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The static and dynamic characteristics of series-connected tunnel diodes and their applications in digital.. 1962

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THE STATIC AND DYNAMIC CHARACTERISTICS OF SERIES-CONNECTED TUNNEL DIODES AND THEIR APPLICATIONS IN DIGITAL CIRCUITS by CLEMENT ANDRE TEWFIK SALAMA B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of EUc^/cg.j Engineering The University of British Columbia, Vancouver 8, Canada. Date y**\uoAy ro1^ y S7*3 ABSTRACT A m u l t i s t a b l e composite volt-ampere c h a r a c t e r i s t i c can be r e a l i z e d u s i n g a number of tunnel diodes. A maximum of 2 st a b l e s t a t e s can be obtained u s i n g n s u i t a b l y chosen tunnel diodes connected i n s e r i e s . The main purpose of t h i s study i s to i n v e s t i g a t e the s t a t i c and dynamic c h a r a c t e r i s t i c s of such a c i r c u i t . P r e l i m i n a r y work deals with the switching behaviour of a s i n g l e tunnel diode and the dependence of the switching time on the f i g u r e of merit and the cur r e n t o v e r d r i v e . This work serves as a background to the study of the m u l t i s t a t e c i r c u i t . The study of the s t a t i c c h a r a c t e r i s t i c s of the composite device determines the c o n d i t i o n s necessary f o r the generation of the r e q u i r e d number of s t a b l e s t a t e s . A d d i t i o n a l c o n d i t i o n s necessary to ensure proper o p e r a t i o n are d e r i v e d from the study of dynamic c h a r a c t e r i s t i c s of a two tunnel diode m u l t i s t a t e c i r c u i t . The dynamic c o n d i t i o n s d e r i v e d i n v o l v e the tunnel diode capacitances and t h e i r r a t i o . The temperature dependence of the c i r c u i t i s a l s o i n v e s t i g a t e d . Experimental r e s u l t s are presented showing an o p e r a t i n g speed of 12.5 ns f o r a four s t a t e c i r c u i t u sing a v a i l a b l e tunnel diodes. The v e r s a t i l i t y of the composite c h a r a c t e r i s t i c obtained, and the inherent high speed of the tunnel diodes combine to make the m u l t i s t a t e device u s e f u l i n high-speed d i g i t a l a p p l i c a t i o n s such as: b i n a r y a d d i t i o n , a n a l o g - t o - d i g i t a l conversion .and counting. These a p p l i c a t i o n s are di s c u s s e d b r i e f l y . ACKNOWLEDGEMENT The author i s indebted to Dr. M.P. Beddoes, the s u p e r v i s i n g p r o f e s s o r of t h i s p r o j e c t , f o r h i s help and guidance throughout the course of t h i s r e s e a r c h . G r a t e f u l acknowledgement i s given to the Northern E l e c t r i c Company L i m i t e d f o r a Fe l l o w s h i p awarded i n 1961, and to the N a t i o n a l Research C o u n c i l f o r a Studentship awarded i n 1962. The work d e s c r i b e d i n t h i s t h e s i s was supported by the Na t i o n a l Research C o u n c i l under Grant (BT<-68). TABLE OF CONTENTS Page L i s t of I l l u s t r a t i o n s v L i s t of Tables v i i i Acknowledgement i x L i s t of S p e c i a l Symbols and Terms x 1. I n t r o d u c t i o n 1 2. The Tunnel Diode as a Negative Resistance Device. 3 2.1 Volt-Ampere C h a r a c t e r i s t i c of a Tunnel Diode . 5 2.2 Temperature Dependence of Tunnel Diode D-C Parameters 10 2.3 Tunnel Diode E q u i v a l e n t C i r c u i t I T 3. S i n g l e Tunnel Diode Switching Behaviour ......... 13 3.1 Dependence of the Figure of M e r i t on the E l e c t r i c a l Parameters of the J u n c t i o n ...... 14 3.2 Tunnel Diode Switching Speed 16 Case I : Current Bias 18 Case I I : Voltage Bias 23 3.3 Approximate Formula f o r the Rise Time of a Tunnel Diode 23 3.4 Experimental R e s u l t s 25 3.5 Summary 27 4. S t a t i c and Dynamic C h a r a c t e r i s t i c s of S e r i e s - Connected Tunnel Diodes 28 4.1 S t a t i c C h a r a c t e r i s t i c of Two Tunnel Diodes Connected i n S e r i e s 28 4.2 S t a t i c C h a r a c t e r i s t i c s of n Tunnel Diodes ... 33 4.3 Experimental Re s u l t s 34 4.4 Dynamic C h a r a c t e r i s t i c s 36 Page 4.5 Two Tunnel Diodes M u l t i s t a t e . C i r c u i t . ....... . 37 4.6 Switching Behaviour ............oo........a... 39 4.7 E f f e c t of Capacitance on T r a n s i e n t Behaviour . 43 4.8 E f f e c t of the D-C Parameters of the Tunnel Diodes on the Switching Behaviour ............ 47 4.9 E f f e c t of Inductance on Tra n s i e n t Behaviour ... 48 4.10 E f f e c t of Input Pulse Rise Time on Transient 4.11 E f f e c t of Temperature on the C i r c u i t Operation 51 4.12 E x t e n s i o n to Three Tunnel Diodes i n S e r i e s ...- 52 4.13 Experimental R e s u l t s 53 4.14 Summary ...,•«.«»»'••..*...». . . o . o . o o . . . . . . . . . . 56 5. A p p l i c a t i o n s 57 5.1 F u l l B i n a r y Adder 57 5.2 A n a l o g - t o - D i g i t a l Converter 59 3*3 C o"u.n."t G i * « • - « - « « # « * * * # « » » » o « » » * « « « o f t » » o » o « « o « e « » # 6 0 - 6 • G one 1 u s i o UH • « • • . • « , * • • * . . o o « < « * • « • « • • o * o o o o « » a * « « » * o « 0 « 62 Appendix I Measurement of Tunnel Diode Parameters -'•••»•... 64 AI.1 Bias C i r c u i t and S t a b i l i t y .................. 64 AI.2 Tunnel Diode Test Mount 66 AI.3 Experimental Measurements and Resu l t s ....... 66 Appendix II Methods of Approximating 'Tunnel Diode Curves 70 A I I . l Polynomial Approximations 71 A l l . 2 Two Term Expo n e n t i a l Approximation 73 Appendix I I I F a c t o r s I n f l u e n c i n g the Choice of the Load Line Resistance R i n a M u l t i s t a t e C X I* C U. 1 ~b e » « o » e - o * e » o « « o o » o » o A « » « o « f r o o o o o » * 73 References ............................................ 77 LIST OP ILLUSTRATIONS Fi g u r e Page 2-1. Energy-band Scheme at J u n c t i o n of a Tunnel Diode at Thermal E q u i l i b r i u m ..<,.... 4 2-2. D-C C h a r a c t e r i s t i c of a Tunnel Diode 5 2-3. Components of the Volt-Ampere C h a r a c t e r i s t i c of a Tunnel Diode 8 2-4. V a r i a t i o n of Tunnel Diode D-C Parameters with Temperature 10 2- 5. E q u i v a l e n t C i r c u i t s of Tunnel Diodes 11 3- 1. a) Si n g l e Tunnel Diode E q u i v a l e n t C i r c u i t ..... 16 b) Tunnel Diode C h a r a c t e r i s t i c I l l u s t r a t i n g Switching Load Line 16 3-2. Switching T r a n s i e n t v ( t ) f o r Var i o u s Overdrive F a c t o r s 20 3-3. Switching Time, Delay Time and Rise Time Versus Overdrive ... 21 3-4. Normalized Switching and Delay Time Versus Overdrive 21 3-5. Timing Delay as a F u n c t i o n of Tunnel Diode Capacitance and Peak Current 22 3-6. Dynamic v - i T r a n s i e n t Behaviour f o r L = 1, 10 and 100 nh 24 3-7. Tunnel Diode Switching Test C i r c u i t 25 3- 8. Experimental Switching Waveforms 26 4- 1. L i n e a r i z e d C h a r a c t e r i s t i c Curve of a Tunnel Diode 30 4-2. a) Tunnel Diodes I n d i v i d u a l C h a r a c t e r i s t i c s ... 30 b) Composite C h a r a c t e r i s t i c 30 4-3. a) Experimental; C h a r a c t e r i s t i c s of the.Negative Resistance Elements 35 b) Two Elements Composite C h a r a c t e r i s t i c 35 F i g u r e , Page c) Three Elements Composite C h a r a c t e r i s t i c ... 35 d) Four Elements Composite C h a r a c t e r i s t i c .... 35 4-4. Two Tunnel Diode{j M u l t i s t a t e C i r c u i t .......... 38 4-5. a) Dynamic v - i C h a r a c t e r i s t i c s of the M u l t i s t a t e C i r c u i t ( 0 0 — Ol) 40 b) Voltage and Current Waveforms (00 — Ol) ... 40 4-6. a) Dynamic v - i C h a r a c t e r i s t i c s of the Multi/state C i r c u i t (00 — 10) 42 b) Voltage and Current Waveforms ( 0 0 — 1 0 ) ... 42 4-7. a) Dynamic v - i C h a r a c t e r i s t i c s of the M u l t i s t a t e C i r c u i t ( 0 0 — 11) 44 b) Voltage and Current Waveforms ( 0 0 — l l ) ... 44 4-8. E f f e c t of Capacitance on T r a n s i e n t Behaviour .. 46 4-9. Dynamic v - i C h a r a c t e r i s t i c f o r (5ly = 0.32 ma . 48 4-10. E f f e c t of Inductance on T r a n s i e n t Behaviour ... 49 4-11. Experimental Voltage Waveforms f o r a Two Tunnel Diode C i r c u i t 54 4- 12. Experimental Voltage Waveforms f o r a Three Tunnel Diode C i r c u i t . 55 5- 1. F u l l B i n a r y Adder 58 AI-1. a) Tunnel Diode Test C i r c u i t 65 b) E q u i v a l e n t C i r c u i t 65 AI-2. Tunnel Diode C o a x i a l Mount .................... 67 AI-3. a) Diagram of the Test C i r c u i t ............... 67 b) Bridge E x t e r n a l Connections ............... - 67 AI-4. Admittance C h a r a c t e r i s t i c s of a IN2939 Diode as a F u n c t i o n of Voltage ...................... 69 AI-5. Capacitance V a r i a t i o n as a F u n c t i o n of Voltage 69 AII-1. T1925 Germanium Tunnel Diode 72 F i g u r e Page a) A c t u a l and C a l c u l a t e d C h a r a c t e r i s t i c s ...... 72 b) Per cent E r r o r Between the A c t u a l and C a l c u l a t e d C h a r a c t e r i s t i c s 72 LIST OF TABLES Table . Page 2.1 P r o p e r t i e s of Tunnel Diodes and Semiconductor: M a t e r i a l s Used i n Their F a b r i c a t i o n 6 4.1 D-C Parameters of the Two Diodes 39 4.2 Temperature C o e f f i c i e n t s of the D-C Parameters f o r the Ge and GaAs Tunnel Diodes 52 4.3 Stable States f o r Three Tunnel Diodes Device .. 53 5.1 Truth Table of F u l l B i n a r y A d d i t i o n 58 A I I . l Polynomial Approximations f o r Tunnel Diodes .... 72 A l l . 2 E x p o n e n t i a l Approximations f o r Tunnel Diodes .. 74 LIST OF SPECIAL SYMBOLS AND TERMS Symbol F i r s t Defined i n S e c t i o n V p = Diode Peak Voltage 2.1 V y = Diode V a l l e y Voltage 2.1 V^ = Diode Forward Voltage 2.1 Ip = Diode Peak Current 2.1 l y - Diode V a l l e y Current ................... 2.1 mP = Reduced E l e c t r o n mass ( r e l a t i v e ) i n a Semiconductor M a t e r i a l ................. 2.1 E = Energy Gap i n a Semiconductor M a t e r i a l . 2.1 e = R e l a t i v e D i e l e c t r i c Constant ........... 2.1 r Z = Tunneling P r o b a b i l i t y 2.1 f ( v ) = Diode v - i C h a r a c t e r i s t i c 2.3 C(v) = Diode Capacitance ...... 2.3 Cy = Diode V a l l e y Capacitance 2.3 |r| = Magnitude of the Diode Negative Resistance 3. C / l p - Figure of merit 3. C|r| = Fig u r e of merit *. 3. T 2 = Precursor Pulse D u r a t i o n 4.6 r ( ) I V = D i f f e r e n c e i n the V a l l e y Currents of the Two Diodes .. •. 4.8 Term Percent Overdrive: 3.2 Rise Time: 3.2 Delay Time: 3.2 Switching Time: 3.2 Prec u r s o r P u l s e : 4.6 1 THE STATIC AND DYNAMIC CHARACTERISTICS OF SERIES-CONNECTED TUNNEL DIODES AND THEIR APPLICATIONS IN DIGITAL CIRCUITS 1. INTRODUCTION A great v a r i e t y of negative r e s i s t a n c e devices have been made from semiconductor m a t e r i a l s ; a recent one i s the tunnel d i o d e . ^ In the past s e v e r a l y e a r s , widespread i n t e r e s t has been shown i n the use of tunnel diodes i n d i g i t a l c i r c u i t s . The reason f o r t h i s i n t e r e s t l i e s i n the tunnel diode's i n h e r e n t , advantages: h i g h switching speed, low power d i s s i p a t i o n , device s i m p l i c i t y , small s i z e , and high s t a b i l i t y w i t h changes i n environmental c o n d i t i o n s such as temperature and n u c l e a r r a d i a t i o n . There are, however, d i f f i c u l t i e s i n o b t a i n i n g g a i n and d i r e c t i o n - a l i t y u s i n g the d e v i c e . The advent of the tunnel diode r e v i v e d the i n t e r e s t i n the use of the volt-ampere c h a r a c t e r i s t i c s of s e v e r a l i n t e r - connected negative r e s i s t a n c e devices to generate- m u l t i s t a b l e composite c h a r a c t e r i s t i c s . The form and the complexity of the composite c h a r a c t e r i s t i c s depend on the i n d i v i d u a l devices and the mode of connection. The complex c h a r a c t e r i s t i c s generated can l e a d to a host of ways of u s i n g negative r e s i s t a n c e devices i n the performance of d i g i t a l l o g i c f u n c t i o n s . One i n t e r e s t i n g combination of n negative r e s i s t a n c e devices connected i n s e r i e s n (2) was found to generate 2 s t a b l e s t a t e s , ' provided the i n d i v i d u a l devices obey c e r t a i n r u l e s . The main purpose of the f o l l o w i n g study i s to i n v e s t i g a t e 2 the s t a t i c and dynamic c h a r a c t e r i s t i c s of such a combination using tunnel diodes as negative r e s i s t a n c e elements. The t h e s i s c o n s i s t s of four main s e c t i o n s . The f i r s t i n t r o d u c t o r y s e c t i o n deals w i t h the tunnel diode as a negative r e s i s t a n c e element. The o p e r a t i o n of the d e v i c e , i t s volt-ampere c h a r a c t e r i s t i c , and i t s e q u i v a l e n t c i r c u i t are d i s c u s s e d b r i e f l y . The second s e c t i o n deals with the behaviour of a s i n g l e tunnel diode and i t s p o s s i b i l i t i e s and l i m i t a t i o n s as a switching d e v i c e . The a n a l y s i s i s c a r r i e d out by computer s o l u t i o n of the c i r c u i t equations. Experimental r e s u l t s are used to v e r i f y these s o l u t i o n s . The i n v e s t i g a t i o n i n t h i s s e c t i o n provides a necessary background to the d i s c u s s i o n i n the next two s e c t i o n s . The t h i r d s e c t i o n d e a l s with the s t a t i c and dynamic c h a r a c t e r i s t i c s of the s e r i e s - c o n n e c t e d tunnel diode c i r c u i t . F i r s t , a g r a p h i c a l a n a l y s i s of the composite c h a r a c t e r i s t i c , supported by experimental r e s u l t s , e s t a b l i s h e s the r e l a t i o n s h i p that the s t a t i c parameters of the i n d i v i d u a l tunnel diodes must s a t i s f y i n order to generate the r e q u i r e d m u l t i s t a b l e composite c h a r a c t e r i s t i c . Next, the dynamic behaviour of a tunnel diode m u l t i s t a t e c i r c u i t i s i n v e s t i g a t e d by a computer s i m u l a t i o n of the n o n - l i n e a r system, and v e r i f i e d by experiment. This i n v e s t i g a t i o n e s t a b l i s h e s some a d d i t i o n a l r e l a t i o n s complementing the c o n d i t i o n s d e r i v e d from the s t a t i c a n a l y s i s to ensure proper o p e r a t i o n . The f o u r t h and l a s t s e c t i o n deals with p o s s i b l e a p p l i c a - t i o n s of the m u l t i s t a t e composite devices i n the performance of d i g i t a l f u n c t i o n s such as b i n a r y - a d d i t i o n , a n a l o g - t o - d i g i t a l conversion,and counting. 3 2. THE TUNNEL DIODE AS A NEGATIVE RESISTANCE DEVICE The tunnel diode i s e s s e n t i a l l y a narrow p-n j u n c t i o n , made of very h i g h l y doped semiconductor m a t e r i a l , which e x h i b i t s a negative r e s i s t a n c e over a l i m i t e d ( s e v e r a l tenths of a v o l t ) v o l t a g e range when b i a s e d i n the forward d i r e c t i o n . The ( 3 ) negative r e s i s t a n c e i s of the non-parametric type and a r i s e s from the so c a l l e d " t u n n e l i n g " mechanism, a s t r i c t l y quantum mechanical e f f e c t . In a normal r e c t i f y i n g p-n diode, conduction i n the forward d i r e c t i o n occurs predominantly by the d i f f u s i o n of m i n o r i t y c a r r i e r s across the p-n j u n c t i o n : these c a r r i e r s c o n s i s t of holes which d i f f u s e from the p - side to the n - s i d e , and e l e c t r o n s which t r a v e l i n the opposite d i r e c t i o n . Both c o n t r i b u t i o n s , because of the d i f f e r e n c e i n the s i g n of the charge, c o n s t i t u t e a conventional c u r r e n t of the same s i g n . However, i n a narrow p-n j u n c t i o n (of the order of 150 Angstroms) which i s h e a v i l y doped wi t h c o n t r o l l e d i m p u r i t i e s , the Fermi>level Ep, i n s t e a d of f a l l i n g w i t h i n the f o r b i d d e n gap, as f o r the r e c t i f y i n g diode, f a l l s w i t h i n the valence band on the p - side and w i t h i n the conduction band on the n - side as shown i n Figure 2-1• This causes an overlap A of the band edges, and i t gives r i s e to a new conduction mechanism: the so c a l l e d . " t u n n e l e f f e c t " or i n t e r n a l f i e l d emission f i r s t (4) considered by Zener v . E l e c t r o n s i n a g i v e n energy stat e i n the valence band can,without changing t h e i r energy s t a t e , " t u n n e l " through the f o r b i d d e n gap i n t o the empty or conduction band and v i c e v e r s a under the a c t i o n of a l a r g e e l e c t r i c f i e l d . 4 Conduction Band E l e c t r o n Energy p - type- Valence Band T r a n s i t i o n Region n - type cn Figure 2-1, Energy-band Scheme at J u n c t i o n of a Tunnel Diode at Thermal E q u i l i b r i u m . As mentioned above, the only e l e c t r o n s i n v o l v e d i n the t u n n e l i n g process are those f a l l i n g i n s t a t e s i n c l u d e d i n the band overlap A . Since charges of one s i g n are i n v o l v e d , the two components of c u r r e n t (due to o p p o s i t e l y moving charges) must be s u b t r a c t e d from each other. In p a r t i c u l a r , with no e x t e r n a l b i a s a p p l i e d , the net current must be zero; t h e r e f o r e exact c a n c e l l a t i o n must take p l a c e , though each component by i t s e l f i s non-zero. On the other hand, the net c u r r e n t vanishes f o r a forward vo l t a g e which j u s t destroys the overlap, but i n t h i s case the i n d i v i d u a l components are a l s o zero. For 5 intermediate v o l t a g e s , a net forward current flows because the rate s of change of these two components with b i a s voltage d i f f e r . Thus f o r i n c r e a s i n g b i a s v o l t a g e , the current i n c r e a s e s , reaches a peak*" and then decreases producing a negative incremental r e s i s t a n c e . 2.1 Volt-Ampere C h a r a c t e r i s t i c of a Tunnel Diode. The low frequency c h a r a c t e r i s t i c of a t y p i c a l tunnel diode i s shown i n Fig u r e 2-2 together with the important d-c parameters. I t d i f f e r s from that of any other type of p-n j u n c t i o n i n that i t e x h i b i t s a "hump" of current g i v i n g r i s e to the negative r e s i s t a n c e . Tunnel diodes have been made with peak cur r e n t s ranging from 10 microamperes to 10 amperes. However, f o r a gi v e n semiconductor m a t e r i a l , the volt a g e sc a l e must remain f i x e d to a la r g e extent, and i t i s determined by the energy gap of the semiconductor. Hence, the tunnel diode i s a low volt a g e d e v i c e , the power range of which can be extended only by i n c r e a s i n g i t s c u r r e n t . V p = Peak Voltage j 1 V„ = V a l l e y Voltage P F i g u r e 2-2. D—C C h a r a c t e r i s t i c of a Tunnel Diode Tunnel diodes have been made from a number of semi- conductor m a t e r i a l s . Table 2.1 l i s t s f i v e of them together with some of t h e i r p r o p e r t i e s . ̂  ^' ' ^ ^ Type of P r o p e r t i e s of Semiconductor Materials E (ev.) g * mr e r I P / I V GaAs 1.35 0.13 11.1 40 1 S i 1.11 0.78 11.2 4 0.7 Ge 0.67 0.44 16 15 0.48 InAs 0.33 0.051 11.7 12 0.25 InSb 0.18 0.028 15.9 10 0.14 P r o p e r t i e s of Tunnel Diodes ( T y p i c a l Values) m̂ . = reduced e l e c t r o n mass ( r e l a t i v e ) E = energy gap o e = r e l a t i v e d i e l e c t r i c constant Table 2.1 P r o p e r t i e s of Tunnel Diodes and Semiconductor M a t e r i a l s Used i n T h e i r F a b r i c a t i o n Tunneling,as a quantum mechanical process, has been t r e a t e d i n c o n s i d e r a b l e d e t a i l i n the l i t e r a t u r e , ̂  ^ ' ' ' 1 (9) and only a b r i e f d i s c u s s i o n w i l l be given here to e x p l a i n the d e t a i l s of the volt-ampere c h a r a c t e r i s t i c curve. At low forward v o l t a g e , a h i g h c u r r e n t flows through the diode due to band-to- band t u n n e l i n g ; at s u f f i c i e n t l y high v o l t a g e , current flows by forward i n j e c t i o n . Between these two v o l t a g e ranges, the current i s unexpectedly high and has been c a l l e d the "excess" c u r r e n t . F i r s t consider the t u n n e l i n g c u r r e n t ; ^ ^ as s t a t e d p r e v i o u s l y , t h i s c urrent c o n s i s t s of two components f l o w i n g across the j u n c t i o n i n opposite d i r e c t i o n s ; I from the valence to the conduction band, and I flowing from the conduction band to the valence band. The current I at any energy l e v e l E i s p r o p o r t i o n a l to the number of e l e c t r o n s i n the conduction band, the number of a v a i l a b l e s t a t e s i n the valence band, and the t u n n e l i n g p r o b a b i l i t y from the conduction band to the valence band. L e t t i n g p (E) and p (E) represent the energy stat e d e n s i t i e s i n the conduction and valence bands r e s p e c t i v e l y , f c ( E ) and f y ( E ) the corresponding Fermi d i s t r i b u t i o n f u n c t i o n s denoting the p r o b a b i l i t y that a given energy stat e i s occupied, and Z v c and Z c v the t u n n e l i n g p r o b a b i l i t i e s i n the two d i r e c t i o n s , Then f (E) P (E) = d e n s i t y of conduction-band e l e c t r o n s t a t e s c c occupied i n dE, ( l - f (E)) P (E) = d e n s i t y of • -^al-easce$'.-band e l e c t r o n s t a t e s unoccupied i n dE. Thus E I = A cv I Z c y p c ( E ) f c ( E ) ( l - f v(E)]P v(E) dE E cn ...(2-1) where A i s the j u n c t i o n area; and the i n t e r v a l of i n t e g r a t i o n i s from the conduction-band edge of the n - side E , to the cn' valence band edge of the p - side E , i . e . , through the band overlap (Figure 2-1). S i m i l a r l y I = A vc E r VP / Z y c P v(E) f y ( E ) [ l - f c(E)]P c(E) dE J E cn ...(2-2) The general shape of the two t u n n e l i n g components as w e l l as the net t u n n e l i n g current' (I - I ) are shown i n Figure 2-3. 6 cv vc 5 rtcultc.nt ~T '1 (1) dependence of the Tunnel Current on v o l t a g e (2) dependence of the D i f f u s i o n Current on v o l t a g e (3) dependence of the Excess Current oh v o l t a g e Figure 2-3. Components of the Volt-Ampere C h a r a c t e r i s t i c of a Tunnel Diode. Next consider the excess c u r r e n t which occurs at forward v o l t a g e s i n the range where the e l e c t r o n s i n the degenerate donor l e v e l s i n the n - side have been r a i s e d to energies g r e a t e r than those of the degenerate acceptor l e v e l s i n the p - s i d e . I d e a l l y , t u n n e l i n g of e l e c t r o n s from the conduction to the valence band, i n a s i n g l e energy-conserving t r a n s i t i o n , should then be impossible and only the d i f f u s i o n current due to the forward i n j e c t i o n of m i n o r i t y c a r r i e r s should flow. In p r a c t i c e as Yajima and E s a k i f i r s t noted, the a c t u a l current at such b i a s e s i s c o n s i d e r a b l y i n excess of the normal d i f f u s i o n c u r r e n t : hence the term excess c u r r e n t . The excess c u r r e n t i s p r i m a r i l y due to the t u n n e l i n g process since i t s behaviour g e n e r a l l y p a r a l l e l s t h a t of the peak cu r r e n t ; the peak and excess cu r r e n t s e x h i b i t much of the same dependence on pressure, on temperature, and on the donor and acceptor concentrations that make up the j u n c t i o n . ^ A few hypotheses have been proposed to e x p l a i n the interband t u n n e l i n g t r a n s i t i o n s i n the excess current range; however, there i s now a growing amount of evidence that the excess current i s caused by e l e c t r o n s t u n n e l i n g not completely through the energy gap, but only part of the way, making use of more or l e s s l o c a l i z e d i m p e r f e c t i o n energy l e v e l s present i n the energy gap. This mechanism was f i r s t suggested by E s a k i and, subsequently, s e v e r a l authors ' ( l ^ ) have obtained strong c o n f i r m a t i o n of i t , by showing that the magnitude of the excess cu r r e n t can be a l t e r e d by d e l i b e r a t e l y changing the i m p e r f e c t i o n content of the c r y s t a l , e i t h e r by s u i t a b l e doping or by r a d i a t i o n damage. Based on a t h e o r e t i c a l a n a l y s i s of such a mechanism, curve (3) Figure 2-3 represents an approximation adopted by (9) Kane f o r the dependence of the excess current on the voltage a p p l i e d to the tunnel diode. 10 2.2 Temperature Dependence of Tunnel Diode D-C Parameters. The tunnel diode d-c parameters are temperature dependent. Fi g u r e 2-4 shows the v a r i a t i o n of the d-c parameters with temperature f o r a T1925 Germanium tunnel diode. As shown the peak cu r r e n t decreases by 4?o as the temperature i s lowered from room temperature to-6D°C. S l i g h t l y beyond room temperature, the peak cu r r e n t decreases w i t h i n c r e a s i n g temperature. The v a l l e y c u r r e n t , as shown, i n c r e a s e s monotonically w i t h temperature. This behaviour i s expected because of the increase i n energy gap and decrease i n t u n n e l i n g as the temperature i s lowered,and because of the smearing out of the Fermi f u n c t i o n at high (13) temperatures. ' The peak v o l t a g e Vp and forward vo l t a g e decrease n e a r l y l i n e a r l y with i n c r e a s i n g temperature at r a t e s of about 0.06 mv/°C and 1.0 mv/°C r e s p e c t i v e l y . v ( m t\ 2.3 Tunnel Diode E q u i v a l e n t C i r c u i t . 11 The tunnel diode i s a v o l t a g e c o n t r o l l e d negative r e s i s t a n c e device (VCNR). For small s i g n a l a p p l i c a t i o n s , the tunnel diode i s s a t i s f a c t o r i l y c h a r a c t e r i z e d by i t s d i f f e r e n t i a l negative r e s i s t a n c e ( - r ) , i t s j u n c t i o n capacitance C, s e r i e s F i g u r e 2-5(a). However, f o r l a r g e s i g n a l and switching mode a p p l i c a t i o n s , which are the main concern of t h i s t h e s i s , the s i g n a l e q u i v a l e n t c i r c u i t , shown i n F i g u r e 2-5(b), c o n s i s t s of three elements i n s e r i e s : an inductance L,, a r e s i s t a n c e r , d' d and a v o l t a g e dependant capacitance C(v) shunted by a n o n - l i n e a r c u r r e n t generator f ( v ) r e p r e s e n t i n g the n o n - l i n e a r d-c c h a r a c t e r i s t i c of the tunnel diode. The t u n n e l i n g phenomenon i s very f a s t and no delay e f f e c t s are i n c l u d e d i n the current generator; the j u n c t i o n capacitance i s assumed to represent a l l the storage mechanism i n t h i s d e v ice. bulk r e s i s t a n c e r , and i t s housing inductance L as shown i n e q u i v a l e n t c i r c u i t of Figure 2-5(a) was extended. (8) The l a r g e 1 (a) Small S i g n a l (b) Large S i g n a l F i g u r e 2-5. E q u i v a l e n t C i r c u i t s of Tunnel Diodes, 12 Measurements of the tunnel diode e q u i v a l e n t c i r c u i t parameters have been d e a l t with e x t e n s i v e l y i n the l i t e r a t u r e ; (14),(15),(16) o n i y a method aimed at measuring the capacitance v a r i a t i o n with v o l t a g e i s considered i n Appendix I (experimental r e s u l t s are i n c l u d e d ) . T h e o r e t i c a l l y , the v a r i a t i o n of C with v o l t a g e may be approximated by: C(v) = C Q s -n n=-2- ...(2-3) where CQ i s the capacitance of the tunnel diode at zero v o l t a g e , and V i s the v o l t a g e gap or the v o l t a g e f o r which the O capacitance would t h e o r e t i c a l l y go to i n f i n i t y . The experimental r e s u l t s i n d i c a t e t h at the exponent n v a r i e s between 0.4 and 0.5. I t w i l l be u s e f u l to d e f i n e here the v a l l e y capacitance Cy as given by the f o l l o w i n g equation: M V = co 1 - — _ _ 1 n - 2 .... (2-4) The most important parameter i n the e q u i v a l e n t c i r c u i t , apart from the capacitance, i s the n o n - l i n e a r current generator f ( v ) r e p r e s e n t i n g the v - i c h a r a c t e r i s t i c . The l a c k of a s a t i s f a c t o r y t h e o r e t i c a l ^ ' ^ 1 7 ^ ' ^ 1 8 ^ e x p r e s s i o n f o r f ( v ) l e d to the use of a n a l y t i c a l approximations which are d e a l t with i n Appendix I I . 13 3. SINGLE TUNNEL DIODE SWITCHING BEHAVIOUR As a f i r s t step, i t was thought worthwhile to analyse the performance of a s i n g l e tunnel diode and i n v e s t i g a t e i t s p o s s i b i l i t i e s as a switching d e v i c e . Against t h i s background i s developed the main subject of the t h e s i s : a treatment of the s t a t i c and dynamic behaviour of an i n t e r e s t i n g combination of ser i e s - c o n n e c t e d tunnel diodes. s i m u l a t i o n of the p e r t i n e n t n o n - l i n e a r system was c a r r i e d out. Experimental r e s u l t s were found to v e r i f y the general behaviour p r e d i c t e d by the computer s o l u t i o n s . The mathematical a n a l y s i s c l a r i f i e s diode o p e r a t i o n and provides a u s e f u l r u l e of thumb f o r the switching time as a f u n c t i o n of the device's f i g u r e of (2l ) (22) mer i t . The f i g u r e of merit / , v ' of the diode i s u s u a l l y expressed as e i t h e r the C / lp or C | r | constant of the diode; where C i s the capacitance, | r | i s the magnitude of the average negative r e s i s t a n c e and I i s the peak c u r r e n t . E m p i r i c a l l y , the r e l a t i o n between the C | r | and C/l-p constants i s given approximately by the f o l l o w i n g equation ( f o r a Ge tunnel d i o d e ) : parameters such as doping, energy gap e t c . , w i l l be considered f i r s t ; the subsequent a n a l y s i s w i l l be p u r e l y mathematical, without any p h y s i c a l c o n s i d e r a t i o n of the d e t a i l e d mechanism of S t a r t i n g with one tunnel diode, a d i g i t a l computer The f i g u r e of merit and i t s dependence on v a r i o u s 14 t u n n e l i n g . This mathematical approach may be e n t i r e l y j u s t i f i e d because the time constant of t u n n e l i n g determined by the (23 ) -13 d i e l e c t r i c r e l a x a t i o n time (10 sec.) i s much smaller than the C | r| time constant ( l 0 ~ ^ s e c ) . In other words the e q u i v a l e n t c i r c u i t of s e c t i o n 2.3 i s v a l i d over the wide frequency range of i n t e r e s t . 3.1 Dependence of the F i g u r e of M e r i t on the E l e c t r i c a l Parameters of the J u n c t i o n . The t u n n e l i n g c u r r e n t through a narrow p-n j u n c t i o n (Figure 2-1) i n any semiconductor i s p r o p o r t i o n a l to the t u n n e l i n g p r o b a b i l i t y Z = Z = Z (see equation 2-1). The V c c v (9) t u n n e l i n g p r o b a b i l i t y Z has been e v a l u a t e d v ' f o r i n d i v i d u a l e l e c t r o n t r a n s i t i o n s across a r e g i o n of constant f i e l d F, i n terms of F, the energy gap E and the reduced mass of the e l e c t r o n m and i s given by: Z = exp *4 TC m E 3/2 n e h F ...(3-1) where h = Planck's constant, e = e l e c t r o n i c charge. D e f i n i n g an average F by: We F = E , where the width W of an abrupt j u n c t i o n i s given by:'2"*' W = N + P NP 2 e E 15 e being the d i e l e c t r i c constant, and N, P the doping l e v e l s on the n and p sides of the j u n c t i o n r e s p e c t i v e l y . S u b s t i t u t i n g f o r F i n equation 3-1 we get: exp V I u 2 eh NP g . . . ( 3 -2 ) The j u n c t i o n capacitance C and the peak current lp are given by: r e A 0 ~ W ' . . . ( 3 -3 ) I p = A Z where / stands f o r the t u n n e l i n g i n t e g r a l (eqs. 2-1, 2-2) and A i s the j u n c t i o n area. Therefore: a C / I p - e(NP) (N + P) E g exp a ( * N + Pa „ i3(m e —Np-) 2 E, E . . . ( 3 -4 ) where a and P are constants. For high l e v e l s of doping, the exponential f a c t o r becomes the dominant f a c t o r i n the above expression. Thus f o r a given semiconductor the f i g u r e of merit i s not a constant but depends on the doping c o n c e n t r a t i o n : the higher the c o n c e n t r a t i o n the lower the f i g u r e of merit and the f a s t e r the swi t c h i n g . For a given doping c o n c e n t r a t i o n , the negative r e s i s t a n c e (-r) i s i n v e r s e l y p r o p o r t i o n a l to the t u n n e l i n g 16 p r o b a b i l i t y Z and depends only to a small extent on the d e n s i t y of s t a t e s . Thus equations 3-2 and 3-3 show t h a t m a t e r i a l s with small band gaps, small d i e l e c t r i c constants, and small e f f e c t i v e Table 2.1 which l i s t s the m a t e r i a l constants of s e v e r a l semi- conductors, we can see t h a t InSb and InAs are the m a t e r i a l s with the lowest f i g u r e s of m e r i t . 3.2 Tunnel Diode Switching Speed. (Figure 3-1(a)) could be an a m p l i f i e r , o s c i l l a t o r or switching element, according to the values of the d-c source v o l t a g e V Q , the e x t e r n a l r e s i s t a n c e B, and the inductance L. This inductance represents the t o t a l s e r i e s inductance of the c i r c u i t i n c l u d i n g the tunnel diode l e a d inductance. masses have the s m a l l e s t C | r j time co n s t a n t s . For the purpose of comparison, r e f e r r i n g back to The diode represented by the e q u i v a l e n t c i r c u i t : \ t—r *low^_ vo l t a g e s t a t e v. high v o l t a g e s t a t e (b). Tunnel Diode C h a r a c t e r i s t i c I l l u s t r a t i n g Switching Load L i n e . 17 (16) The s t a b i l i t y c r i t e r i a of the c i r c u i t d i c t a t e that the negative r e s i s t a n c e p o r t i o n of the tunnel diode c h a r a c t e r i s t i c i s an unstable r e g i o n i f the t o t a l c i r c u i t p o s i t i v e r e s i s t a n c e i s l a r g e r than the minimum magnitude of the negative r e s i s t a n c e of the diode. Consequently the tunnel d'iode can only switch through t h i s r e g i o n provided ^ ^ l ^ m i n * Figu r e 3-1(b) shows the s w i t c h i n g load l i n e superimposed on the tunnel diode c h a r a c t e r i s t i c . The load l i n e i n t e r s e c t s the c h a r a c t e r i s t i c i n three p o i n t s , two of which (x,y) are st a b l e and one i s unstable ( z ) . D e f i n i n g the c u r r e n t i , f ( v ) and i as shown i n Fig u r e 3-1 ( a ) ; the equations d e s c r i b i n g the l a r g e s i g n a l '.. behaviour of the tunnel diode c i r c u i t are: C f | = i - f ( v ) . . . (3 -5 ) L ~ = E - B i - v . . . (3 -6 ) i c = i - f ( v ) . . . (3 -7 ) where v = the t e r m i n a l v o l t a g e of an i d e a l diode excluding the v o l t a g e drop across the diode's s e r i e s r e s i s t a n c e , E = V n + v ; 0 s' = d-c source v o l t a g e , = t r i g g e r i n g source v o l t a g e , 18 f ( v ) = a seventh order polynomial (given by equation A I I - l ) which c l o s e l y f i t s the observed v - i c h a r a c t e r i s t i c of a 1 ma Ge tunnel diode (T1925), C = tunnel diode capacitance assumed constant and equal to the v a l l e y capacitance. I t should be mentioned t h a t equations 3-5, 3-6 are i n v a r i a n t to the f o l l o w i n g s u b s t i t u t i o n s : C = aC L' = - ; R' = - i ' = a i Q , a a c c • • • \J—O) and i ' = a i or C = bC L' = bL and t ' = bt ...(3-9) where a and b are numerical constants. Therefore we may deal with a diode f o r which C = 10 pf and l p = 1 ma without l o s s of g e n e r a l i t y . In the f o l l o w i n g a n a l y s i s , the tunnel diode was biased i n two d i f f e r e n t modes: a cur r e n t mode R = R » | r l c y ' 'mm ( h o r i z o n t a l l o a d l i n e ) , and a volt a g e mode R = R "> I r I . . The » e V / \ \ m i n s w i t c h i n g i n both cases was i n v e s t i g a t e d . However, before a n a l y z i n g the r e s u l t s i t w i l l be u s e f u l to d e f i n e the " o v e r d r i v e " * a s s o c i a t e d with a given input s i g n a l . The percent of o v e r d r i v e i s the excess i n current at the diode peak vol t a g e as a percent of the diode peak c u r r e n t . Case I : Current Bias The tunnel diode was b i a s e d by a constant d-c current source at approximately h a l f the peak cu r r e n t . P r i o r to 19 s w i t c h i n g , the diode was i n i t s low v o l t a g e s t a t e . Switching was accomplished by the a p p l i c a t i o n of a p o s i t i v e t r i g g e r c u r r e n t , d e f i n e d here as a s t e p - f u n c t i o n current source i n p a r a l l e l with the d-c power supply. When the diode i s d r i v e n from a current source, the " s m a l l " s e r i e s inductance of the tunnel diode i s swamped by the source r e s i s t a n c e and does not a f f e c t the dynamic behaviour of the o v e r a l l c i r c u i t . In t h i s case (L=0, R=RC)> the equations d e s c r i b i n g the network may be expressed i n the s i m p l i f i e d form: C = i - f ( v ) ...(3-10) i c = i - f (v) where i = I„ + i ; 1^ = d-c b i a s c u r r e n t , 0 s' 0 ' i = t r i g g e r c u r r e n t . (28) Equation 3-10 was solved by the Runge-Kutta-Gill method using the IBM 1620 computer. Figure 3-2 shows the waveforms of the v o l t a g e across the diode f o r v a r i o u s overdrive f a c t o r s . The switching time i s c u s t o m a r i l y d e f i n e d as the time f o r e i t h e r the v o l t a g e - i o r current-change to reach 90$ of the t o t a l change expected. For the case under c o n s i d e r a t i o n only the v o l t a g e s w i t c h i n g time i s s i g n i f i c a n t . A c l o s e r i n v e s t i g a t i o n of the switching t r a n s i e n t s i n Figure 3-2 shows that the switching time can be d i v i d e d i n t o a "delay time", which i s the time 20 r e q u i r e d f o r a 0 to 10% change i n output v o l t a g e ; and a " r i s e time", which i s the time r e q u i r e d f o r a 10% to 90% change i n output v o l t a g e . Figure 3-3 shows the v a r i a t i o n of switching time, r i s e time and delay time w i t h o v e r d r i v e . T.D. Voltage (v) # Overd. 1 3 . 6 % ' 2 S.6% 3 1 3 . 6 % 4 1 8 . 6 % 5 23„6ti 6 3 3 . 6 % 5 6 Nanoseconds 10 F i g u r e 3-2. Switching T r a n s i e n t v ( t ) f o r Various Overdrive F a c t o r s . (C •= 10 pf, I 1 ma, B = B ) ' c I t i s observed t h a t the v a r i a t i o n s i n overdrive have the apparent e f f e c t of changing the delay of the output waveform to (oo) a much grea t e r extent than the r i s e t i m e v . This i s due to the f a c t t h a t f o r small o v e r d r i v e the c a p a c i t i v e current i s small near the diode peak v o l t a g e , and t h i s r e s u l t s i n a slow rate of change of vol t a g e i n the i n i t i a l stage of the t r a n s i e n t . To make the curves of Figure 3-3 more g e n e r a l l y a p p l i c a b l e , the switching and delay times were normalized with r e s p e c t to the c h a r a c t e r i s t i c C | r | time constant and p l o t t e d iTime (ns) 61 IO'/- 207- 3 0 / 46'/- % Current Overdrive F i g u r e 3-3. Switching Time, Delay Time and Rise Time versus Overdrive (C = 10 pf, I p = 1 ma, R = R,) t k Normalized Time C Irl 22 versus o v e r d r i v e on a l o g - l o g s c a l e i n Figure 3-4. Given the C | r| time constant of a Ge diode, we can deduce from F i g u r e 3-4, i t s s witching and delay times f o r a given overdrive f a c t o r . The importance of the delay time i n the o p e r a t i o n of tunnel diode switching c i r c u i t s i s evi d e n t . Figure 3-5 shows the delay time f o r 10fo o v e r d r i v e as a f u n c t i o n of the diode capacitance and peak c u r r e n t . For example a 10 pf, 1 ma diode has a d e l a y of 2 ns. (ns = nanosecond = 10 ^ sec.) T>dofj hm* (ns) . 10 9,1 0,61 0,00/ 10 i / — — Jr i". — -A — / --jA- — - _. / /oe Capacitance F i g u r e 3-5. Timing Delay as a F u n c t i o n of Tunnel Diode Capacitance and Peak Current. 23 Case I I : Voltage Bias In t h i s case, both R and L were assumed f i n i t e and equations 3-5 and 3-6 were solved f o r v a r i o u s values of R and L. Switching was accomplished by the a p p l i c a t i o n of a p o s i t i v e t r i g g e r v o l t a g e d e f i n e d here as a modified s t e p - f u n c t i o n (0.5 ns r i s e time) v o l t a g e source i n s e r i e s with the d-c power supply. The v o l t a g e v ( t ) across the diode was found to e x h i b i t the same general behaviour as f o r the c u r r e n t - b i a s case. The inductance was found to a f f e c t the switching time and delay time and, to a smaller extent, the r i s e time. Figure 3-6 shows the dynamic v - i behaviour f o r v a r i o u s values of L. I t i s seen that the dynamic v - i curve f o l l o w s the load l i n e only f o r L = 1 nh. (nh = nanohenry = 10~^ henry). The l a r g e r values of inductance increase the delay time and, t h e r e f o r e , the switching time by an appreciable amount. I t should be noted that the capacitance was assumed constant throughout the a n a l y s i s . This i s a v a l i d assumption since i t was found that t a k i n g the capacitance v a r i a t i o n i n t o account had n e g l i g i b l e e f f e c t on the switching time. 3.3 Approximate Formula f o r the Rise Time of a Tunnel Diode. An approximate formula f o r the r i s e time can be e a s i l y d e r i v e d under the f o l l o w i n g assumptions: 1. The load l i n e i s assumed h o r i z o n t a l and tangent to the peak of the c h a r a c t e r i s t i c . 2. The v - i c h a r a c t e r i s t i c of the diode i s l i n e a r i z e d . ,.Current (ma) • one+-^10<?o of F i n a l Voltage Input Voltage v =0.35^ s B i a s Voltage V n = 0.44v Fig u r e 3-6. Dynamic v - i T r a n s i e n t Behaviour f o r L = 1, 10 and 100 hh. (C = 10 pf, I p = 1 ma, R = 500 ohms). From equation 3-10 we haves i = e f t c dt f o r switching we have: AV i c = C --jr where % i s the r i s e time, and i = I_ - I T r c P V 25 AV ^ V p - V p r = c d p - i v ) c_ I , V F - V P 1 IT, ...(3-11) This equation shows the dependence of the r i s e time on the f i g u r e of- merit, the forward v o l t a g e , and the peak to v a l l e y c urrent r a t i o . 3.4 Experimental R e s u l t s . The experimental c i r c u i t used to t e s t the switching time of tunnel diodes i s shown i n Fig u r e 3-7. A l l the r e s i s t o r s used were high-frequency non-inductive r e s i s t o r s and the whole c i r c u i t was b u i l t i n a c o a x i a l form. TEKTRONIX TYPE 111 PULSE GEN. - v v w - IK - r v W ( fe*,5X) 50"-- Pulse Rise Time = 0.5nsZ Pulse D u r a t i o n = 20ns T.D. under Test TEKTRONIX TYPE N SAMPLING PLUG-IN P l u g - i n Rise Time = = 0. 6ns :- Figure 3-7. Tunnel Diode Switching Test C i r c u i t . F i g u r e 3-8 shows the waveforms obtained f o r a T1925 ( l p = 1 ma, Cy = 10 pf) tunnel diode u s i n g the switching t e s t c i r c u i t . To i l l u s t r a t e the delay between input and output; one of the input s i g n a l s (waveform l ) as seen across the output 26 te r m i n a l s with the diode o p e n - c i r c u i t e d , was superimposed on the output waveforms 2 and 3 obtained with the diode i n the c i r c u i t . These output ..wavef orms were obtained f o r cur r e n t o v e r d r i v e s of 20$ and 40$ r e s p e c t i v e l y , and were d i s p l a y e d using a Tektronix type N sampling p l u g - i n . The input s i g n a l corresponds to the 20$ overdrive case. The r i s e time of the sampling p l u g - i n . i s 0.6 ns and should be accounted f o r i n the measurement, since the t o t a l r i s e time i s the square root of the sum of the squares of the scope and c i r c u i t r i s e times. F i g u r e 3-8. Experimental Switching. Waveforms ( l ns/div h o r i z o n t a l , 10 mv/dxr v e r t i c a l ) (1) Input V o l t a g e , f o r 20$ Overdrive (2) Output V o l t a g e , f o r 20$ Overdr ive (3) Output V o l t a g e , f o r 40$ Overdrive. The d i s p l a y e d r i s e times are of the order of 3.5 ns This agrees f a i r l y w e l l with the values p r e d i c t e d from.the computer s o l u t i o n s , and the value given by the approximate formula i n equations 3-11. 3.5 Summary, lo The switching time of the tunnel diode c o n s i s t s of 27 a delay time and r i s e time. Ignoring the e f f e c t of s e r i e s inductance, the delay time i s dependent on the f i g u r e of merit and on the c u r r e n t o v e r d r i v e ; the r i s e time i s dependent on the f i g u r e of merit and, to a smaller extent, on the o v e r d r i v e . 2. A l a r g e s e r i e s inductance i n the c i r c u i t i n c r e a s e s the delay time and changes the dynamic v - i path c o n s i d e r a b l y (voltage b i a s c a s e ) . 3. For a g i v e n semiconductor, the f i g u r e of merit i s dependent on the doping concentrations the higher the doping the b e t t e r the f i g u r e of m e r i t , 4. For a given l e v e l of doping, the f i g u r e of merit i s dependent on the energy gap, the d i e l e c t r i c constant, and the reduced mass of the e l e c t r o n . Semiconductors w i t h small energy gaps, d i e l e c t r i c constants, and reduced masses have the best f i g u r e s of m e r i t . Though the preceding a n a l y s i s i s based on the Ge tunnel diode, i t can be extended with minor m o d i f i c a t i o n s to other m a t e r i a l s . 28 4. STATIC AND DYNAMIC CHARACTERISTICS OP SERIES-CONNECTED TUNNEL DIODES When a number of tunnel diodes are connected i n s e r i e s w i t h a common source, a m u l t i s t a b l e v o l t a g e - c u r r e n t composite c h a r a c t e r i s t i c i s obtained. The form and complexity of the (29) c h a r a c t e r i s t i c w i l l depend on the i n d i v i d u a l elements. (2) A t t e n t i o n w i l l be confi n e d to the most i n t e r e s t i n g combination, which r e s u l t s i n 2 n s t a b l e s t a t e s generated using n negative r e s i s t a n c e elements, provided the i n d i v i d u a l elements obey c e r t a i n r u l e s d e r i v e d i n the f o l l o w i n g s e c t i o n s . As a f i r s t step, a study of the s t a t i c c h a r a c t e r i s t i c of such a composite device was undertaken to e s t a b l i s h the r e l a t i o n s h i p t h at the s t a t i c parameters of the i n d i v i d u a l elements must s a t i s f y . Experimental curves, obtained using tunnel diodes i n s e r i e s , were found c o n s i s t e n t w i t h the e s t a b l i s h e d r e l a t i o n s h i p s . Because of the i m p o s s i b i l i t y of making accurate p h y s i c a l measurements of the c i r c u i t c u rrent and vol t a g e s at high frequency, and the d i f f i c u l t y of a p u r e l y a n a l y t i c a l approach, a computer s o l u t i o n of the c i r c u i t equations was c a r r i e d out. The computer s o l u t i o n l e d to a b e t t e r understanding of the c i r c u i t o p e r a t i o n , 4.1 S t a t i c C h a r a c t e r i s t i c of Two Tunnel Diodes i n S e r i e s . device Simple g r a p h i c a l c o n s t r u c t i o n s based on the i n d i v i d u a l c h a r a c t e r i s t i c - c u r v e enable one to p r e d i c t the nature of 29 the composite c h a r a c t e r i s t i c s . F i g u r e 4-1 shows the l i n e a r i z e d c h a r a c t e r i s t i c curve of a t y p i c a l tunnel diode. For the purpose of the f o l l o w i n g d i s c u s s i o n , the range 0 < V \ V p i s d e f i n e d as a "0" or "low v o l t a g e " s t a t e , and the range v < V p i s d e f i n e d as a "1" or "high v o l t a g e " s t a t e . The negative r e s i s t a n c e r e g i o n Vp <̂  v <̂  i s not counted as a s t a t e since i t i s unstable i n most cases, M u l t i s t a b l e composite c h a r a c t e r i s t i c s are obtained when the tunnel diodes have p r o g r e s s i v e l y o v e r l a p p i n g m u l t i v a l u e d ranges and are c o n s t r a i n e d to have the m u l t i v a l u e d q u a n t i t y ( c u r r e n t ) the same at every i n s t a n t of time. To i l l u s t r a t e the composite c h a r a c t e r i s t i c s , we consider the case of two tunnel diodes connected i n s e r i e s and d r i v e n from a low impedance source. Let I p 2 > I p i ...(4-1) ^ 1 > *V2 ...(4-2) V p 2 > V p i ...(4-3) Figure 4-2(a) shows the l i n e a r i z e d i n d i v i d u a l tunnel diode c h a r a c t e r i s t i c s ( i n t h i s F i g u r e V p 2 = 2 V p ^ ) . Figure 4-2(b), the piecewise l i n e a r composite c h a r a c t e r i s t i c , was obtained u s i n g the f a c t t h a t at any p o i n t the c u r r e n t i s i d e n t i c a l i n both diodes while the t e r m i n a l v o l t a g e i s the sum of the v o l t a g e s across each. I n s p e c t i o n of the superimposed i n d i v i d u a l c h a r a c t e r i s t i c s r e v e a l s which p o r t i o n s overlap i n c u r r e n t . Each 30 4* Figure 4-1. L i n e a r i z e d C h a r a c t e r i s t i c Curve of a Tunnel Diode. (a). (b) Fi g u r e 4-2(a) (b) Tunnel Diodes I n d i v i d u a l C h a r a c t e r i s t i c s Composite C h a r a c t e r i s t i c . p a r t of an i n d i v i d u a l c h a r a c t e r i s t i c combines with every p a r t of the other, having c u r r e n t values i n common with i t , to form a l i n e a r segment i n the composite c h a r a c t e r i s t i c . In the f o l l o w i n g d i s c u s s i o n , the v a r i o u s s t a b l e s t a t e s w i l l be assigned a b i n a r y n o t a t i o n . When a s s i g n i n g a p a i r of b i n a r y d i g i t s to a . s t a t e , the d i g i t to the r i g h t (the l e s s s i g n i f i c a n t d i g i t ) w i l l r e f e r to the f i r s t tunnel diode D-̂ , and the d i g i t to the l e f t (the more s i g n i f i c a n t d i g i t ) w i l l r e f e r to the second tunnel diode D 2 « La t e r as more diodes are added, the sequence w i l l be preserved w i t h the l e a s t s i g n i f i c a n t b i t to the extreme r i g h t . R e f e r r i n g to F i g u r e 4—2(b), a short d i s c u s s i o n of the composite c h a r a c t e r i s t i c , i t s s t a b l e s t a t e s , and the order i n which they are generated, w i l l l e a d to a b e t t e r understanding of the c i r c u i t behaviour. 00 S t a t e : I n i t i a l l y both diodes are i n t h e i r low voltage s t a t e . As the a p p l i e d v o l t a g e i s i n c r e a s e d , the current and volt a g e across the diodes i n c r e a s e u n t i l the cur r e n t reaches the value of Ip^« This i s the f i r s t peak of the combined c h a r a c t e r i s t i c . Segment AW i s given a 00 b i n a r y r e p r e s e n t a t i o n , 01 S t a t e : As the input v o l t a g e i n c r e a s e s , the current decreases f o r c i n g the v o l t a g e across to increase and th a t across D 2 to decrease ( s i n c e I p 2 > I p ^ ) * Hence, D 2 remains i n i t s low vo l t a g e s t a t e while D^ switches to i t s high v o l t a g e s t a t e and segments BW, CW, DW are t r a c e d . The b i n a r y r e p r e s e n t a t i o n of CW and DW i s 01. 32 10 S t a t e : I f the voltage i s i n c r e a s e d f u r t h e r , the current and voltage i n both diodes i n c r e a s e u n t i l I p 2 i s reached (second peak of the composite c h a r a c t e r i s t i c ) . Beyond t h i s p o i n t , i n c r e a s i n g the v o l t a g e must r e s u l t i n a decrease i n current through the diodes, Along segment DX the v o l t a g e across D 2 i n c r e a s e s but t h a t across D^ decreases u n t i l p o i n t " f " i s reached. The system then switches to p o i n t " i 1 " f o r an i n f i n i t e s i m a l v o l t a g e i n c r e a s e v i a the v i r t u a l path " f g h i i ' " . This path i s the unique path from " f " to " i 1 n , : having the property of i d e n t i c a l c u r r e n t across the diodes at a l l p o i n t s . However, over t h i s path the v o l t a g e i s always l e s s than i t i s at " f " . I t i s c l e a r t h a t i n the switching process the excess (2) voltage must be taken up by a storage element ( i n t h i s case the tunnel diode c a p a c i t a n c e ) . During t h i s process, the current decreases to the low value of Iy2$ diode D 2 ends up i n i t s high-voltage s t a t e while diode D^ i s switched back to i t s low- vo l t a g e sta t e because D^ cannot support the low value of I y 2 unless i t i s near i t s zero v o l t a g e l e v e l . F u r t h e r increase i n vo l t a g e generates segments AT and AZ with a b i n a r y r e p r e s e n t a t i o n of 10. 11 S t a t e : As the c u r r e n t reaches again the value Ip-^? the t h i r d peak of the composite c h a r a c t e r i s t i c i s brought f o r t h . For any a p p l i e d v o l t a g e beyond t h i s , tunnel diode D 2 remains i n the h i g h v o l t a g e s t a t e while the v o l t a g e across D^ i n c r e a s e s and segments BZ, CZ and DZ are t r a c e d . CZ and DZ have the b i n a r y r e p r e s e n t a t i o n 11. 33 The above d i s c u s s i o n i s v a l i d only f o r i n c r e a s i n g a p p l i e d v o l t a g e . I f the process i s reversed, the path t r a c e d i n coming back i s " n m l k j i c 1 c b a " . The b i n a r y s t a t e 01 i s missing i n t h i s case, showing an i r r e v e r s i b l e e f f e c t e x h i b i t e d by the c h a r a c t e r i s t i c . 4.2 S t a t i c C h a r a c t e r i s t i c s of n Tunnel Diodes. The r e s u l t s of the preceding s e c t i o n can be extended to n elements of the type d i s c u s s e d by g e n e r a l i z i n g c o n d i t i o n s 4 r r l j 4T2, 4T3 to become: XPn > ^ ( n - l ) " - - > IP1 ...(4-4) t h This c o n d i t i o n s t a t e s t h a t the n tunnel diode cannot switch to i t s "1" s t a t e u n t i l a l l the preceding ( n - l ) tunnel diodes are i n t h e i r "1" s t a t e . XVn < Y( n-i,)"" '<*Y1 ...(4-5) When a tunnel diode switches to i t s "1" s t a t e i t o b v i o u s l y must pass through i t s v a l l e y . Then c o n d i t i o n 4r-5 ensures that i t w i l l cause a l l lower order diodes.(which must be i n the "1" s t a t e from 4-4) to switch back to the zero s t a t e . For the b i n a r y regions to have equal v o l t a g e ranges the requirement i s : VFm = ^ 1 V p i ^ m = 2 to n ...(4-6) 34 Also f o r a l l b i n a r y s t a t e s to be d i s t i n c t * n-1 V V 1 < V F „ - V p m " ^ • " ( 4 " 7 ) m=l The above c o n d i t i o n nsure the existence of 2 d i s t i n g u i s h a b l e p o s i t i v e regions d e r i v e d from n tunnel diodes. In a s p e c i a l case r e s u l t i n g from r e l a x a t i o n of c o n d i t i o n s 4-6 and 4-7,(n+l) s t a b l e p o s i t i v e regions can be generated u s i n g n d e v i c e s ; t h i s case a r i s e s from the use of diodes s a t i s f y i n g c o n d i t i o n s 4-4 and 4-5 and having i d e n t i c a l v o l t a g e c h a r a c t e r i s t i c s . By a l t e r i n g the f o u r c o n d i t i o n s , e i t h e r s i n g l y or together, i t i s p o s s i b l e to achieve c e r t a i n numbers^of p o s i t i v e r e s i s t a n c e regions between (n+l) and 2 n. 4.3 Experimental R e s u l t s . A curve t r a c e r ^ ^ was used to o b t a i n the s t a t i c c h a r a c t e r i s t i c s of s e r i e s - c o n n e c t e d tunnel diodes. The f o l l o w i n g f e a t u r e s i n c o r p o r a t e d i n the curve t r a c e r were e s s e n t i a l to minimize the p o s s i b l e o s c i l l a t i o n s i n the negative r e s i s t a n c e r e g i o n : 1. a low s e r i e s r e s i s t a n c e sweep c i r c u i t , 2. a low inductance diode t e s t mount. R e f e r r i n g to Table 2.1, i t i s seen t h a t a combination of GaAs, Ge, InAs, and InSb diodes can be made to s a t i s f y c o n d i t i o n 4-6 approximately. However, at the time of t h i s T*7 These c o n d i t i o n s are s i m i l a r to the ones obtained by R a b i n o v i c i and Renton(2) f o r n c u r r e n t c o n t r o l l e d negative r e s i s t a n c e d e v i c e s . 35 w r i t i n g , the InAs and InSb diodes were not yet commercially a v a i l a b l e and t h e i r c h a r a c t e r i s t i c v - i curve had to be simulated. The p a r a l l e l combination of a Ge tunnel diode and an o r d i n a r y Ge diode simulates the InAs diode with a forward v o l t a g e VT, = 0.275v. The p a r a l l e l combination of a Ge tunnel diode and F (*) a S i l i c o n backward diode simulates the InSb diode with a forward vo l t a g e V C = 0.15v. GCLAS Gt 6c <k Si Ce I N 653 IN 294: H5-I0o\wi^l tfo-lto\iN2<tft (b), ( c ) , (d) T.D. r.D. B. ' r.z> B,b, I T.Z> © © ® ©• F i g u r e 4.3(a) Experimental C h a r a c t e r i s t i c s of the Negative Resistance Elements (b) Two Elements Composite C h a r a c t e r i s t i c (c) Three Elements Composite C h a r a c t e r i s t i c (d) Four Elements Composite C h a r a c t e r i s t i c (2$ ma/dry v e r t i c a l ; 0.25 v/div h o r i z o n t a l ) . The backward diode i s "a tunnel diode, with a r e l a t i v e l y ' l o w i m p u r i t y c o n c e n t r a t i o n . I t i s used with reverse a p p l i e d v o l t a g e and e x h i b i t s , over that range, c h a r a c t e r i s t i c s s i m i l a r to those of a Zener reference diode but with a lower v o l t a g e drop. The vo l t a g e drop I s 0.28v f o r S i and 0.08v f o r Ge backward diodes. F i g u r e 4-3(a) shows the separate i n d i v i d u a l c h a r a c t e r - i s t i c s ( s a t i s f y i n g c o n d i t i o n s 4-5, 4-6, 4-7, and 4-8) of four negative r e s i s t a n c e elements, and the diodes used to o b t a i n these c h a r a c t e r i s t i c s . F i g u r e s 4-3(b), ( c ) , and (d), show the composite c h a r a c t e r i s t i c s having 4, 8, and 16 s t a b l e p o s i t i v e regions; these c h a r a c t e r i s t i c s were obtained using two, three, and f o u r devices r e s p e c t i v e l y . A 32 s t a t e composite c h a r a c t e r - i s t i c could be obtained by the a d d i t i o n of a f i f t h element c o n s i s t i n g of the p a r a l l e l combination of a Ge tunnel diode and a Ge backward diode. This combination would have a forward v o l t a g e of 0.08v. 4.4 Dynamic C h a r a c t e r i s t i c s . The study of the s t a t i c c h a r a c t e r i s t i c s e s t a b l i s h e d the necessary c o n d i t i o n s f o r the r e a l i z a t i o n of 2 n s t a b l e s t a t e s f o r n tunnel diodes. Tire1 se c o n d i t i o n s h o l d when volt a g e and current v a r i a t i o n s occur at speeds w e l l below the switching speed of the d e v i c e . However, at h i g h f r e q u e n c i e s , the e f f e c t s of storage and s t r a y elements a s s o c i a t e d w i t h the device must be taken i n t o c o n s i d e r a t i o n ; ' ^ and the c o n d i t i o n s d e r i v e d from s t a t i c c o n s i d e r a t i o n s become necessary but not s u f f i c i e n t f o r proper o p e r a t i o n . I t i s the purpose of the f o l l o w i n g i n v e s t i g a t i o n t o : (1) determine the exact i n f l u e n c e of the c i r c u i t parameters on the response, (2) formulate some e m p i r i c a l c o n d i t i o n s which w i l l ensure c o r r e c t o p e r a t i o n and f a s t response. 3 7 Most of the i n v e s t i g a t i o n was c a r r i e d out by computer s o l u t i o n of the c i r c u i t '\. equations. Only a two tunnel diode c i r c u i t was i n v e s t i g a t e d and the r e s u l t s g e n e r a l i z e d wherever p o s s i b l e . 4 . 5 Two Tunnel Diode M u l t i s t a t e C i r c u i t . The c i r c u i t i n v e s t i g a t e d i s shown i n Figure 4 - 4 ( a ) a n d . i t s e q u i v a l e n t c i r c u i t i n F i g u r e 4 - 4 ( b ) . With minor a l t e r a t i o n s , t h i s c i r c u i t could represent a f u l l b i n a r y adder, a two-bit a n a l o g - t o - d i g i t a l converter or a counter. I t c o n s i s t s of a Ge ( 1 N 2 9 4 1 ) and a GaAs ( 1 N 6 5 3 ) tunnel diode i n s e r i e s . The s t a t i c c h a r a c t e r i s t i c s of the i n d i v i d u a l diodes s a t i s f y the c o n d i t i o n s d e r i v e d i n s e c t i o n 4 . 2 . The diodes are b i a s e d from a constant current source I Q J where Ip^ yIQ yIy^. The t r i g g e r i n g source v g i s an i d e a l v o l t a g e source. The l o a d l i n e r e s i s t a n c e R i s chosen to be s l i g h t l y l a r g e r than the magnitude of the l a r g e r of the two negative r e s i s t a n c e s of the diodes (see Appendix I I I ) . This choice ensures t h a t : (a) R i s small enough f o r a l l the s t a b l e o p e r a t i n g p o i n t s to be a c c e s s i b l e , and (b) the negative r e s i s t a n c e regions of the composite c h a r a c t e r i s t i c are u n s t a b l e . I f v^ and v 2 are the v o l t a g e drops across the f i r s t and second diode r e s p e c t i v e l y ( n e g l e c t i n g s e r i e s r e s i s t a n c e ) , and i i s the instantaneous c u r r e n t , then the equations d e s c r i b i n g the behaviour of the c i r c u i t are: 38 ~L C M l + (X ^ O Gads (a) ft A c r |4, /ft | i . I f t - - i rh £ 2 ( ^ J * ^ | r 1 (b) F i g u r e 4-4. Two Tunnel Diode: M u l t i s t a t e C i r c u i t . L f t = E - R i " V l " V 2 .,.(4-8) dv k k dt = ( i + I Q ) - f k ( v k ) k = 1, 2 ...(4-9) ck ( i + I Q ) - f f c ( v k ) k = 1, 2 ...(4-10) where L = t o t a l inductance i n the c i r c u i t , E = V. + v , b s' = v o l t a g e across the b l o c k i n g c a p a c i t o r C^ (C^ i s assumed a short c i r c u i t at the ck f r e q u e n c i e s of i n t e r e s t ) c a p a c i t i v e c u r r e n t , and 39 f 1 ( v ^ ) , ^2^ v2^ a r e S i v e n by equations AII-3 and AII-4 r e s p e c t i v e l y . These equations c l o s e l y approximate the e x p e r i - mental v - i c h a r a c t e r i s t i c s of the diodes, the d-c parameters 6£ which are given i n Table 4.1. Comparison of the c i r c u i t behaviour w i t h C, h e l d constant r a t h e r than v a r i a b l e i n d i c a t e d k only t r i v i a l d i f f e r e n c e s . Consequently most of the computation was c a r r i e d out with C, constant. I t should be noted that equations 4-8 and 4-9 are i n v a r i a n t to the s u b s t i t u t i o n s given i n equations 3-8 and 3-9. h V P V V V P |r| 6lV ~ TV1 ~ IV2 ma ma V V V ohms . ma Diode D 1 4.56 1.16 0.06 .0.3 0.48 30 Ge 1N2941 0.82 Diode D 2 4.86 0.34 0.10 0.6 1.0 40 GaAs 1N653 Table 4.1 D-C Parameters of the Two Diodes 4.6 Switching Behaviour. The tunnel d i o d e s ' switching behaviour, under the a p p l i c a t i o n of modified step v o l t a g e s with 0.5 ns r i s e time, was i n v e s t i g a t e d . Three input v o l t a g e s of amplitudes 0.45, 0.90 and 1.35v, cause the diodes to switch from the 00 sta t e to the 01, 10, and 11 s t a t e s r e s p e c t i v e l y . The superimposed s t a t i c and dynamic v - i c h a r a c t e r i s t i c s of the two diodes are shown i n F i g u r e s 4-5(a), 4-6(a) and 4-7(a). The parameters of the c i r c u i t under i n v e s t i g a t i o n are a l s o l i s t e d i n these F i g u r e s . In t h i s i n i t i a l study, the inductance L was assumed zero. This s i m p l i f i e s ^Current (ma) 01 STATE C± = 10 pf c 2 = 10 pf L = 0 B = 50-n- v. = 0.45v (0.5ns r i s e time) = 2.41 ma Voltag e (v) o.l \ 0.8 Current.. ( Voltage (v) (a) (b) Figure 4-5(a) Dynamic v - i C h a r a c t e r i s t i c s of the M u l t i s t a t e C i r c u i t ( 0 0—01) (b) Voltage and Current Waveforms ( 0 0 — 0 1 ) . 41 the s o l u t i o n , cuts down on computation time and makes the behaviour of the c i r c u i t r e l a t i v e l y e a s i e r to analyse. The e f f e c t s of inductance on the switching t r a n s i e n t w i l l be t r e a t e d i n a l a t e r s e c t i o n . 01 S t a t e : I n i t i a l l y both diodes are i n t h e i r low voltage s t a t e . A t r i g g e r s i g n a l w i t h 0.45v amplitude i s a p p l i e d to the c i r c u i t (see F i g u r e 4 . 5 ( a ) ) . During the r i s e time of the s i g n a l , the curr e n t and vo l t a g e across both diodes i n c r e a s e . When the step reaches i t s steady value, a t r a n s i e n t r e l a x a t i o n process occurs. Both diodes switch towards t h e i r high v o l t a g e state and the s e r i e s current decreases to f o l l o w the t r a n s i e n t load l i n e . Unbalance i n the inherent switching-time constants causes the f a s t e r diode D^ to race diode D^ with r e s p e c t to voltage i n c r e a s e . E v e n t u a l l y , diode D^ can cause the current i to decrease to such an extent t h a t the c a p a c i t i v e current i ^ becomes zero and the vol t a g e across D 2 ceases to i n c r e a s e . F u r t h e r decrease i n i , discharges capacitance causing diode D 2 to switch back to i t s low v o l t a g e s t a t e while D^ assumes i t s hig h v o l t a g e counterpart. The f i n a l adjustment to the quiescent c o n d i t i o n relaxes very r a p i d l y . A p l o t of the t r a n s i e n t v o l t a g e s and current versus time i s shown i n Fi g u r e 4-5(b). 10 S t a t e : A t r i g g e r s i g n a l with 0.9v amplitude i s a p p l i e d to the c i r c u i t . When the steady s t a t e i s reached, diode D^ i s i n i t s "0" s t a t e while diode D i s i n i t s "1" s t a t e . The switching C u r r e n t (ma) Voltage (v) 'ma) Voltage (v) (a) (b) F i g u r e 4-6(a) Dynamic v - i C h a r a c t e r i s t i c s of the M u l t i s t a t e C i r c u i t ( 0 0 — i (b) Voltage and Current Waveforms ( 0 0 — 1 0 ) . 4̂ 43 behaviour d i f f e r s r a d i c a l l y from the preceding case. When the step reaches i t s steady v a l u e , each diode r e l a x e s towards i t s h i g h v o l t a g e s t a t e (see F i g u r e 4-6(a))« The c u r r e n t decreases u n t i l a p o i n t i s reached where the c a p a c i t i v e current i ^ becomes zero and the v o l t a g e across stops i n c r e a s i n g ; t h i s occurs when i s i n i t s h i g h v o l t a g e s t a t e . Any f u r t h e r decrease i n i discharges while i s charged; t h i s e f f e c t i s due to the c o n s t r a i n t s e s t a b l i s h e d by the d-c c h a r a c t e r i s t i c s of the diodes|. Therefore diode i s momentarily turned on, and then r e l a x e s to i t s low v o l t a g e s t a t e , g i v i n g r i s e to what may be termed a "precursor" p U i s e 0 f d u r a t i o n (see Figure 4.6(b)). The p o r t i o n of the r e l a x a t i o n process during which the dynamic op e r a t i n g p o i n t of t r a v e r s e s i t s v a l l e y , makes up the l a r g e s t p o r t i o n of 11 S t a t e : A t r i g g e r v o l t a g e w i t h 1.35v amplitude i s a p p l i e d to the c i r c u i t . In the steady s t a t e , both diodes end up i n t h e i r high v o l t a g e s t a t e . Quiescent c o n d i t i o n s are reached more r a p i d l y than i n the previous two cases. Figure 4-7(a) and Figure 4-7(b) show the dynamic behaviour and the current and voltages as a f u n c t i o n of time f o r t h i s p a r t i c u l a r case. From the above d i s c u s s i o n and the t r a n s i e n t waveforms, i t i s obvious that the d u r a t i o n of the p r e c u r s o r pulse can be considered as a measure of the speed of o p e r a t i o n of the c i r c u i t . For the case considered here, the steady s t a t e s corresponding to a l l the output p o s s i b i l i t i e s can be achieved i n T 2 = 8 ns. 4.7 E f f e c t of Capacitance on T r a n s i e n t Behaviour. (a) (J) F i g u r e 4-7(a) Dynamic v - i C h a r a c t e r i s t i c s of the M u l t i s t a t e C i r c u i t ( 0 0 — l l ) (b) Voltage and Current Waveforms (00 — l l ) . 4̂ For L = 0, the s w i t c h i n g times are dependent ori the figures of merit of both diodes: the smaller the f i g u r e s of merit the smaller the o v e r a l l switching times. However, i t was found t h a t the capacitance r a t i o ; :C 2 a f f e c t s the switching time and the o p e r a t i o n of the c i r c u i t . F igure 4-8 shows the t r a n s i e n t v o l t a g e waveforms f o r capacitance r a t i o s of 1:2, 1:1 and 1:0.75. The waveforms f o r the three s t a t e s are shown; the v o l t a g e input being a step i n each case. A comparison of the waveforms shows that f o r a capacitance r a t i o of 1:2, the general response i s slowed down and the d u r a t i o n and peak amplitude of the p r e c u r s o r pulse are q u i t e l a r g e . The r a t i o 1:0.75 o f f e r s a b e t t e r response and a reduced peak amplitude f o r the precursor p u l s e . This p a r t i c u l a r r a t i o corresponds to equal C | r | time constants f o r the two d i o d e s . F u r t h e r i n c r e a s e i n the capacitance r a t i o C^:C 2 does not improve the response and may cause f a l s e o p e r a t i o n . An example of f a l s e o p e r a t i o n (waveforms not shown) i s obtained when the capacitance r a t i o i s i n c r e a s e d to 1:0.5. In t h i s case, an input corresponding to the 01 s t a t e , causes the c i r c u i t to end up i n the 10 s t a t e ; tunnel diode i s switched p a r t i a l l y on and then r e s e t to the low v o l t a g e s t a t e while diode P 2 s e t t l e s i n i t s v a l l e y v o l t a g e r e g i o n . The above d i s c u s s i o n suggests the use of the capacitance r a t i o as an a d d i t i o n a l parameter which can be adjusted i n d e s i g n i n g f o r f a s t response. The diode capacitances should be chosen as small as p o s s i b l e , and the diode with the smaller band gap should have the l a r g e r c a p a c i t a n c e . Furthermore, the Figure 4-8» E f f e c t of Capacitance on T r a n s i e n t Behaviour (R = 50-n.) 47 capacitance r a t i o should f a l l i n the range 1< ( C ^ : ^ ) <1.9 f o r f a s t response and proper o p e r a t i o n . The r a t i o (C-^:C^) = 1:0.75 (equal time constants) seems to o f f e r optimum response. 4.8 E f f e c t of the D-C Parameters of the Tunnel Diodes on the Switching Behaviour, For f i x e d c a p a c i t a n c e s , the d i f f e r e n c e between the v a l l e y currents of the diodes (5ly = (^y^ ~ \2^ determines to a great extent the d u r a t i o n of the p r e c u r s o r pulse 1^' the l a r g e r £)Ly, the l a r g e r the c a p a c i t i v e c u r r e n t d i s c h a r g i n g capacitance C-̂  and t h e r e f o r e the f a s t e r the r e l a x a t i o n t r a n s i e n t towards a s t a b l e s t a t e . To show the e f f e c t of 6ly on the d u r a t i o n of T^> the Ge diode v - i c h a r a c t e r i s t i c used i n the preceding a n a l y s i s was r e p l a c e d by a new c h a r a c t e r i s t i c i d e n t i c a l to the f i r s t , but with a v a l l e y current of 0,656 ma. i . e . , (5ly was reduced from 0,82 ma to 0,32 ma. This r e d u c t i o n in(5ly i n c r e a s e d the p r e c u r s o r pulse d u r a t i o n to 19 ns ( i t was p r e v i o u s l y 8 ns.? see F i g u r e 4-8(b) f o r the case ^ = 0 , 2 = 10 p f ) . F i g u r e 4-9 shows the dynamic v - i c h a r a c t e r i s t i c f o r the 10 s t a t e with(5lv = 0.32 ma. The change i n (^ly d i d not a f f e c t the values of the switching time f o r the 01 and 11 s t a t e s by any a p p r e c i a b l e amount. The d i f f e r e n c e between the peak currents of the two diodes i s h o t c r i t i c a l as f a r as s w i t c h i n g time or proper o p e r a t i o n i s concerned, however," i t i s w e l l worth mentioning that the l a r g e r the I /I r a t i o f o r each i n d i v i d u a l diode, the f a s t e r the p v c i r c u i t response. 48 STATE 10 Current (ma) r _ i n n f C 2 = 10 pf Voltage (v) Figu r e 4-9. Dynamic v - i C h a r a c t e r i s t i c f o r 6lv = 0.32 m a (R = 50 a). The e f f e c t s of Vp , Vy, V p on the switching behaviour were not i n v e s t i g a t e d since these v o l t a g e s were assumed to be f i x e d f o r a given semiconductor m a t e r i a l . 4.9 E f f e c t of Inductance on T r a n s i e n t Behaviour. Only small values of inductance were considered i n t h i s i n v e s t i g a t i o n . F i g u r e 4-10 shows the output voltage waveforms f o r two values of induetance?10 and 30 nh. With the low r e s i s t a n c e l o a d l i n e s and l a r g e input voltages encountered i n the m u l t i s t a t e c i r c u i t , the main e f f e c t s of inductance on the switching behaviour are: F i g u r e 4-10. E f f e c t of Inductance on Tr a n s i e n t Behaviour c1 = 1 4 . 4 ( 1 - 576)"V; c 2 = 1 4 . 4 ( 1 - R = 50^- , v g ( 0 . 5 n s r . t . ) = 0 . 4 5 , 0.90, 1 . 3 5 v r e s p e c t i v e l y f o r 0 1 , 1 0 , 1 1 states. 50 1. The inductance introduces a small delay i n the output v o l t a g e waveform. 2. The higher value of inductance (30 nh) causes an overshoot i n the i n i t i a l r i s e of the v o l t a g e wave- form. This overshoot decays i n a damped o s c i l l a t o r y manner towards the steady s t a t e . This i s shown i n F i g u r e s 4-10(a), 4-10(b). 3. The higher value of inductance causes a f a s t e r r e l a x a t i o n of the diode t r a n s i e n t towards i t s s t a b l e s t a t e . This shortens the d u r a t i o n of the precursor pulse and i n c r e a s e s the speed of o p e r a t i o n (see Figure 4-10 (b)). Hence, by proper choice of the inductance and load l i n e r e s i s t a n c e , the i n d u c t i v e t r a n s i e n t can be made to c o n t r i b u t e r e l a t i v e l y f l a t top output waveforms, and to speed up the c i r c u i t o p e r a t i o n . 4.10 E f f e c t of Input Pulse Rise Time on Transient Behaviour. The behaviour of the c i r c u i t f o r input steps with 3, 5, 10 ns r i s e time was i n v e s t i g a t e d . The f o l l o w i n g observations were made? 1. The dynamic v - i c h a r a c t e r i s t i c s i n a l l cases are q u i t e s i m i l a r to those ; shown i n Figures 4-5(a), 4-6(a), 4-7(a) except f o r the f a c t t h at the slower the pulse r i s e time, the c l o s e r i s the dynamic path to the d-c c h a r a c t e r i s t i c s of the .diodes,i.e., the smaller the 51 c a p a c i t i v e c u r r e n t s , 2, The d u r a t i o n of the p r e c u r s o r pulse i s to a l a r g e extent independent of the r i s e time, 3, Over the p o s i t i v e r e s i s t a n c e regions, the instantaneous dynamic oper a t i n g p o i n t f o l l o w s the input pulse r i s e time, 4, Over the negative r e s i s t a n c e regions, the diodes switch, and the switching speed depends mainly on the C | r | time constant and the c a p a c i t i v e current; the l a t t e r depends on the r i s e time, 4,11 E f f e c t of Temperature on the C i r c u i t Operation, Table 4,2 gives the temperature c o e f f i c i e n t s of the important d-c parameters f o r the Ge and GaAs tunnel diodes. Prom t h i s t a b l e and the d i s c u s s i o n i n s e c t i o n 4.8 i t i s obvious that the d i f f e r e n c e i n the v a l l e y - c u r r e n t temperature c o e f f i c i e n t s w i l l s e r i o u s l y a f f e c t the switching speed. I t was estimated that a change of + 25°C i n temperature produces a corresponding change of + 15$ i n s w i t c h i n g speed ( T 2 ) . The d i f f e r e n c e i n the forward—voltage temperature c o e f f i c i e n t s w i l l a f f e c t both the s w i t c h i n g time and the s t a b l e steady s t a t e s ; however, t h i s e f f e c t i s not very c r i t i c a l . S i m i l a r i l y , the d i f f e r e n c e i n peak-voltage c o e f f i c i e n t s does not a f f e c t the o p e r a t i o n of the c i r c u i t to any a p p r e c i a b l e extent. Hence, i n choosing tunnel diodes f o r a p p l i c a t i o n i n a m u l t i s t a t e c i r c u i t , i t i s advisable to p i c k diodes having the same v a l l e y - c u r r e n t temperature c o e f f i c i e n t s . 52 Coef f i c i ent Symbol Germanium (1N2941) Gallium Arsenide (1N653) Peak Point voltage Temp. Coeff . ** AV p /AT -60 (xV/°C -120 [J,V/°C V a l l e y Point voltage Temp. Coeff . ** AV y /AT -1 mv/°C Forward Point voltage Temp. Coeff . ** AV F /AT -1 mv/°C -1 mv/°C -1 .5 mv/°C V a l l e y Point current Temp. Coeff . * I y AT 0.75%/°C 0.6/o/°C AT = T - T R * Measured T = operating temperature ** Manufacturer's Data T R = room temperature Table 4 .2 Temperature Coeff ic ients of the D-C Parameters for the Ge and GaAs Tunnel Diodes 4.12 Extension to Three Tunnel Diodes i n Ser ies . If three tunnel diodes are connected i n s er i e s , eight stable states resu l t (see section 4 . 3 ) . Table 4.3 gives the eight states of the composite device i n terms of the ind iv idua l diode's states . 53 State D 3 D 2 1 0 0 0 2 0 0 1 3 0 1 0 4 0 1 1 5 1 0 0 6 1 0 1 7 1 1 0 8 1 1 1 Table 4.3 Stable States f o r Three Tunnel Diodes Device I t i s seen that f o r every s t a t e except 5, the switching process i n v o l v e s only two tunnel diodes and t h e r e f o r e the preceding a n a l y s i s can be used to o b t a i n an estimate of the switching time. For s t a t e 5, the t h i r d tunnel diode switches to i t s "1" state and f o r c e s the two preceding diodes to switch back to t h e i r "0" s t a t e . The switching behaviour i s complicated i n t h i s case and the d u r a t i o n of the r e l a x a t i o n t r a n s i e n t s f o r diodes D^ and D^ w i l l determine to a c e r t a i n extent the speed of o p e r a t i o n of the c i r c u i t (see s e c t i o n 4.13). 4.13 Experimental R e s u l t s . F i g u r e 4-11 shows the experimental response of the c i r c u i t of Figure 4-4 to the a p p l i c a t i o n of pulses of amplitudes 0.45, 0.90 and 1.35v. The pulses were obtained from a Tek t r o n i x 54 111 pulse generator and have a r i s e time of 0.5 ns, a f a l l time of 1 ns, and a d u r a t i o n of 20 ns. The response was observed on a Tek t r o n i x 581 o s c i l l o s c o p e ( r i s e time 3.3 n s ) . The parameters of the experimental c i r c u i t ( 0 ^ = 2 0 pf, C y 2 = 18 pf, R = 50 ohms) were approximately the same as those of the c i r c u i t analysed on the computer i n s e c t i o n 4.9 (waveforms of Fi g u r e 4-10). No attempt was made to evaluate the.inductance i n the experimental c i r c u i t . The experimental r e s u l t s ( t a k i n g i n t o account the r i s e time of the o s c i l l o s c o p e ) show r e l a t i v e l y good agreement with the computer s o l u t i o n s . The d u r a t i o n of the prec u r s o r pulse shown i s approximately 12.5 ns; t h i s would l i m i t the p o s s i b l e pulse r e p e t i t i o n r a t e to 80 Mc/s. 01 State •4- 10 State A 11 State m t 1 IS. X I Diode D. Diode D, Figure 4-11. Experimental Voltage Waveforms f o r Two Tunnel Diode C i r c u i t (0.5 v/diY v e r t i c a l ; 10 ns/diiv h o r i z o n t a l ) 55 001 11 111 r nif 010 H I ' i 1111 3 111 A : N T* I' | Diode D3 Diode D. Diode D, Figure 4-12, Experimental Voltage Waveforms f o r a Three Tunnel Diode C i r c u i t (0.5 v/div v e r t i c a l 5 lOns/clvV h o r i z o n t a l ) 56 Fi g u r e 4-12 shows the response of the eigh t s t a t e c i r c u i t (see s e c t i o n s 4.3 and 4.12) to pulses of magnitude 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4v. The loa d l i n e r e s i s t a n c e was 25 ohms and the b i a s c u r r e n t I Q = 4 ma. The time necessary f o r the diodes to r e l a x to t h e i r s t a b l e stateswas approximately 20 ns. The c i r c u i t was slowed down i n t h i s case because of the recovery time of the germanium diode (Q5-100) used to shunt the f i r s t tunnel diode to o b t a i n a low forward v o l t a g e . 4.14 Summary. 1. The speed of o p e r a t i o n of the c i r c u i t i s l i m i t e d by the d u r a t i o n of the pr e c u r s o r p u l s e . 2. The switching times of the c i r c u i t are dependent on the f i g u r e s of merit of the i n d i v i d u a l diodes and on the r a t i o of the capacitances C ^ > the l a t t e r should be i n the range 1^(0-^ S C 2 X I .9 f o r best response. 3. The d u r a t i o n of the prec u r s o r pulse i s i n v e r s e l y p r o p o r t i o n a l to (5ly« The temperature dependence thus a f f e c t s the switching time. To reduce the e f f e c t of temperature on the switching time, i t i s a d v i s a b l e to p i c k diodes having equal v a l l e y - c urrent temperature c o e f f i c i e n t s . 4. A proper choice of s e r i e s inductance can be made to co n t r i b u t e r e l a t i v e l y f l a t - t o p output waveforms and to speed up the c i r c u i t o p e r a t i o n . 57 5. APPLICATIONS The m u l t i s t a t e composite devices d i s c u s s e d i n the preceding chapter can be used to f u l f i l l a v a r i e t y of d i g i t a l (2) f u n c t i o n s . Some of the p o s s i b l e a p p l i c a t i o n s w i l l be d i s c u s s e d b r i e f l y 7 p o i n t i n g out the advantages and disadvantages of these d e v i c e s . 5.1 F u l l B i n a r y A d d e r . ^ 2 ^ ' ^ 3 5 ^ The c i r c u i t i s shown i n Figure 4-13 and has three inputs d r i v e n from v o l t a g e sources. The c i r c u i t r e c e i v e s "1" or "0" signals on each of i t s three input channels and y i e l d s on i t s two outputs: Sum and Carry a 0,0; 0,1; 1,0; or 1,1 according to whether none, one, two, or a l l three input s i g n a l s are 1. The d e t a i l e d a r i t h m e t i c f u n c t i o n performed by t h i s c i r c u i t i s shown i n Table 5.1, where X,T,Z denote the three v a r i a b l e inputs r e p r e s e n t i n g the b i n a r y numbers to be added, and the Sum and Carry i n d i c a t e the r e s u l t a n t outputs. There are eig h t p o s s i b i l i t i e s corresponding to the four output s t a t e s . One notes from Table 5.1 that i f any of the inputs are bias e d "ON" ( l ) or "OFF" (0) a v a r i e t y of u s e f u l l o g i c a l f u n c t i o n s (35) can be performed. I f input Z i s so adjusted as to be always "OFF"; the top of Table 5.1 shows t h a t : Sum i s assigned "1" i f and only i f "1" i s assigned to e i t h e r X or T but not both. The b i n a r y e x c l u s i v e "or" i s thus achieved. Furthermore Carry i s assigned "1" j u s t Gt 0 Sum CJI, M C a r r y - 1 Figure 5-1. F u l l B inary Adder. Inputs X Y z 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 CARRY Table 5.1 Truth t a b l e of F u l l B i n a r y A d d i t i o n 59 i n t h a t s i n g l e case where "1" are assigned to both X and Y. The connective "and" i s thus r e a l i z e d . Suppose however that Z i s always "ON"; the bottom h a l f of the t a b l e r e v e a l s that "1" i s assigned to Sum j u s t i n those cases where both X and Y r e c e i v e l i k e assignments of "1" or "0". The b i n a r y f u n c t i o n " i f and only i f " i s achieved., On the other hand, Carry i s assigned "0" j u s t i n that case where both X and Y are assigned "0"; the " i n c l u s i v e or" i s t h e r e f o r e r e a l i z e d . There are a number of u n d e s i r a b l e f e a t u r e s to be noted i n the f u l l adder d e s c r i b e d above. F i r s t l y , the adder has no gain so t h a t a m p l i f i c a t i o n at the output i s necessary f o r fan-out. Secondly, the c i r c u i t does not provide u n i d i r e c t i o n a l i t y i n the flow of i n f o r m a t i o n and i s o l a t i o n from the f o l l o w i n g stages. This problem i s encountered i n most tunnel diode c i r c u i t s and any (36) of the e x i s t i n g methods of c a s c a d i n g v ' can be used. T h i r d l y , the Sum output i s above ground. The adder, however, o f f e r s the advantages of c i r c u i t s i m p l i c i t y , s t a b i l i t y , high speed of o p e r a t i o n and compactness whieh make i t w e l l s u i t e d f o r m i c r o m i n i a t u r i z a t i o n and c i r c u i t i n t e g r a t i o n . (37) 5.2 A n a l o g - t o - D i g i t a l Converter. ' The f u l l b i n a r y adder d e s c r i b e d above becomes a two-bit a n a l o g - t o - d i g i t a l converter by removing the steady s t a t e current I Q and t r i g g e r i n g from a s i n g l e source. The analog v o l t a g e i s sampled f i r s t h a n d the sampling pulses are f e d to the diode composite c i r c u i t . The output of the a n a l o g - t o - d i g i t a l converter i s a v a i l a b l e i n two forms. The voltage across the tunnel diode 60 stack i s a s t a i r c a s e with incremental steps equal to the forward v o l t a g e of the lowest order diode,.. Binary output i s a l s o a v a i l a b l e by sensing the vol t a g e s across the i n d i v i d u a l diodes. The c i r c u i t can be made to encode the analog s i g n a l without i n i t i a l sampling; however, timing problems may a r i s e i n t h i s The r e s o l u t i o n of the converter can be extended to four b i t s by using f o u r negative r e s i s t a n c e devices (see s e c t i o n 4.3). The extension of the converter's r e s o l u t i o n i s l i m i t e d by the p r e s e n t l y a v a i l a b l e d i odes. The f o l l o w i n g u n d e s i r a b l e f e a t u r e should be noted. An accurate encoder n e c e s s i t a t e s very c a r e f u l l y c o n t r o l l e d v - i c h a r a c t e r i s t i c s f o r each diode. This imposes a severe r e s t r i c t i o n on the generation of a p r e c i s e predetermined composite c h a r a c t e r i s t i c u s ing diodes. Means of o b t a i n i n g a negative r e s i s t a n c e device that avoids t h i s shortcoming have been (2) d i s c u s s e d by R a b i n o v i c i and Renton . When the accuracy i s not c r i t i c a l , however, the tunnel diode c i r c u i t has the advantage of s i m p l i c i t y and high speed of o p e r a t i o n . A t h r e e - b i t encoder was found to have an inherent encoding time of 20 ns. 5.3 Counter. Counting to a base n can be performed by any device that possesses n d i s t i n g u i s h a b l e s t a t e s which can be a t t a i n e d i n a d e f i n i t e sequence. The counter i s designed to r e c e i v e a set of pu l s e s . The system changes stat e a f t e r each incoming pulse, and remains i n i t s hew s t a t e u n t i l the a r r i v a l of the next p u l s e . 61 t h The n i n c i d e n t p u l s e , i n a counter to the base n, i s made to t r i g g e r a r e s e t c i r c u i t which ret u r n s the counter to zero, and at the same time d e l i v e r s a c a r r y output to the next stage. The m u l t i s t a t e tunnel diode c i r c u i t can be used as a counter. The diodes are bia s e d from a constant d-c current source I Q such t h a t Iy-^^ I Q ̂  l p ^ . A p p l i c a t i o n of pulses of proper amplitude enables the stack to switch from one state to the next. The stack v o l t a g e waveform takes the appearance of a s t a i r c a s e . When t h i s s t a i r c a s e reaches the appropriate l e v e l , i t t r i g g e r s the reset c i r c u i t . R e s e t t i n g i s accomplished by removal of I Q , e i t h e r by shunting the diodes, or by o p e n - c i r c u i t i n g the source. A f t e r the r e s e t p u l s e , I Q flows again, and the tunnel diodes are a l l i n t h e i r "0" s t a t e , ready to form another s t a i r c a s e upon f u r t h e r a p p l i c a t i o n of the input p u l s e s . 62 6. CONCLUSION The main object of the study i s to i n v e s t i g a t e the behaviour of a simple combination of s e r i e s - c o n n e c t e d tunnel diodes from the s t a t i c and dynamic p o i n t s of view. 2 n s t a b l e s t a t e s are obtained by i n t e r c o n n e c t i n g n tunnel diodes. While i n p r i n c i p l e , t h i s technique i s v a l i d f o r any number n, i t i s at present l i m i t e d by the semiconductor m a t e r i a l s a v a i l a b l e f o r the f a b r i c a t i o n of tunnel diodes. In the course of t h i s study, the dynamic behaviour of a s i n g l e tunnel diode, and the dependence of i t s switching time on the f i g u r e of merit and current overdrive (see Summary, S e c t i o n 3.5) i s d i s c u s s e d to provide a background f o r the study of the m u l t i s t a t e c i r c u i t . A study of the s t a t i c c h a r a c t e r i s t i c of the composite device r e v e a l s the necessary c o n d i t i o n s f o r the generation of the r e q u i r e d number of s t a b l e s t a t e s . The dynamic behaviour of a two tunnel diode c i r c u i t was i n v e s t i g a t e d by d i g i t a l computer s o l u t i o n s of the n o n - l i n e a r system to o b t a i n an estimate of the s w i t c h i n g time. Experimental r e s u l t s compatible with the computer s o l u t i o n s are presented, The most important conclusions drawn from the dynamic i n v e s t i g a t i o n are: ( l ) The r a t i o of the capacitances of the two tunnel diodes i s an important parameter i n determining the o p e r a t i o n and speed of the c i r c u i t . This r a t i o must l i e w i t h i n a c e r t a i n range to ensure the existence of the s t a b l e s t a t e s p r e d i c t e d from the s t a t i c a n a l y s i s . 63 (2) The d i f f e r e n c e i n the diodes' v a l l e y c u r r e n t s , and i t s dependence on temperature a f f e c t the switching speed of the composite d e v i c e . The m u l t i s t a t e c i r c u i t has many a p p l i c a t i o n s i n the performance of d i g i t a l l o g i c f u n c t i o n s such as b i n a r y a d d i t i o n , a n a l o g - t o - d i g i t a l conversion,and counting. These a p p l i c a t i o n s , together with many u s e f u l features,make the m u l t i s t a t e c i r c u i t w e l l worth f u r t h e r i n v e s t i g a t i o n . Some of these f e a t u r e s a re: 1. inherent h i g h speed of op e r a t i o n , 2. s i m p l i c i t y of design, 3. small s i z e and small number of components, which make the c i r c u i t w e l l s u i t e d f o r m i c r o m i n i a t u r i z a t i o n . APPENDIX I 64 A l . Measurement of Tunnel Diode Parameters The tunnel diode e q u i v a l e n t c i r c u i t parameters need to be known f o r n e a r l y every a p p l i c a t i o n . For h i g h frequency applica-^ t i o n s ^ t h e j u n c t i o n capacitance and l e a d inductance are b e l i e v e d to set the upper l i m i t on usable frequency, and to determine the a t t a i n a b l e s w i t c h i n g speeds. A method of measuring these parameters using a modified General Radio 1602 -A U.H.F. Admittance Bridge i s d e s c r i b e d here. A I . l Bias C i r c u i t and S t a b i l i t y To measure the tunnel diode parameters, some means must be provided to b i a s the tunnel diode to the d e s i r e d operating p o i n t . I f t h i s o p e r a t i n g p o i n t i s i n the p o s i t i v e conductance r e g i o n , then i n s t a b i l i t y does not occur and any convenient b i a s i n g arrangement can be used. However, i f the d e s i r e d o p e r a t i n g p o i n t i s i n the negative conductance r e g i o n , then the network may be u n s t a b l e , and s p e c i a l precautions must be taken to o b t a i n the s t a b i l i t y necessary f o r a c c u r a t e , s t e a d y - s t a t e , s m a l l - s i g n a l measurements. I t cannot be overemphasized that the measuring a-c s i g n a l across the tunnel diode must be l e s s that 5 mv to prevent e r r o r s due to the n o n - l i n e a r i t y of the diode c h a r a c t e r - i s t i c s . The 1602-A Admittance Bridge presents a n e g l i g i b l e impedance to the unknown,and has a d-c path; t h e r e f o r e , t h e b i a s i n g 65 c o n f i g u r a t i o n shown i n Figure A l - l a was used. From the e q u i v a l e n t c i r c u i t of F i g u r e A l - l b , with the diode b i a s e d i n i t s negative r e s i s t a n c e r e g i o n , the necessary and s u f f i c i e n t c o n d i t i o n s f o r s t a b l e o p e r a t i o n a r e : ^ ^ r j ^ r < R, + r d < | r | and L < | r | 2 C where | r [ i s the absolute magnitude of the negative r e s i s t a n c e at the p o i n t under c o n s i d e r a t i o n , and L i s the t o t a l s e r i e s inductance i n the c i r c u i t . . . . A l - l a . . . A l - l b T.D. under Test , — A A / W - R, « fit C(v) •-r Tunnel Diode E q u i v a l e n t C i r c u i t Bridge Input Impedance Neglected (a) (b) F i g u r e AI-1 (a) Tunnel Diode Test C i r c u i t (b) E q u i v a l e n t C i r c u i t 66 AI.2 Tunnel Diode Test Mount The t e s t mount f o r the diode must be designed to minimize s t r a y inductance and capacitance. F i g u r e A i r 2 shows a t e s t mount made of r e a d i l y a v a i l a b l e General Radio connectors* R^ i s a d i s c r e s i s t o r which i s e s s e n t i a l l y n o n - i n d u c t i v e , has n e g l i g i b l e shunt capacitance and i s chosen to s a t i s f y the i n e q u a l i t i e s AI-1 f o r the diode under t e s t with L = 2L^ ( i t has been estimated that the bridge and mount inductance i s of the same order as L , ) . The d i s c r e s i s t o r R, i s l o c a t e d as c l o s e to a 1 "the diode as p o s s i b l e , i n order to minimize s t r a y inductance. AI.3 Experimental Measurements and R e s u l t s F i g u r e AI-3a shows the complete diagram of the t e s t c i r c u i t and F i g u r e AI-3b shows the bridge e x t e r n a l connections. The s e r i e s inductance, being constant f o r . a given package design, need be measured only once,using a dummy diode. The s e r i e s r e s i s t a n c e may be measured at c o n d i t i o n s of very l a r g e reverse v o l t a g e . For the measurement of the diode admittance,the procedure i s as f o l l o w s ; l ) The bridge i s connected as shown i n F i g u r e AI-3b. The connections from the generator and the s i g n a l d e t e c t o r have been interchanged. Normally, the s i g n a l i s a p p l i e d at the generator end, and i t appears without a t t e n u a t i o n across the unknown impedance*, only a p o r t i o n of the s i g n a l i s detected. However, a p p l y i n g the s i g n a l from the generator at the d e t e c t o r 67 874-QNJ Connector -874-QUP Connector D i s c R e s i s t o R, «.562 B. Tunnel Diode Fig u r e AI-2 Tunnel Diode C o a x i a l Mount h-p 608-D VHF SIGNAU GENERATOR Detector Input 80 Mc/s 3 50 mv Co a x i a l mount -»-To Bridge ̂ Unknown— Generator Inpu RADAR RECEIVER Conductance Standard Susc eptance Standard Tuned to 80 Mc/s (a) (b) Figure AI-3(a) Diagram of the Test C i r c u i t (b) Bridge E x t e r n a l Connections 68 te r m i n a l ensures that t h i s s i g n a l i s attenuated (40 db) before being f e d to the r e s t of the bri d g e c i r c u i t s the s i g n a l appearing across the tunnel diode can then be of the order of a few m i l l i v o l t s while"" the d e t e c t e d s i g n a l i s attenuated by the same amount as f o r normal o p e r a t i o n . Another m o d i f i c a t i o n was introduced i n the bridge to allow f o r measurement of negative conductance/ t h i s was achieved by r o t a t i n g the c o u p l i n g loop a s s o c i a t e d with the m u l t i p l i e r l e v e r through 90°. 2) The admittance of the diode mount must be measured wit h the d i s c r e s i s t o r , a n d with a dummy s h o r t - c i r c u i t e d diode r e p l a c i n g the a c t u a l diode. This admittance value i s subtr a c t e d from the readings taken with the diode i n place to o b t a i n the true diode values f o r v a r i o u s b i a s v o l t a g e s . Figure AI-4 shows the t y p i c a l admittance c h a r a c t e r i s t i c s of a Ge 1N2939 diode as a f u n c t i o n of b i a s v o l t a g e . Measurements were made at 80 Mc/s. Fig u r e AI-5 shows the experimental v a r i a t i o n of capacitance with b i a s as obtained from the above measurements. The capacitance v a r i a t i o n f o r t h i s s p e c i f i c diode can be expressed ass C(v) = n pf n = 0.42 1 - 0.6 The t o t a l inductance of the mount and diode was found to be 10 nh, from the s h o r t - c i r c u i t e d diode reading. G-Cmmho) 3( m mho) Y s h o r t C r c u i f -. 51.5 m h o • 10 Figure AI-4 Admittance C h a r a c t e r i s t i c s of a 1N2939 Diode as a F u n c t i o n of Voltage / / i (v\ >• " * — • • • — * - • — » 4 c . >3.Z Pt x 4.1' 10 Figure. AI-5 Capacitance V a r i a t i o n as a F u n c t i o n of Voltage 70 APPENDIX I I A l l . Methods of Approximating Tunnel Diode Curves In the study of the tunnel diode switching c i r c u i t s i t i s necessary to represent the tunnel diode c h a r a c t e r i s t i c i = f ( v ) i n terms of an a n a l y t i c a l e x p r e s s i o n . Most of the t h e o r e t i c a l expressions f o r the tunnel diode c h a r a c t e r i s t i c are q u i t e elaborate and u n s u c c e s s f u l i n t h e i r r e p r e s e n t a t i o n of the observed c h a r a c t e r i s t i c . This i s mainly due to the f a c t t h a t the r e l a t i o n s h i p between the v a r i o u s c u r r e n t s flowing through t h e diode i s not y e t w e l l understood. The only t h e o r e t i c a l e xpression found to approximate ( w i t h i n 20fo) the observed Germanium tunnel diode c h a r a c t e r i s t i c i s the one developed by (9) (2l) Kane, ' I t c o n s i s t s of three components* the t h e o r e t i c a l e xpression f o r the t u n n e l i n g c u r r e n t ( d i r e c t t u n n e l i n g c a s e ) , one tenth of the peak c u r r e n t f o r the excess c u r r e n t , and an expression of the form Kexp (40v) f o r the d i f f u s i o n c u r r e n t , where K i s a constant. However, t h i s type of r e p r e s e n t a t i o n was found to be quite inadequate f o r the case of GaAs diodes. The l a c k of a s a t i s f a c t o r y t h e o r e t i c a l expression f o r the c h a r a c t e r i s t i c curve l e d to the c o n s i d e r a t i o n of a n a l y t i c a l approximations. Two types of approximations were used, the f i r s t i s a polynomial approximation, and the second i s a two term exponential approximation. Both approximations are concerned with the c h a r a c t e r i s t i c i n the forward v o l t a g e r e g i o n . In the reverse v o l t a g e r e g i o n , the conduction c u r r e n t can be approximated by an expr e s s i o n of the form I = exp (-0v) where *J and 0. are 71 c onstants. i A I I . l Polynomial Approximation The polynomial, approximation was obtained by the method of l e a s t mean squares u s i n g the IBM 1620 computer. The polynomial contains as many terms as necessary i n order to b r i n g the standard e r r o r of the dependent v a r i a b l e (current) w i t h i n the range s p e c i f i e d . I t should be noted however, that each polynomial generated contains a constant term,and t h e r e f o r e the approximate curve does not go through the o r i g i n i n the i - v plane. This e r r o r i s n e g l i g i b l e since the p o r t i o n of the curve near the o r i g i n i s of l i t t l e p r a c t i c a l importance i n most cases. An attempt to make the curve go' through the o r i g i n r e s u l t e d i n a very poor o v e r a l l accuracy. F i g u r e AII-1 shows, f o r a t y p i c a l Ge tunnel diode (T1925), the experimental c h a r a c t e r i s t i c and the c h a r a c t e r i s t i c c a l c u l a t e d from a seventh order p o l y n o m i a l . The cur r e n t d i f f e r e n c e between the a c t u a l and c a l c u l a t e d curves i s normalized with respect to the peak c u r r e n t and p l o t t e d on the lower p o r t i o n of the graph. I t i s seen that the approximation i s accurate to w i t h i n + 5$ over the whole range. A s i m i l a r approximation was obtained f o r a t y p i c a l GaAs tunnel diode (1N653). In t h i s case a n i n t h order polynomial ensured a maximum e r r o r of + 2$,except at the or i g i n 7 w h e r e i t i s 4.3$. Table A I I . l gives the polynomials f o r the two diodes. 72 4-y. -4X Experimental v v \\ \\ ... — — Approximation 1 A V '/ if if ll II if % ^*****»^ • E r r o r / i o.t 0.3 • * • • » • • • 1 • • • • • • • • • 1 • • F i g u r e AII-1 T1925 Germanium Tunnel Diode (a) A c t u a l and C a l c u l a t e d C h a r a c t e r i s t i c s (b) P e r c e n t-Error Between the A c t u a l and C a l c u l a t e d C h a r a c t e r i s t i c s Diode I Polynomial Approximation ( l p = 1 ma) Ge f(v).= 0.03281 + 44.6474v - 683.013v 2 + 3744.82v 3 - 7211.73V 4 - 5624.81v 5 + 36259.62v 6 - 33114.68v 7. ...AII-1 GaAs f ( v ) = -0.043518 + 25.372706v - 207.321040v 2 + 676.02962v 3 - 1086.3072v 4 + 899.342610v 5 - 497.832v 6 + 490.08273v 7 - 459.487v 8 + I60i9966v 9. ...AII-2 Table A I I . l Polynomial Approximations f o r Tunnel Diodes 73 A l l . 2 Two Term Expon e n t i a l Approximation The second method of approximation uses a two term exponential expression. (19)»(20) approximation i s of the form: f ( v ) = h + I 2 = A v exp (-av) + B(exp(bv) - l ) where a, b, A, B are c o n s t a n t s . The f i r s t term represents the t u n n e l i n g c u r r e n t and the second i s the d i f f u s i o n current (see Figure 2-3). The four constants are determined by measuring f o u r sets of c u r r e n t and v o l t a g e values at f o u r p i l o t p o i n t s to be f i t t e d a c c u r a t e l y . Two of the p o i n t s should be i n the tunnel c u r r e n t r e g i o n and the other p o i n t s i n the d i f f u s i o n current r e g i o n . The accuracy of t h i s type of approximation i s + 10$. Table A l l . 2 gives the expressions f o r a Ge (IN2941) and a GaAs (IN 653) d i o d e s . The main advantage of t h i s method of approximation i s that i t does not r e q u i r e a C o m p u t e r L a n d d i f f e r e n t I p / l y r a t i o s can be represented f o r the same type of tunnel diode by changing the two constants B and b. This l a s t f e a t u r e was found q u i t e u s e f u l i n the i n v e s t i g a t i o n of the series-connected tunnel diode c i r c u i t . I t should be noted t h a t Tables A I I . l and AH.2 give the approximations f o r diodes with peak cu r r e n t s normalized to one ma. 74 Diode E x p o n e n t i a l Approximation (Ip = 1 ma) Ge f ( v ) = 0.044v exp(-l6.8v) + 5.4x10 7 ( e x p ( l 5 . 4 v ) - l ) ...AII-3 GaAs f ( v ) = 0.026v. exp(_-10v) + 1. 5xlO~ 7 (exp (8.76v)-l) ...AII-4 Table AII.2 E x p o n e n t i a l Approximations f o r Tunnel Diodes 75 APPENDIX I I I A I I I . F a c t o r s I n f l u e n c i n g the Choice of the Load Line Resistance R i n a M u l t i s t a t e C i r c u i t In choosing a value f o r R i n a two tunnel diode m u l t i - s t a t e c i r c u i t , we must compromise between the n e c e s s i t y of making a l l the s t a b l e s t a t e s a c c e s s i b l e (i»e., R as small as p o s s i b l e ) , and the requirement t h a t the negative r e s i s t a n c e regions be unstable (switching l o a d l i n e ; R l a r g e ) . A s t a b i l i t y t e s t can be a p p l i e d to give the minimum l o a d r e s i s t a n c e f o r switching, provided L and C f e(k :yi>'2) *are known. The problem of determining whether a given p o i n t on the s t a t i c c h a r a c t e r i s t i c i s s t a b l e when the composite device i s connected i n a given e x t e r n a l c i r c u i t , c a n be answered i n terms of l i n e a r network a n a l y s i s . The device i t s e l f i s r e p l a c e d i n the c i r c u i t by a small s i g n a l l i n e a r e q u i v a l e n t c i r c u i t , which i s v a l i d f o r small v a r i a t i o n s of c u r r e n t and volt a g e about the p o i n t i n q u e s t i o n . The e q u i v a l e n t c i r c u i t must include a l l s t r a y (32) r e a c t i v e elements. The c h a r a c t e r i s t i c equation ' of the system can be d e r i v e d from the set of d i f f e r e n t i a l equations r e p r e s e n t i n g the l i n e a r i z e d c i r c u i t . I f the roots of the c h a r a c t e r i s t i c equation have p o s i t i v e r e a l p a r t s , the p o i n t i n v e s t i g a t e d i s u n s t a b l e . In other words, the t r a n s i e n t f o l l o w i n g any small displacement from the p o i n t i s a sum of i n c r e a s i n g e x p o n e n t i a l s . For the set of two tunnel diodes, the c h a r a c t e r i s t i c r o o t s are given as eigenvalues a n , a , a of the matrix: 7 6 evaluated at the p o i n t of i n t e r e s t . This matrix i s obtained, by (33) adapting,., to our s p e c i f i c case, Moser's i n v e s t i g a t i o n of the s t a b i l i t y of two i d e n t i c a l tunnel diodes i n s e r i e s . With L and C ^ f i x e d , one c o n d i t i o n f o r i n s t a b i l i t y i n the negative r e s i s t a n c e r e g i o n i s that the load l i n e r e s i s t a n c e R should be l a r g e r than the magnitude of the l a r g e r of the two negative r e s i s t a n c e s of the di o d e s . I t was found that a value of R = 50 ohms, f o r the c i r c u i t under c o n s i d e r a t i o n i n s e c t i o n 4.5,gave a f a i r margin of i n s t a b i l i t y - s u f f i c i e n t f o r s a t i s f a c t o r y s w i t c h i n g . 77 REFERENCES 1. E s a k i , L., "New Phenomenon,in Narrow P-N J u n c t i o n s " , Phys. Rev. L e t t e r s , V o l . 109, p. 603, 1958. 2. Renton, C , and R a b i n o v i c i , B., "Composite C h a r a c t e r i s t i c s of Negative Resistance Devices and Th e i r A p p l i c - a t i o n s i n D i g i t a l C i r c u i t s " , Proc. IRE, V o l . 50, pp. 1648-55, J u l y , 1962. 3. Reed, D.E., "The V a r i a b l e Capacitance Parametric A m p l i f i e r " , IRE Trans. PGED, V o l . ED-6, pp. 216-21, A p r i l , 1959. 4. Zener, C , "Theory of the E l e c t r i c a l Breakdown of S o l i d D i e l e c t r i c s " , Proc. Royal Soc., V o l . 145, pp. 523- 528, 1934. 5. Kleenknecht, H., "Indium Arsenide Tunnel Diodes", S o l i d State E l e c t r o n i c s , V o l . 2, pp. 133-142, 1961. 6. E s a k i , L», and Tajima, T,, "Excess Noise .in Narrow Germanium P-N J u n c t i o n s " , J . Phys. Soc. Jap., V o l . 13, pp. 1281-1287, November, 1958. 7. 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S c h u l l e r , M. , and Gartner, W.M., "Large S i g n a l C i r c u i t Theory f o r Negative Resistance Diodes i n P a r t i c u l a r Tunnel Diodes", Proc. IRE, V o l . 49, pp. 1268-78, August, 1961. 26. Gummel, H.K., and Smith F.M., "Margin C o n s i d e r a t i o n s f o r an E s a k i Diode OR Gate", B e l l Sys. Tech. J . . V o l . 40, pp. 230-32, January, 1961. 27. S a r a f i a n , G.P., "Tunnel Diode Threshold L o g i c " , Wescon IRE Conv. Rec., V o l . 9, P a r t 2, pp. 271-76, 1961. 28. R a l s t o n , A., a n d W i l f , H.S., E d i t o r s , "Mathematical Methods f o r D i g i t a l Computers", Wiley, New York, 1960. 29. Johnston, R.A., and Harbourt, CO., " S t a t i c C h a r a c t e r i s t i c s of Combinations of Negative Resistance Devices", Proc. N a t l . E l e c t r o n i c s Conf., Chicago, V o l . 16, pp. 427-37, I960. 79 30. Harbourt, C , "The Dynamic Behaviour of Negative Resistance Devices", Trans. AIEE, Communications and E l e c t r o n i c s , pp. 216-22, J u l y , 1962. 31. Herzog, G.B., "Tunnel Diode Balanced P a i r Switching A n a l y s i s " , RCA Rev., V o l . 23, pp. 187-214, June, 1962. 32. Cunningham, W.J., " I n t r o d u c t i o n to Nonlinear A n a l y s i s " , McGraw H i l l , New York, Ch. 10, 1958. 33. Moser, K.J., " B i s t a b l e Systems of D i f f e r e n t i a l Equations w i t h A p p l i c a t i o n s to Tunnel Diode C i r c u i t s " , IBM J . of Res, and Dev.. V o l . 5, PP. 226-40, J u l y , 1961. 34. Rutz, R.F., "Two C o l l e c t o r T r a n s i s t o r f o r Binary F u l l A d d i t i o n " , IBM J . of Res, and Dev., V o l . 1, pp. 212-22, J u l y , 1957. 35. Dunham, B., "The Multipurpose B i a s Device", IBM J . of Res. and Dev.. V o l . 1, pp. 117-129, March, 1957. 36. Sims, R . C , Beck, E.R. , and Kamm, V.C., "A Survey of Tunnel- Diode D i g i t a l Techniques", Proc. IRE, V o l . 49, pp. 136-148, January, 1961. 37* Beddoes, M.P., and Salama, C.A., "Tunnel Diodes", U.B.C. Engineer, V o l . 2, pp. 18-21, 1962.

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