100 v o l t s at 2 The I-V c h a r a c t e r i s t i c s were displayed on a Tektronix  C.R.O. type 581, with Y - a m p l i f i e r p l u g - i n u n i t type 82.  The  rent s i g n a l was applied to the Y - a m p l i f i e r while the voltage applied to the X - a m p l i f i e r .  The value of the current  curwas  monitoring  resist'or was 1 ohm  f o r diodes with r e s i s t a n c e s smaller than 100  ohms , and 5 ohms  otherwise.  4.1.2  Observations Figure 4.2  (a) and  (b) show the I-V c h a r a c t e r i s t i c s , of two  diodes, E-122, and K-103.•Table I I contains information about the diodes used i n t h i s work. The sub-threshold  Both diodes had evaporated Au-Ge contacts.  c h a r a c t e r i s t i c s were almost s t r a i g h t l i n e s ,  until  the current dropped and began to decrease slowly with the bias.  The  amount of the current drop was about 30$.for the f i r s t diode and about 10$ f o r the second.  This d i f f e r e n c e could be accounted f o r by  32  DISC  HEW LET PACKARD PULSE GENERATOR 214 A  RESISTOR  SAWTOOTH GENERATOR  EXTERNAL  Fig.  TRIGGERING  4-1 C i r c u i t Arrangement f o r Measuring I-V C h a r a c t e r i s t i c s ..  the f a c t that n l product of the f i r s t diode was l a r g e r than that of the second.  The threshold voltage was quite near the expected  value i n d i c a t i n g that the resistances of the ohmic contacts were small i n comparison to the bulk r e s i s t a n c e . Figure 4.3(a), (b), ( c ) , (d) shows the unsymmetricai chara c t e r i s t i c s of two diodes, E-98 and E-112.  The f i r s t diode had t i n -  n i c k e l contacts, the second had evaporated Au-Ge contacts.  On one  bias p o l a r i t y both diodes had threshold voltages quite near to the expected values.  On the other bias p o l a r i t y the f i r s t diode had  a curved prethreshold c h a r a c t e r i s t i c , then a current drop at a bias of 65 v o l t s .  The second diode d i d not show any current drop f o r  bias up to 80 v o l t s ; instead the current increased  monotonically,  although at a lower rate than the other p o l a r i t y . A l s o the current had not reached the threshold value.  Testing t h i s diode i n both  r e s i s t i v e and resonant c i r c u i t s revealed that i t d i d not o s c i l l a t e  33  (a)  (b)  200 ma/Div, X = 10 v / D i v  Y = 200 ma/Div, X = 10 v / D i v  (c) 200 ma/Div, X = l O v / D i v  (d) Y = 2 0 % ma/Div, X = 10 v/Div  Fig. 4.3 I-V C h a r a c t e r i s t i c of Diode E098 a) One B i a s D i r e c t i o n b) Reversed, and Diode E-112 c) One b i a s d i r e c t i o n d) Reversed  34  when biased i n the second d i r e c t i o n up to a bias of 100 v o l t s . An explanation of such behaviour could be that one of the contacts was nonohmic, the forward d i r e c t i o n being that f o r which the semiconductor Section 4-3)•  was negative r e l a t i v e t o the metal (see  I f t h i s nonohmic contact happened to be the anode,  the diode would o s c i l l a t e at the normal threshold.  In the other  d i r e c t i o n , the voltage drop at t h i s c o n t a c t — t h e cathode—would l i m i t the e f f e c t i v e voltage across the bulk m a t e r i a l and the apparent threshold would be high or even inaccessible before breakdown would occur. 4.2  O s c i l l a t i o n of the Diodes i n the R e s i s t i v e C i r c u i t In t h i s experiment  diodes  was measured.  the free running frequency of the  The c i r c u i t arrangement i s shown i n F i g . 4.4.  DISC  PULSE GENERATOR  PULSE AM PL IF IER  RES/STOR  SPECTRUM ANALYSER DIODE  4.4  Schematic Diagram of the C i r c u i t used i n Operating the Diode i n a R e s i s t i v e C i r c u i t  Table 4.1 Diode No.  Type of GaAs Material  Low F i e l d Resistance  Gunn Diodes and Their C h a r a c t e r i s t i c s  Threshold Voltage  2 Area u  Thickness u  Estimated Carrier Concentration i/i _^ x 10^ cm 0  Estimated n l Product , 12 -2 x 10 cm A  E-98  E  80  40  350  X  350  110  1.4"  1.55  E-112  E  60  35  300  X  300  110  1.9  2.1  E-122  E  58  35  275  X  275  110  2.5  2.75  E-125  E  40  150  X  250  110  1.02  1.13  0.455  0.505  E-125 at - i o ° c '  160 360  K-102  K  140  40  300  X  300  110  0.82  0.9  K-103  K  235  42  350  X  350  110  0.32  0.385  K-105  K  130  40  300  X  300  110  0.87  0.96  K-107  K  110  36  250  X  250  110  1.4  1.55  K-124  K  150  36  275  X  275  0.74  0.67  90  36.  Fig. 4 . 6  Spectrum of the Output from Cavity Mounted Diodes a) Diode K-101 b) Diode K-107  37 The pulse a m p l i f i e r used was capable of d e l i v e r i n g 2 p.sec. pulses of 200 v o l t s at 1.5 amperes. times of the pulse were l e s s than 50 n.sec.  The r i s e and f a l l The- spectrum analyser  was Panoramic, type RF-4a. 4.2.1  Observations The free running frequencies of most of the diodes were  between 700 and 850 MHz.  The frequency was found to decrease w i t h  the bias i n agreement w i t h the theory (see Appendix I I ) . Inspection of the current gives an i n d i c a t i o n of the homogeneity of the GaAs and the contacts. Incoherent o s c i l l a t i o n i s expected where m u l t i p l e domain formation occurs at several s i t e s of inhomogeneity.  Such o s c i l l a t i o n would have a spectrum  scattered over a wide range of frequencies rather than concentrated over a frequency range of the order of the r e c i p r o c a l of the pulse width used to bias the diode. A case of incoherent o s c i l l a t i o n was found i n diode K 101. Pigure 4.5.a shows the current spectrum i n the r e s i s t i v e c i r c u i t . The  sweep width was 5 MHz.  A c t u a l l y the spectrum was scattered  over the frequency range from 500 to 750 MHz, with some peaks i n between.  Pigure 4.5.b  diode K 107. 2 MHz apart as sec.  shows the spectrum of the current of the  The spectrum was concentrated and i t s minima•occurred w ould  Figures 4.6.a  be  and 4-6.b  expected f o r the pulse width of 1 LI show the spectrum of the output of  the two diodes K 101 and K 107 when mounted i n a c o a x i a l c a v i t y tuned at about 950 4.3  MHz.  H a l l E f f e c t and Conductivity Measurements H a l l e f f e c t and c o n d u c t i v i t y measurements were performed  38 on the E-GaAs m a t e r i a l (see Table l ) f o r temperatures between -40°C and 80°C.  A rectangular piece of the semiconductor about  2 mm wide, 6 mm long, and 0.45 mm t h i c k was used f o r the H a l l e f f e c t measurements.  The sample was mounted on a brass holder  which had a small heater i n s i d e i t .  The holder was suspended by  a t h i n s t e e l tube of a 1/8 inch diameter i n s i d e a dewar which was cooled from outside by l i q u i d nitrogen.  The temperature of  the holder was adjusted by changing the power supplied to the heater i n s i d e the holder.  The temperature was measured by a ther-  mocouple attached to the holder. The Van der Pauw method was used i n the c o n d u c t i v i t y measurement.  The sample used had a shape s i m i l a r to the Clover (33)  Leaf c o n f i g u r a t i o n u s u a l l y used i n t h i s method. w  mounted i n s i d e  a  y/  The sample was  temperature regulated chamber.  Knowing the H a l l voltage V^ and the current I, the H a l l constant was c a l c u l a t e d as f o l l o w s : R  = w.V .B  4  H  '  1  At the same time we have, R = -— n e  .  4.2  Prom the c o n d u c t i v i t y measurement we have, 6  - u n e  4.3  Using both equations 4.2 and 4-3> t i o n and the m o b i l i t y were obtained. of temperature are shown i n F i g . 4.7•  the c a r r i e r concentra-  P l o t s of n and p, as functions  MOBILITY AND C A R R I R  6£  CONCENTRATION  40  4.4  Determination of the Minimum n l Product A t e s t was performed on the diode K 102 to determine  the minimum value of n l product f o r o s c i l l a t i o n i n r e s i s t i v e circuits.  The c a r r i e r concentration was reduced by c o o l i n g the  diode i n a temperature-regulated pressed COg.  chamber (Statham) cooled by com-  No detectable output was found below -40°C ( s e n s i -  t i v i t y of the spectrum analyser was -95 dbm. .. 1000  MHz) .  i n the range of 500 -  The low f i e l d , r e s i s t a n c e at t h a t c r i t i c a l temperature  was measured by an ohmmeter and found to be about 1300  ohms.  Using  the same method used before to c a l c u l a t e the c a r r i e r concentration, we assumed the low f i e l d m o b i l i t y to be the same as the s p e c i f i e d value at room temperature; was 4.5  a minimum n l product of about 2.0 x  10"^  obtained. Ohmic Contacts The main f u n c t i o n of ohmic contacts used i n semi-conduc-  tor devices i s to act as an i n l e t to the current from m e t a l l i c electrodes to the semiconductor body, without a c t i v e l y a f f e c t i n g any mechanism i n the device,  low r e s i s t a n c e contacts are always  required, n o n - i n j e c t i n g contacts are also p r e f e r r e d .  A low r e s -  istance contact may be defined as one with a r e s i s t a n c e comparable to that of a l a y e r of the semiconductor whose thickness i s small r e l a t i v e to the body of the device.  L i n e a r i t y i n the character-  i s t i c s of the contact i s d e s i r a b l e , although often not e s s e n t i a l . With devices l i k e Gunn diodes, where the current density i s very high  almost equal to the s a t u r a t i o n current density i n the bulk  material- the requirement of the contact becomes more c r i t i c a l . Not only i s low r e s i s t a n c e e s s e n t i a l , but a l s o the contact should be able, to d e l i v e r a high current density w e l l below i t s s a t u r a t i o n  41  limit. D i r e c t contact between a metal and an n-type semiconductor may have one of the two configurations shown i n Figure 4.8 (a) and (b). The semiconductor has i t s bulk Fermi l e v e l at ^ below the conduction band edge.  n  ev  In (a) jz^ i s l a r g e r than the n  contact b a r r i e r height to ^6 ; i n (b) 6 H  s  i s smaller than 6 . ^  N  S  The bar-  r i e r height <h i s u s u a l l y taken to be the d i f f e r e n c e between the s work f u n c t i o n of the metal and the e l e c t r o n a f f i n i t y semiconductor.  X of the  I t depends on the type of the metal and also the  conditions of the semi-conductor surface and the a l l o y i n g process. In case ( a ) , an e l e c t r o n d e p l e t i o n l a y e r i s formed near the boundary i n s i d e the semiconductor. The e l e c t r o n concentration at the boundary n , w i l l be equal t o , n  s  = n  B  -rf )/kT  e  where n-g i s the bulk e l e c t r o n concentration.  The d e p l e t i o n l a y e r  acts as a high r e s i s t a n c e l a y e r at which a voltage drop, comparable to that across the whole bulk, might occur.  At zero bias the num-  ber of electrons crossing the contact from the semiconductor to the  metal i s exactly the same as that crossing the contact from  the  metal to the semiconductor (exchange current). Now i f the semi-  conductor i s biased negative r e l a t i v e to the metal, l a r g e r current from the semiconductor w i l l flow, a c t u a l l y equal to the exchange eV/kT current m u l t i p l i e d by the f a c t o r e  '  , where V i s the p o t e n t i a l  by which the conductor band minimum i n the bulk r i s e s r e l a t i v e to that at the boundary, and i s equal to the e f f e c t i v e bias across the  contact. With the semiconductor biased p o s i t i v e , the current  42 CONDUCTION  BAND  FERMI  VALENCE  BAND  FERMI  METAL  LEVEL  LEVEL  SEMICONDUCTOR  F i g . 4.8 Energy Level.. Diagram f o r D i r e c t Contact between Metal and n-Type Semiconductor.  CONDUCTION  BAND  FERMI TNTRTNSIC  FERMI  METAL INVERSION  LAYER  VALENCE  BAND  F i g . 4.9 Contact with Inversion Layer  LEVEL LINE  43  —eV/kT flowing from the semiconductor decreases by a f a c t o r of e On the other hand the current flowing from the metal to the semiconductor i s s t i l l the same, as the electrons i n the metal have to cross a barrier' of height ^  independent on the b i a s .  Thus  the net current at any bias i s I  - i (e  =  e T  ^  T  - 1),  JL  where I  g  = n§ ( k T / ) :  2 m  2  exp (- ( $  Q  - ^ )/ ) n  k T  Although the r e c t i f i c a t i o n r a t i o i s independent of (6-6), o  t h e • s a t u r a t i o n current and the r e s i s t a n c e (the d i f f e r e n -  Ii  t i a l at the o r i g i n ) are l a r g e l y determined by i t .  The smaller  the value of ( 6 - 6 ) the more the contact approaches the desired ohmic contact  requirements.  I n j e c t i n g contact may  occur due to the formation of an  i n v e r s i o n l a y e r near the.boundary f o r the case where (6^-$ ) i s l a r g e r than about'half the value of the band gap 6 •  The region  between the boundary and the point where the i n t r i n s i c Fermi l i n e crosses the Fermi l e v e l (see Figure 4. 9 ) , w i l l contain more holes than e l e c t r o n s .  With the contact forward biased,, holes w i l l d i f -  fuse i n t o bulk semiconductor f o r quite a distance which might be comparable to the bulk width.  At the same time, l o c a l increase  i n e l e c t r o n concentration w i l l occur to n e u t r a l i z e the holes. the c o n d u c t i v i t y of the bulk may the normal value.  Thus  increase several times l a r g e r than  D r a s t i c reduction i n the e f f i c i e n c y of the Gunn  diode might occur due to the c o n t r i b u t i o n of holes i n the conduct i o n current. Now  i n case (b), the boundary accumulation l a y e r acts as  a pool of electrons supplying conduction, and i t s r e s i s t a n c e i s lower  44 than a l a y e r i n the bulk of comparable width.  Saturation current  w i l l be much higher than i n case (a), a l s o higher than the satura t i o n current i n the bulk. A contact of type (a), might work s a t i s f a c t o r i l y i n Gunn e f f e c t devices i f followed by a h e a v i l y doped region.  In that  case the s a t u r a t i o n current w i l l be much higher, and the contact can supply high current density without c l o s e l y approaching the s a t uration l i m i t .  This h e a v i l y doped region could be formed because  of d i f f u s i o n of dopant from the contact metal i n t o the semiconductor. A c o n f i g u r a t i o n which might be the most s u i t a b l e f o r Gunn and LSA diodes, would be the sandwich s t r u c t u r e . ^ ^ 2  In t h i s  structure, the a c t i v e region i s sandwiched between two h e a v i l y doped l a y e r s to which the ohmic contacts are applied.  I t has the f o l -  lowing advantages: i)  The ohmic contacts could be formed e a s i l y on the h e a v i l y doped l a y e r s ,  at the same time i t can work s a t i s f a c t o r i l y  since the current density w i l l be r e l a t i v e l y small. ii)  Gunn and LSA diodes work more e f f i c i e n t l y i f the contacts (28) cover a l l the diode area.  Thus non-uniformity i n the  contacts w i l l be masked by the existence of the h e a v i l y doped l a y e r s which w i l l act as an i d e a l ohmic contact. i i i ) Since the minority c a r r i e r d i f f u s i o n length i n the h e a v i l y doped region i s quite short, i n j e c t e d holes from the anode contact w i l l not d i f f u s e much i n t o the a c t i v e region.  45 5. 5.1  GUNN DIODES IN MICROWAVE CIRCUITS  Theory of Operation of Gunn Diode i n a Resonant C i r c u i t The operation in- a resonant c a v i t y of a Gunn e f f e c t diode  with n l product l a r g e r than the c r i t i c a l value i s characterized by a large ac voltage amplitude across the diode i n both the LSA mode and the tunable domain modes.  In the latter mode, quenching of  the domain before reaching the anode i s necessary f o r operation at frequencies higher than the t r a n s i t time frequency.  The  quenching  i s accomplished by the drop of the voltage below threshold f o r a part of the r . f c y c l e .  For frequencies lower than the t r a n s i t , time  frequency, the voltage swings under threshold and thus delays the domain formation f o r part of the cycle. (23 24) The theory of the LSA mode has been elaborated a n a l y t i c a l l y assuming small s i g n a l c o n d i t i o n . *  '  However, since i n  the tunable domain- mode such a c o n d i t i o n i s not v a l i d , the a n a l y s i s of t h i s mode i s only a v a i l a b l e using numerical methods.  In the  treatment of a problem numerically one needs to consider a l a r g e number of cases with d i f f e r e n t parameters i n order to get a reason(17) ably complete p i c t u r e of the problem.  Copeland  has considered  a number of cases and t r i e d to draw a general theory of operation of Gunn diodes i n resonant c i r c u i t s .  However, h i s r e s u l t s were  not i n complete agreement with the experimental r e s u l t s obtained i n t h i s work.  A b r i e f , mainly q u a l i t a t i v e d i s c u s s i o n of the theory of  the tunable domain mode i s given. 5.1.2 Quenched Domain Mode, f>f The quenched domain mode i s s i m i l a r to the LSA mode i n *  See Appendix I I I f o r the theory of the LSA mode,  46 t h a t the space charge c o n t r o l c o n d i t i o n should be s a t i s f i e d , t h a t i s , the domain formed i n one c y c l e should be c o m p l e t e l y b e f o r e the next c y c l e b e g i n s .  quenched  T h i s c o n d i t i o n i s not easy t o handle  numerically. Large s i g n a l ac n e g a t i v e r e s i s t a n c e o r i g i n a t e s as f o l l o w s : i)  As the v o l t a g e i n c r e a s e s above t h r e s h o l d the diode  switches  to a low c u r r e n t s t a t e because of the f o r m a t i o n of a h i g h f i e l d domain.ii)  W i t h the domain p r o p a g a t i n g  i n s i d e the diode the c u r r e n t dec-  reases s l o w l y w i t h the v o l t a g e a l t h o u g h a t a low r a t e . i i i ) As the v o l t a g e swings below t h r e s h o l d the diode s w i t c h e s  back  to a h i g h c u r r e n t s t a t e as the domain i s quenched. Thus a net harmonic component of the c u r r e n t i n phase o p p o s i t i o n t o the v o l t a g e i s o b t a i n e d , which i s r e s p o n s i b l e f o r the dc t o ac c o n v e r s i o n .  Although  the o r i g i n of the r . f;,negative  r e s i s t a n c e i s the - d i f f e r e n t i a l n e g a t i v e c o n d u c t i v i t y , i t s v a l u e i s q u i t e dependent upon the domain mechanism and  consequently  upon the  n l p r o d u c t , u n l i k e the LSA mode where t h i s . f a c t o r i s not of much importance.  To get a l a r g e ac component i n the diode c u r r e n t , i t s  wave form should be as s y m m e t r i c a l  as p o s s i b l e , which c o u l d  be (17  obtained by u s i n g diodes w i t h r e l a t i v e l y low n l product. found t h a t the.optimum v a l u e of the n l product As the frequency t r a n s i t time frequency,  Copeland 12 -2  i s about 2-5x10  cm  of o p e r a t i o n d e v i a t e s more from the  more v o l t a g e swing under t h r e s h o l d d u r i n g  a l a r g e r p a r t of the r . f . c y c l e i s needed to quench the domain. the power output drops as the frequency  Thus  increases, u n t i l eventually  the n e g a t i v e r . f . r e s i s t a n c e i s not s u f f i c i e n t t o match the l o s s e s i n the c i r c u i t and coherent o s c i l l a t i o n w i l l not be p o s s i b l e .  47 5.1.3  Delayed Domain Mode Perhaps the best way t o e x p l a i n the o p e r a t i o n of t h i s  mode i s t o use t h e time dependent I-V c h a r a c t e r i s t i c s suggested by (ic)  Gunn  , P i g . 5.1. B e g i n n i n g a t time t-^ a t which t h e domain from  the p r e v i o u s c y c l e has been c o m p l e t e l y  quenched, once t h e v o l t a g e  exceeds t h e t h r e s h o l d v a l u e a t A, domain f o r m a t i o n w i l l occur i n a r e l a t i v e l y s h o r t time and the c u r r e n t drops t o p o i n t B on t h e s t e a d i l y propagating  domain c h a r a c t e r i s t i c s .  the domain propagates through t h e d i o d e .  From time t ^ t o  The o p e r a t i n g p o i n t  will  f o l l o w the l i n e CB up t o p o i n t D a t which t h e domain i s c o m p l e t e l y quenched.  The c u r r e n t then w i l l  jump up t o p o i n t E on t h e p r e -  t h r e s h o l d c h a r a c t e r i s t i c s and f o l l o w i t f o r t h e r e s t of the c y c l e up t o time t ^ . The optimum o p e r a t i o n a t t h i s mode would be such t h a t t h e domain a r r i v e s a t t h e anode j u s t when t h e v o l t a g e reaches t h e t h r e s h o l d v a l u e , i . e . , p o i n t D w i l l c o i n c i d e w i t h B.  S i m p l i f y i n g the  c h a r a c t e r i s t i c s t o s o l v e f o r t h e e q u i v a l e n t r . f i m p e d a n c e — P i g . 5-2. We have t h e r . f power P „ = rf  2K  Jf  2 it  I  ac  V ac  doot  o  = &  (  ( I  th- V ^  9  - ^  ( 8 - ^ ) }  The diode n e g a t i v e conductance i s , o V  o  L  o  i s r e l a t e d t o t h e b i a s V^ by t h e e q u a t i o n , Vo  =  (V, 0, b - V., t h)/cos ' '  i  5.1  49  and 0 i s r e l a t e d to the frequency by f 29  =  -f 5.4  %  o At a c e r t a i n frequency, the output power increases with V  q  and consequently with the b i a s , then begin to drop u n t i l i t  reaches zero at a value of the bias equal to V _ bmax  =  V •+ th  2(I -I ) R . th v o + V  S i n  Q  ~ /0 - s i n 0\ cos 0 {  — 2 — '  This value decreases with f u n t i l i t reaches V,, at f = f /2. th o' 5.2  Experimental I n v e s t i g a t i o n of the Gunn E f f e c t Diodes i n Resonant C a v i t i e s The experimental i n v e s t i g a t i o n aimed f o r an understanding  of the operation of Gunn e f f e c t diodes i n resonant c i r c u i t s , and comparing the experimental r e s u l t s with the t h e o r e t i c a l and experimental r e s u l t s obtained by other workers.  Diodes w i t h d i f f e r e n t  values of n l product were used, also c o n t r o l l a b l e change of this parameter was a f f e c t e d by c o o l i n g the diode to reduce i t s c a r r i e r concentration. The experimental procedure consisted of three p a r t s , the f i r s t t o determine the frequency ranges i n which the diodes can o s c i l l a t e coherently.  The second part was to measure the maximum  a v a i l a b l e power as a f u n c t i o n of frequency.  The t h i r d part was to  determine the e f f e c t of bias on the output at d i f f e r e n t frequencies, and a l s o to determine i t s e f f e c t on the l i m i t s of frequency of o s c i l l a t i o n i n both the tunable domain mode and the LSA mode. T h e o r e t i c a l i n t e r p r e t a t i o n of the experimental observat i o n is'presented, .followed by a b r i e f d i s c u s s i o n of the observed  50 f a i l u r e of some of the diodes while being tested. 5.2  Experimental Set-up The arrangement used i n these experiments i s shown i n  P i g . 5.3.  A double stub tuner c o n s i s t i n g of two c o a x i a l shorts  6 cm apart was used to match the output l i n e to the c a v i t y .  The  output s i g n a l was passed through a f i x e d attenuator to reduce i t to a l e v e l s u i t a b l e f o r the operation of the sampling o s c i l l o s c o p e , whose dynamic range was 2 v o l t s peak to peak.  The sampling o s c i l l o s -  cope was a Hewlett-Packard type 140A, with sampling v e r t i c a l amplif i e r u n i t type 1425A and sampling time base u n i t type 1411A.  The.  s i g n a l was then fed to the sampling u n i t type 1430A which was cascaded with a t r i g g e r count-down u n i t type 1104A to generate s i g n a l subharmonic pulses of about 100 MHz r e p e t i t i o n rate f o r t r i g g e r i n g the time base of the scope. R.f. power measurement was done by measuring the output voltage on the screen of the sampling o s c i l l o s c o p e , then estimating the power to a load of 50 ohms.  Input power to the diode was  measured as f o l l o w s : i)  The bias voltage was measured on a second o s c i l l o s c o p e  (Tek-  t r o n i x type 581, w i t h v e r t i c a l a m p l i f i e r u n i t type 82). ii)  The current was then obtained from the I-V c h a r a c t e r i s t i c s measured before, on the assumption that i t would be the same i n the c a v i t y as i t was i n the ' r e s i s t i v e c i r c u i t . measurement  Frequency  was done by counting the number of cycles i n a  c e r t a i n i n t e r v a l of time on the screen of the scope. also checked by the spectrum analyser.  I t was  1  COUNTDOWN U  N  ,  T  -j  1  !  SAMPLING UNIT  1  ATTENUA- 1 I TOR  1  DOUBLE STUB  TUNER  TRIGGERING  O  SAMPLINGOSCILLOSCOPE  F i g . 5.3  i  PULSE AMPLIFIER  PULSE GENERATOR  CAVITY  Experimental Set-up f o r T e s t i n g Diodes i n Resonant Cavity. H  52  5.2.2  E x p e r i m e n t a l S u r v e y i n g o f t h e O p e r a t i o n of Gunn Diodes a t D i f f e r e n t Frequencies 12 Three diodes w i t h n l p r o d u c t s from 0.4 t o 3 x 10  were used.  -2 cm  A r e p r e s e n t a t i v e diagram o f the ranges o f f r e q u e n c i e s  i n which t h e diodes were found t o o s c i l l a t e i s shown i n F i g . 5-4. The b i a s was kept c o n s t a n t a t about 120 v o l t s , u n l e s s i t was nece s s a r y t o reduce i t t o get coherent o s c i l l a t i o n s .  E  125 a r -W°C  E-125  _____  K- 107_ E-122  I  I  1  1  1  .3  "  1  M  I  1  1.  3. FREQUENCY  Fig. 5.4  GHz  Frequency Ranges of O s c i l l a t i o n  The f o l l o w i n g i s a d e s c r i p t i o n of t h e e x p e r i m e n t a l obs e r v a t i o n s f o r each d i o d e . Diode E-112: Thid diode o s c i l l a t e d from about 360 t o 1500 MHz, w i t h o u t a break.  I t was observed t h a t the b i a s r e q u i r e d f o r coherent  oscil-  l a t i o n a t f r e q u e n c i e s l o w e r than 700 MHz, c o u l d not exceed a c e r t a i n maximum v a l u e .  This maximum b i a s decreased w i t h the frequency  u n t i l i t reached a v a l u e of about 5% l a r g e r than t h e t h r e s h o l d a t 360 MHz.  The e f f i c i e n c y  o f t h i s diode was r e l a t i v e l y low, t h e power  output remained almost c o n s t a n t from about 600 t o 1200 MHz, a f t e r  which i t began t o drop s h a r p l y .  A l s o t h e power decreased a t  f r e q u e n c i e s lower than 600 MHz. Diode K-107 The l o w e r l i m i t o f o s c i l l a t i o n was about 680 MHz, w i t h s u b s t a n t i a l l y u n i f o r m output from 800 t o 1200 MHz.  The upper  fre-  quency branch was observed on t h e 3/4 wave mode o f the c a v i t y .  The  lower f r e q u e n c y l i m i t of t h i s branch was w e l l determined, but i t was n o t p o s s i b l e t o operate t h e diode a t f r e q u e n c i e s h i g h e r than 2800 MHz as the c o n d i t i o n f o r o s c i l l a t i o n a t t h e t u n a b l e domain mode became more f a v o r a b l e .  The b e h a v i o u r of t h e diode a t t h i s  upper f r e q u e n c y branch suggested t h a t the diode was o p e r a t i n g on the LSA mode, f o r t h e f o l l o w i n g r e a s o n s : i)  The l o w e r l i m i t o f t h e frequency c o i n c i d e d w e l l w i t h t h e t h e o r e t i c a l l i m i t s e s t i m a t e d by Copeland  ( 2 3 ) , and B o t t and  Hilsum (24) which a r e e x p l a i n e d i n Appendix I I I . ii)  The onset o f o s c i l l a t i o n i n t h i s mode d i d n o t occur u n l e s s t h e b i a s was i n c r e a s e d beyond a c e r t a i n v a l u e , a t which another t h r e s h o l d o c c u r r e d ( o b s e r v i n g t h e b i a s v o l t a g e t o jump up i n d i c a t e d a sudden decrease i n t h e c u r r e n t ) .  T h i s second  t h r e s h o l d decreased from about 120 v o l t s a t 2.4 GHz t o 80 v o l t s a t 2.8 GHz.  At l o w e r b i a s t h e diode e i t h e r  oscillated  a t t h e Gunn domain mode o r i n a mixed i n c o h e r e n t o s c i l l a t i o n . iii)  T h i s mode c o u l d not be j u s t an e x t e n s i o n t o t h e t u n a b l e quenched domain mode, o t h e r w i s e we would"get  continuous t u n a b i l i t y  over  the whole frequency range from 0.68 t o 2.8 GHz, which was not the case.  I n s t e a d , the power was found t o decrease  rapidly  beyond 1.2 GHz, u n t i l i t became u n d e t e c t a b l e a t about 1.6 GHz (the minimum s i g n a l v o l t a g e which c o u l d be d i s p l a y e d on the  54 scope w i t h o u t e x t e r n a l t r i g g e r i n g was about 0.1 v o l t s ) .  By chan-  ging the tuning of the c a v i t y , r e l a t i v e l y strong o s c i l l a t i o n occurred from 2.4 t o 2.8 GHz. I t i s worth m e n t i o n i n g t h a t i t was p o s s i b l e t o suppress t h i s mode by changing the p o s i t i o n of t h e diode i n s i d e t h e c a v i t y , by p u s h i n g i t from near the movable s h o r t towards t h e c e n t e r of t h e cavity. Diode E-125 T h i s diode had a s l i g h t l y l o w e r v a l u e of n l product than the p r e v i o u s one, but i t had about t h e same behaviour a t room temp e r a t u r e . Measurements were done a t room temperature and a t -10°C where t h e c a r r i e r c o n c e n t r a t i o n was reduced t o about h a l f i t s p r e v ious value.  The c a r r i e r c o n c e n t r a t i o n was e s t i m a t e d i n b o t h cases  as d e s c r i b e d i n Chapter 4.  5he m o b i l i t y used i n t h e c a l c u l a t i o n  was taken t o be equal t o the v a l u e o b t a i n e d from the H a l l  effect  measurements. At -10°C t h e upper frequency l i m i t o f t h e t u n a b l e domain mode was 2.05 GHz, and t h e l o w e r frequency l i m i t of t h e LSA mode became 2.25 GHz, both a t t h e 3/4 wave mode o f the c a v i t y . 5.2.2  Power Versus  Frequency  Three s e t s o f r e s u l t s were t a k e n f o r diode K-102, and diode E-125 a t room temperature and at-.10°C  The measurements were  done under a f i x e d b i a s except i n a few p o i n t s where I t was n e c e s s a r y t o reduce i t f o r f r e q u e n c i e s below 700 MHz. M a x i m i z a t i o n of power was o b t a i n e d by u s i n g t h e double stub t u n e r and a l s o by u s i n g c o u p l i n g l o o p s w i t h d i f f e r e n t s i z e s . Maximum power output as a f u n c t i o n of frequency i s shown i n F i g . 5-5  FREQUENCY  Fig. 5.6  Output. versus Frequency  GHZ  57  f o r the 3 d i f f e r e n t c a s e s .  E f f i c i e n c y was  f o r the case of d i o d e K-107  and diode E-125  Maximum e f f i c i e n c y was  plotted  a t room temperature.  o b t a i n e d from d i o d e K-107  be 2 . 5 7 $ at a f r e q u e n c y of about 1.1 5.2.3  e s t i m a t e d and  and was found t o  GHz-(see F i g . 5 - 6 ) .  The E f f e c t of B i a s The diode used i n t h i s experiment was K-124  improved  i n which an  t e c h n i q u e f o r i n s u l a t i n g the s i d e s of the diode w i t h  vacuum s i l i c o n grease was used (see S e c t i o n 5-2.6). 12 n l product of 0.67x10  —2 cm  .  The b i a s was v a r i e d from the t h r e s -  h o l d v a l u e of 36 v o l t s up t o 260 v o l t s .  The output was  as a f u n c t i o n of b i a s at a f i x e d c a v i t y t u n i n g . r e p e a t e d f o r a number of c a v i t y t u n i n g p o s i t i o n s . h i g h e r than 1.04  GHz  b i a s range mentioned  The diode had  The procedure  At f r e q u e n c i e s  before.  The output i n c r e a s e d r n o n o t o n i c a l l y s a t u r a t i o n at higher values.  f r e q u e n c y i n c r e a s e d , t h e output became s m a l l e r but s t i l l with bias,although i t saturated e a r l i e r . GHz,  was  the diode o s c i l l a t e d w i t h o u t a break over the  w i t h the b i a s and approached  than 1.04  measured  As the  increasing  At f r e q u e n c i e s l o w e r  the diode o s c i l l a t e d c o h e r e n t l y from the. t h r e s h o l d  up t o a c e r t a i n v a l u e of the b i a s where o s c i l l a t i o n s were no l o n g e r c o h e r e n t ; then i t began o s c i l l a t i n g a g a i n w i t h about the same f r e quency as the b i a s i n c r e a s e d beyond a n o t h e r c r i t i c a l v a l u e .  The  upper l i m i t of the b i a s f o r the l o w e r branch decreased as the f r e q u e n c y decreased u n t i l i t became j u s t over the t h r e s h o l d a t about 0.78  GHz.  T h i s k i n d of b e h a v i o u r was  experiments mentioned  before.  observed i n the o t h e r  On the o t h e r hand the l o w e r l i m i t  of the b i a s f o r the h i g h e r branch i n c r e a s e d as the f r e q u e n c y decreased.  At 0.96  GHz  i t was about 160 v o l t s , and i n c r e a s e d t o  about 260 v o l t s at 0.86  GHz.  The upper branch of o s c i l l a t i o n was  .  59  not observed before, since the bias d i d not 'exceed 120 v o l t s , The r e s u l t s are shown i n F i g . 5.7-  '  'Forthe ISA mode an experiment was done to determine the e f f e c t of bias on the lower l i m i t of o s c i l l a t i o n i n t h i s mode. A. 12 -2 diode K-105 of n l product of 1.03x10  cm . was used.  The c a v i t y  was set at a c e r t a i n tuning p o s i t i o n and the bias was r a i s e d u n t i l the diode s t a r t e d to o s c i l l a t e .  The procedure was repeated at d i f -  ferent c a v i t y tuning p o s i t i o n s . At the maximum bias of 260 v o l t s the frequency was 1.58 GHz. As the tuning frequency increased the threshold bias f o r the s t a r t of o s c i l l a t i o n decreased u n t i l i t reached a value of 65 at 2.4 GHz. The r e s u l t s are shown i n F i g . 5.8. The upper frequency of o s c i l l a t i o n of t h i s diode i n the tunable domain mode was 15-2 GHz.  100  BIAS  200  300  IN VOLTS  F i g . 5.8 Lower Limit of the Frequency of O s c i l l a t i o n i n the LSA Mode versus Bias.  60.  f  A  A •  •  V  (a)  / \  V \  f = 1GHz  /  \  f = 1.6 Ghz  f = 2.6 GHz P i g . 5.9  Waveforms of the Output of Diode K-107 at D i f f e r e n t Frequencies  F i g . 5.10  Distorted Waveform of Output of Diode E-112,f = 0.8 GHz  61 5.2.4  O b s e r v a t i o n s of t h e Waveform and Harmonic D i s t o r t i o n Diodes w i t h r e s i s t a n c e s g r e a t e r than 100 ohms seemed i n  g e n e r a l to. have a waveform w i t h low harmonic  content.  F o r diodes  w i t h l o w e r r e s i s t a n c e and h i g h e r n l product the output was more d i s t o r t e d , e s p e c i a l l y at lower f r e q u e n c i e s .  T h i s may be a t t r i b u t e d  t o t h e f a c t t h a t as t h e r e s i s t a n c e decreases the l o a d e d Q - f a c t o r d e c r e a s e s , a l s o a t h i g h e r v a l u e s of n l product the c u r r e n t waveform became more s p i k e d w i t h h i g h e r harmonic  c o n t e n t , as d i d t h e output.  F i g u r e s 5-9, a, b, c, show t h e output from diode K-107 a t d i f f e r e n t frequencies.  F i g u r e 5.10 shows t h e waveform o f t h e output from  diode E-112 a t 800 MHz, which seems t o have r e l a t i v e l y h i g h t h i r d harmonic  content. The harmonic  analyser.  d i s t o r t i o n was measured u s i n g t h e spectrum  On t h e Assumption  t h a t t h e s e n s i t i v i t y of t h i s  instru-  ment was u n i f o r m over t h e range of f r e q u e n c i e s from 1000 t o 4400 MHz., i t was found t h a t the t h i r d harmonic  i n t h e output was about  30 and 34" db. below the fundamental a t about 1 GHz f o r t h e diodes K-107 and E-125 r e s p e c t i v e l y . 5.2.5  T h e o r e t i c a l I n t e r p r e t a t i o n o f the R e s u l t s The problem of t u n a b l e domain mode was worked out num(17)  e r i c a l l y by Copeland . H i s r e s u l t s i n d i c a t e d continuous tunab i l i t y from l / 2 f t o about 2f , independent of t h e n l p r o d u c t . He o o a l s o e s t i m a t e d the output power t o be s u b s t a n t i a l l y c o n s t a n t over 0  the range of frequency from 0 . 6 f  Q  t o 1.4f , w i t h a maximum a t about  0.71' o as shown i n F i g . 5-10. At f r e q u e n c i e s h i g h e r than fo the output i n c r e a s e s w i t h the b i a s and s a t u r a t e s a t h i g h e r v a l u e s , and a f t e r t h a t begins t o decrease s l o w l y .  To the a u t h o r ' s knowledge,  62 t h e o r e t i c a l r e s u l t s about the e f f e c t of bias on the operation of the diode at frequencies lower than the t r a n s i t time frequency are not a v a i l a b l e i n the l i t e r a t u r e .  o  VO  N  I  UJ  —i  s  0-5  o  0-5 o  1>4  fo FREQUENCY  Fig.  5.11 Maximum Power versus Frequency (Copeland  (ll) 1  In our experimental r e s u l t s , i t was found that the upper frequency l i m i t i s dependent on the n l product and varied from 1.8f to 2.5f as n l v a r i e d from 4x10 12 to .3x10 12 cm -2 . Other workers obtained o s c i l l a t i o n up to 2.Of 12 -2 10  cm  .  or even more f o r n l product around  For frequencies higher than f , the output was found to  increase s t e a d i l y with b i a s , s a t u r a t i n g at f a i r l y high values of b i a s , and sometimes began to decrease again. the output was smaller but s t i l l i t saturated somewhat e a r l i e r .  had  At higher frequencies  s i m i l a r behaviour, although  The r e s u l t s of the frequency and  bias dependence of the output agreed f a i r l y w e l l with Copeland's t h e o r e t i c a l r e s u l t s and with the r e s u l t s obtained by Hobson !28)  63  For frequencies lower than the t r a n s i t time frequency, the diodes o s c i l l a t e d coherently r i g h t from the threshold up to a c e r t a i n value of the bias.  In that range the output increased f i r s t ,  and then decreased sharply with i n c r e a s i n g bias u n t i l i t stopped o s c i l l a t i n g coherently. When the bias was increased f u r t h e r , the diode began to o s c i l l a t e again at about the same frequency. The operation of the diode i n the lower bias range could be explained as a delayed domain mode.  In the theory presented i n  Section 5.1.2, i t was shown that the above mentioned behaviour i n the lower range of bias should be expected.  Also i t was shown that  both the output and the upper l i m i t of the bias decreased w i t h decreasing tuning frequency.  This has been observed experimentally.  Now f o r diodes w i t h lower value of the n l product, the current drop (I-j-k  _  I-y) i  n  equation 5.1 i n Section 5-1.2  - i s lower and thus the  output and the upper l i m i t and lower l i m i t of the bias both decrease . This i s explained i n F i g . 5.10, where with alower value of n l ' the  curves moved down.  P r a c t i c a l l y , the lower frequency of o s c i l -  l a t i o n w i l l be l i m i t e d to some value i n between l / 2 f and f . o o  This  value approaches the t r a n s i t time frequency as n l product decreases u n t i l the c r i t i c a l value of n l i s reached where no delayed domain mode i s p o s s i b l e . The o s c i l l a t i o n at the higher bias range could be explained as operation i n a m u l t i p l e domain mode, where the domain s t a r t i n g at the beginning of the cycle reaches the anode before the voltage drops below the threshold.  A new domain forms at the cathode, t r a v e l s f o r  some distance through the diode, and then gets quenched by the drop of the voltage below threshold.  64  nl  nl > nl  1 1  nI  f < f <f.  i  1  2 2  2 2  CL I—  O  B I A S  F i g . 5.12  V a r i a t i o n of Output w i t h B i a s a t D i f f e r e n t F r e q u e n c i e s , and a t D i f f e r e n t V a l u e s of n l  T h i s k i n d of o p e r a t i o n might not be p o s s i b l e a t l o w e r v a l u e s of t h e b i a s , s i n c e t h e v o l t a g e sw.ing under t h r e s h o l d would be r e l a t i v e l y h i g h e r than t h a t needed w i t h h i g h e r v a l u e s of b i a s and l a r g e r ac a m p l i t u d e . In the LSA mode t h e l o w e r l i m i t of f r e q u e n c y d i d not n seem t o f o l l o w the r e l a t i o n ^ = 5 x 10  5 - 2 secern exactly.  the frequency decreased from.2.4 t o 1.9 GHz as n decreased 4 x 10  1 5  However, from  t o 0.3 x 1 0 c m ~ , w i t h an. a p p l i e d b i a s of 120 v o l t s . 1 5  5  The  l o w e r frequency l i m i t was found t o decrease w i t h i n c r e a s i n g b i a s , which i s c o n s i s t e n t w i t h the t h e o r y ^ ^ . (23) The o p e r a t i o n i n t h e h y b r i d mode p r e d i c t e d by Copeland 12 -2 f o r d i o d e s w i t h n l product lower than 10 cm , has not been r e a l i z e d i n t h e b i a s range up t o seven, times the t h r e s h o l d . However,  65  the gap between the lower and the upper l i m i t s of the frequency of o s c i l l a t i o n at the tunable domain and the LSA modes, r e s p e c t i v e l y ]2  reduced to l e s s than 0.2 GHz, at n l equal to 0.4 x 10' 5.2.5  cm  -2  Diode F a i l u r e Some of the diodes were found to undergo various kinds of  failure.  Few broke down suddenly under what could be c a l l e d normal  conditions of operation; t h e i r performance deteriorated w i t h time u n t i l f u l l breakdown occurred.  I t was u s u a l l y found that the r e s -  istance of the diode had dropped to a very low value.  Etching the  sides of the diode was not h e l p f u l as a r e p a i r process i n d i c a t i n g that a pUnch-through  i n the bulk m a t e r i a l had happened.  Another type o f . f a i l u r e was due to thermal runaway because of excessive heating.  This i s p a r t i c u l a r l y l i a b l e to occur with ma-  t e r i a l s having deep donor l e v e l s .  This made i t s r e s i s t i v i t y very  s e n s i t i v e to the v a r i a t i o n i n temperature, as was shown i n F i g . 4 . 8 . To prevent t h i s type of f a i l u r e a low pulse r e p e t i t i o n rate was used. A t h i r d type of f a i l u r e was due to the a p p l i c a t i o n of r e l a t i v e l y high bias, corresponding to more than 12 KV/cm.  Breakdown  on the side surfaces seemed to be the main cause of f a i l u r e .  Con-  ducting channels on the side surfaces of the diode were observed under the microscope.  Some of the diodes d i d recover t h e i r prev-  ious c h a r a c t e r i s t i c s on etching, although breakdown occurred again a f t e r a short time. To minimize the r i s k of f a i l u r e and i n order to be able to apply higher values of b i a s , the experiment was t r i e d of coating the diode with polystyrene, or vacuum s i l i c o n grease.  A droplet of  the i n s u l a t o r dissolved i n t r i c h l o r o e t h e l e n e was put on the diode  66 and l e f t to dry. Diodes coated with polystyrene were able to withstand bias up to 1 6 0 v o l t s .  Others coated with the grease had  breakdown voltages over 3 0 0 v o l t s corresponding to about 3 0 KV/cm.  67 6.  CONCLUSIONS  Reproducible Gunn e f f e c t diodes with c o n t r o l l a b l e chara c t e r i s t i c s have been f a b r i c a t e d from s i n g l e c r y s t a l GaAs.. Improving the diode  r e s i s t a n c e to side surface breakdown was achieved  by coating the diode with i n s u l a t i n g m a t e r i a l .  T y p i c a l l y , the  breakdown voltage increased from about 100 volts, to more than 280 v o l t s which was about eight times the threshold. The experimental work involved: i)  The preliminary t e s t i n g of the diodes i n r e s i s t i v e c i r c u i t s f o r the purpose of measuring some of the diode c h a r a c t e r i s t i c s and f o r assessing t h e i r performance.  The important r o l e  played by the ohmic contacts on the q u a l i t y of the diodes has been demonstrated. ii)  H a l l e f f e c t and c o n d u c t i v i t y measurements on the GaAs c r y s t a l s used i n f a b r i c a t i n g the diodes.  The measurements were made  at temperatures between -50°C and 80°C.  From these measure-  ments the c a r r i e r concentration at d i f f e r e n t temperatures was estimated. i i i ) Experimental i n v e s t i g a t i o n of the operation i n microwave cavi t i e s of Gunn diodes with d i f f e r e n t c h a r a c t e r i s t i c s under d i f ferent operating conditions.  For t h i s purpose c o a x i a l c a v i t i e s  having wide tuning range from 500 MHz to 2000 MHz were designed and b u i l t f o r operating the Gunn diodes as harmonic  oscillators.  The r e s u l t s obtained showed that the upper l i m i t of the frequency of o s c i l l a t i o n i n the tunable domain mode v a r i e d from about 1.8 to 2.5 times the t r a n s i t time frequency as n l product varied from 3 . 0 x l 0 . to 0.3xl0 cm~ . 12  12  2  68  The operation of the diode  at frequencies lower than  the t r a n s i t time frequency showed what could be two d i f f e r e n t modes. One was the delayed domain mode at lower bias.  For t h i s mode a  theory has been developed which i s i n f a i r agreement with the experimental r e s u l t s .  The other mode was observed at higher bias and  was considered to be a m u l t i p l e domain mode, i n which the domain reaches the anode before the voltage swings below threshold and a new domain forms at the cathode and then gets quenched before a new cycle, begins. For the delayed domain mode i t was found that the lower l i m i t of the frequency of o s c i l l a t i o n v a r i e d from about 0.55f t o about f as n l v a r i e d from 3-0x1012 to 0.3x1012 cm-2 . The range of Q  bias i n which the diodes could o s c i l l a t e coherently was found to be very s e n s i t i v e to the frequency and decreased w i t h decreasing the frequency and the n l product. For the ISA mode the e f f e c t of the c a r r i e r concentration and the bias on the lower l i m i t of the frequency of o s c i l l a t i o n was investigated.  I t was found that as n increased the lower l i m i t of  the frequency decreased. ing bias. theory.  However, the l a t t e r decreased with i n c r e a s -  Both of these observations are i n agreement w i t h the  APPENDIX I Theory of the D i f f e r e n t i a l ' N e g a t i v e C o n d u c t i v i t y i n Two-Valley Semiconductors Methods have been developed  f o r d e s c r i b i n g the f i e l d  induced e l e c t r o n t r a n s f e r mechanism i n t w o - v a l l e y  semiconductors.  Among these t h e o r i e s a r e those of McCumber and Chynoweth ( 9 ) , and Butcher and Fawcett  , D  .  Although r e c e n t e x p e r i m e n t a l i n v e s -  (22) tigations  have demonstrated t h a t the Butcher and Fawcett  model  i s more a c c u r a t e , the treatment below f o l l o w s c l o s e l y the l i n e s of the McCumber and Chynoweth model, s i n c e t h i s model does not need much mathematical  m a n i p u l a t i o n , and i n the same time l i t t l e i s l o s t  i n u n d e r s t a n d i n g the p r i n c i p l e s of d i f f e r e n t i a l n e g a t i v e conduct i v i t y i n v o l v i n g the t r a n s f e r of e l e c t r o n s between the two v a l l e y s of the c o n d u c t i o n band. I.1  The Band S t r u c t u r e The band s t r u c t u r e o f the m a t e r i a l s e x h i b i t i n g the  Gunn e f f e c t h a s the f o l l o w i n g c h a r a c t e r i s t i c s : i)  The c o n d u c t i o n band has a minimum energy ( v a l l e y ) f o r GaAs a t the c e n t e r of t h e B r i l l o u i n zone w i t h other h i g h e r energy minima or ( s a t e l l i t e v a l l e y s ) a l o n g the <^100_)> d i r e c t i o n s .  ii)  The e l e c t r o n e f f e c t i v e mass i n the c e n t r a l minimum i s s m a l l e r than t h a t i n t h e s a t e l l i t e minima, r e f l e c t i n g h i g h e r  carrier  m o b i l i t y and lower d e n s i t y of s t a t e s . iii)  The energy s e p a r a t i o n of the s a t e l l i t e minima from the c e n t r a l one i s of the order of a f r a c t i o n of an e l e c t r o n v o l t :  too  h i g h a v a l u e of t h i s energy d i f f e r e n c e would r e s u l t i n a v e r y h i g h v a l u e of the t h r e s h o l d f i e l d . On the other hand, too low.  70 a v a l u e would r e s u l t i n h a v i n g the  satellite valleys  occupied a t room temperature and no d i f f e r e n t i a l  normally  conductivity  would appear. The at the figures  band s t r u c t u r e  can be s i m p l i f i e d t o two v a l l e y s , one  o r i g i n and another i n the are  those of GaAs (see  Intervalley  energy s e p a r a t i o n  (l00)> d i r e c t i o n .  Fig.  The f o l l o w i n g -  I.l)  0.36 e.v.  =  E f f e c t i v e mass i n the l o w e r v a l l e y  m*  E f f e c t i v e mass i n the upper v a l l e y  m. u  .08 =  m  1.2  o  mo '  2 5000 cm / v o l t sec. 2 C a r r i e r m o b i l i t y i n the upper v a l l e y u^- = 100 cm / v o l t sec. R a t i o between the d e n s i t y o f s t a t e s i n the two v a l l e y s a = 60 Carrier mobility  i n the l o w e r v a l l e y  u.^  =  * 3/2 One  should n o t i c e t h a t a - (—) 1  <000>  (34)  (see  e.g.  Shockley ^  pp.464)  <ioo>  F i g . I i i Band S t r u c t u r e o f GaAs I.2  The T r a n s f e r Mechanism At low v a l u e s o f the  t r o n s occupy the lower v a l l e y . to the  e l e c t r i c f i e l d most of the  elec-  The d r i f t v e l o c i t y i s n e a r l y e q u a l  e l e c t r i c f i e l d times the lower v a l l e y m o b i l i t y .  As the  field  71  increases the electrons acquire higher energies (heated) and t r a n s - . f e r to the higher v a l l e y at an i n c r e a s i n g rate which i s favoured because of the higher density of states per unit e l e c t r o n energy i n the upper v a l l e y .  I f the t r a n s f e r i s complete enough a decrease  i n the average d r i f t v e l o c i t y may r e s u l t g i v i n g a d i f f e r e n t i a l negative conductivity.  At s u f f i c i e n t l y high e l e c t r i c f i e l d most of  the electrons w i l l occupy the upper v a l l e y and the e f f e c t i v e c a r r i e r m o b i l i t y i s asymptotic to the upper v a l l e y m o b i l i t y . To solve the problem q u a n t i t a t i v e l y the f o l l o w i n g assumpt i o n s are introduced: ,i) ' Electrons i n both v a l l e y s have the same e f f e c t i v e temperature. ii)  The m o b i l i t y of electrons i n a given v a l l e y i s independent of of the e l e c t r i c f i e l d .  i i i ) The e l e c t r o n d i s t r i b u t i o n among the two v a l l e y s obeys nondegenerate Eermi s t a t i s t i c s (Maxwellian d i s t r i b u t i o n ) .  In a  non-degenerate quantum system characterised by a s i n g l e parab o l i c energy minimum, the average e l e c t r o n energy above the minimum i s equal to ^ kT.  In mulxi-minima system, the average  e l e c t r o n energy i s ,  2 1  2  i  J o J  o  f (e)ede i  i = 12 , ...n  .00  1.1  f (e)de  where n i s the number of energy minima, f(e) = N(e) e  - e / /  ^,  density of populated states per unit e l e c t r o n  energy. N(e) = density of states per u n i t e l e c t r o n energy, and e /k^ = the Maxwellian. d i s t r i b u t i o n f a c t o r . e  72  From equation 1.1 we get ,  -e/kT I - e / k T 1 + e ' A  2 ^  +  where T i s the e l e c t r o n temperature, which i s d i f f e r e n t from the l a t t i c e temperature T . Now the number of electrons i n the two v a l l e y s are, n^  =  n  =  u  J  f  J  f  o  A  (e) d e  9  (e) de  From which we get,  n  l  =  ^ ^ A / k l j  1  ne -A/kT ' ^ 1 + ae  nu  = X 7 1  1  ^  - (1 t _ f  )  e  "  A  /  k  T  1  ( I  (a + 1 + A/kT)  -  2 )  ( I - 3)  7  and 1^2 n  a +  =  l  1 + A/kT  1 + (1+  A/kT)e"  ( I - 4) A / k T  The average e l e c t r o n d r i f t v e l o c i t y i s , v  =  (n u + n (i )E — ^ 3__L_ n + n . u 1  (j \  n  _  5  )  JI  The electron temperature T i s r e l a t e d to the e l e c t r i c f i e l d through the energy transport equation,  JU> = -&<"> -  1 ^  + ^  ( I  "  6 )  73 Where 7^ i s the energy r e l a x a t i o n time constant of the c o o l i n g of the electrons by the l a t t i c e . In the steady state case, where the e l e c t r i c f i e l d i s constant i n space and time, equation 1-6 reduces to, T  (I - 7)  2/3 \ vE  Solving equations 1-5, 1-7, numerically -12 ?y = 2 x 10 Chynoweth.  for T  Q  = 300 K,  sec. ; the f i g u r e shown was .obtained by McCumber and The d i f f e r e n t i a l negative c o n d u c t i v i t y occurs beyond  a threshold of 3-2 KV/cm; the d r i f t v e l o c i t y minimum occurs at about 35 KV/cm.  2  4  8  10 20  40  ELECTRIC FIELD P i g . 1.2  60  80  WO  (KV/cm)  v-E C h a r a c t e r i s t i c s of GaAs (McCumber and Chynoweth)  74 APPENDIX I I A Theory of Gunn Domain Dynamics The d i f f e r e n t i a l e q u a t i o n s a s s o c i a t e d w i t h the Gunn domain dynamics a r e h i g h l y n o n l i n e a r and a n a l y t i c s o l u t i o n i s almost i m p o s s i b l e .  F o r the purpose of u n d e r s t a n d i n g the o p e r a t i o n ,  of Gunn o s c i l l a t o r s and a l s o to get some i n s i g h t i n t o the. c h a r a c t e r i s t i c s of the h i g h f i e l d domain, we w i l l p r e s e n t here a semia n a l y t i c treatment o f the s t e a d i l y p r o p a g a t i n g domain i n a b u l k m a t e r i a l extending a long distance.  T h i s treatment i s e s s e n t i a l l y  (28 27) the same as t h a t of Butcher and'Fawcett ' . II.1  S t e a d i l y P r o p a g a t i n g Domain Assuming no c a r r i e r t r a p p i n g or g e n e r a t i o n i n s i d e the  m a t e r i a l d u r i n g the passage o f the domain', we have the c a r r i e r tinuity  con-  equation  where J i s the c o n d u c t i o n c u r r e n t d e n s i t y . We a l s o have P o i s s o n ' s e q u a t i o n , |f  =  *  ( n  - o n  ( I I - 2)  }  We assume the p o s i t i v e d i r e c t i o n o f E a l o n g the n e g a t i v e x - a x i s . The t h i r d e q u a t i o n i s t h a t of the c a r r i e r t r a n s p o r t due to the e f f e c t of the e l e c t r i c f i e l d and d i f f u s i o n .  Taking i n t o  c o n s i d e r a t i o n the c o n t i n u i t y of the t o t a l c u r r e n t — c o n d u c t i o n p l u s d i s p l a c e m e n t — i n s i d e and o u t s i d e the domain, t h i s e q u a t i o n i s , n,v o o  =  nv  -  -Dn j3 n - ox  E  -  e  3E r3t  (/I-f-T I - 3) _  x  75  We assume  that the d r i f t v e l o c i t y i s an instantaneous f u n c t i o n  of the e l e c t r i c f i e l d , and the d i f f u s i o n constant independent the e l e c t r i c  on  field.  Since the domain i s s t e a d i l y propagating,invariant with time, we can express the q u a n t i t i e s E, n, v i n the form Y(x,t)  = Y(x-ct)  where c i s the v e l o c i t y of the domain.  Now  transforming the above  equations to a new coordinate X = x - c t , equations I I - 2 and I I - 3 become dE dX n vo o  e (n-.no) c  -  -r, dn  r = n v - D -dXr ^ - c  u  e dE  •  —e ^dX  ,  ~  ±  4 j  /  . v  (\ I I - >/ 5)  Equation I I - 5 can be reduced f u r t h e r t c J) ^% aX  =  n v - n ov o - c (n-no)  ( I I - 6)  D i v i d i n g equation I I - 6 by equation I I - 4 we get /  ( n  ~  n o )  \ dn dE  iD  =  e n(v-c)  - eD  e n  , o^ -  \  /  c )  ( I 1  0  Tx  „  -  N  7 )  I n t e g r a t i n g the above equation from-ooto x, and again from +c*>to x, we get the two equations E ^n° - I n n— 0  - 1  =  0  en D 0  //)  E  E,x (v(E)_c)dE _- cD ^ ( vo--c)) ;j (v(E)-o)dE E , -oo v  y  c  0  0  - In n^ 0  - 1  =  ^-y: en D 0  'J  E •  0  (II-9)  0  E ^n  n-S dE  E,x (v(E)-c)dE - \eD (v o -c) 'J E , +oo 0  n^-dE 0  (11-10)  76 In  the above two e q u a t i o n s , b o t h of t h e l a s t terms i n the R.H.S.  should be e q u a l .  However, we can n o t i c e from F i g u r e I I - 1,  t h a t f o r t h e same v a l u e s of E on e i t h e r s i d e of t h e domain, we get d i f f e r e n t v a l u e s of n.  Thus t h e v a l u e of t h e i n t e g r a l from - ooto  x i s n o t the same as t h a t from +00to x, u n l e s s both of these two terms a r e equal t o z e r o , but s i n c e n i s always p o s i t i v e , we should have (v -c) equal t o z e r o .  I n t h i s case t h e domain i s p r o p a g a t i n g  w i t h a v e l o c i t y equal t o the o u t s i d e d r i f t  velocity.  X  Fig.  I I . 1 E l e c t r i c F i e l d and t h e C a r r i e r C o n c e n t r a t i o n i n s i d e t h e Domain E q u a t i o n I I - 9 now becomes E  — n  - inn  "I =  E  (v(E)-c)dE  en D E o At the p o i n t i n s i d e t h e domain where n = n , t h e maximum 0  0  Q  field  (H-li: electric  occurs . S u b s t i t u t i n g t h a t i n e q u a t i o n I I - 11, we g e t E max (v(E)-c)dE = o  I I - 12)  E The above e q u a t i o n i s known as the equal areas r u l e , which i m p l i e s  77 that the area A^, should be equal to the area  i n Figure I I - 2 .  The dotted curve i s known as the dynamic c h a r a c t e r i s t i c s .  It  r e l a t e s the maximum e l e c t r i c f i e l d i n s i d e the domain to the outside drift velocity.  o  E  th  E  E  max  F i g . II-2 The Equal Area Rule and the Dyanmic C h a r a c t e r i s t i c II.2  C h a r a c t e r i s t i c s of the Domain Some of the c h a r a c t e r i s t i c s of s t e a d i l y propagating  domains and hence of G-unn diodes under o s c i l l a t i o n conditions, follow from the above, i)  From the dynamic c h a r a c t e r i s t i c s , the peak e l e c t r i c f i e l d decreases with i n c r e a s i n g outside f i e l d .  Since the voltage across  the domain i s p r o p o r t i o n a l to the peak e l e c t r i c f i e l d , we can conclude that f o r a c e r t a i n diode, the outside d r i f t v e l o c i t y decreases as the bias increases, and consequently the frequency of o s c i l l a t i o n decreases too. ii)  For the same outside f i e l d the voltage across the domain i s i n v e r s e l y proportional to the doping.  Thus f o r a c e r t a i n bias  on the diode the outside f i e l d should decrease with i n c r e a s i n g  78  doping as more voltage i s required across the domain. percentage current drop a f t e r threshold  The  i s larger for larger  n l products as shown i n Pigure I I - 3-  n  1 1< 2 2 l  n  -J5*1  Pig. 2.3  E f f e c t of n l on the Dynamic I-V C h a r a c t e r i s t i c s of Gunn Diodes  l  < 3*3 n  79 APPENDIX I I I The Theory of t h e L i m i t e d Space Charge A c c u m u l a t i o n Mode (LSA) The l i m i t e d space charge a c c u m u l a t i o n mode' i s c h a r a c t e r i z e d by: i)  Space charge a c c u m u l a t i o n i s not p e r m i t t e d t o grow i n t o h i g h f i e l d domains.  Thus most o f t h e diode i s under u n i f o r m e l e c -  t r i c f i e l d p r o p o r t i o n a l t o t h e v o l t a g e a c r o s s the d i o d e . ii)  Quenching of t h e space charge a c c u m u l a t i o n by the drop of t h e v o l t a g e under t h r e s h o l d f o r p a r t of t h e n . f . c y c l e i s n e c e s s a r y i n o r d e r t h a t coherent o s c i l l a t i o n c o u l d be e s t a b l i s h e d .  i i i ) A n e g a t i v e l a r g e s i g n a l r . f . impedance i s n e c e s s a r y a l o n g s i d e w i t h t h e s a t i s f a c t i o n of c o n d i t i o n III.l  (ii)..  Conditions f o r O s c i l l a t i o n In t h e a n a l y t i c treatment c o n s i d e r e d h e r e , s m a l l s i g n a l  theory w i l l be used on the assumption t h a t space charge i s very small.  accumulation  N e g l e c t i n g the d i f f u s i o n c u r r e n t , the c a r r i e r  con-  t i n u i t y e q u a t i o n becomes dn  dt  =  9 "dx  (nv)  =  _n 3_v v 9n dx ~ dx  T r a n s f o r m i n g t o c o o r d i n a t e s moving w i t h t h e c a r r i e r X  =  CTTT U  ±  ±  "  I ±  }  stream  x - vt  We get e q u a t i o n I I I - 1 transformed t o §T where u = ^  =  " ^ If  the d i f f e r e n t i a l m o b i l i t y .  We have P o i s s o n ' s e q u a t i o n  <m  ">)  80 9_E  (n - n )  (III  Q  dx  Denoting 3(n-n ) by (n-no)  n  Q  - 3)  the space charge d e n s i t y , and assuming  we s o l v e e q u a t i o n s I I I - 2 and I I I - 3 t o g e t h e r t o get t <5(t) = r j ( t ) e x p (- n e o  Fig.  III.l  Q  J  u (r)dr)  ( I I I - 4)  The O p e r a t i n g C h a r a c t e r i s t i c s of C a r r i e r s i n LSA Diode.  D u r i n g p a r t of the r . f . c y c l e i n which the diode i s over t h r e s h o l d , any i n i t i a l space charge i r r e g u l a r i t y w i l l grow a c c o r d i n g to the e q u a t i o n tf(t ) 2  = ( 5 ( t ) exp (-h/ 1  ) =6(t )  h  ±  n  .G  n  ( I I I - 5)  81 n Where h = ~f  t,  (  ,' h n =- eT (—  ^J *1  l  and T i s the period  u(t) dt)  -1 ,  of the r . f . c y c l e .  During the under threshold part of the c y c l e , the space charge accumulation decays to 6 (t ) = 6 ( t ) exp (-h/ 2  ) = <J(t ) G  h  ( I I I - 6)  2  p  s  where h  = (~  P  J  u (z)  dt)  1  Condition ( i ) w i l l be s a t i s f i e d , i f Gn i s r e s t r i c t e d to (23)  be smaller than a c e r t a i n value; Copeland e  as an upper l i m i t to G^, G <e  suggested a value of  so that,  5  n  ( I I I - 7)  Condition ( i i ) w i l l be s a t i s f i e d i f , G .G <1 n or hp  p  N  h , a c o n d i t i o n which can be w r i t t e n as: n  hh  z  n  <i  (i<i) Condition ( I I I - 7) implies a higher l i m i t f o r ^. R  y  ( r a - 8) Condition  ( I I I - 8) c o n t r o l s the r e l a t i o n between the ac and dc voltages across the diode, i n other words a condition f o r a minimum r . f . impedance.  However, there i s an optimum value f o r the load imped-  ance at a c e r t a i n bias f o r maximum e f f i c i e n c y of dc to ac conversion. III.2  Power and E f f i c i e n c y Assuming the e l e c t r i c f i e l d d i s t o r t i o n due to space charge  82 accumulation w i t h i n the diode to be small, we w i l l have the curr e n t - v o l t a g e - c h a r a c t e r i s t i c s f o l l o w i n g the v - E c h a r a c t e r i s t i c s of the bulk m a t e r i a l .  Assuming the voltage across the diode to  be V(t) = V, + V s i n OJ t b o where  i s the bias voltage, and V  Q  i s the r . f . voltage amplitude  the e l e c t r i c f i e l d w i l l be, E(t) = E, b +o E The r . f . power i s equal t o , 2rc P „ = ~ r. i . 2% J f  s i n oo t  l(-o.t)V(ojt)- dojt  o  2%  E oLA e no  f  0  v(E(cut)) s i n art dart  '  2%  ( I I I - 9)  o where L and A are length and the area of the diode, r e s p e c t i v e l y . The dc power input to the diode i s P  dc  =  E Menp. o I-K  ^  Q  J  V  ( (,^) E  _  d w t  1 0  )  The r . f . r e s i s t a n c e w i l l be equal to E  R  _P  r , 1 #  =  2  .  L  2  _r 2 Pr . f,.  At a c e r t a i n bias one can c a l c u l a t e the maximum e f f i c i e n c y taking i n t o consideration both of the conditions of space charge growth l i m i t ( I I I - 7 ) , and space charge c o n t r o l ( I I I - 8 ) .  A plot  of maximum e f f i c i e n c y versus bias i s shown i n Figure I I - 2 — s o l i d c u r v e — t a k i n g only i n t o consideration the space charge c o n t r o l con( \ d i t i o n (Bott and-Hilsum ). For d i f f e r e n t values of ^ we would n  83 have a minimum "bias below which the r e s t r i c t i o n on G i s v i o l a t e d , n ' and the e f f i c i e n c y drops very sharply as shown by the dotted curves-Copeland (23). A maximum value of ^ estimated by Copeland to be 5 -3 equal to 2.0 x 10 secern. - seems to be p o s s i b l e only at f a i r l y high valuesof b i a s . At bias i n the range (2.0-5.0) times the 5 -3 threshold, a value of 0.7 x 10  secern.  was found by Bott and  Hilsum. III.3  n On the Minimum Value of ^ A minimum value of j predicted by Copeland( ^) 2  a s  a  r e s u l t of the c o n d i t i o n ,  V 1 was taken to be —. e 4 -3 10 .secern .  G  p<  This gave a minimum value of in of about 2.0 x  However, i f we consider that at low values of j the growth f a c t o r i s decreasing f a s t , we might introduce a s i m i l a r cond i t i o n f o r space charge c o n t r o l which i s l e s s r e s t r i c t e d , v i z .  ^ <\ n  1 <i.o  n In such a case no minimum l i m i t i s imposed on the value of -r^, and h are independent cf. -z. Once becomes smaller than p n i . i the c r i t i c a l value which s a t i s f y the space charge growth l i m i t , n since h  the e f f i c i e n c y becomes independent of ^ and w i l l follow the s o l i d l i n e i n Pigure I I I - 2.  84  * A B  2-4 x 10 4-0  / sec  3 fcm J  8-0 D 10  C  10 BIAS  F i g . II-2  15 FIELD  Maximum•Efficiency v e r s u s B i a s (Bott and Hilsum ( 2 4 ) ) .  (KV/cm)  85 REFERENCES  1.  Gunn, J . B. "Microwave O s c i l l a t i o n of C u r r e n t i n I I I - V Semiconductors" S o l i d S t a t e Comm., V o l . I , pp. 8 8 - 9 1 , 1963-  2.  Kreomer, H. "Theory of t h e Gunn E f f e c t " P r o c . IEEE "Correspondence", V o l . 52, pp. 1936, 1964.  3.  R i d l e y , B. K., Watkins, T. B. "The P o s s i b i l i t y o f N e g a t i v e R e s i s t a n c e E f f e c t s i n Semiconductors" P r o c . Phys. Soc. (London), V o l . 78, pp. 293-304, 1961.  4.  H i l s u m , C. " T r a n s f e r r e d E l e c t r o n a m p l i f i e r s and O s c i l l a t o r s " P r o c . IRE, V o l . 50, pp. 954-966, 1962. .  5.  R i d l e y , B. K. " S p e d i f i c N e g a t i v e R e s i s t a n c e i n S o l i d s " P r o c . Phys. Soc. (London), V o l . 82, pp. 954-966, 1963.  6.  Hutson, A. R., Jayaraman, A., Chynoweth, A. G., C o r e i l l , A. S., Eeldman, ¥. L. "Mechanism of t h e Gunn E f f e c t from a P r e s s u r e Experiment" Phys.. 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