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The Detection of alpha particles with superconducting tunnel junctions Wood, Gordon Harvey 1969

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THE DETECTION OF ALPHA PARTICLES WITH SUPERCONDUCTING TUNNEL JUNCTIONS by GORDON HARVEY WOOD B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1963 M.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f PHYS ICS  The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , Canada i i ABSTRACT A superconducting t h i n f i l m tunnel j u n c t i o n (Sn-SnO -Sn) of o -4 2 t o t a l t h i c k n e s s 4000 A, area 7 x 10 cm and normal (4.2 K) r e s i s t a n c e 77 mft was prepared on a g l a s s substrate. When cooled to 1.2 K the j u n c t i o n was biased at 0.3 mV where, the Josephson supercurrent having been suppressed w i t h a magnetic f i e l d , the j u n c t i o n dynamic r e s i s t a n c e had i t s maximum value of 9.3ft . The j u n c t i o n was then bombarded w i t h 5.1 MeV alpha p a r t i c l e s and the r e s u l t i n g pulses induced i n the t u n n e l i n g current were observed to have amplitudes up to 19 times the preamplifier-dominated rms output noise l e v e l . | For purposes of a n a l y s i s , i t was assumed that the induced current pulse had the form i ( t ) = i e x p ( - t / t ) , t >, 0. With t h i s form of the ! current pulse and the known t r a n s f e r f u n c t i o n of the transmission l i n e -a m p l i f i e r system, i t was c a l c u l a t e d that f o r a l l pulses T = (1.38±.33)xl0 ^ sec and t h a t f o r the l a r g e s t amplitude p u l s e s , corresponding to an energy l o s s AE £ 2.75 MeV, i l a y i n the range 20 $ i " £ 26 uA w i t h a most probable a ' o ° o r value of 22 uA. With t h i s value of i and AE = 2.75 MeV, an upper l i m i t of -3 ° ot 8.2 x 10 eV has been assigned to the'value .of'w(Sn), the average energy expended by the alpha p a r t i c l e to e x c i t e a q u a s i p a r t i c l e p a i r i n super-conducting t i n at 1.2 K. A t e n t a t i v e theory of the superconducting tunnel j u n c t i o n charged p a r t i c l e detector i s given and the cryogenic and e l e c t r o n i c apparatus required f o r the measurements are described. D e t a i l s r e l a t e d to t h i n f i l m j u n c t i o n f a b r i c a t i o n technology and i n t e r p r e t a t i o n of dc experimental r e s u l t s are discussed i n four appendices. i i i TABLE OF CONTENTS Page Chapter 1 - INTRODUCTION . . . . . . 1 A. M o t i v a t i o n 1 B. P r i n c i p l e of Operation . . , , . . . 4 1. Phenomenological Treatment 4 2. Mi c r o s c o p i c Viewpoint 6 3. D i f f e r e n c e from Intermediate State Bolometers . . 7 C. Expected S i g n a l 7 D. Noise , . 9 E. Preview of Results 9 F. Thesis O u t l i n e 11 Chapter 2 - THEORETICAL ASPECTS OF.THE SUPERCONDUCTING TUNNEL JUNCTION 13 A. I n t r o d u c t i o n 13 B. Tunneling Between Normal Metals 13 1. Tunneling Current . . . . . 15 2. Dependence of Tunnel Current upon B a r r i e r Parameters 18 C. Review of Superconductivity Theory 21 1. Two F l u i d Model . . 21 2. Energy Gap . . . 22 3. D i s t r i b u t i o n Function f o r Q u a s i p a r t i c l e s 27 4. Density of States 27 D. Si n g l e Q u a s i p a r t i c l e Tunneling . . 28 i v Page 1. Tunneling Between a Normal Metal and a Superconductor 28 2. Tunneling Between Two Superconductors . . 29 3. The S 1~S 2 Tunneling Current 30 E. Josephson Tunneling 32 1. Unfavourable Aspects of the Josephson E f f e c t . . . 33 2. Theory of the dc Josephson E f f e c t 35 3. E f f e c t of Magnetic F i e l d on dc Josephson Current . 36 4. Suppression of Josephson Supercurrent . . . . . . . 39 F. M u l t i p a r t i c l e Tunneling Phenomena . . 39 Chapter 3 - OPERATING PRINCIPLES OF THE SUPERCONDUCTING CHARGED PARTICLE DETECTOR . . . 40 A. I n t r o d u c t i o n 40 1. Present Methods of Charged P a r t i c l e Spectrometry . 41 2. Charged P a r t i c l e Detection A p p l i c a t i o n s of Superconducting Devices . . 42 B. Detection of Microwave R a d i a t i o n w i t h Tunnel Junctions . 45 1. Microwave and I n f r a r e d Photon Detection . . . . . . 45 2. Microwave Phonon Detection 48 i C. Detection of I o n i z i n g R a d i a t i o n . . . . . . . . 49 D. Optimum J u n c t i o n Type and Constituent M a t e r i a l . . . . . 50 1. Type of J u n c t i o n to be Used as a Detector 50 2. Type of Superconductor i 58 E. E x c i t a t i o n s i n the Tunnel J u n c t i o n Charged P a r t i c l e Detector 62 F. Small S i g n a l Equivalent C i r c u i t . 63 1. General Treatment of S i g n a l 64 2. Small S i g n a l A n a l y s i s 66 3. D e r i v a t i o n of Small S i g n a l Parameters 67 v Page G. Estimate of the S i g n a l S i z e 68 1. Macroscopic o r Phenomenological Approach 69 2. "M i c r o s c o p i c " Approach 71 H. Tunneling P r o b a b i l i t y 73 I . P r a c t i c a l Operation of Detector 76 J . Noise ; • • i 77 1. Shot Noise on Bi a s i n g ; Current 78 2. Johnson Noise 78 3. Generation-Recombination Noise ... 78 4. V a r i a t i o n s i n I n s u l a t o r Thickness . 78 5. F l u c t u a t i o n s i n S i g n a l 79 K. Summary 79 Chapter 4 - CRYOGENIC APPARATUS . . 81 A. I n t r o d u c t i o n .| , 81 B. Dewars and Dewar Cap 81 C. Pumps 84 D. Pressure-Temperature Measurements 84 E. Sample Mounts r . . . 84 Chapter 5 - ELECTRICAL MEASUREMENT TECHNIQUES 89 A. dc Measurements 89 1. I n t r o d u c t i o n . . . 89 2. Power Supply 89 3. S h i e l d i n g ;. . 89 4. D i s p l a y of I-V C h a r a c t e r i s t i c s 91 B. Helmholtz C o i l s f o r Magnetic B i a s i n g . . . . 91 C. Pulse Detection E l e c t r o n i c s . 93 v i Page 1. B i a s i n g and Pulse C i r c u i t r y 93 2. P r e a m p l i f i e r Design 93 3. A n c i l l a r y E l e c t r o n i c s 96 D. In t e r f e r e n c e Vetoing System 96 E. E l e c t r i c a l P r o p e r t i e s of Transmission L i n e 99 1. I n t r o d u c t i o n . . : 99 2. E f f e c t i v e Line Impedance 101 3. Line Input Impedance w i t h P r a c t i c a l Loads 102 4. C h a r a c t e r i s t i c Impedance . . . . . 106 5. Consistency Check of Measured Values w i t h Theory. . 107 6. Propagation V e l o c i t y 107 F. Summary 110 Chapter 6 - RESULTS I l l A. dc C h a r a c t e r i s t i c s I l l 1. Temperature Dependence of Tunneling Current . . . . I l l 2. Magnetic F i e l d Dependence of Tunneling Current . . I l l i 3. Determination of Dynamic Resistance (8V/8I) as a Function of Voltage 115 4. D e t e r i o r a t i o n of Specimens a f t e r Thermal C y c l i n g . 117 5. Magnetic F i e l d Dependence of Energy Gap 119 B. Observation of Pulses from Alpha P a r t i c l e s . . . . . . . . 121 1. Counting of Pulses 121 2. Pulse C h a r a c t e r i s t i c s . . 125 3. Pulse Height Spectrum . . 127 C. Noise 129 1. Observed Noise . ; ' 129 2. O r i g i n of Noise . t 129 D. Determination of J u n c t i o n Capacitance 136 v i i Page 1. P a r a l l e l P l a t e Model. 136 2. Noise Measurements 138 E. Summary 143 Chapter 7 - ANALYSIS OF RESULTS 145 A. I n t r o d u c t i o n ; 145 B. D e r i v a t i o n of Detector-Transmission-Line-Amplifier Transfer Function 146 1. Small S i g n a l Equivalent C i r c u i t 146 2. Laplace Transform Representation . . 149 3. Inverse Transform 150 4. J u s t i f i c a t i o n of Approximate Transmission L i n e Treatment 152 C. Determination of Current Pulse Parameter x 152 1. E x t r a c t i o n of the x^ and o\ from Photographs . . . 153 2. M* C a l c u l a t i o n . . 154 3. V a l i d i t y of the Assumed Input Current Pulse . . . . 157 D. Estimate of Current Pulse Amplitude ( i ^ ) - 157 1. Transfer Function f o r Input Current Step Pulse . . 159 2. E v a l u a t i o n of GR' . . . . . , . 160 3. E v a l u a t i o n of i . . :. . 160 o E. Energy Loss per Q u a s i p a r t i c l e . . . 161 1. Number of Q u a s i p a r t i c l e s Produced 161 2. Energy Loss Corresponding to Maximum Amplitude Pulse . . . 164 3. Estimate of w 165 F. Energy Loss and D i f f u s i o n . . . 166 1. Energy Loss Processes . . . . . 166 2. Energy Transfer . . . . . . . 168 3. Evidence f o r Heat C o n t r i b u t i o n from Substrate . . . 170 v i i i Page G. Summary 172 Chapter 8 - CONCLUSIONS 174 A. Major Results 174 B. E v a l u a t i o n of the Device as a Nuclear Spectrometer . . . 175 1. Energy R e s o l u t i o n * 175 2. L i n e a r i t y 176 3. Stopping E f f i c i e n c y 177 4. Advantages over Conventional Spectrometers . . . . 177. 5. Disadvantages f o r Operation as a Spectrometer . . . 178 C. Future Work . 178 1. Use of Junctions w i t h Larger Dynamic Resistance . . 178 2. Thermal Decoupling of J u n c t i o n from Substrate . . . 179 3. P r e a m p l i f i e r Improvements 179 Appendix A - JUNCTION PREPARATION .- 180 A. Substrate P r e p a r a t i o n . 180 B. Evaporation Procedure . . . . 180 1. Base F i l m 180 2. Oxi d a t i o n 182 3. Top F i l m Evaporation 182 4. Attachment of E l e c t r i c a l Leads and Mounting . . . . 182 C. Evaporation Apparatus 183 1. Masks 183 2. Substrate Holder . . . . . . 183 3. Evaporation Sources ;. 185 4. F i l m Thickness Measurement 185 i x Page Appendix B - dc CHARACTERISTICS OF LEAKY SPECIMENS 188 A. I n t r o d u c t i o n 188 B. Results 188 1. I-V C h a r a c t e r i s t i c 188 2. Magnetic F i e l d Dependence of Supercurrent 190 C. A n a l y s i s of Results 190 1. I-V C h a r a c t e r i s t i c s f o r V > 0.1 mV 190 2. Supercurrent 193 3. I-V C h a r a c t e r i s t i c , f o r 0 < V j 0.1 mV . 196 D. Unresolved Problems i n the A n a l y s i s . . 197 1. Excess Currents . -. 197 2. S t r u c t u r e near V = 0 199 Appendix C - OBSERVATIONS ON Pb-Pb TUNNEL JUNCTIONS 201 A. Tunnel J u n c t i o n Results , 201 1. J u n c t i o n P r e p a r a t i o n 201 2. Low Temperature dc C h a r a c t e r i s t i c s 202 3. Conclusions From Pb-Pb J u n c t i o n Studies 202 B. I n v e s t i g a t i o n of Nodule Growth on Lead Films 203 1. D e s c r i p t i o n of Nodules. . . i 203 2. O r i g i n of Nodules—The Result of Thermal Treatment. 203 3. Tests Made to I s o l a t e Nodule-Producing Parameters . 205 4. Conclusions from Tests; 207 C. Oxide Growth on Lead Films , 208 Appendix D - EFFECT OF FINITE FILM RESISTANCE ON TUNNEL JUNCTION CHARACTERISTICS 210 A. I n t r o d u c t i o n . . 210 B. Experimental Observations of E f f e c t s w i t h Tunnel Junctions 210 x Page 1. "Negative Resistance" 210 2. S t r i p I n t e r s e c t i o n Angle-Dependence of Slope. . . . 213 C. T h e o r e t i c a l I n v e s t i g a t i o n s 213 D. Experimental Simulation of Crossed-Film Junctions . . . . 217 1. Graphite Coated Paper 217 2. Soldered Manganin S t r i p s 217 3. Compressed Nichrome S t r i p s 219 E. Angular Dependence of Thin F i l m J u n c t i o n Resistance . . . 223 B i b l i o g r a p h y ; 226 x i LIST OF TABLES Table Page 1- 1 P a r t i c l e D e t e c t i o n Parameters f o r Several M a t e r i a l s . . . 3 2- 1 Comparison of T h e o r e t i c a l and Experimental Values of 2A(0) 26 3- 1 Comparison of Parameters f o r M-S and S-S J u n c t i o n s . . . . 57 3-2 Comparison of Figures of M e r i t f o r Sn and Pb 61 3- 3 Tunneling P r o b a b i l i t y per sec f o r Several Tunneling Thicknesses 76 4- 1 Comparison of Transmission Line C h a r a c t e r i s t i c s 88 5- 1 E l e c t r o n i c U n i t s used on the Experiment 99 5- 2 Comparison of Experimental and T h e o r e t i c a l Transmission L i n e Parameters 108 6- 1 Comparison of Experimental and T h e o r e t i c a l Maximum Supercurrent 113 6- 2 Specimen C h a r a c t e r i s t i c s Before and A f t e r Thermal C y c l i n g 118 7- 1 Estimate of i f o r Maximum Amplitude Pulses 161 o D-l Dependence of J u n c t i o n Resistance on I . . 213 D-2 Least Squares F i t Parameters. 221 D-3 J u n c t i o n Resistance Asymmetry Observed 225 x i i LIST OF FIGURES Figure Page 2-1 14 2-2 P o t e n t i a l Energy Diagram f o r I d e a l M-I-M S t r u c t u r e . . . . 16 2-3 I n s u l a t o r Thickness vs Tunnel J u n c t i o n Resistance . . . . 20 2-4 Temperature Dependence of Energy Gap Parameter A(T) . . . 25 2-5 Q u a s i p a r t i c l e Energy Diagram.for S^-I-S2 J u n c t i o n . . . . 31 2-6 31 2-7 I-V C h a r a c t e r i s t i c f o r J u n c t i o n D i s p l a y i n g dc Josephson 34 2-8(a) 37 2-8(b) Dependence of I m a x Upon Applied Magnetic F i e l d B . . . . 37 3-1 Schematic Energy Diagrams D e p i c t i n g E f f e c t s of Microwave Photon or Phonon Absorption and Phonon Generation . . . . 47 3-2 52 3-3 Energy l o s s of 5.1 MeV a - P a r t i c l e Traversing F i l m 60 3-4 T y p i c a l dc I-V Curves f o r Tunnel J u n c t i o n f o r Various 65 3-5 Small S i g n a l Equivalent C i r c u i t f o r Tunnel J u n c t i o n . . . 65 4-1 82 4-2 85 4-3 86 5-1 90 5-2 92 5-3 Schematic of J u n c t i o n B i a s i n g and Pulse Detection C i r c u i t 94 5-4 95 x i i i F i g u r e Page < 5-5 P r e a m p l i f i e r Input Impedance 97 5-6 Block Diagram of E l e c t r o n i c U n i t s 98 5-7 Schematic of Veto D i s c r i m i n a t o r 100 5-8 Resistance and Reactance of Transmission L i n e 103 5- 9 Impedance of Transmission Line w i t h P r e a m p l i f i e r as Load. 105 6- 1 I-V C h a r a c t e r i s t i c s , Sn-Sn0 2"Sn Tunnel J u n c t i o n , B=100 G. 112 6-2 I-V C h a r a c t e r i s t i c s of Sn-SnO^-Sn Tunnel J u n c t i o n f o r B i n Plane of J u n c t i o n 114 6-3 V a r i a t i o n of Maximum Dynamic Resistance w i t h Magnetic F i e l d ; 116 6-4 Dynamic Resistance vs Voltage f o r Specimen J-5. . . . . . 116 6-5 Magnetic F i e l d Dependence of Energy Gap 120 6-6 Detector B i a s i n g Conditions 122 6-7 Junction-Source Geometry 124 6-8 Pulses Observed at Optimum Bandwidth 124 6-9 Current S e n s i t i v i t y C a l i b r a t i o n 126 6-10 Pulse Amplitude vs Dynamic Resistance . . . . . 126 6-11 Pulse Height Spectrum 128 6-12 Output Noise vs J u n c t i o n Dynamic Resistance 130 6-13 Output Noise vs Ex t e r n a l : Capacitance 130 6-14 Equivalent C i r c u i t s of Detector and P r e a m p l i f i e r w i t h Noise Generators Included 131 6-15 O v e r a l l Equivalent C i r c u i t f o r Detector and P r e a m p l i f i e r With Noise 134 6-16 Equivalent C i r c u i t f o r S i g n a l to Noise Ratio Estimate . . 134 6-17 Equivalent C i r c u i t of J u n c t i o n - P r e a m p l i f i e r System with Noise Sources Only (no alpha pulses) 139 6- 18 T h e o r e t i c a l and Measured P r e a m p l i f i e r Noise Output vs Dynamic Resistance of J u n c t i o n . . 142 7- 1 Equivalent C i r c u i t of Pulse Detection System 147 x i v Figure Page 7-2 M* as Function of x, t and C T 155 o J 7-3 90% Confidence Volume i n C J f t , T Space 156 7-4 M* (minimum) vs R e l a x a t i o n Time x 158 7-5 P a r t i c l e Track Geometry 167 7-6 T h e o r e t i c a l Pulse Height ,Spectrum 167 A - l J u n c t i o n P r e p a r a t i o n 181 A-r2 Evaporation Apparatus . 184 A-3 Photomicrograph of Sn-Sn Tunnel J u n c t i o n 184 B - l T y p i c a l I-V C h a r a c t e r i s t i c s f o r "Leaky" Specimens . . . . 189 B^ 2 T y p i c a l I . P l o t f o r "Leaky" Specimens . 191 J t r c r i t B-3 I-V C h a r a c t e r i s t i c f o r Model of I d e a l J u n c t i o n i n P a r a l l e l w i t h M e t a l l i c Filaments. . 192 B-4 Josephson Current P e r i o d vs (Junction Width) i n Ap p l i e d Magnetic F i e l d . . . 195 B-5 S t r u c t u r e i n I-V C h a r a c t e r i s t i c Near V = 0 195 C-1 Photomicrographs of Pb-Pb Tunnel J u n c t i o n 204 C-2 Photomicrograph Showing Nodules 204 D-l Crossed- and P a r a l l e l - F i l m J u n c t i o n s ; T y p i c a l I-V C h a r a c t e r i s t i c s of 2 Sn Tunnel Junctions 211 D-2 Four-Terminal Equivalent C i r c u i t of Tunnel J u n c t i o n . . . 212 D-3 E f f e c t of F i l m Resistance (R) on Measured J u n c t i o n Resistance (Rj) , 216 D-4 Graphite Coated Paper " J u n c t i o n " . . . 218 D-5 Pressed Nichrome S t r i p " J u n c t i o n s " 220 D-6 Comparison of P a r a l l e l F i l m Theory with Crossed Nichrome " S t r i p " Data 222 D-7 p vs Load f o r Nichrome S t r i p " J u n c t i o n s " ' 224 X V ACKNOWLEDGEMENTS S p e c i a l thanks are due Dr. B. L. White f o r h i s i n s p i r i n g s u p e r v i s i o n and generous a s s i s t a n c e throughout the du r a t i o n of t h i s p r o j e c t , e x p e c i a l l y during times of despair when j u n c t i o n a f t e r j u n c t i o n had f a i l e d and during the e a r l y hours of numerous mornings when many of the measurements were made. Valuable d i s c u s s i o n s w i t h the other members of my supervisory committee, Drs. R. E. Burgess, G. Jones and P. W. Matthews, are acknowledged w i t h g r a t i t u d e . The a s s i s t a n c e of members of the departmental,Van de Graaff and Low Temperature shops, p a r t i c u l a r l y Mr. P. Haas and Mr. C. Sedger, i n the design and c o n s t r u c t i o n of apparatus i s much appreciated. My h e a r t f e l t thanks go to my wif e Linda f o r her s u s t a i n i n g encouragement and her i n v a l u a b l e a s s i s t a n c e both i n the typ i n g of the complete manuscript and the f i n a l p r e p a r a t i o n of many of the f i g u r e s . Mr. K. Taylor i s a l s o deserving of thanks f o r h i s help i n the prep a r a t i o n of s e v e r a l drawings. The f i n a n c i a l a s s i s t a n c e received from the N a t i o n a l Research C o u n c i l i n the form of one bursary and three studentships as w e l l as the con t i n u i n g support from the Van de Graaff group i n the form of a research a s s i s t a n t s h i p i s g r a t e f u l l y acknowledged. The loan of instrumentation from the B. C. Vo c a t i o n a l School helped g r e a t l y w i t h some of the impedance measurements. i -1-CHAPTER 1 INTRODUCTION A. M o t i v a t i o n Much of the present knowledge of the s t r u c t u r e of the atomic nucleus has been obtained through the a n a l y s i s of the energy d i s t r i b u t i o n of n u c l e a r r e a c t i o n products. Because of the c o n t i n u i n g need f o r more p r e c i s e data, considerable e f f o r t has gone i n t o the development of p a r t i c l e d e t e c t o r s , r e s u l t i n g i n the e v o l u t i o n of a s o p h i s t i c a t e d technology of n u c l e a r p a r t i c l e spectrometry. (See eg. Yuan, 1961; Ajzenberg-Selove, 1960; Dearnaley & Northrop, 1966). From the nuclear p h y s i c i s t ' s point of view, the i d e a l d etector (or spectrometer) i s one w i t h high energy r e s o l u t i o n — the a b i l i t y to d i s t i n g u i s h between p a r t i c l e s having very n e a r l y the same e n e r g y — w i t h sharp time r e s o l u t i o n — t h e a b i l i t y to d i s t i n g u i s h between p a r t i c l e s a r r i v i n g at the detector at very n e a r l y the same time, and w i t h high d e t e c t i o n e f f i c i e n c y to maximize the r a t e of p a r t i c l e d e t e c t i o n . (Note: i n subsequent d i s c u s s i o n the term "de t e c t o r " i s o f t e n used synony-mously w i t h "spectrometer".) This t h e s i s describes the theory of and p r e l i m i n a r y r e s u l t s from a new, f a s t , h i g h l y s e n s i t i v e p a r t i c l e detector i n the form of a superconducting tunnel j u n c t i o n — a "sandwich" c o n s i s t i n g of two t h i n superconducting f i l m s separated by a very t h i n i n s u l a t i n g l a y e r (see f i g u r e 2-1). The detector shows promise of improved energy r e s o l u t i o n , and comparable time r e s o l u t i o n , but reduced d e t e c t i o n e f f i c i e n c y compared w i t h l i t h i u m d r i f t e d germanium d e t e c t o r s , which are c u r r e n t l y the best nuclear detectors a v a i l a b l e . When an energetic p a r t i c l e such as a proton or alpha p a r t i c l e penetrates a medium, i t loses energy v i a a s e r i e s of i o n i z i n g c o l l i s i o n s w i t h the atoms clo s e to i t s path. For ; each given substance, (gas or s o l i d ) , i t i s convenient to define a q u a n t i t y w, the average energy which the charged p a r t i c l e must l o s e i n that substance to produce a p a i r of e x c i t a t i o n s such as an i o n p a i r or e l e c t r o n - h o l e p a i r . Consequently, i f an energetic p a r t i c l e -2-l o s e s energy AE i n a substance, the number N of e x c i t a t i o n s u l t i m a t e l y produced (a measure of the s i g n a l amplitude) i s N = AE/w. I f the energetic p a r t i c l e always had to l o s e e x a c t l y w to produce an e x c i t a t i o n i n a given substance, then a p e r f e c t detector made from that substance could d i s t i n g u i s h between energies AE^ = Nw and AE^ = (N ± l ) w , and the energy r e s o l u t i o n of the detector would be w. However, si n c e w i s an average over a very l a r g e number of e x c i t a t i o n s , most of which w i l l i n v o l v e energy l o s s e s d i f f e r i n g from w, the number of e x c i t a t i o n s produced i n r e a l d e t e c t i n g media by a c e r t a i n energy input i s subject to f l u c t u a t i o n s . In the simplest case where the e x c i t a t i o n s are considered to be independently produced and where i N i s l a r g e , the f l u c t u a t i o n may be taken to be the s t a t i s t i c a l f l u c t u a t i o n i • j N 2. Hence, u n c e r t a i n t y of order v^/N i s introduced i n determining the energy of a p a r t i c l e , f o r now AE = wN(l ± v^/N). In a more r e a l i s t i c case, which very o f t e n occurs, the e x c i t a t i o n processes are not independent, but are c o r r e l a t e d by the f a c t that the p a r t i c l e must los e a l l i t s energy l i n the d e t e c t o r ; the s t a t i s t i c a l f l u c t u a t i o n i s then given by F N 2 r a t h e r than N 2, where F, the Fano' f a c t o r (Dearnaley and Northrop, 1966) l i e s between 0 and 1 and, f o r example, has the value F $ 0.16 f o r e x c i t a t i o n s produced by 1-10 MeV e l e c t r o n s i n germanium at 78 K. (Mann et a l , 1966). The u n c e r t a i n t y decreases w i t h i n c r e a s i n g N so t h a t , from t h i s v i ewpoint, the best energy r e s o l u t i o n w i l l be obtained from the detector i n which f o r given energy l o s s , N i s ' l a r g e s t or, i n other words, i n which w i s s m a l l e s t . The m o t i v a t i o n f o r c o n s i d e r i n g the superconducting tunnel j u n c t i o n as a p a r t i c l e detector i s apparent from the comparison of 7N/N f o r s e v e r a l media i n t a b l e 1-1. Two other q u a n t i t i e s of i n t e r e s t f o r comparing d i f f e r e n t d e t e c t i n g media have been included i n the t a b l e . The s p e c i f i c r a t e of energy l o s s (-dE/dx) and the range, c a l c u l a t e d f o r a 5 MeV alpha p a r t i c l e , are optimal i n Pb which means that to stop p a r t i c l e s of a given energy a smaller thickness of Pb would be needed than of any conventional d e t e c t i n g m a t e r i a l . Smaller s i z e , of course, g e n e r a l l y i m p l i e s f a s t e r response. The idea of using superconductors to detect nuclear p a r t i c l e s i s not new; what i s novel i s the e x p l o i t a t i o n of the favourable character-i s t i c s of the superconducting tunneling j u n c t i o n f o r t h i s purpose. -3-Andrews et a l (1949) bombarded a t h i n s t r i p of niobium n i t r i d e , held very c l o s e to i t s superconducting t r a n s i t i o n temperature, w i t h alpha p a r t i c l e s N f o r -dE/dx Range M a t e r i a l w(eV) E=l MeV (MeV/cm) (5 MeVa) Hydrogen-gas 37 2.7 x 10 4 6.1xl0~ 3 .18 15.5 cm Krypton-gas 24 4.2 x 10 4 4 . 9 x l 0 ~ 3 1.25 2.81 S i l i c o n - s e m i - c o n . 3.6 2.8 x 1 0 5 1 . 9 x l 0 ~ 3 1500 2.5 x 10" 3 Germanium-semi-con 2.9 3.5 x 1 0 5 1 . 7 x l 0 - 3 2000 1.5 x 10" 3 Tin*-super-con. .003* 3.3 x 10 8 5 . 5 x l 0 ~ 5 2300 1.6 x 10~ 3 Lead*-super-con. .005* 2.x 10 8 7.1xl0" 5 2600 1.3 x 10" 3 Table 1-1: P a r t i c l e D e t e c tion Parameters f o r Several M a t e r i a l s * The magnitude of w used f o r the superconductors i s based on the assumption t h a t , i n analogy w i t h semiconductors, (Sherman, 1964) w may be 3-5 times the energy gap w i d t h — an assumption l a t e r shown to be c o n s i s t e n t w i t h the r e s u l t s of t h i s experiment. (see s e c t i o n D of t h i s chapter and chapter 7.) and observed the r e s u l t i n g v o l tage pulses 1 developed along the length of the s t r i p . A s i m i l a r experiment using evaporated narrow t h i n f i l m s r a t h e r than bulk s t r i p s were l a t e r proposed by Sherman (1962,B) and performed by S p i e l et a l (1965). When an alpha p a r t i c l e impinged on the superconductor a s m a l l , c y l i n d r i c a l r e g i o n surrounding the: t r a c k was d r i v e n normal. The diameter of t h i s r e g i o n increased as a f u n c t i o n of time at a rate determined by the ambient current c a r r i e d by the f i l m and the energy l o s t by the alpha p a r t i c l e , u n t i l i t became equal to f i l m width; at t h i s p o i n t , because of the f i n i t e r e s i s t a n c e due to t h i s transverse normal r e g i o n , a pulse was observed i n the voltage between the ends of the f i l m . The b a s i c d i f f e r e n c e s between the superconducting s i n g l e s t r i p and the tunnel j u n c t i o n detectors w i l l be o u t l i n e d at the end of the next s e c t i o n ; more d e t a i l on the super-conducting s t r i p detectors i s given i n s e c t i o n A, chapter 3. I t was B u r s t e i n et a l (1961) who f i r s t proposed and analyzed the use of the superconducting tunnel j u n c t i o n s , which had r e c e n t l y been studied i n the p i o n e e r i n g work of Giaever (1960,61), as detectors of e l e c t r o -magnetic r a d i a t i o n . Such a detector i s of great p r a c t i c a l i n t e r e s t because the frequency range to which i t i s s e n s i t i v e , approximately 85 to 650 GHz, -4-i s v i r t u a l l y i n a c c e s s i b l e by conventional d e t e c t i o n techniques. The two processes of i n t e r e s t i n t h i s device are the " o p t i c a l " e x c i t a t i o n of q u a s i p a f t i c l e s across the superconducting energy gap and "photon-assisted" t u n n e l i n g . I t turns out that tunnel j u n c t i o n s may al s o serve as generators and detectors of microwave sound (Lax and Vernon, 1965; G o l d s t e i n et a l , 1965; and Eisenmenger and Dayem, 1967) v i a an analogous process known as "phonon-a s s i s t e d " t u n n e l i n g . A b r i e f review of these a p p l i c a t i o n s i s given by Taylor (1968). The scene was now set f o r work to begin on the superconducting tunnel j u n c t i o n charged p a r t i c l e d e t e c t o r . On one si d e was the nuclear s p e c t r o s c o p i s t w i t h h i s continuing demand f o r more s e n s i t i v e d e t e c t o r s ; on the other was the low temperature p h y s i c i s t w i t h a new device having i n t e r e s t i n g p h y s i c a l p r o p e r t i e s and e x c i t i n g p o t e n t i a l f o r a p p l i c a t i o n s . The f i r s t steps i n an attempt to adapt t h i s p o t e n t i a l and supply the demand are described i n t h i s t h e s i s . B. P r i n c i p l e of Operation 1 1. Phenomenological Treatment For a given superconducting t h i n f i l m tunnel j u n c t i o n at temperature T whose i n s u l a t i n g l a y e r (see f i g u r e 2-1) i s s u i t a b l y t h i n , there e x i s t s a w e l l - d e f i n e d r e l a t i o n s h i p I = I ( V , T ) — c a l l e d the I-V c h a r a c t e r i s t i c — b e t w e e n the dc current I passing from one f i l m to the other v i a e l e c t r o n t u n n e l i n g and the voltage V developed across the i n s u l a t i n g l a y e r . At b i a s voltages 0 < V < 2A(T)/e, where e i s the e l e c t r o n i c charge and 2A(T) i s the temperature dependent superconducting energy gap width (discussed i n chapter 2 ) , the current i s p r o p o r t i o n a l to the den s i t y of thermally e x c i t e d q u a s i p a r t i c l e s which depends approximately upon the temperature according to exp(-A(T)/k f iT). (See al s o f i g u r e 6-1; k g i s Boltzmann's constant.) This statement neglects tunnel current c o n t r i b u t i o n s a r i s i n g from the tunneling processes i n v o l v i n g more than one p a r t i c l e ; such a d d i t i o n a l c o n t r i b u t i o n s are discussed i n Appendix B. Neglecting f o r the present the temperature dependence of A, i t i s evident that any stimulus, (eg. the energy l o s t by a nuclear charged p a r t i c l e i n t r a v e r s i n g or coming to r e s t i n a tunnel j u n c t i o n ) , which increases the f i l m temperature w i l l i n crease the tunneling current. That such an e f f e c t e x i s t e d was not i n question. What was i n question, however, was whether or not the change i n tu n n e l i n g current due to a 5 MeV alpha p a r t i c l e , f o r example, was observable -5-and, i f so, d i d the e f f e c t show promise f o r charged p a r t i c l e energy measurements? Before going f u r t h e r , the r o l e played by the energy gap should be s t r e s s e d . As mentioned e a r l i e r , low s t a t i s t i c a l l y l i m i t e d energy r e s o l u t i o n r e q u i r e s that A should be as small as p o s s i b l e (A . ). On the mxn other hand, the "leakage" or thermal q u a s i p a r t i c l e t u n n e l i n g current through the j u n c t i o n should be minimal to reduce shot noise and t h i s i m p l i e s that A should be l a r g e (A ). C l e a r l y , f o r a given lowest operating temperature, a superconductor w i t h a compromise value (A ) such that A . < A < A i s c min c max r e q u i r e d , but A should favour A f o r the energy r e s o l u t i o n i s only n c max & 1 degraded by a f a c t o r (1 + A /'A^ whereas the thermal current i s increased from the optimal value by a f a c t o r e x p f H A ^ - A )/kgT]. When a tunnel j u n c t i o n i s bombarded w i t h charged p a r t i c l e s , the p a r t i c l e s t r a v e r s e one or both of the f i l m s g i v i n g up some of t h e i r energy to the f i l m s predominantly v i a : a s e r i e s of i o n i z i n g c o l l i s i o n s . The amount of energy deposited d i r e c t l y i n the j u n c t i o n AE i s given by AE = jf " (dE/dx)dx where i s the path length through the superconducting l a y e r s and (dE/dx) i s the s p e c i f i c r a t e of energy l o s s which depends on the i n i t i a l energy and type of p a r t i c l e and the m a t e r i a l from which the f i l m s are prepared. For purposes of a n a l y s i s , i t i s assumed that the r e s u l t i n g disturbance of the temperature d i s t r i b u t i o n away from i t s e q u i l i b r i u m value i s a maximum immediately f o l l o w i n g the passage of the alpha p a r t i c l e and that i t r e l a x e s back to the e q u i l i b r i u m c o n d i t i o n i n a c h a r a c t e r i s t i c time T . During t h i s p e r i o d an increase i n the t u n n e l i n g current occurs which, a f t e r appropriate a m p l i f i c a t i o n and shaping, appears as a s i g n a l pulse superimposed on the ambient thermal tunneling c u r r e n t . This pulse may be analyzed f o r i n f o r m a t i o n about the energy l o s s which produced i t and the time at which i t was produced, using standard nuclear data a c q u i s i t i o n e l e c t r o n i c s . I t i s found experimentally (chapter 6 ) , i n agreement w i t h t h e o r e t i c a l p r e d i c t i o n s (chapter 3) that the s i g n a l to noise r a t i o f o r the o v e r a l l system (detector plus a m p l i f i e r s ) i s a maximum when the dynamic r e s i s t a n c e r = ( 3 V / 9 I ) T i s as l a r g e as p o s s i b l e . The dc operating current of the j u n c t i o n was t h e r e f o r e chosen so that ( 3 V / 8 I ) T was a maximum; the magnitude of I was not c r i t i c a l , as tjhe j u n c t i o n noise sources were s m a l l compared w i t h p r e a m p l i f i e r n o i s e . Questions concerning the amplitude and d u r a t i o n of such a pulse are d e a l t w i t h b r i e f l y i n the f o l l o w i n g s e c t i o n -6-and discussed at length i n chapters 3 and 7. 2. M i c r o s c o p i c Viewpoint The purely phenomenological d i s c u s s i o n given i n the preceding s e c t i o n i s n e c e s s i t a t e d by the extreme complexity of the microscopic p i c t u r e sketched below. U l t i m a t e l y , the energy l o s t by a charged p a r t i c l e as i t passes through the f i l m s of the tunnel j u n c t i o n leads to the generation of d e n s i t i e s of e l e c t r o n and phonon e x c i t a t i o n s along the p a r t i c l e t r a c k which exceed those present when the f i l m s are i n thermal e q u i l i b r i u m w i t h a constant temperature bath. The energy i s transported away from the t r a c k by these e x c i t a t i o n s and, from the work of C r i t t e n d e n (1968), i t seems that the tr a n s p o r t processes might be adequately described by c l a s s i c a l heat d i f f u s i o n theory. As a consequence of the temperature d i s t r i b u t i o n which obtains' i n the superconducting f i l m s , e x c i t e d q u a s i p a r t i c l e d e n s i t i e s N q are produced which are i n excess of the thermal e q u i l i b r i u m v a l u e . The excess i s maximum immediately f o l l o w i n g the p e n e t r a t i o n of the alpha p a r t i c l e and i t decays back to zero w i t h the same c h a r a c t e r i s t i c time as does the temperature. D e t a i l e d c a l c u l a t i o n of N i s made d i f f i c u l t by at l e a s t o J two c o m p l i c a t i n g f a c t o r s . F i r s t of a l l , the superconducting energy gap (2A(T)) at a point i n the superconductor i s u n l i k e the band gap i n semi-conductors i n that i t i s s t r o n g l y temperature dependent and vanishes, of course, at the superconducting-normal t r a n s i t i o n temperature T . Now large temperature excursions are a n t i c i p a t e d i n the v i c i n i t y of the t r a c k so that f o r periods of tens of nanoseconds ( i n d i c a t e d by p r e l i m i n a r y c a l c u l a t i o n s c a r r i e d out by Mr. G. May of t h i s l a b o r a t o r y ) the metal i n t h i s very small region i s , i n f a c t , i n the normal s t a t e — n o f i x e d value of A may therefore be assumed. Secondly, the phonons emitted when two e x c i t e d q u a s i p a r t i c l e s combine to form a Cooper p a i r may not be ignored f o r they, i n t u r n , may break up other p a i r s . Rothwarf and Taylor (1967) have shown that the r a t e at which these recombination phonons are l o s t from the energy range nu) £ 2A(T) o f t e n determines the r a t e at which excess e x c i t e d q u a s i p a r t i c l e d e n s i t i e s w i l l decay. In a small volume l i k e the tunnel j u n c t i o n , i t i s expected that transmission of the phonons to the glass substrate and super-f l u i d helium w i l l play an important p a r t . (For f u r t h e r d i s c u s s i o n , see chapter 7). -7-C e r t a i n l y , the s i t u a t i o n i n superconducting tunnel j u n c t i o n s i s even more complicated than i n semiconductor detectors of p r a c t i c a b l e geometry where, although the energy gap may be considered constant and the e f f e c t of boundaries ignored, the a n a l y s i s i s qu i t e i n v o l v e d (see eg. Shockley, 1961). 3. D i f f e r e n c e from Intermediate State Bolometers At t h i s p o i n t a d i s t i n c t i o n can be made between the tunnel j u n c t i o n detector and the s i n g l e superconducting s t r i p used by previous workers. E s s e n t i a l l y , the s i n g l e s t r i p acts l i k e a switch. P a r t i c l e s w i t h energy below some threshold value (E ^ ')—determined by the b i a s current and s t r i p dimensions, m a t e r i a l and temperature—are unable to convert a s u f f i -c i e n t l y l a r g e region of the s t r i p i n t o the normal s t a t e so that no voltage pulse occurs; p a r t i c l e s w i t h energy E > E ^ do generate a pulse but i t s amplitude i s i n s e n s i t i v e to the a c t u a l magnitude of IE and i s not s u i t a b l e f o r multichannel spectrometry. In the j u n c t i o n d e t e c t o r , on the other hand, an excess q u a s i p a r t i c l e d e n s i t y i s generated as long as E i s greater than a few times 2A(T), say, so that the energy threshold i s set by the j u n c t i o n -a m p l i f i e r n o i se and f l u c t u a t i o n s i n the s i g n a l . ( I n s u f f i c i e n t t h e o r e t i c a l or experimental evidence i s a v a i l a b l e yet to make any statements concerning the l i n e a r i t y between pulse amplitude and energy deposited i n the j u n c t i o n . ) By analogy, i t might be s a i d that the s i n g l e " superconducting s t r i p i s to the tunnel j u n c t i o n detector as the geiger counter i s to the p r o p o r t i o n a l counter. C. Expected S i g n a l As a s t a r t i n g p o i n t , the time dependence of the current pulse r e s u l t i n g from the passage of an alpha p a r t i c l e through the j u n c t i o n i s assumed to be • • • i ( t ) = i exp(-t/-r) , t * o (1-1) o where the peak current i i s p r o p o r t i o n a l to the maximum excess e x c i t e d q u a s i p a r t i c l e d e n s i t y and T i s the c h a r a c t e r i s t i c time mentioned e a r l i e r . To evaluate i i n equation 1-1, i t i s convenient to define o i = 61 (9I/8T)„6T o v N = o 5n = On(T)/3T)6T = 2AE/w (Vol) -8-where n(T) i s the thermal e q u i l i b r i u m q u a s i p a r t i c l e d e n s i t y and V o l i s the volume i n which the energy AE i s d i s t r i b u t e d (see chapter 3). E l i m i n a t i n g 6T between these expressions gives 1 o 91 9T 9T 9n 2AE w- (Vol) U L ) Rough estimates of the v a r i o u s parameters i n equation 1-2 i n d i c a t e that i Q - 15 uA f o r AE = 0.5 MeV i n t i n at 1.2 K. ( I t should be pointed out that the values of (91/91)^ and (9T/9n) used i n t h i s estimate are constant and do not represent i n any way the t r u e , complicated behaviour of these two q u a n t i t i e s which are s t r o n g l y temperature dependent.) In s p i t e of these gross s i m p l i f i c a t i o n s , t h i s estimate of i Q i s reasonably c o n s i s t e n t w i t h the most probable value of 22 yA deduced experimentally (see s e c t i o n E f o l l o w i n g and chapter 7). Estimates of the time constant T are on an even l e s s f i r m f o o t i n g . Decay of the excess q u a s i p a r t i c l e d e n s i t y occurs e i t h e r by tunneling (under s u i t a b l e b i a s c o n d i t i o n s ) w i t h p r o b a b i l i t y W^  per second per excess q u a s i -p a r t i c l e p a i r or recombination w i t h p r o b a b i l i t y W per second per excess is. q u a s i p a r t i c l e p a i r . (Recombination, see chapter 7, i s taken to i n c l u d e those processes which i r r e v e r s i b l y reduce the q u a s i p a r t i c l e d ensity i n the superconducting f i l m s , i t does not i n c l u d e merely the recombination of q u a s i p a r t i c l e s to form a Cooper p a i r w i t h the emission of a phonon, since such a phonon may i n turn create a new p a i r of q u a s i p a r t i c l e s from a Cooper p a i r , i n which case the q u a s i p a r t i c l e d e n s i t y has not been i r r e v e r s i b l y reduced.) Hence the r a t e at which the excess q u a s i p a r t i c l e density decays i s W = W_ + W = T " 1 . I K > W^  may be estimated v i a a method due to Ginsberg (1962). For r e a l i s t i c j u n c t i o n geometries and b a r r i e r t h i c k n e s s e s , see chapter 3, W 5 6 -1 ranges from 2 x 10 to 3 x 10 sec . Because of the i n f l u e n c e of the recombination phonons mentioned e a r l i e r , i t i s expected that W should not be estimated s o l e l y from the theory of q u a s i p a r t i c l e recombination (see eg. Woo and Abrahams, 1968). Measurements of an e f f e c t i v e recombination p r o b a b i l i t y W which includes K the e f f e c t of phonon l o s s from the f i l m s , have been made i n t h i n f i l m t r i o d e s t r u c t u r e s by s e v e r a l workers (see eg. Levine and Hsieh, 1968) who f i n d that -9-f o r superconductors and temperatures of i n t e r e s t 2 x 10^ sec ^ > WR' > 5 - 1 5 x 10 sec . Although the steady s t a t e c o n d i t i o n s under which W ' was R measured d i f f e r from those i n tunnel j u n c t i o n s bombarded w i t h charged p a r t i c l e s , one has l i t t l e recourse but to use W 1 + W f o r an order of _1 R T magnitude estimate of T . . W i t h i n the r e l i a b i l i t y of these e s t i m a t e s , ! ranges from 4 x 10 —6 to 1.4 x 10 sec. In summary, the expected current i s assumed to be of the form i ( t ) = i expC-t/x) w i t h i = 15uA (for energy AE = 0.5 MeV deposited i n the t i n j u n c t i o n at 1.2 K) and 4 x 10~ 8 < T < 1.4 x 1 0 ~ 6 sec. i D. Noise I t was found (chapter 6) that the dominant source of noise i n the present experiment was the current s e n s i t i v e p r e a m p l i f i e r and t h a t ! t h e magnitude of the noise was 3.5 uV(rms) r e f e r r e d to the p r e a m p l i f i e r input.' (This noise l e v e l turned out to be equivalent to the pulse amplitude produced by a 100 keV energy l o s s i n the p a r t i c u l a r j u n c t i o n used i n the d e t e c t i o n experiment.) When tunnel j u n c t i o n s having higher output impedances than the u n i t employed i n t h i s work (~ 9.3fi) are f a b r i c a t e d , noise sources i n the j u n c t i o n i t s e l f , as o u t l i n e d i n chapter 3, may assume an important r o l e i n determining j u n c t i o n performance. The requirement that the a m p l i f i e r s e n s i t i v i t y had to meet was that pulses having the parameters p r e d i c t e d above should produce an a m p l i f i e r output equal to or greater than three times the a m p l i f i e r noise l e v e l . An a m p l i f i e r was constructed which met t h i s c r i t e r i o n . E. Preview of Results The experiment reported i n t h i s t h e s i s demonstrates f o r the f i r s t time that the superconducting t h i n f i l m tunnel j u n c t i o n may indeed be used as'a detector of charged p a r t i c l e s . S p e c i f i c a l l y , i t shows that the pulse of t u n n e l i n g current r e s u l t i n g from the bombardment of a Sn-SnO^-Sn j u n c t i o n by 5.1 MeV alpha p a r t i c l e s i s r e a d i l y observable. An upper l i m i t of 8.2 x -3 10 eV has been placed on w(Sn), the average energy l o s s by a charged i p a r t i c l e r e q u i r e d to e x c i t e a q u a s i p a r t i c l e p a i r i n superconducting t i n at 1.2 K which, w i t h reference to t a b l e 1-1, may be seen to agree reasonably w e l l w i t h the value derived from a very naiilve analogy w i t h semiconductors. -10-(Because of ambiguity caused by heat d i f f u s i n g back from the substrate to c o n t r i b u t e to the pulses as discussed i n chapter 7 — t h i s agreement may perhaps be f o r t u i t o u s , and i n f a c t w(Sn) may be smaller than the naive analogy would suggest.) The pulses observed ( f i g u r e 6-8) had amplitudes up to 19 times the rms noise l e v e l . Because t h e i r amplitude decreased w i t h decreasing j u n c t i o n dynamic r e s i s t a n c e as predicted,because they could be turned o f f w i t h a mechanical s h u t t e r placed between the j u n c t i o n and source, and because the pulse count ra t e agreed w i t h that c a l c u l a t e d from the source strength and the experimental geometry, i t was concluded that the pulses c e r t a i n l y were the r e s u l t of alpha p a r t i c l e bombardment of the j u n c t i o n i t s e l f and d i d not a r i s e from bombardment of the l e a d - i n f i l m s , or the substrate alone. A l e a s t squares f i t of the t h e o r e t i c a l pulse shape (modified by the t r a n s f e r f u n c t i o n of the a m p l i f i e r system) to that recorded p h o t o g r a p h i c a l l y y i e l d s x = 138 ± 33 nsec where the e r r o r s correspond to a 90% confidence l i m i t . With t h i s range of j and the known s e n s i t i v i t y of the a m p l i f i e r system, i f o r the l a r g e s t amplitude pulses observed i s deduced to l i e i n the range 20 $ i $ 26 uA w i t h a most probable value of 22 uA. The value of AE corresponding to t h i s value of i Q i s discussed below. The t o t a l number N Q of q u a s i p a r t i c l e p a i r s produced by the alpha p a r t i c l e i s c a l c u l a t e d to be W N = — — o W„,e i ( t ) dt = i /We « 3.A x 1 08 o T where W and W^  have been defined p r e v i o u s l y and e i s the e l e c t r o n i c charge. Considerable u n c e r t a i n t y i s i n v o l v e d i n e s t i m a t i n g the t o t a l amount of energy AE a that the alpha p a r t i c l e deposits 'in the j u n c t i o n . Taking 2.75 MeV as an upper bound (see the d i s c u s s i o n i n chapter 7) gives _3 w(Sn) $ AE /N = 8.2 x 10 eV as mentioned^'above. o o These r e s u l t s must be regarded as p r e l i m i n a r y , both i n s o f a r as the understanding of the b a s i c physics i n v o l v e d i s concerned, and i n s o f a r as an e v a l u a t i o n of these j u n c t i o n s as worthwhile nuclear r a d i a t i o n detectors i s concerned. Further t h e o r e t i c a l and experimental work, which i s being continued i n t h i s l a b o r a t o r y by Mr. G. May and Dr. B. White, i s required to e l u c i d a t e the p h y s i c a l d e t a i l s of the processes involved and probe such subjects as energy r e s o l u t i o n , l i n e a r i t y of response and r e p r o d u c i b i l i t y of performance (see chapter 8). In p a r t i c u l a r , i n v e s t i g a t i o n s are being made -11-concerning the d e t a i l e d mechanism f o r charged p a r t i c l e energy l o s s i n superconductors and the manner i n which the heat spike d i f f u s e s throughout the s u b s t r a t e and f i l m s ; t h i s t o p i c i s ther e f o r e t r e a t e d very b r i e f l y i n t h i s t h e s i s . F. Thesis O u t l i n e Chapter 2 introduces the subject of tunneling w i t h a b r i e f d i s c u s s i o n of t u n n e l i n g between normal metals. This i s followed by a:short review of t o p i c s from s u p e r c o n d u c t i v i t y theory which are of relevance to the tunnel j u n c t i o n charged p a r t i c l e d e tector. An o u t l i n e of p e r t i n e n t t o p i c s from q u a s i p a r t i c l e and dc Josephson tunneling theory concludes the chapter. The op e r a t i n g p r i n c i p l e s of the superconducting charged p a r t i c l e detector are given i n chapter 3 . 'To place the tunnel j u n c t i o n detector i n pe r s p e c t i v e three commonly used devices f o r measuring charged p a r t i c l e energy s p e c t r a , the g a s - f i l l e d , s c i n t i l l a t i o n and semiconductor d e t e c t o r s , are q u i c k l y reviewed. Other d e t e c t i o n a p p l i c a t i o n s of tunnel j u n c t i o n s , p a r t i -c u l a r l y microwave photons and phonons are examined because of t h e i r analogy to charged p a r t i c l e d e t e c t i o n . I t i s e s t a b l i s h e d that the j u n c t i o n having the most favourable c h a r a c t e r i s t i c s f o r p a r t i c l e d e t e c t i o n i s one composed of i d e n t i c a l superconductors which have high stopping power and a large i i energy gap. The nature of the q u a s i p a r t i c l e e x c i t a t i o n s u l t i m a t e l y generated i S i n the superconductor by a t r a v e r s i n g charged p a r t i c l e i s discussed a f t e r which the parameters f o r a small s i g n a l equivalent c i r c u i t f o r the detector are derived and an estimate of the s i g n a l s i z e obtained. This i s followed by notes on the importance of the q u a s i p a r t i c l e tunneling p r o b a b i l i t y and the p r a c t i c a l o p e ration of the de t e c t o r . Concluding the chapter i s a survey of the noise sources i n the tunnel j u n c t i o n detector which may u l t i m a t e l y l i m i t i t s performance. Chapters 4 and 5, which describe the cryogenic apparatus and e l e c t r i c a l measurement techniques r e s p e c t i v e l y , are p r i m a r i l y of t e c h n i c a l i n t e r e s t and may be overlooked by the reader u n i n t e r e s t e d i n these d e t a i l s . A d i s c u s s i o n of the experimental r e s u l t s may be found i n chapter 6. Of c h i e f concern are the dc c h a r a c t e r i s t i c s of the Sn-Sn j u n c t i o n s used, the d e t a i l s connected w i t h the observation and p h y s i c a l c h a r a c t e r i s t i c s of pulses i n the tu n n e l i n g current which were induced by alpha p a r t i c l e -12-bombardment and the e s t i m a t i o n of the j u n c t i o n capacitance from noise measurements. Chapter 7 i s an account of the a n a l y s i s of the r e s u l t s d e a l i n g p a r t i c u l a r l y w i t h the determination of the parameters of i n t e r e s t T , i and w. A d i s c u s s i o n of problems p e r t a i n i n g to a rigorous c a l c u l a t i o n of the amount of energy deposited i n the j u n c t i o n by a 5.1 MeV alpha p a r t i c l e c l o s e s the s e c t i o n . F i n a l l y , i n chapter 8, the important r e s u l t s are summarized and placed i n p e r s p e c t i v e by e v a l u a t i n g the tunnel j u n c t i o n as a p o s s i b l e charged p a r t i c l e spectrometer. Guidelines f o r f u t u r e work b r i n g the chapter and the main body of the t h e s i s to a c o n c l u s i o n . The four appendices deal w i t h matters of s p e c i f i c t e c h n i c a l i n t e r e s t . Appendix A o u t l i n e s t h i n f i l m tunnel j u n c t i o n p r e p a r a t i o n techniques; Appendix B analyzes the I-V c h a r a c t e r i s t i c s of s o - c a l l e d " l e a k y " j u n c t i o n s ; Appendix C i s a synopsis of phenomena observed i n the p r e p a r a t i o n of Pb-Pb j u n c t i o n s ; Appendix D i s an account of studies made to determine the e f f e c t of f i n i t e f i l m r e s i s t a n c e on the apparent s i g n of the r e s i s t a n c e of the j u n c t i o n measured w i t h a standard 4-terminal technique. CHAPTER 2 THEORETICAL ASPECTS OF THE SUPERCONDUCTING TUNNEL JUNCTION A. I n t r o d u c t i o n In i t s most common form, the superconducting tunnel j u n c t i o n c o n s i s t s of two superimposed evaporated t h i n metal f i l m s separated by an extremely t h i n i n s u l a t i n g l a y e r , u s u a l l y a thermally grown oxide on the bottom f i l m (see f i g u r e 2-1). The " j u n c t i o n " i s defined as the region of overlap of the f i l m s . T y p i c a l l y , the f i l m s are a few tenths of a m i l l i m e t e r wide and s e v e r a l thousand angstroms t h i c k ; the i n s u l a t i n g l a y e r thickness i s u s u a l l y 10 to 20 angstroms. The mechanism f o r current flow from one superconducting f i l m to the other i s , as the name i m p l i e s , e l e c t r o n t u n n e l i n g . To introduce the e s s e n t i a l s of the subject of t u n n e l i n g , the problem of t u n n e l i n g between normal metals i s considered f i r s t ( s e c t i o n B). Before going on to discus s tunneling between superconductors, a pause i s made ( s e c t i o n C) f o r the purpose of reviewing b r i e f l y those t o p i c s from the theory of s u p e r c o n d u c t i v i t y which are r e l e v a n t to the superconducting tunnel j u n c t i o n ( e s p e c i a l l y as a charged p a r t i c l e d e t e c t o r ) . The l a s t three s e c t i o n s are devoted to a d i s c u s s i o n of tunneling between super-conductors: s e c t i o n D considers s i n g l e q u a s i p a r t i c l e t u n n e l i n g , s e c t i o n E i n v e s t i g a t e s Josephson or p a i r tunneling (because of i t s nuisance rather than i n t r i n s i c value to the tunnel j u n c t i o n detector) and, f o r the sake of completeness, s e c t i o n F looks b r i e f l y at m u l t i p a r t i c l e t u n n e l i n g . The reader i n t e r e s t e d i n studying tunneling phenomena i n general i s r e f e r r e d to a very recent and comprehensive review e d i t e d by B u r s t e i n and Lundqvist (1969). B. Tunneling Between Normal Metals I t was Sommerfeld and Bethe (1933) who f i r s t c a l c u l a t e d the e l e c t r o n tunnel current through a m e t a l - i n s u l a t o r - m e t a l (M-I-M) tunneling -15-diode. During the past 36 years, considerable t h e o r e t i c a l work on t u n n e l i n g s t r u c t u r e s has been c a r r i e d o u t — b i b l i o g r a p h i e s to the l i t e r a t u r e on t h i s subject can be found i n S t r a t t o n (1962), Simmons (1963A,B), Kuhn (1966) and Duke (1969)—but the a c t u a l p h y s i c a l preparation and study of t h i n f i l m tunnel j u n c t i o n s was delayed u n t i l 1960 when high vacuum and t h i n f i l m technology had s u f f i c i e n t l y advanced f o r Giaever (1960) to make Att-AA^O^-Pb tunneling j u n c t i o n s . In t h i s s e c t i o n , the general r e s u l t f o r the tunneling current between normal metals i s quoted followed by an approximation due to Simmons which f i n d s a p p l i c a t i o n i n r e l a t i n g the observed tunnel current to the i n s u l a t o r t h i c k n e s s . ( D e t a i l s concerning the c a l c u l a t i o n s may be found i n Harrison (1961) and the references c i t e d above.) 1. Tunneling Current The p o t e n t i a l energy diagram f o r an i d e a l M-I-M tunnel j u n c t i o n i s shown i n f i g u r e 2-2. As long as one i s i n t e r e s t e d only i n the gross features of the I-V c h a r a c t e r i s t i c (of the order of hundreds of meV), i t i s p e r m i s s i b l e to use a n o n - i n t e r a c t i n g or f r e e e l e c t r o n model of the t u n n e l i n g process (see eg. Duke, 1969). (In such a model the i n t e r a c t i o n s of the e l e c t r o n s both with themselves and the phonons i n the metal and the b a r r i e r are neglected; the e f f e c t of the p e r i o d i c p o t e n t i a l of the l a t t i c e i s approximated by an e f f e c t i v e mass.) I f a one dimensional b a r r i e r and specular transmission through i t are assumed ( i e . the transverse wave number k^ i s the same f o r the i n i t i a l and f i n a l e l e c t r o n s t a t e s ) and i f i t i s f u r t h e r assumed that the matrix element f o r 2 2 2 the t r a n s i t i o n from metal 1 to metal 2 i s such that |Mi 2| = | M 2 1 1 = M , the tunnel current d e n s i t y i n the x d i r e c t i o n i s (Kuhn,..1966 or H a r r i s o n , 1961) | M | 2 p l x ( E ) p 2 x ( E + eV)[f (E) - f^E + eV)]dE (2-1) In equation 2-1, e i s the e l e c t r o n i c charge, h = "h2ir i s Planck's constant, f (E) i s the Fermi d i s t r i b u t i o n f u n c t i o n and p (E) °= (8E(k)/3k ) 1 i s a X X d e n s i t y of s t a t e s g i v i n g the number of e l e c t r o n s having x d i r e c t e d energy between E and E + dE w i t h f i x e d transverse momentum k . Within the x x x t approximations s t a t e d , equation 2-1 i s the fundamental equation f o r t u n n e l i n g through a one-dimensional b a r r i e r and a p p l i e s e q u a l l y w e l l to 4ire I k. -16-vacuum Figure 2-2: P o t e n t i a l Energy Diagram f o r I d e a l M-I-M Structure ( a f t e r Simmons, 1963A) -17-M-I-M s t r u c t u r e s , whether the metal i s normal or superconducting. A d i s t i n c t i o n between the two cases must be made, as w i l l be seen s h o r t l y , when the matrix element i s evaluated. , , 2 E v a l u a t i n g |M| by assuming that the e l e c t r o n s were e s s e n t i a l l y l o c a l i z e d to one side of the b a r r i e r or the other and using a WKB approximation through the b a r r i e r r e g i o n , H a r r i s o n (1961) found |M|2 = T ( E x ) t ^ 2 P l x ( E ) P 2 x ( E + e V ) ] _ 1 S u b s t i t u t i o n i n equation 2-1 gives CO T(E ) [f(E) - f ( E + eV)]dE (2-2) o where T(E ) i s the WKB r e s u l t f o r the b a r r i e r transmission of a f r e e x e l e c t r o n i n c i d e n t on the b a r r i e r . Examination of equation 2-2 rev e a l s that the den s i t y of st a t e s f a c t o r s appearing i n 2-1 have c a n c e l l e d w i t h those contained i n the mat r i x element so that the tunneling current no longer depends on them. This r e s u l t i s c o n s i s t e n t w i t h the observed behaviour of normal M-I-M tunnel j u n c t i o n s but contrary to the behaviour of j u n c t i o n s i n which one or both of the metals are superconducting (see eg. f i g u r e 2-6). B r i e f l y , t h i s absence of the den s i t y of s t a t e s from the r e s u l t f o r superconductors a r i s e s from the f a i l u r e of the independent, p a r t i c l e model to adequately represent superconductors. The discrepancy may be removed by assuming (Ha r r i s o n , 1961) that the d e n s i t y of s t a t e s e n t e r i n g equation 2-1 i i 2 « •: corresponds to that of q u a s i p a r t i c l e s but that |M| i s s t i l l s a t i s f a c t o r i l y given by the normal d e n s i t y of s t a t e s . (The v a l i d i t y of t h i s assumption has been e s t a b l i s h e d from microscopic theory, see eg. Douglas & F a l i c o v , 1964.) As a r e s u l t , c a n c e l l a t i o n of these f a c t o r s i s removed and i i agreement w i t h experiment obtained. Further d i s c u s s i o n of tunneling between superconductors i s reserved f o r s e c t i o n D of t h i s chapter. With equation 2-2, the problem of f i n d i n g the tunnel current between two normal metals i s solved i n p r i n c i p l e but f u r t h e r approximations are required to obt a i n an e x p l i c i t current voltage r e l a t i o n -18-f o r comparison to experiment. Of p a r t i c u l a r i n t e r e s t to the present experiment i s the r e l a t i o n of the tunnel current to the thickness of the i n s u l a t o r . 2. Dependence Of Tunnel Current UpOn B a r r i e r Parameters I t w i l l be shown i n chapter 3 that one c o n d i t i o n f o r optimum performance of the tunnel j u n c t i o n as a charged p a r t i c l e detector i s that the b a r r i e r be as t h i n as p o s s i b l e . Consequently, from an e x p e r i m e n t a l i s t ' s p o i n t of view, an expression r e l a t i n g e f f e c t i v e b a r r i e r thickness to the measurable I-V c h a r a c t e r i s t i c w i l l serve as a u s e f u l guide i n f a b r i c a t i n g tunnel j u n c t i o n s . Such c a l c u l a t i o n s have been given by Simmons (1963A,B) who has r e l a t e d the current d e n s i t y to the t h i c k n e s s , average value of the i n s u l a t o r energy gap and d i e l e c t r i c constant of the b a r r i e r . I t i s the purpose of t h i s paragraph to give h i s r e s u l t s f o r b a r r i e r parameters of i n t e r e s t to t h i s experiment. I f the sum over k^ i s changed to an i n t e g r a l over the appropriate surface i n k space and i t i s assumed that the e f f e c t i v e mass of the e l e c t r o n i s the same whether i n the conduction band of the metal or t u n n e l i n g through the i n s u l a t o r , equation 2-2 may be r e - w r i t t e n i n the form (Kuhn, 1966; Simmons, 1963A—equation 7) J = 4irme E m T(E )dE x x [f(E) - f ( E + eV)]dE (2-3) where m i s the e l e c t r o n mass, E i s the maximum energy of the e l e c t r o n s 2 2 m and E = im(v + v ). E v a l u a t i o n of equation 2-3 to determine the current r y z voltage r e l a t i o n e x p l i c i t l y i s made d i f f i c u l t by the complicated behaviour of T(E ) which, w i t h reference to f i g u r e 2-2, may be w r i t t e n x T(E ) = exp x -4TT h rs, s. {2m(<f>(x) - E ) } 2 dx x Simmons' c o n t r i b u t i o n was to express T(E^) i n an approximate form f o r various b a r r i e r shapes <{> (x) and subsequently c a l c u l a t e the tunneling current d e n s i t y (J) at T = 0 K f o r low, intermediate and high voltages between elect r o d e s of s i m i l a r and d i s s i m i l a r metals. For s i m i l a r m e t a l s — -19-the type of j u n c t i o n used i n t h i s e x p e r i m e n t — h i s expression f o r a rec t a n g u l a r b a r r i e r , w i t h image f o r c e s i n c l u d e d , i n the r e g i o n of low voltag e s (V) i s J = (2m) V s <|>L2 exp(-A<j>L) (2-4) where and 1.15As S 2 " S 1 £n s. = is[l-(l-1.15X/K(f. ) 2 ] * 3(eV A)/K<j> , J. O O K<t> » 1.15X r o A a (4irs/h) (2m) e l e c t r o n mass e l e c t r o n charge m e s h = Planck's constant thickness of i n s u l a t i n g region = s 2 + s i To X = z = "o K = mean b a r r i e r height e2£n(2)/167rKe s p e r m i t t i v i t y of i n s u l a t i n g f i l m p e r m i t t i v i t y of empty space d i e l e c t r i c constant = e/e Equation 2-4 has been evaluated f o r s e v e r a l values of K and .<)> and p l o t t e d i n f i g u r e 2-3. Such curves provide a convenient estimate of the i n s u l a t o r thickness once the room (or lower) temperature experimental j u n c t i o n conductance i s known. (X =4 the q u a n t i t y used by Simmons, 1963A) I t i s important to note that the thickness S^ , = s derived from tunneling r e s i s t a n c e measurements (R ) and equation 2-4 (V/J = R 'A) n n i s the thi c k n e s s of an i d e a l , uniform b a r r i e r whose area A i s the geometrical area of the j u n c t i o n . The thickness of a r e a l b a r r i e r , however, i s not constant but v a r i e s s t o c h a s t i c a l l y from point to point (Hurych, 1966). -20-i o 1 7 r J D i e l e c t r i c Constant (K) H h — • — 1 — 1 — j — • — « — i — i i i • i • 10 20' 30 I n s u l a t o r Thickness (s) - A Figure 2-3: I n s u l a t o r Thickness vs Tunnel Ju n c t i o n J/V -21-Now the t h i n n e r regions of the i n s u l a t o r w i l l be emphasized because of the exponential dependence of r e s i s t a n c e upon thickness so that the e f f e c t i v e area of the b a r r i e r f o r t u n n e l i n g i s not n e c e s s a r i l y equal to A(geometrical) nor i s the average thickness <S>A = S(x,y)dx dy n e c e s s a r i l y equal to S^. This p o i n t i s of relevance £ 0 chapter 3 where the dependence of t u n n e l i n g p r o b a b i l i t y on i n s u l a t o r thickness i s discussed 1 and to chapter 6 where the j u n c t i o n capacitance i s estimated. C. Review of Superconductivity Theory As mentioned e a r l i e r , t h i s s e c t i o n i s devoted to those aspects of s u p e r c o n d u c t i v i t y theory which are germane to an understanding of the superconducting tunnel j u n c t i o n and i t s u t i l i z a t i o n as a detector of charged p a r t i c l e s . Topics such as zero e l e c t r i c a l r e s i s t a n c e and the Meissner e f f e c t , w h i l e of i n t e r e s t i n general, w i l l t h erefore be ignored (apart from the f a c t that the standard 4-terminal network f o r measuring I-V c h a r a c t e r i s t i c s of tunnel j u n c t i o n s i s exact only i n the l i m i t of zero f i l m r e s i s t a n c e ; see appendix D). Many reviews are now a v a i l a b l e d e a l i n g w i t h the subject of s u p e r c o n d u c t i v i t y i n great d e t a i l so that no attempt w i l l be made to j u s t i f y or d e r i v e the r e s u l t s given i n the succeeding paragraphs. The i n t e r e s t e d reader i s r e f e r r e d t o : Kuper (1968), Rickayzen (1965), B l a t t (1964), S c h r i e f f e r (1964) or Bogoliubov (1962); concise presentations are given by Tinkham (1965) and Lynton (1964). The prominent part played by the energy gap, as f a r as the tunnel j u n c t i o n detector i s concerned, was s t r e s s e d i n chapter 1. This s e c t i o n t h e r e f o r e w i l l be concerned p r i m a r i l y w i t h the energy gap and i t s dependence upon temperature, specimen s i z e and magnetic f i e l d . (A comprehensive t h e o r e t i c a l and experimental review of the energy gap i n superconductors has been given by Douglass and F a l i c o v (1964).) Other important concepts such as the two f l u i d model, the q u a s i p a r t i c l e density of s t a t e s and the q u a s i p a r t i c l e d i s t r i b u t i o n f u n c t i o n are introduced at t h i s time f o r c l a r i t y and subsequent reference. 1. Two F l u i d Model The very s u c c e s s f u l Bardeen, Cooper, S c h r i e f f e r (BCS, 1957) theory views the superconductor as c o n s i s t i n g of two i n t e r p e n e t r a t i n g f l u i d s : a s u p e r f l u i d and a normal f l u i d . Making up the s u p e r f l u i d are -22-Cooper p a i r s which are p a i r s of e l e c t r o n s , l o o s e l y "bound" v i a a phonon-mediated i n t e r a c t i o n and occupying a p a i r of e l e c t r o n s t a t e s (k>, -k+) w i t h i n a t h i n s h e l l (+(1^0 / E ) ^ = + .004k F) s t r a d d l i n g the Fermi s u r f a c e . (0 = 200 K i s a t y p i c a l Debye temperature, k i s D B Boltzmann's constant) I t i s the Cooper p a i r s which give the superconductor i t s c h a r a c t e r i s t i c long range order and i t s a b i l i t y to c a r r y current without d i s s i p a t i o n . The normal f l u i d c o n s i s t s of q u a s i p a r t i c l e e x c i t a t i o n s from the superconducting ground s t a t e , having c h a r a c t e r i s t i c s r e l a t e d to e l e c t r o n and hole e x c i t a t i o n s i n a normal metal. Much as i n a normal metal, the q u a s i p a r t i c l e s are r e a d i l y s c a t t e r e d and are able to change t h e i r energy by a r b i t r a r i l y small amounts. The main d i f f e r e n c e between e x c i t a t i o n s i n superconductors and i n metals l i e s i n the r e l a t i o n s h i p i between energy and momentum, as described below. As the temperature of a metal capable of becoming a superconductor i s lowered below a c h a r a c t e r i s t i c t r a n s i t i o n temperature T^, the Fermi sea becomes unstable against the formation of Cooper p a i r s . ' Further decreasing of the temperature permits more and more q u a s i p a r t i c l e s to condense i n t o the p a i r e d s t a t e u n t i l at absolute zero no unpaired q u a s i p a r t i c l e s , whose energies are w i t h i n approximately k 9 of the Fermi D U energy are present. This h i g h l y c o r r e l a t e d BCS ground s t a t e of the superconductor leads n a t u r a l l y to an energy gap or band of forbidden energies i n the s i n g l e q u a s i p a r t i c l e energy spectrum of a superconductor as w i l l be seen s h o r t l y . 2. Energy Gap At 0 K, the lowest e x c i t e d s t a t e of the superconductor may be found by c o n s i d e r i n g the e f f e c t of adding to the system an e l e c t r o n i n the s t a t e let where i t s mate -k+ i s assumed to be empty. The e f f e c t of t h i s process i s to remove the s t a t e (k+, -k-t-) from the manifold of s t a t e s p a r t i c i p a t i n g i n the p a i r i n g i n t e r a c t i o n r e s p o n s i b l e f o r b i n d i n g together the Cooper pairs;removing t h i s s t a t e r e q u i r e s a f i n i t e energy A^. I f the Bloch energy of the added e l e c t r o n i s measured w i t h respect to the Fermi energy, the energy required to create a q u a s i p a r t i c l e e x c i t a t i o n i n let i n the superconductor i s then given by. • 9 1 9 i E k = + \ >' (2-5) -23-In c o n t r a s t to the s i t u a t i o n i n normal metals at 0 K, the e f f e c t of the p a i r i n g i n t e r a c t i o n i n superconductors i s to e s t a b l i s h a f i n i t e p r o b a b i l i t y that an e l e c t r o n may be added to a s t a t e below as w e l l as above the Fermi su r f a c e . In e i t h e r case, the e x c i t a t i o n energy E, i s p o s i t i v e . S i m i l a r l y , ->-k the energy r e q u i r e d to e x t r a c t an e l e c t r o n from k t i n the ground s t a t e (or create a hole i n let ) i s found to be given by equation 2-5 and to be p o s i t i v e whether |ic| i s greater or l e s s than the Fermi momentum k„. The r minimum energy r e q u i r e d to i n j e c t or e x t r a c t an e l e c t r o n i s then - 4 - 3 Afc|, = A = 10 -10 eV (see eg. t a b l e 2-1). This gap i n the s i n g l e q u a s i p a r t i c l e e x c i t a t i o n spectrum i s that which would be observed i n experiments which deposit or withdraw e l e c t r o n s from the superconductor (as i n normal metal-insulator-superconductor tunnel j u n c t i o n s ) . In experiments which do not i n j e c t or e x t r a c t e l e c t r o n s (as i n electromagnetic absorption work) the energy gap i n the observed spectrum i s E = 2A f o r now two q u a s i p a r t i c l e s must be created: (a) an e l e c t r o n i s e x t r a c t e d from k^t (or a quasihole i s created i n k^t) r e q u i r i n g energy E^l.; ^ a n e l e c t r o n i s created i n s t a t e k^t r e q u i r i n g energy The t o t a l e x c i t a t i o n energy required i s then E ^ + E^2 ^ 2A = E ^ Paragraphs (a) to (c) f o l l o w i n g consider b r i e f l y the temperature, specimen s i z e and magnetic f i e l d dependence of the energy on the average, the t o t a l number of e l e c t r o n s , i t i s tempting to consider these e x c i t a t i o n s as e l e c t r o n - l i k e and h o l e - l i k e i n analogy w i t h the s i t u a t i o n i n normal metals. Such an analogy must be used w i t h caution however as the e x c i t a t i o n s from the ground s t a t e are r e a l l y l i n e a r combinations of "hole s " and " e l e c t r o n s " becoming h o l e - l i k e or e l e c t r o n -l i k e only i n the l i m i t of I £ - l L I > (A/2e„) k„ . (The r r r e x c i t a t i o n s from the ground s t a t e are conveniently described by the Bogoliubov operators gap-Before going on to these t o p i c s , one important point should be s t r e s s e d . Since q u a s i p a r t i c l e s must be created i n p a i r s to conserve, Y + kt V : + kt (2-6) Y + -k+ = V - k + + + v. c k i t -24-->- ->• which create q u a s i p a r t i c l e s i n s t a t e s k+ and -k+ r e s p e c t i v e l y . In equations 2-6, c , + and c^ are the c r e a t i o n and a n n i h i l a t i o n operators f o r Fermions, v f c = i(l-e^/E, ) i s the p r o b a b i l i t y that (kt , -k+ ) i s 9 2 ••-»•-»• occupied and ur = 1-v, i s the p r o b a b i l i t y that (k+ , -k+ ) i s unoccupied. C l e a r l y , f o r k » k p, v£ •> 0 and y k + •* c f c + ). The importance of t h i s d i s t i n c t i o n w i l l become evident i n s e c t i o n D where the use of the so c a l l e d semiconductor model to represent superconductors i s b r i e f l y d iscussed. (a) Temperature Dependence of the Energy Gap At T = 0 K, BCS f i n d the magnitude of the energy gap to be 2A(0) =3.5 k g T c where T^ i s the superconducting-normal t r a n s i t i o n temperature mentioned e a r l i e r . The v a r i a t i o n of the r a t i o A(T)/A(0) w i t h temperature i s p l o t t e d i n f i g u r e 2-4 i n d i c a t i n g that the gap decreases very slowly f o r T < T c/3 so that 2A(T) = 3.5 L I 0 $ T $ 0.3T B c c For temperatures near T^, the gap width i s approximated by 2A(T) =3.2 k T (1-T/T )* B e c Tabulations of A(T)/A(0) are given by Parmenter and Berton (1964) and Muhlschlegel (1959). Table 2-1 gives the t r a n s i t i o n temperatures as w e l l as the t h e o r e t i c a l and experimental values f o r the energy gap f o r four commonly used superconductors ( c f . a l s o Douglass & F a l i c o v , 1964). For Sn at 1.2 K i t i s c l e a r , from f i g u r e 2-4, that the gap width may be taken to be 2A(0) w i t h n e g l i g i b l e e r r o r . (b) Specimen-Size Dependence of the Energy Gap Expressions derived from the BCS microscopic theory are a l l f o r systems of i n f i n i t e volume; f o r t h i n f i l m s proper boundary c o n d i t i o n s should be considered. Anderson's theory of the " d i r t y " - 2 5 -0 .2 .4 .6 .8 1.0 T/T Figure 2-4: Temperature Dependence of Energy Gap parameter A(T) -26-superconductor (Anderson, 1959) and the good agreement of energy gap Superconductor T (°K) c 2A(0) meV BCS Exp't. (T < |T ) A£ 1.19 0.36 0.35 In 3.41 1.03 1.05 Sn 3.72 1.13 1.12 Pb 7.18 2.18 2.54 Table 2-1: Comparison of T h e o r e t i c a l and Experimental Values of 2A(0). values whether determined experimentally from bulk or t h i n f i l m samples (Lynton, 1964), i n d i c a t e that i n zero magnetic f i e l d , the boundary c o n d i t i o n problem i s not a s e r i o u s one so that i t i s assumed henceforth that A ( f i l m s ) - A ( b u l k ) , H = 0 When H ^ 0, however, the s i t u a t i o n becomes more complicated as seen i n the f o l l o w i n g s e c t i o n . (c) Magnetic F i e l d Dependence of Energy Gap I t i s w e l l known (see eg. Lynton (1964) or Kuper (1968)) that f o r T < T^, s u p e r c o n d u c t i v i t y i n a bulk specimen may be quenched by applying a magnetic f i e l d of s u f f i c i e n t strength so t h a t , i f H = H f o r T = 0, c o H (T) c H o = 1 -As i t has been shown, the presence of an energy gap i s fundamental to the superconducting s t a t e which i m p l i e s , that i f su p e r c o n d u c t i v i t y can be m a g n e t i c a l l y quenched f o r T < T^, the energy gap width must depend on the magnetic f i e l d present. No complete microscopic theory of the e f f e c t of a magnetic f i e l d on the energy gap i s yet a v a i l a b l e . Experimental -27-evidence w i t h t h i n f i l m s (Meservey and Douglass, 1964) taken at T = 0.12T i n d i c a t e s that the gap goes continuously to zero w i t h magnetic f i e l d H according to ^ = 1 - (H/H ) 2 (2-7) A(0) c For t h e i r f i l m s , the r a t i o d/X =1.5 was l e s s than the approximate value of 4 which i s appropriate to the f i l m s used i n the present experiment but, as described i n s e c t i o n A, chapter 6, the 7% gap decrease f o r H = 100 Oe p r e d i c t e d by 2-7 agrees favourably w i t h that which we measured f o r t i n . (Here, d i s the f i l m t hickness and X i s a c h a r a c t e r i s t i c p e n e t r a t i o n depth of the order of 500 A i n t i n at 1.2 K.) 3. D i s t r i b u t i o n Function f o r Q u a s i p a r t i c l e s BCS assume the q u a s i p a r t i c l e e x c i t a t i o n s are independent and obey-Fermi Dirac s t a t i s t i c s w i t h the r e s u l t that the d i s t r i b u t i o n f u n c t i o n f o r the q u a s i p a r t i c l e s i s the f a m i l i a r Fermi d i s t r i b u t i o n g(E) = [1 + e x p ( B E ) ] " 1 (2-8) where 8 = 1/k^T and a l l the other symbols have been defined p r e v i o u s l y (Note: E >, M 0) 4. Density of States The s i n g l e q u a s i p a r t i c l e d e n s i t y of s t a t e s per u n i t energy per u n i t volume i s given by BCS as N Q(E) = N (0) |E| ( E 2 - A 2 ( T ) ) " * , E ^ 4(1) (2-9) b m where N (0) i s the d e n s i t y of e l e c t r o n s t a t e s of one s p i n per u n i t energy m per u n i t volume at the Fermi surface of the normal metal. From equation 2-9, i t i s evident that N_(E) reduces to N (0) f o r E >> A(T) as i t should. b m (The i n t e r a c t i o n r e s p o n s i b l e f o r s u p e r c o n d u c t i v i t y a f f e c t s only those s t a t e s w i t h i n a few kT of the Fermi energy and the q u a s i p a r t i c l e and normal Bloch s t a t e s should c o i n c i d e at energies f a r from the Fermi s u r f a c e ) . The s i n g u l a r i t y at E - ± A(T) a r i s e s from the f a c t that s t a t e s "squeezed out" -28-of the gap must be accommodated at the edges, to s a t i s f y the requirement that the t o t a l number of s t a t e s be conserved. D. S i n g l e Q u a s i p a r t i c l e Tunneling In the d i s c u s s i o n of t u nneling between normal metals given i n s e c t i o n B i t was pointed out t h a t , i n order to o b t a i n agreement w i t h the r e s u l t s obtained when one or both of the metals are superconducting, i t was necessary only to s u b s t i t u t e the q u a s i p a r t i c l e d e n s i t y of s t a t e s f o r that of the e l e c t r o n s i n the normal metal. The purpose of t h i s s e c t i o n i s to b r i e f l y o u t l i n e why t h i s a s s e r t i o n i s v a l i d and why the semiconductor energy diagram may be s a f e l y used to i n t e r p r e t s i n g l e q u a s i p a r t i c l e t u n n e l i n g . Concluding the s e c t i o n i s an expression f o r the s i n g l e q u a s i p a r t i c l e tunneling current between superconductors derived on the b a s i s of the semiconductor model. The f i r s t experiments d e a l i n g w i t h tunneling currents between a normal metal and a superconductor were reported by Giaever (1960). Soon afterward, N i c o l et a l (1960) and Giaever and Megerle (1961) extended the study to the case of tunneling between two superconductors. Since then a v a s t experimental and t h e o r e t i c a l e f f o r t has gone i n t o understanding and e x p l o i t i n g t h i s phenomenon; the present s t a t e of the subject i s summarized by B u r s t e i n and Lundqvist (1969). 1. Tunneling Between a Normal Metal and a Superconductor Cohen et a l (1962) computed the current I g ^ i n an M-S tunnel s t r u c t u r e at f i n i t e temperature and found t h a t , because of the twofold degeneracy of energy l e v e l s i n superconductors, the coherence f a c t o r s u^ and v^ from equation 2 _6 disappeared y i e l d i n g the simple r e s u l t XSM = C |M|2 [ f M ( E - eV). - f s ( E ) ] N M(E-eV) N g(E)dE (2-10) (The degeneracy a r i s e s from the f a c t that f o r each|k^|> k p there i s a | k21 < k p such that E^ = E^ 2 and u f c 2 + u^.2 = 1 = v ^ 2 + v k 2 . In summing over a l l k, terms of the form £ u^ h(k) + h'(k) reduce to I h(k) + h'(k).) In equation '2-10, C i s a constant, |M|2 i s the matrix element r e l a t i n g normal e l e c t r o n s on each side of the i n s u l a t o r , f i s the Fermi f a c t o r f o r the normal metal, N M ( E ) i s the d e n s i t y of e l e c t r o n -29-s t a t e s i n the normal metal, N g(E) i s the d e n s i t y of q u a s i p a r t i c l e s t a t e s i n the superconductor (equation 2-9) and V i s the d i f f e r e n c e i n the Fermi l e v e l s . The occupation p r o b a b i l i t y f c ( E ) i s defined such that f_(E) = g(E) f o r E > 0 ( 2 - 1 D = l-g(|E|) f o r E < 0 where g(E) i s the q u a s i p a r t i c l e occupation p r o b a b i l i t y given i n equation 2-8. The s i g n i f i c a n c e of t h i s r e s u l t obtained by Cohen et a l i s that i t could have been derived from.the general t u n n e l i n g equation 2-1 simply by i n s e r t i n g the q u a s i p a r t i c l e d e n s i t y of s t a t e s f o r metal 1 say, remembering that the one dimensional 1 d e n s i t y of s t a t e s i s p r o p o r t i o n a l to the t o t a l d e n s i t y of s t a t e s , and summing over with the assumption of s p h e r i c a l symmetry of the dependence of E upon wave number. In other words, the only e s s e n t i a l d i f f e r e n c e i n the tunneling current when one of the metals i s superconducting i s i n the d e n s i t y of s t a t e s f a c t o r — t h e u and v^ may be ignored and the q u a s i p a r t i c l e e x c i t a t i o n s i n the super-conductor regarded i n one-to-one correspondence w i t h the e l e c t r o n and hole e x c i t a t i o n s i n the normal s t a t e . This view; of the superconductor i s , of course, the s o - c a l l e d semiconductor model f i r s t introduced by Giaever and Megerle (1961) to s u c c e s s f u l l y e x p l a i n t h e i r experimental r e s u l t s . Taking a s l i g h t l y d i f f e r e n t approach to Cohen et a l , Bardeen (1962) confirmed t h e i r r e s u l t , p r o v i d i n g f u r t h e r j u s t i f i c a t i o n f o r using the semiconductor model i n c a l c u l a t i n g simple s i n g l e q u a s i p a r t i c l e tunneling c u r r e n t s . (When the simultaneous t u n n e l i n g of two or more q u a s i p a r t i c l e s i s considered (see eg. W i l k i n s , 1969) the coherence f a c t o r s u^ and v^ do not cancel and the simple independent q u a s i p a r t i c l e model w i t h a modified d e n s i t y of s t a t e s i s inadequate.) 2. Tunneling Between Two Superconductors Fol l o w i n g the approach of Cohen et a l (1962), Josephso'n (1962) computed the tunneling current between two superconductors (S^-S2) and 1 found i t to be of the form JSS * J o + * J 1 S 1 + S 2 + * J ! S 2 + S 1 ( 2 " 1 2 ) -30-where S i s an operator defined i n such a way that when a c t i n g on the ground s t a t e of superconductor i w i t h N e l e c t r o n s i t creates a new super-conducting ground s t a t e of the system w i t h N + 2 e l e c t r o n s . J i s s i m i l a r o to the expression obtained by Cohen et a l and reduces because of c a n c e l l a -t i o n of the coherence f a c t o r s to a form l i k e equation 2-10 w i t h N M ( E ) ->- N (E) and f M ( E ) + f S L ( E ) . The other terms i n equation 2-12 represent a new e f f e c t , now known as the Josephson e f f e c t , which i s discussed f u r t h e r i n s e c t i o n E of t h i s chapter. The important r e s u l t from the foregoing d i s c u s s i o n of s i n g l e p a r t i c l e t u n n e l i n g i n M-S or S^-S^ s t r u c t u r e s i s that the tunnel current due to thermally e x c i t e d q u a s i p a r t i c l e s may be s a t i s f a c t o r i l y derived from the simple semiconductor-type s i n g l e p a r t i c l e approximation of a super-conductor. Such a d e r i v a t i o n i s o u t l i n e d i n the next paragraph f o r an S^-S^ j u n c t i o n as t h i s i s the s i t u a t i o n of relevance to the superconducting tunnel j u n c t i o n to be used f o r charged p a r t i c l e d e t e c t i o n . 3. The S -S- Tunneling Current 1 2 a  When a superconducting tunnel j u n c t i o n l i k e that of f i g u r e 2-1 i s biased at a voltage V, i t may be represented by f i g u r e 2-5. In the semiconductor model sketched there, the thermally e x c i t e d q u a s i p a r t i c l e s i n each superconductor are represented by the occupied s t a t e s above and the unoccupied s t a t e s below the gap. The d e n s i t y of the q u a s i p a r t i c l e s t a t e s i s given by ( c f . equation 2-9) N S ± ( E ) = N.(0) ti.(E) = N i ( 0 ) | E | / ( E 2 - A . 2 ( T ) ) 2 ' ,|E| >A.(T) (2-13) = 0 , |E|< A.(T) and they are occupied w i t h p r o b a b i l i t y fg(E) given by equation 2-11. The s i n g l e q u a s i p a r t i c l e current f l o w i n g from side 1 to side 2 may then be w r i t t e n (see eg. Giaever and Megerle (1961) or Shapiro et a l , (1962)) as h-2 - A ' |M|2 N S 1 ( E ) f g l ( E ) N g 2 (E + e V ) ( l - f s 2 ( E + eV))dE where A' i s a constant, JMI i s the matrix element f o r the t r a n s i t i o n and -31-reler.tron current k l F2» ' A 2 ( T ) _ t . Fermi surface |A2(T) N S 2 ( E ) . 1 ATITT E=0 (£„, ) A1(T) F l ' +j.- i T > 0 K - ^ N S 1 ( E ) Figure 2-5: Q u a s i p a r t i c l e Energy Diagram f o r S^-I-S Ju n c t i o n 3r-..Calculated current . Experimental points ( f i t t e d at point marked by arrow) 1 2 voltage (mV) Figure 2-6: I-V C h a r a c t e r i s t i c f o r Asymmetric Junction. ( a f t e r Shanim kt al . 1Q67^ -32-e_„ i s the Fermi energy of metal 2. A s i m i l a r r e s u l t obtains f o r I„ , 2 so that by assuming the d e n s i t y of s t a t e s 1^(0) and | M| (see Bardeen, 1961) are constant near the Fermi s u r f a c e , the net current may be w r i t t e n 1 " I 1 _ 2 ~ V l = A | M | n x ( E ) n 2 ( E + e V ) [ f s ( E ) - f g(E+eV)]dE (2-14) where A i s a constant. (For mathematical convenience, the lower l i m i t of i n t e g r a t i o n has been extended to -<*>. Because of the a c t i o n of the Fermi f u n c t i o n s the important region of i n t e g r a t i o n i s confined to an i n t e r v a l of a few k T about the Fermi l e v e l . ) The s i m i l a r i t y between the r e s u l t s of equation 2-14 d e r i v e d from the phenomenological semiconductor p i c t u r e and those of equation 2-10 derived from an exact microscopic approach i s evident. The agreement of equation 2-14 w i t h experiment has been demonstrated by Shapiro et a l (1962) who evaluated the equation numerically and obtained a c l o s e f i t w i t h t h e i r experimental data (see f i g u r e 2-6). E. Josephson Tunneling The t u n n e l i n g current between two superconductors as c a l c u l a t e d by Josephson (1962) was given i n equation 2-12. Two new e f f e c t s i m p l i c i t i n t h i s r e s u l t were i n d i c a t e d by Josephson at that time and l a t e r confirmed exp e r i m e n t a l l y : (1) at zero b i a s v o l t a g e , the s i n g l e q u a s i p a r t i c l e current J = 0 , but a dc supercurrent up to a maximum of | j . I can occur; o 1 (2) at f i n i t e v o l t a g e s the usual dc current J q appears but there i s a l s o a high frequency ac current of amplitude | j | and frequency 2eV/h (luV corresponds to 483.6 MHz). The l a t t e r e f f e c t — t h e ac Josephson e f f e c t — i s not r e l e v a n t to the superconducting tunnel j u n c t i o n charged p a r t i c l e d etector and w i l l not be discussed f u r t h e r ; the i n t e r e s t e d reader i s r e f e r r e d to the book e d i t e d by B u r s t e i n and Lundqvist (1969). The other e f f e c t — t h e dc Josephson e f f e c t — c o r r e s p o n d s to the n o n d i s s i p a t i v e t r a n s f e r of Cooper p a i r s across the i n s u l a t o r i n which the q u a s i p a r t i c l e d i s t r i b u t i o n i n e i t h e r superconductor i s not d i s t u r b e d . P h y s i c a l l y , the e f f e c t a r i s e s when two superconductors are arranged o s u f f i c i e n t l y c l o s e together (=10 A) as to become weakly coupled i n the sense that the r e l a t i v e phase between the Cooper p a i r s i n each of the super--33-conductors i s no longer a r b i t r a r y . In other words, the Cooper p a i r c o r r e l a t i o n s extend through the i n t e r v e n i n g d i s t a n c e so that the two superconductors behave i n some respects as a s i n g l e block. The Josephson e f f e c t i s thus not n e c e s s a r i l y a tunneling phenomenon but tunnel j u n c t i o n s have been used as a means of studying the e f f e c t because t h i s i s the p h y s i c a l s i t u a t i o n f o r which d e t a i l e d c a l c u l a t i o n s are most r e a d i l y made. As mentioned i n the i n t r o d u c t i o n to t h i s chapter, the dc Josephson supercurrent i s of i n t e r e s t to the present experiment only because of i t s nuisance value. The way i n which t h i s e f f e c t i s undesirable i s o u t l i n e d i n paragraph 1. Paragraphs 2 and 3 b r i e f l y review the theory of the dc Josephson supercurrent and paragraph 4 considers how the supercurrent may be suppressed w i t h a magnetic f i e l d . I n a d d i t i o n to the B u r s t e i n , Lundqvist book (1969) a recent concise review of the subject has been given by Anderson (1967); probably the most recent comprehensive b i b l i o g r a p h y i s given by Schroen (1968). 1. Unfavourable Aspects of the Josephson E f f e c t With reference to f i g u r e 2-7 i t i s seen that the I-V c h a r a c t e r i s t i c of a j u n c t i o n e x h i b i t i n g Josephson tunneling features a l a r g e supercurrent which may exceed the magnitude of the thermally-e x c i t e d q u a s i p a r t i c l e t u n n e l i n g current discussed i n s e c t i o n D and shown as the curve l a b e l l e d " i d e a l " (V < 2A/e) i n f i g u r e 2-7), by a f a c t o r of 5 to 10. When the supercurrent reaches a c r i t i c a l value (I . ) i t i s c r i t replaced by t u n n e l i n g q u a s i p a r t i c l e s and a voltage V = 2A/e appears across i the j u n c t i o n . Now i t i s evident that a Josephson j u n c t i o n biased at I < I . might be made to switch from V = 0 to V = 2A/e when bombarded c r x t ° i w i t h a charged p a r t i c l e and so be u s e f u l as a p a r t i c l e sensor and energy th r e s h o l d d i s c r i m i n a t o r but i t s n o n - l i n e a r nature makes t h i s response u n s u i t a b l e f o r multichannel spectrometry. I t i s shown i n chapter 3 that the d e s i r e d operating point of the tunnel j u n c t i o n detector (denoted by Q i n f i g u r e 2-8(b)) i s i n the region of high d i f f e r e n t i a l r e s i s t a n c e (r = SW/DI)^ on the q u a s i p a r t i c l e t u n n e l i n g curve. As t h i s b i a s i n g p o i n t cannot be reached reproducibly i n a j u n c t i o n e x h i b i t i n g a Josephson supercurrent, a way must be found to prevent p a i r t u n n e l i n g from occuring. Such a technique i s o u t l i n e d i n paragraph 4 f o l l o w i n g . - 3 4 -I L c r i t z load l i n e switching dc Josephson supercurrent , _ E_xp_e rimen t a l Q t Idea r " " j _L 1.1 (2A/e) Figure 2-7: I-V C h a r a c t e r i s t i c f o r J u n c t i o n D i s p l a y i n g dc Josephson Current -35-2. Theory of the dc Josephson E f f e c t The d e t a i l e d microscopic theory of the Josephson e f f e c t (Scalapino (1969), Josephson (1965), Anderson (1964)) shows that the super-current d e n s i t y J passing between two superconductors comprising a tunnel j u n c t i o n i s given by ( c f . equation 2-12) J = J s i n <j> (2-15) where <J> i s the quantum mechanical phase d i f f e r e n c e between the p a i r wave-f u n c t i o n s i n the two superconductors and J ^ i s a measure of the tunneling p r o b a b i l i t y through the b a r r i e r . J ^ v a r i e s e x p o n e n t i a l l y w i t h the b a r r i e r t h i c k n e s s and als o depends on the temperature. Ambegoakar and Ba r a t o f f (1963) have c a l c u l a t e d J ^ f o r a symmetric j u n c t i o n — i d e n t i c a l superconductors on e i t h e r s i d e of the b a r r i e r — a n d f i n d i t to be A.(T) : n 2k BT where e i s the e l e c t r o n i c charge, i s the low temperature, low voltage r e s i s t a n c e of the j u n c t i o n when i n i t s normal s t a t e and A i s the j u n c t i o n overlap area. When T = 0 K, J ^ i s a maximum given by , n -and f o r t y p i c a l t i n specimens used i n t h i s experiment, -4 2 (R - 0.1 n, A = 7 x 10 cm ) I = J 'A - 10 mA. n 1 1 In p r a c t i c e , the current passing through a j u n c t i o n i s c o n t r o l l e d by an e x t e r n a l power supply—perhaps a b a t t e r y i n s e r i e s w i t h a v a r i a b l e r e s i s t o r (see eg. f i g u r e 5-1)—and the phase d i f f e r e n c e <j> adjusts i t s e l f as i n equation (2-15) to car r y t h i s value of cur r e n t . Consequently as the current determined by the e x t e r n a l c i r c u i t i s increased, <j> increases u n t i l i t reaches i n at which point the j u n c t i o n i s c a r r y i n g the maximum or c r i t i c a l dc Josephson current I c = J^(max) A. For currents greater than I , the j u n c t i o n switches to the q u a s i p a r t i c l e I-V curve as shown i n the recorder t r a c i n g of f i g u r e 2-7. The dotted l i n e on the c h a r a c t e r i s t i c of the f i g u r e i n d i c a t e s r e t r a c e behaviour common to almost a l l specimens. -36-r f 2 - i J = J ^ s i n x 2e * - t r A* 'dt 1 -1 When the current (I) through the j u n c t i o n i s decreasing, the j u n c t i o n becomes unstable f o r J. < I and switches over to the zero-voltage or supercurrent c o n d i t i o n w i t h the s l i g h t e s t e l e c t r i c a l or mechanical disturbance. The o r i g i n and d e t a i l e d shape of t h i s h y s t e r e s i s loop are s t i l l an open question (Schroen, 1968). B a s i c a l l y , i t i s governed by the q u a s i p a r t i c l e t u n n e l i n g current and some workers (March and B l a c k f o r d , 1967) suggest that the power d i s s i p a t e d i n the j u n c t i o n may play an important r o l e . 3. E f f e c t of Magnetic F i e l d on dc Josephson Current A f u r t h e r r e s u l t of the microscopic theory i s that the dc Josephson current should depend on the vector p o t e n t i a l s i n c e the quantum mechanical phase d i f f e r e n c e depends on the vector p o t e n t i a l . Hence, i n mks u n i t s , equation 2-15 should be ( J a k l e v i c et a l , 1965) (2-17) where the l i n e i n t e g r a l term from side 1 to side 2 i s included to ensure that the current d e n s i t y i s i n v a r i a n t under a gauge transformation. Equation 2-17 has been evaluated by J a k l e v i c et a l , (1965) f o r a symmetric t u n n e l i n g j u n c t i o n placed i n a uniform magnetic f i e l d B which i s p a r a l l e l to the oxide l a y e r ; they f i n d J ( x ) = J s i n '[<f>(6) +C2e/h)Bx(t + 2X)] (2-18) where t i s the oxide t h i c k n e s s , X i s the pene t r a t i o n depth of the magnetic f i e l d and x i s the di s t a n c e along the j u n c t i o n perpendicular to B (x = 0 at the center of the j u n c t i o n , see f i g u r e 2-8(a).) Equation 2-18, which gives the magnetic f i e l d dependence of the Josephson c u r r e n t , i n d i c a t e s that the current at one poi n t can flow opposite to the way i t flows at another. I t f o l l o w s from equation 2-18 ther e f o r e that f o r a j u n c t i o n of given s i z e there are c e r t a i n values of B* f o r which the maximum dc Josephson current I(max) c a r r i e d by the j u n c t i o n w i l l be zero. This may be seen by i n t e g r a t i n g 2-18 from -£W to, +£W (W i s the len g t h of the j u n c t i o n perpendicular to S) -37-Figure 2-8 (a): Schematic Cross-Section of Tunneling.Junction I ( A r b i t r a r y u n i t s ) max ] -38-and s e t t i n g I = J • A so that I - \ ^ X E l I . s i n ,,,(0) (2-19) where p = $ . /$ 3 ° B(2X + t)W = f l u x enclosed by e f f e c t i v e c r o s s -s e c t i o n a l area of j u n c t i o n -7 2 h/2e = 2.07 x 10 G cm = quantum u n i t of magnetic f l u x and $. = 3 $ = As mentioned above, the phase d i f f e r e n c e <f> adjusts to the experimental c o n d i t i o n s so that maximum p o s i t i v e current flow i s obtained w i t h <J>(0) = +?TT making I(max) = 1^ s i n (pip pir (2-20) I t i s c l e a r from (2-20) that I(max) = 0 whenever $. equals some i n t e g r a l m u l t i p l e of h/2e and a p l o t of I(max) against B w i l l resemble the f a m i l i a r Fraunhoffer d i f f r a c t i o n p a t t e r n (see f i g u r e 2-8(b)). T y p i c a l l y , W = 0.03 cm and X =500 A so that the period of I(max) i n B i s about 0.7 G. The existence of t h i s e f f e c t was f i r s t confirmed by Rowell (1963) using symmetrical Pb-Pb j u n c t i o n s . In j u n c t i o n s of small dimension or low c r i t i c a l c u rrent, the magnetic f i e l d due to the tunneling current can be neglected. Such i s not the case i f the j u n c t i o n width becomes comparable to or l a r g e r than a c h a r a c t e r i s t i c length 2X^, the Josephson p e n e t r a t i o n depth, given by ( i n mks u n i t s ) X T = (h/4ey A J . ) * - (2-21) J o 1 where u i s the p e r m e a b i l i t y of f r e e space and i t i s assumed that o 2X + t - 2X . The s i g n i f i c a n c e of Xj i s that when the j u n c t i o n width W i s la r g e compared to X^, the a p p l i e d magnetic f i e l d i s screened from the i n t e r i o r of the j u n c t i o n due to the flow of the Josephson currents themselves w i t h the r e s u l t that most of the d i r e c t current flows w i t h i n a distance 2X T of the j u n c t i o n edges ( F e r r e l l and Prange, 1963). Estimates of X f o r -39-t y p i c a l j u n c t i o n s i n the l i t e r a t u r e range from 0.1 to 1.0 mm which i s s u f f i c i e n t l y l a r g e compared to the j u n c t i o n dimensions used i n t h i s e x p e r i -ment, that uniform p e n e t r a t i o n of the magnetic f i e l d may be s a f e l y assumed. Instances where Xj i s important have been discussed by Rowell (1963), Owen and Scalapino (1967) and P r i t c h a r d and Schroen (1968). 4. Suppression of Josephson Supercurrent Equation 2-20 and f i g u r e 2-8(b) suggest two a l t e r n a t i v e approaches f o r m a g n e t i c a l l y suppressing the supercurrent: e i t h e r apply a p r e c i s e l y determined f i e l d such that p i s an i n t e g e r ( i e . B = n$ Q(W(2X + t ) ) , n = 1,2 ) making I(max) = 0 or apply a s u f f i c i e n t l y l a r g e f i e l d that the (pir) ^  dependence reduces I (max) to n e g l i g i b l e p r o p o r t i o n s . Immediately, the f i r s t o p t i o n . i s r u l e d out because of the f a c t t h a t , i n the j u n c t i o n s used i n t h i s experiment, no sharp minima i n which the supercurrent vanished were ever observed. (Even i f such minima had been obtained, an extremely w e l l - r e g u l a t e d magnetic f i e l d would, of course, be re q u i r e d to maintain that c o n d i t i o n . ) E x p e r i m e n t a l l y , i t turns out (see s e c t i o n 1, chapter 6) that the second approach i s the one to take i n that magnetic f i e l d s of 30-100 G are s u f f i c i e n t to e f f e c t i v e l y suppress the supercurrent w i t h a maximum 'reduction i n the energy gap width of l e s s than about.6%. F. M u l t i p a r t i c l e Tunneling Phenomena For the sake of completeness, i t should be mentioned here that higher order or m u l t i p a r t i c l e t u n n e l i n g processes may a l s o occur i n super-conducting t u n n e l i n g j u n c t i o n s . As the name i m p l i e s , m u l t i p a r t i c l e t u n n e l i n g i s the name given to the simultaneous t r a n s f e r of more than one q u a s i p a r t i c l e through the tunnel j u n c t i o n , b a r r i e r . (Probably no higher order than t w o - p a r t i c l e t u n n e l i n g has ever been observed ( W i l k i n s , 1969).) This i 14 second order process ( p r o p o r t i o n a l to |M| ) goes only i f there i s an i n t e r -a c t i o n between the p a r t i c l e s (as i n a superconductor) and i s manifested as a sharp i n c r e a s e i n the current at eV = A (see Taylor and B u r s t e i n , 1963). An e x c e l l e n t review of t h i s t o p i c has r e c e n t l y been given by W i l k i n s (1969); f u r t h e r d i s c u s s i o n may als o be found i n appendix B of t h i s t h e s i s where d e f e c t i v e or "l e a k y " j u n c t i o n s are considered. -40-CHAPTER 3 OPERATING PRINCIPLES OF THE SUPERCONDUCTING CHARGED PARTICLE DETECTOR A. I n t r o d u c t i o n This chapter o u t l i n e s the p r i n c i p l e s and p r a c t i c a l operation requirements of the superconducting tunnel j u n c t i o n as a charged p a r t i c l e d e t e c t o r . To place t h i s a p p l i c a t i o n of tunnel j u n c t i o n s i n pe r s p e c t i v e , i t i s convenient to review b r i e f l y : (1) three of the present techniques f o r measuring charged p a r t i c l e energy spectra and (2) charged p a r t i c l e d e t e c t i o n a p p l i c a t i o n s of other superconducting devices. Because of i t s r e l a t i o n to the d e t e c t i o n of charged p a r t i c l e s , previous work on the d e t e c t i o n of microwave r a d i a t i o n i s a l s o reviewed ( s e c t i o n B). The d e t e c t i o n of i o n i z i n g r a d i a t i o n i s considered b r i e f l y i n s e c t i o n C f o r the purpose of s t i p u l a t i n g the a t t r i b u t e s which the tunnel j u n c t i o n ' s response to such r a d i a t i o n should have i f the j u n c t i o n i s to be u s e f u l as a spectrometer. In s e c t i o n D i t i s e s t a b l i s h e d that the most favourable tunnel j u n c t i o n c o n f i g u r a t i o n i s the symmetric one with l e a d f i l m s as the superconductor. The nature of the q u a s i p a r t i c l e e x c i t a t i o n s u l t i m a t e l y generated by an alpha p a r t i c l e i n the superconducting tunnel j u n c t i o n i s considered i n s e c t i o n E. A small s i g n a l equivalent c i r c u i t f o r the detector i s given i n s e c t i o n F f o l l o w i n g which ( s e c t i o n G) the assumed form of the r e s u l t i n g s i g n a l current pulse i s st a t e d along w i t h estimates of i t s parameters. Because of the r e s t r i c t i o n i t places on the maximum acceptable i n s u l a t i n g l a y e r t h i c k n e s s , the tunneling p r o b a b i l i t y i s d e a l t w i t h i n s e c t i o n H. A note on the p r a c t i c a l operation of the detector i s given i n s e c t i o n I and, concluding the chapter, i s a short survey of tunnel j u n c t i o n noise sources ( s e c t i o n J ) . -41-1. Present Methods of Charged P a r t i c l e Spectrometry Broadly speaking, the methods can be d i v i d e d i n t o three major c a t e g o r i e s : the measurement of momentum, energy l o s s or v e l o c i t y . General reviews of these methods can be found i n books by Yuan (1961) and Ajzenberg-Selove (1960); a short survey i s given by Wood (1965). Since the superconducting " d e t e c t o r " i s e s s e n t i a l l y an energy-loss device, only those detectors of the "energy l o s s measurement" type w i l l be discussed here. When a charged p a r t i c l e passes through matter, i t loses energy through the e x c i t a t i o n and i o n i z a t i o n of atoms c l o s e to the path of the p a r t i c l e . This property has been e x p l o i t e d i n the development of s e v e r a l types of p a r t i c l e spectrometers. Three i n common use are the g a s - f i l l e d , s c i n t i l l a t i o n , and semiconductor d e t e c t o r s . B a s i c a l l y , the g a s - f i l l e d detector i s a metal or glass chamber, equipped w i t h an anode and cathode, and f i l l e d w i t h an accu r a t e l y defined volume of gas. The passage of an energetic charged p a r t i c l e through a s u i t a b l e "window" i n t o the gas volume causes e l e c t r o n s to be removed from some of the gas molecules thereby producing e l e c t r o n - p o s i t i v e - i o n p a i r s . Under the i n f l u e n c e of the e l e c t r i c f i e l d , the e l e c t r o n s migrate to the anode and the p o s i t i v e ions to the cathode. Because the average amount of energy l o s t by the charged p a r t i c l e to produce an i o n p a i r i s approximately independent of the energy of the p a r t i c l e , the amplitude of the r e s u l t i n g charge pulse i s p r o p o r t i o n a l to the energy l o s t i n the gas by the i o n i z i n g p a r t i c l e . A t y p i c a l s c i n t i l l a t i o n detector c o n s i s t s of a s c i n t i l l a t i n g p h o s p h o r — c r y s t a l , l i q u i d , p l a s t i c s o l i d or g a s — o p t i c a l l y coupled to a p h o t o m u l t i p l i e r tube. When an i o n i z i n g p a r t i c l e s t r i k e s the phosphor, the energy d i s s i p a t e d causes l o o s e l y bound e l e c t r o n s i n the m a t e r i a l to be e x c i t e d i n t o the conduction band. Then, at imp e r f e c t i o n s i t e s i n the c r y s t a l , a l a r g e f r a c t i o n of the e x c i t e d e l e c t r o n s f a l l back to t h e i r ground s t a t e and emit a photon i n a process c a l l e d fluorescence. The photons emitted during fluorescence s t r i k e the photocathode of the p h o t o m u l t i p l i e r tube whose output s i g n a l i s p r o p o r t i o n a l to the number of photons generated or the energy l o s t by the charged p a r t i c l e i n t r a v e r s i n g the phosphor. In t h i s case, p r o p o r t i o n a l i t y a r i s e s from the approximate constancy of the -42-average energy l o s s per photon generated. During the past ten years, semiconductor detectors have become almost u n i v e r s a l l y p r e f e r r e d f o r charged p a r t i c l e d e t e c t i o n . Books by Dearnaley and Northrop (1966) and Taylor (1963) deal thoroughly w i t h these detectors g i v i n g d e t a i l s on theory and a p p l i c a t i o n ; a recent review by Tavendale (1967) surveys the present s t a t e of detector technology. E s s e n t i a l l y , the semiconductor detector c o n s i s t s of a p-n j u n c t i o n which i s formed c l o s e to one face of a slab of germanium or s i l i c o n . T y p i c a l l y the j u n c t i o n i s w i t h i n 0.5 p of the one face and i s very abrupt. When the j u n c t i o n i s reverse b i a s e d , a d e p l e t i o n region i s created which forms the s e n s i t i v e volume of the d e t e c t o r . The e l e c t r o n s and holes generated by the passage of a charged p a r t i c l e through t h i s region d r i f t to the electrodes under the a p p l i e d e l e c t r i c f i e l d thence g i v i n g r i s e to a charge pulse whose amplitude i s l i n e a r l y r e l a t e d to the p a r t i c l e energy, the l i n e a r i t y being due to the approximate energy independence of the average energy l o s s per e l e c t r o n - h o l e p a i r . I t i s of i n t e r e s t to ;note that the average energies w r e q u i r e d to produce e l e c t r o n - h o l e p a i r s i n s i l i c o n and germanium are 3.6 and 2.9 eV r e s p e c t i v e l y , compared to an average value of about 30 eV per i o n p a i r i n gases and about 300 eV per photoelectron at the photocathode of a s c i n t i l l a t i o n counter. (Dearnaley and Northrop, 1966). This i m p l i e s , t h e r e f o r e , that f o r a given p a r t i c l e energy, the a v a i l a b l e s i g n a l i n a semiconductor i s s i g n i f i c a n t l y l a r g e r than f o r the other counters and that the s t a t i s t i c a l f l u c t u a t i o n s , expressed as a percentage of the s i g n a l (see chapter 1) are correspondingly smalle r . 2. Charged P a r t i c l e D e t e ction A p p l i c a t i o n s of Superconducting Devices Although the d e t e c t i o n of charged p a r t i c l e s with supercon-ducting tunneling j u n c t i o n s has never been reported p r i o r to the present experiment, s e v e r a l workers have proposed .and/or performed experiments i n v o l v i n g the d e t e c t i o n of a l p h a ; p a r t i c l e s using other superconducting devices. Andrews et a l (1949) bombarded a s t r i p of niobium n i t r i d e (3.5 x 0.4 x 0.006 mm, T^ = 15.6 K ) , w i t h alpha p a r t i c l e s from a polonium source and found that countable pulses were produced, one f o r each impact. The NbN s t r i p was maintained at 15.5 K, j u s t below i t s t r a n s i t i o n tem--43-perature, a d i r e c t current was ap p l i e d and the p o t e n t i a l between the ends of the s t r i p monitored w i t h a pulse a m p l i f i e r and o s c i l l o s c o p e . I t was found that the counting r a t e was a maximum when the ambient current was 40 mA and the specimen was maintained at the mid point of i t s t r a n s i t i o n between the normal and superconducting s t a t e s — a change of ± 0.1 K reduced the s i g n a l to the same l e v e l as the noise . I t was estimated that the -4 pulses l a s t e d f o r about 10 sec w i t h an amplitude of about 0.1 uV. The maximum s i g n a l to noise r a t i o was 3 to 1. No t h e o r e t i c a l explanation of t h e i r f i n d i n g s was given by the authors but, s i n c e the s t r i p was presumably i n the intermediate s t a t e , the temperature r i s e accompanying the passage of an alpha p a r t i c l e would be manifested as an increase i n the s t r i p r e s i s t a n c e and therefore (assuming constant current b i a s i n g ) as an increase i n the voltage across the specimen. A proposal to use the c r y o t r o n , or superconducting s w i t c h , to detect i o n i z i n g p a r t i c l e s was put forward by Sherman (1962,A). The cr y o t r o n , i t was suggested, should c o n s i s t of two gate loops, made of d i s s i m i l a r superconductors such as lead and t i n w i t h a c o n t r o l loop of niobium on the lead gate. I f the device were maintained at a temperature j u s t below the lowest of the three c r i t i c a l temperatures (T =3.72 K f o r Sn), c i t was argued that the c r y o t r o n could be kept i n a metastable s t a t e w i t h the o t i n loop (1000 A t h i c k , 10 u wide) c a r r y i n g a p e r s i s t e n t current s l i g h t l y l e s s than the c r i t i c a l current at which s u p e r c o n d u c t i v i t y would be quenched. The passage of a charged p a r t i c l e i n t o that loop might then e s t a b l i s h a small r e g i o n of normal metal across the t i n f i l m and cause the current to be suddenly switched i n t o the lead loop which, p r i o r to t h i s i n s t a n t , would be i n the superconducting s t a t e but not c a r r y i n g any cur r e n t . A pulse would then r e s u l t on the niobium c o n t r o l winding which could be used to r e g i s t e r the count and then r e f l e c t e d back to d r i v e the Pb gate normal thereby s w i t c h i n g the current back i n t o the t i n loop ready f o r the next p a r t i c l e . Sherman (1962,B) made a second proposal f o r a superconducting nuclear p a r t i c l e detector which was to c o n s i s t of a narrow (lOu), t h i n o (1000 A) f i l m of c u r r e n t - c a r r y i n g superconductor cooled w e l l below i t s t r a n s i t i o n temperature and placed i n s e r i e s w i t h a small r e s i s t o r . I t was argued that an alpha p a r t i c l e impinging on a t i n f i l m of such dimensions would cause the r e s i s t a n c e to change from zero to about 0.2 by i n s e r t i n g a s mall transverse s t r i p of normal metal i n the path of the current. The -44-r e d u c t i o n of current would be detected across the load r e s i s t o r as a voltage pulse estimated to be about 1 mV f o r a 5 mA t r a n s p o r t current and 10 ft loa d . The r i s e time of the pulse was estimated to be roughly l O - " ^ sec and the f a l l time to be about 10 ^ sec. Recently, S p i e l et a l (1965), s u c c e s s f u l l y detected 5.3 MeV alpha p a r t i c l e s using t h i n superconducting f i l m s i n an experiment s i m i l a r i n s p i r i t to that o u t l ined by Sherman (1962,B) but d i f f e r e n t i n d e t a i l i n that the p a r t i c l e induced t r a n s i t i o n s were observed by means of the IR drop produced by the ambient current along the length of the f i l m rather than i n a load r e s i s t a n c e . The e f f e c t was observed i n both indium and t i n f i l m s ; t y p i c a l f i l m s were 1000 A t h i c k and 34 u wide. S i m i l a r l y to the experiment reported by Andrews (1949), voltage pulses occurred only f o r temperatures near the c r i t i c a l temperature and transport currents near the c r i t i c a l c u r r e n t . For an In f i l m c a r r y i n g a current of 5.5 mA at 3.3 K, the v o l t a g e pulses had a peak amplitude of approximately 200 pV r i s i n g i n about 15 nsec and decaying i n roughly 70 nsec. The radius of the c y l i n d r i c a l volume d r i v e n normal by the passage of an alpha p a r t i c l e was estimated to be 14 p under these c o n d i t i o n s . The s w i t c h i n g a c t i o n i n these devices could be used as a p a r t i c l e detector and energy threshold d i s c r i m i n a t o r but i t i s not s u i t e d to multichannel spectrometry. A subject that might be re--investigated, f o l l o w i n g Andrews' l e a d , i s the p o s s i b i l i t y of employing intermediate s t a t e bolometers as charged p a r t i c l e spectrometers, f o r a l l o y s : having high s e n s i t i v i t y f o r heat pulse d e t e c t i o n are now being developed. For example, Ackerman et a l (1969) o , r e p o r t the development of 400 A t h i c k Zn-Cd f i l m s f o r temperature pulse d e t e c t i o n w i t h a s e n s i t i v i t y (AR/RAT) = 400near the centre of the normal-superconducting t r a n s i t i o n (T - 0.5 K). (For comparison, (AR/RAT) - 5 f o r carbon f i l m s . ) I t i s conceivable that such a f i l m , mounted on a s u i t a b l e s ubstrate and biased- e l e c t r i c a l l y , m a g n e t ically, and thermally at i t s m i d - t r a n s i t i o n p o i n t , could be made to respond l i n e a r l y to heat pulses r e s u l t i n g from energetic charged p a r t i c l e s impinging upon the s u b s t r a t e . As the c r o s s - s e c t i o n a l area of these f i l m s must be kept small to make the r e s i s t a n c e h i g h , the volume a v a i l a b l e f o r d i s s i p a t i o n of a concentrated heat pulse i s l i m i t e d . Consequently, to avoid switching the f i l m completely normal, i t would be necessary to d i s s i p a t e the o r i g i n a l heat pulses i n a r e l a t i v e l y l a r g e volume which, of course, increases the response -45-time. The stopping of low energy r a d i a t i o n i n the f i l m i t s e l f i s another a l t e r n a t i v e but now the heating would no longer be macroscopic and the complex behaviour of mixed s t a t e domains i n the presence of e l e c t r i c a l , magnetic and thermal gradients must be considered. B. Detection of Microwave R a d i a t i o n With Tunnel Junctions Since the pioneering work of Giaever (1960,1961) on tunneling between superconductors, much a t t e n t i o n has been given to using t h i s phenomenon to study the more fundamental aspects of s u p e r c o n d u c t i v i t y . In a d d i t i o n , however, the p o t e n t i a l p r a c t i c a l a p p l i c a t i o n s of superconductive t u n n e l i n g were q u i c k l y r e a l i z e d w i t h the r e s u l t that many attempts have been made to develop u s e f u l devices based on the t u n n e l i n g p r i n c i p l e . A recent review by Taylor (1968) summarizes work that has been done i n t h i s area. Of p a r t i c u l a r i n t e r e s t , because the processes in v o l v e d are somewhat analogous to those t a k i n g place i n the d e t e c t i o n of charged p a r t i c l e s w i t h tunnel j u n c t i o n s , i s the work which has been done on the d e t e c t i o n of microwave photons and phonons. The aim of t h i s s e c t i o n i s to acknowledge the h i s t o r i c a l p r i o r i t y of these tunnel j u n c t i o n a p p l i c a t i o n s and to i l l u s t r a t e thereby those ideas and r e s u l t s from previous workers which have b e n e f i t e d the present experiment. For s i m p l i c i t y , a symmetric supercon-ducting j u n c t i o n w i l l be considered. 1. Microwave and I n f r a r e d Photon Detection The f e a s i b i l i t y of using tunneling j u n c t i o n s as microwave and f a r i n f r a - r e d quantum detectors was f i r s t proposed and analyzed by B u r s t e i n et a l (1961A,B). The process in v o l v e d i n c r e a s i n g the q u a s i p a r t i c l e d e n s i t y by " o p t i c a l l y " e x c i t i n g q u a s i p a r t i c l e s across the energy gap of the superconductor i n a manner analogous t o the quantum d e t e c t i o n of v i s i b l e and near i n f r a - r e d r a d i a t i o n by p-n semiconductor diodes. In the former case, the lower frequency (a)) l i m i t of detectable r a d i a t i o n would be governed by the requirement that the energy of the r a d i a t i o n absorbed i n the j u n c t i o n f i l m s i s hw >, 2A(T) ; which i s the energy required to break up a Cooper p a i r i n t o two q u a s i p a r t -i c l e s , which could subsequently t u n n e l — a s i n d i c a t e d s c h e m a t i c a l l y i n -46-f i g u r e 3-1(a). F o l l o w i n g a short burst of r a d i a t i o n , the e x t r a current due to the t u n n e l i n g of the " o p t i c a l l y " e x c i t e d q u a s i p a r t i c l e s would be i n the form of a pulse superimposed on the background tunneling current which, f o r b i a s v o l t a g e s eV < 2A(T) (as seen i n chapter 2 ) , i s propor-t i o n a l to the d e n s i t y of thermally e x c i t e d q u a s i p a r t i c l e s . The magnitude and d u r a t i o n of t h i s pulse would depend upon such f a c t o r s as the q u a s i p a r t i c l e t u n n e l i n g p r o b a b i l i t y and l i f e t i m e . An important f e a t u r e of such detectors i s that the frequency range they cover, approximately 85 GHz (A£) to 650 GHz (Pb), i s one which has h i t h e r t o been v i r t u a l l y i n a c c e s s i b l e . By the same token, because of the paucity of l a b o r a t o r y generators i n t h i s range, the p r o p e r t i e s of the B u r s t e i n device have not been widely s t u d i e d ; the only c l a i m to the e f f e c t being found experimentally seems to be the unpublished work of Dayem and M i l l e r (see T a y l o r , 1968). I f the i n t e r a c t i o n shown i n : f i g u r e 3-l(a) were the only one p o s s i b l e , no change i n the tunneling current would be observed f o r ha) < 2A(T). Dayem and M a r t i n (1962) n o t i c e d experimentally, however, that considerable i n t e r a c t i o n between the microwave f i e l d and the j u n c t i o n d i d take place f o r fiu < 2A(T) v i a a process which has come to be known as photon-assisted t u n n e l i n g (see f i g u r e 3-1(b)). As i n the case f o r e x c i t a t i o n across the gap described above, the r a d i a t i o n breaks up a Cooper p a i r i n t o two q u a s i p a r t i c l e s one of which remains i n the normal f l u i d i n S w h i l e the other tunnels through the b a r r i e r to become part of the normal a f l u i d of S^. I t i s c l e a r that the p r o c e s s - i n d i c a t e d i n f i g u r e 3-1(b) can proceed f o r r a d i a t i o n energies l e s s than 2A(T) and, i n f a c t , f o r a j u n c t i o n biased at voltage V, the threshold f o r a one-photon process i s fun = 2A(T)-eV . In a d d i t i o n , multiphoton processes, having, a threshold given by nfuo = 2A(T)-eV. (n an i n t e g e r ) were observed by Dayem and M a r t i n and more, r e c e n t l y by Cook and Everett (1967). .,. Perhaps the most u s e f u l reference at the present to the -47-Figure 3-1: Schematic Energy Diagrams D e p i c t i n g E f f e c t s of Microwave Photon.or Phonon Absorption (a,b). and Phonon Generation (c) -48-subject of photon-assisted tunneling i s the work of Cook and Everett c i t e d above who give extensive experimental r e s u l t s — i n c l u d i n g , f o r example, the d e t e c t i o n of as l i t t l e as 2 uW of power at 36 GHz—and improve on the e a r l i e r t h e o r e t i c a l i n t e r p r e t a t i o n s of Tien and Gordon (1963) and Cohen, et a l (1963). 2. Microwave Phonon Detection In the t w o - f l u i d p i c t u r e of the superconductor (0 < T < T ), a dynamic e q u i l i b r i u m e x i s t s between the q u a s i p a r t i c l e s and the Cooper p a i r s . P a i r s are c o n t i n u a l l y being broken up by absorbing thermal phonons of energy greater than or equal to 2A(T) and q u a s i p a r t i c l e s are c o n t i n u a l l y recombining i n t o p a i r s v i a the emission of phonons of energy equal to 2 A ( T ) — a c c o r d i n g to B u r s t e i n , et a l (1961) and Rothwarf and Cohen (1963) phonon emission r a t h e r than photon emission i s the dominant decay mode. Because of t h i s property, tunnel j u n c t i o n s can be used as e i t h e r detectors or generators of very high frequency phonons—several hundreds of GHz. Detection takes place v i a a process analogous to " o p t i c a l " e x c i t a t i o n as shown i n f i g u r e 3-1(a). The absorption of phonons of energy greater than 2A(T) i n a j u n c t i o n r e s u l t s i n the d i s s o c i a t i o n of Cooper p a i r s i n t o q u a s i p a r t i c l e s which tunnel and give r i s e to an e x t r a c u r r e n t . U n f o r t u n a t e l y , t h i s e f f e c t i s d i f f i c u l t to study as phonons i n t h i s frequency range are p r o h i b i t i v e l y d i f f i c u l t to generate by conventional means. Eisenmenger and Dayem (1967) overcame t h i s d i f f i c u l t y and were able to observe the e f f e c t by using another tunnel j u n c t i o n as the source of phonons. Ac o u s t i c coupling was provided by evaporating the two Sn-SnO^-Sn j u n c t i o n s on opposite ends of a 1 cm long sapphire c r y s t a l ; the generating u n i t was biased at eV £ 2A(T) and the r e c e i v i n g one at eV <2A(T). The d e t a i l s of phonon generation are i l l u s t r a t e d s chematically i n f i g u r e 3-1(c). Q u a s i p a r t i c l e s . t u n n e l i n g from S i n t o S undergo i n 3 D general two r e l a x a t i o n processes. F i r s t , there i s a r e l a x a t i o n to the top of the energy gap i n a c h a r a c t e r i s t i c time T ^ , w i t h the emission of a phonon of energy fuo = eV-2A followed by a recombination w i t h another e x c i t e d q u a s i p a r t i c l e i n a c h a r a c t e r i s t i c time x^ accompanied by the emission of a phonon of energy "hco = 2A. Recent experimental r e s u l t s ( M i l l e r & Dayem, (1967) and Levine and Hsieh (1968)) i n d i c a t e 5 x 10 < T r < 2 x 10 sec, -1 depending on the temperature, and x < 10 x . T R Microwave phonons can a l s o i n t e r a c t w i t h tunnel j u n c t i o n s to produce phonon-assisted tunneling which i s analogous to the photon-assisted t u n n e l i n g described above and shown i n f i g u r e 3-1 (b). Eisenmenger and Dayem (1967 , see a l s o Eisenmenger, 1969) found the e f f e c t to be n e g l i g i b l e at the frequencies w i t h which they were working but other workers, eg. Lax and Vernon (1965), Abeles and G o l d s t e i n (1965) and G o l d s t e i n et a l (1966), using frequencies i n the X band (9 GHz) have demonstrated the existence of the e f f e c t . Because the frequencies were too low to produce current steps at v o l t a g e s eV = 2A(T)-nnu) as i n the quantum photon-assisted tunneling experiments, the process was manifested simply as an excess current dependent on both the j u n c t i o n b i a s and a c o u s t i c power i n c i d e n t on the j u n c t i o n . Agreement w i t h theory i s only q u a l i t a t i v e and s e v e r a l serious d i s c r e p a n c i e s between experiment and theory remain. C. Detection of I o n i z i n g R a d i a t i o n In nuclear p a r t i c l e spectroscopy, a p a r t i c l e i s "detected" by measuring i t s k i n e t i c energy and e s t a b l i s h i n g an i n t e r v a l of time during which the p a r t i c l e i n t e r a c t e d w i t h the d e t e c t i n g medium. Thus, a d i s t i n c -t i o n i s made between devices which respond to nuclear r a d i a t i o n i n the sense j u s t defined and a device l i k e the geiger counter which merely s i g n a l s the presence of r a d i a t i o n , g i v i n g no measure of i t s energy other than i t must exceed some threshold value. Photographic emulsions, on the other hand, y i e l d i n f o r m a t i o n concerning the energy of i n d i v i d u a l p a r t i c l e s but are not able to d i s t i n g u i s h i n time between 1 the a r r i v a l of one p a r t i c l e and any other. To detect charged p a r t i c l e s , then, the superconducting tunnel j u n c t i o n must respond i n a manner that i s not only p r o p o r t i o n a l to the energy of the p a r t i c l e s impinging on the j u n c t i o n but i s s u f f i c i e n t l y r a p i d and r e v e r s i b l e as to minimize the time r e s o l u t i o n — a measure of the smallest time i n t e r v a l which can occur between the a r r i v a l of two p a r t i c l e s that are d i s t i n g u i s h e d and recorded as separate e n t i t i e s . The p r i n c i p l e upon which the superconducting i o n i z i n g p a r t i c l e detector i s based has been o u t l i n e d i n chapter 1. I t i s shown there and confirmed experimentally (see chapter 7) that x, the c h a r a c t e r i s t i c time during which the s i g n a l current pulse 1 relaxes back to zero, i s of the order 1 ^ of 10 - 10 sec. As the current pulse i s assumed to r i s e instantaneously, -50-(instantaneously meaning i n times much shorter than x ) the o v e r a l l pulse width i s expected to be s u f f i c i e n t l y short to make p o s s i b l e the r e s o l u t i o n i n time of i n d i v i d u a l events i n f l u x e s as high as 1 0 6 p a r t i c l e s / s e c . In the three types of p a r t i c l e detector described e a r l i e r i n s e c t i o n A, there was some t y p i c a l energy w required to generate the i o n p a i r s , photoelectrons or e l e c t r o n - h o l e p a i r s which c o n s t i t u t e d the d e t e c t i o n s i g n a l . This concept was c a r r i e d over to the superconducting tunnel j u n c t i o n i n chapter 1 where, i f energy AE i s deposited i n the j u n c t i o n by a nuclear p a r t i c l e , the number of q u a s i p a r t i c l e p a i r s e x c i t e d i s taken to be N = AE/w o Since i n conventional p a r t i c l e spectrometers w does not vary s t r o n g l y w i t h p a r t i c l e energy, i t i s hoped (although there i s no evidence yet to sub-s t a n t i a t e t h i s hope) that the amplitude of the excess current pulse may be l i n e a r l y dependent upon the p a r t i c l e energy l o s s i n the detector. From t h i s b r i e f h e u r i s t i c d i s c u s s i o n , i t can be seen that i n p r i n c i p l e the superconducting tunnel j u n c t i o n may p o s s i b l y s a t i s f y the c r i t e r i a e s t a b l i s h e d above f o r i t to be considered a p a r t i c l e detector. The f o l l o w i n g s e c t i o n s w i l l examine the a t t r i b u t e s of the tunnel j u n c t i o n which determine i t s performance as a detector and describe how such a j u n c t i o n may be used i n p r a c t i c e . D. Optimum J u n c t i o n Type and Constituent M a t e r i a l 1. Type of J u n c t i o n to be Used as a Detector There are four types or c o n f i g u r a t i o n s of tunnel j u n c t i o n which could be of i n t e r e s t as charged p a r t i c l e d e t e c t o r s : the normal metal-normal metal j u n c t i o n (M-M), the normal metal-superconductor j u n c t i o n (M-S), the symmetric j u n c t i o n (S-S) w i t h the same superconductor on e i t h e r side of i the b a r r i e r and the asymmetric j u n c t i o n (S^-S^) composed of two d i f f e r e n t superconductors. (Because of the d i f f i c u l t y i n d e a l i n g w i t h the Sj-S^ j u n c t i o n mathematically and because the M-S and S-S j u n c t i o n s represent l i m i t i n g cases of the S^-S 2 j u n c t i o n , i t i s not considered e x p l i c i t l y i n the f o l l o w i n g d i s c u s s i o n . ) I t turns out that the S-S j u n c t i o n i s the most favourable. -51-The purpose of t h i s s e c t i o n i s to e s t a b l i s h a q u a n t i t a t i v e b a s i s f o r s e l e c t i n g the S-S c o n f i g u r a t i o n . F i r s t of a l l , general c r i t e r i a f o r a c h i e v i n g maximum s i g n a l - t o - n o i s e r a t i o i n the coupling of tunnel j u n c t i o n s to a common base p r e a m p l i f i e r are introduced (paragraph a) a f t e r which the r e l e v a n t parameters f o r each c o n f i g u r a t i o n are estimated from theory (paragraph b) and inter-compared (paragraph c ) . (A common base p r e a m p l i f i e r was chosen over the common emitter c o n f i g u r a t i o n because the lower input impedance of the former was more compatible w i t h the j u n c t i o n dynamic r e s i s t a n c e r = (SV/31)^, which had been measured on various specimens and could be expected t h e o r e t i c a l l y . ) I t must be emphasized that t h i s a n a l y s i s i n no way purports to be a general theory of the optimum coupling of tunnel j u n c t i o n s to an a r b i t r a r y a m p l i f i e r nor i s i t i m p l i e d that the common base t r a n s i s t o r pre-a m p l i f i e r used was the optimum p r e a m p l i f i e r . " While such a general theory of s o u r c e - a m p l i f i e r coupling i s of i n t e r e s t f o r - t h e u l t i m a t e , e f f i c i e n t use of tunnel j u n c t i o n d e t e c t o r s , i t i s of l i t t l e value a t the present rudimentary stage of i n v e s t i g a t i o n because of the l a c k of d e t a i l e d i n f o r m a t i o n about s i g n a l pulse r i s e and f a l l times and because of the present i n a b i l i t y to p r e d i c t r a c c u r a t e l y even f o r a j u n c t i o n made under s p e c i f i e d c o n d i t i o n s . (a) Performance C r i t e r i a Consider the small s i g n a l equivalent c i r c u i t of f i g u r e 3-2 where the tunnel j u n c t i o n detector i s represented as a current generator i n p a r a l l e l w i t h the j u n c t i o n dynamic r e s i s t a n c e r . (The equivalent c i r c u i t f o r the detector i s considered i n more d e t a i l i n a subsequent p o r t i o n of t h i s chapter; the o v e r a l l d e t e c t o r - p r e a m p l i f i e r c i r c u i t i s examined c l o s e l y i n chapters 6 and 7.) In a n t i c i p a t i o n of the r e s u l t s from the noise c a l c u l a t i o n and measurements of chapter 6, noise sources associated w i t h the d e t e c t o r , such as Johnson noise i n the bi a s r e s i s t a n c e and shot noise on the b i a s i n g current I , have been omitted as they are n e g l i g i b l e i n comparison to the t r a n s i s t o r noise sources. The s i g n a l current i g i s trea t e d as being due to the momentary heating of the m a t e r i a l on both sides of the i n s u l a t i n g b a r r i e r from i t s ambient (bath) temperature T to T + ST wbere 6T << T, then -52-Figure 3-2: Equivalent C i r c u i t of Detector and P r e a m p l i f i e r -53-where I = I(V,T) i s the thermal tunnel current f o r the j u n c t i o n biased at v o l t a g e V. (Note: throughout t h i s and subsequent d i s c u s s i o n , only those cases are considered i n which both sides of the j u n c t i o n are at the same temperature. The r e s t r i c t i o n that 6T << T c l e a r l y does not hold i n the p a r t i c l e t r a c k v i c i n i t y immediately f o l l o w i n g the passage of the p a r t i c l e . Hence, t h i s small s i g n a l approach i s a p p l i c a b l e only f o r times s u f f i c i e n t l y long f o l l o w i n g a p a r t i c l e bombardment that the energy l o s t by the p a r t i c l e may be considered to have d i f f u s e d throughout a l a r g e enough volume that the average temperature T + 6T i n t h i s volume s a t i s f i e s the r e s t r i c t i o n . ) I f one were concerned w i t h a detector to be used with n o i s e l e s s a m p l i f i e r s , the q u a n t i t y of i n t e r e s t f o r comparing one type of j u n c t i o n w i t h the other would be I = T I 2 Here, £ i s a measure of the fundamental s i g n a l - t o - n o i s e r a t i o of the detector and, of course, i t i s d e s i r a b l e that £ should be as l a r g e as p o s s i b l e . In p r a c t i c e , however, one deals w i t h noisy a m p l i f i e r s so t h a t , w h i l e the c r i t e r i o n that £ should be l a r g e i s unchanged, the r o l e played by the j u n c t i o n r e s i s t a n c e r becomes important. With reference to the noise f i g u r e F f o r the d e t e c t o r - p r e a m p l i f i e r system (see equation 6-6) i t can be seen t h a t , remembering M i s p r o p o r t i o n a l to r l , F decreases monotonically w i t h r , decreasing to i t s lowest value i n the l i m i t of r-*». Thus, from the p o i n t of view of minimizing F, the d e s i r e d type of j u n c t i o n and b i a s i n g point are those where r i s as l a r g e as p o s s i b l e . However, a d d i t i o n a l f a c t o r s , ( r e q u i r i n g considerations too d e t a i l e d to be given i n t h i s p r e l i m i n a r y work) having to do w i t h the r e q u i r e d value of the system response time and the s p e c i f i c value of the energy l o s s AE, enter i n t o the determination of an optimum (presumably f i n i t e ) r . Nonetheless, s i n c e r ^ i s probably f a r i n excess of ' opt r J what one i s able to o b t a i n p r e s e n t l y w i t h tunnel j u n c t i o n s , the performance c r i t e r i a may be adequately summarized by the statement that r and £ should be maximum. -54-(b) D e r i v a t i o n of the Parameters Determining Performance ( i ) M-M J u n c t i o n The tunnel current i n the M-M j u n c t i o n may be found from the general r e s u l t of equation 2-14 s i n c e the matrix element 2 |M| i s taken to be the same whether the metals on e i t h e r side of the i n s u l a t o r are normal or superconducting. When both metals are normal, n 1 ( E ) = n 2 ( E + eV) = 1 so that I = A|M|2 [f(E) - f ( E + eV)]dE 2 I t i s convenient to redefine the constant A|M| to be G/e so t h a t , upon i n t e g r a t i o n , the current i s I B i | * n ( e & e V ) = G V where $ = 1/kT and e i s the e l e c t r o n i c charge. The constant G may thus be i n t e r p r e t e d as the low temperature normal s t a t e conductance of the tunnel j u n c t i o n g i v i n g rMM = ( 3 V / 9 I > T = G _ 1 ( 3 " 1 } ( 3 I / 3 T ) V = 0 Immediately, t h i s l a s t r e s u l t removes the M-M j u n c t i o n from contention as, to f i r s t order at l e a s t , the tunnel current i s i n s e n s i t i v e to temperature change making i = 0. ( i i ) M-S J u n c t i o n Again, the tunnel current i s given by equation 2-14 w i t h say n 2 ( E + eV) = 1 but w i t h n^(E) given by equation 2-13. Hence -55-MS e J 1 9 i [f(E) - f(E + eV)]dE (E Z - A 2 ) ? G e n 1(E)[f (E-eV) - f (E + eV)]dE To go further requires some approximations. At T = 1.2 K (3 = 10 (meV)-1) the minimum value of E is A(1.2 K) = 0.55 meV (for Sn) so that, to a good approximation f(E + eV) = exp(-8(E + eV)), V > 0 arid, to within about 10%, f(E - eV) = exp(-B(E - eV)), 0 <eV < A/2. Thus I M S = (2G/e) n T sinh (3eV) where, for future reference, the thermal equilibrium density of quasiparticles in the superconductor is n o(T) N (0)n m i 2 N (0) m = 2N (0)e m A -BA(T) n(E) exp(-BE)dE P + A(T) (p^ + 2 PA(T))' x -e 3 p dp (3-2) = 2 N m(0) A(T) KX(BA(T)) with a f i r s t order modified Bessel function of the second kind (Erd£lyi, 1954). For purposes of this analysis, i t is convenient to set K^(x) = a (x)exp(-x) where a(x) varies slowly with x (for x = BA >> 1) so that 1MS = ( 2 G / e ) A ( T ) a< A( T>/ k T) exp(-A(T)/kT) sinh(eV/kT), 0 <eV < A/2 T = 1.2 K. As mentioned in chapter 2, for T < T /3, A(T) = constant so that for small c temperature changes near T Q = 1.2 K one may take A(T) =A = constant. Since a varies only slightly with i t s argument, i t may also be treated approximately as a constant giving -56-I M S = (2GAa/e) exp(-BA) sinh(BeV) MS (3-3) = 2GAag exp(-gA) cosh(3eV) and MS f-2GAcx] 8T V e k l 2 exp(-BA)[-Asinh(BeV) + Vcosh(3eV)] ( i i i ) S-S J u n c t i o n Once more, the tunnel current i s given by equation 2-14 w i t h n 1 ( E ) and n 2 ( E + eV) given by 2-13 w i t h the r e s u l t SS e |EJ |E 4- eV| [ f ( E ) - f ( E + eV)]dE ( E 2 - A 2 ( T ) ) ^ [(E + e V ) 2 - A 2 ( T ) ] ^ Using s i m i l a r reasoning and n o t a t i o n to the M-S case, one f i n d s I (see s e c t i o n E (equation 3-8) of t h i s chapter) to be given approximately by I s s = ( 2 G A a / e ) [ l - exp(-BeV)] exp(-3A) 0 < T < 2 K i A <eV < 2 A The other q u a n t i t i e s of i n t e r e s t are - = 2GAct3 exp(-8(A + eV)) r s s (3-4) and SS 81 f -2GActl w V 2 ekT exp(-3A)[exp(-8eV)(eV + A) -A ] (c) Comparison of Parameters Because of the r e s t r i c t i o n s on V i n equations 3-3 and 3-4, i t i s convenient to evaluate the MS parameters at eV = A/2 and the SS -57-parameters at eV = A. (This corresponds to t h e i r r e s p e c t i v e s o - c a l l e d mid b i a s p o i n t s where the b i a s v o l t a g e i s halfway between 0 and the l e v e l at which the l a r g e current jump occurs.). I t i s then easy to show that I (MS) I(SS) r (MS) r(SS) |exp(i3A) 2exp(-3(3A/2) and S(MS) S(SS) *exp(iBA) - i 1 ^ 1 K S S ) These r a t i o s , which have been evaluated f o r Pb (A = 1.25 meV) and Sn (A = 0.55 meV), are given i n t a b l e 3-1. Parameter Pb Sn I (MS) K S S ) 260 7.8 S(MS) . S(SS) 130 3 .9 r(MS) r(SS) 1 . 4 x l 0 ~8 ; 5 . 2 x l 0 " 4 5 (MS) 5(SS) 8.1 1.4 Table 3-1: Comparison of Parameters f o r M^S and S-S Junctions The important c o n c l u s i o n to be drawn from these r e s u l t s i s that w h i l e the i n t r i n s i c s i g n a l - t o - n o i s e r a t i o £ i s s l i g h t l y greater i n the M-S j u n c t i o n t h i s advantage i s overwhelmingly o f f s e t by the r e l a t i v e l y much lower dynamic r e s i s t a n c e of the M-S j u n c t i o n . Balancing a s l i g h t l o s s of s i g n a l against more e f f i c i e n t coupling of that s i g n a l to the a m p l i f i e r along w i t h a s i g n i f i c a n t r e d u c t i o n i n a m p l i f i e r n o i s e , one i s compelled to choose the symmetric j u n c t i o n as the most promising device. As confirmed -58-i n the next s e c t i o n , i t i s a l s o evident that the j u n c t i o n should be made of Pb f i l m s (or a superconductor having an even l a r g e r energy gap) f o r optimal performance. 2. Type of Superconductor To provide a q u a n t i t a t i v e b a s i s f o r comparing the response of d i f f e r e n t superconductors to nuclear p a r t i c l e bombardment, i t i s convenient to define a f i g u r e of merit F' = N / (N + N ) 2 o T o' where N q i s the number of q u a s i p a r t i c l e p a i r s generated i n a j u n c t i o n by the r a d i a t i o n and N_ = n (T) 'Vol,.where Vol=Junction Volume, i s the thermal T o e q u i l i b r i u m number of q u a s i p a r t i c l e s present i n the j u n c t i o n at the ambient temperature T. For a good s i g n a l - t o - n o i s e r a t i o , F' should be as l a r g e as p o s s i b l e f o r the s i g n a l i s p r o p o r t i o n a l to N and the noise i s p r o p o r t i o n a l i o to (N^ + N q ) 2 . To deal w i t h F' i n general, however, i s inconvenient so that two l i m i t i n g forms of F' w i l l be considered i n s t e a d , (a) N q « N T This case, which corresponds to the s i t u a t i o n when a very s m a l l amount of energy i s i n j e c t e d i n t o the d e t e c t o r , may be studied w i t h the use of F = N Q/N t^ (3-5) I f , as suggested p r e v i o u s l y , an average energy l o s s by a charged p a r t i c l e w i s r e q u i r e d to d i s s o c i a t e a Cooper p a i r i n t o two q u a s i p a r t i c l e s , then f o r a nuclear p a r t i c l e l o s i n g energy AE i n . the j u n c t i o n , N = AE/w - AE/vA(T) o under the assumption ( f o l l o w i n g Sherman, 1962,A) that w i s some m u l t i p l e of the gap energy. (In a semiconductor l i k e s i l i c o n , f o r example, w(Si) = average energy l o s s per e l e c t r o n - h o l e p a i r = 3.6 eV = 3.2 E , Dearnaley gap and Northrop, 1966.) N i s given by equation 3-2, -59-n o(T) = N T/Vol. = 2N m(0) A(T) (3A(T)) At the t y p i c a l o p e rating temperature of 1.2 K, 3 - 10 (meV) ^  so that BA= 5.5 and 12.5 r e s p e c t i v e l y f o r Sn and Pb, the two superconductors of i n t e r e s t . (see t a b l e 3-2.) As before, f o r arguments of t h i s magnitude, i t i s u s e f u l to w r i t e K^(x) = a(x) exp (-x) which gives F AE exp(£BA(T)) YA 3 /^(T) (2N (0) • Vol • a ( 3 A ( T ) ) ) ? m The energy AE gained by a superconductor i n slowing down a t r a v e r s i n g nuclear p a r t i c l e depends on the p a r t i c l e ' s s p e c i f i c energy l o s s i n that m a t e r i a l . Considerable data i n t h i s area are a v a i l a b l e i n nuclear data t a b l e s (eg. Marion, 1960). From these and other t a b l e s , Blackmore (1967) constructed a semi-empirical expression from which could be c a l c u l a t e d , v i a a program w r i t t e n f o r the Van de Graaff PDP-8 computer, the energy l o s s of p a r t i c l e s i n passing through a t h i n f o i l . This program was used to estimate the energy l o s s of 5.1 MeV alpha p a r t i c l e s i n t h i n Sn and Pb f i l m s ; the r e s u l t s , along w i t h those f o r A£, are p l o t t e d i n f i g u r e 3-3. (For p r a c t i c a l reasons having to do w i t h t h e i r ease of pre p a r a t i o n as t h i n f i l m s and the working temperatures conveniently a v a i l a b l e , Pb and Sn were the only superconductors given s e r i o u s c o n s i d e r a t i o n . ) No evidence i s a v a i l a b l e yet concerning the v a r i a t i o n of y = w/A(T) from one superconductor to another but perhaps some guidance may be sought from the s i t u a t i o n i n semiconductors. Dearnaley and Northrop (1966) l i s t the gap energy E and experimental values of w f o r s i x d i f f e r e n t s o l i d s (see t a b l e 1-1 f o r Ge and S i ) ; the r a t i o of w/E v a r i e s only between 2.9 and 4.6. There appears t h e r e f o r e to be some j u s t i f i c a t i o n i n assuming the change i n y from superconductor to superconductor may be s u f f i c i e n t l y small that y can be taken as a constant. N (0), the energy density of st a t e s at the Fermi l e v e l when m the metal i s i n the normal s t a t e , i s s i m i l a r l y not very w e l l known but, as Ziman (1965) p o i n t s out, N m ( 0 ) f° r most metals except the t r a n s i t i o n ones i s not f a r d i f f e r e n t from the f r e e e l e c t r o n value n(E) = const x (E ) 2 r where E i s the Fermi energy. I t w i l l be assumed therefore that N (0).«n(E ) F m r where E_, = 4.4 eV f o r both Pb and Sn. -60--61-The r e l a t i v e values of F are s u f f i c i e n t f o r a comparison of the two superconductors; the rel e v a n t f a c t o r s are summarized i n t a b l e 3-2 f o r f i l m s of i d e n t i c a l t hickness and area. Superconductor A E ( r e l a t i v e u n i t s ) F i g . 3-3 A(T)meV 3 A(T) a ( 0 A ) F ( r e l a t i v e to Sn) Sn Pb 3 4 .55 1.25 5.5 12.5 .57 .36 1 16.5 Table 3-2: Comparison of Figures of Merit f o r Sn and Pb. C l e a r l y , from t h i s p o i n t of view, Pb i s superi o r to Sn as a m a t e r i a l from which to make a superconducting i o n i z i n g p a r t i c l e d e tector. The temperature 1.2 K was chosen as a b a s i s on which to compare the super-conductors because t h i s i s a temperature reached w i t h r e l a t i v e ease e x p e r i -mentally. Lower temperatures would serve only to increase t h i s f i g u r e of merit i n favour of Pb whose t r a n s i t i o n temperature (T c) i s higher than S n — because F <* exp (ISA) = exp(.875 T /T) I c This l a s t r e s u l t could give the f a l s e impression that f o r optimum performance, the energy gap should be as l a r g e as p o s s i b l e and one should perhaps use semiconductors. That such i s not true may be seen from co n s i d e r i n g the case when AE i s very l a r g e so that N Q >> N^ ,. (b) N Q » N T In t h i s i n s t a n c e , the"general f i g u r e of merit approaches F" = N * = ( A E / Y A ( T ) ) ^ : so that F"(Sn) F"(Pb) 1.25 .55 = 1.3 -62-and Sn i s seen to be s l i g h t l y favoured over Pb because of the former's smaller energy gap. (c) Conclusions I t i s evident t h a t , i n general, the choice of the optimum superconductor w i l l depend upon the expected energy l o s s i n the superconduc-t o r s and the a c c e s s i b l e operating temperatures. On these grounds, a symmetric Pb-Pb j u n c t i o n cooled to 1.2 K would appear to be the b e t t e r choice f o r the p a r t i c l e energies of t h i s experiment. Unfortunately, as discussed i n Appendix C, Pb-Pb j u n c t i o n s w i t h s a t i s f a c t o r y t u n n eling c h a r a c t e r i s t i c s proved p r o h i b i t i v e l y d i f f i c u l t to make so that an unhappy compromise was necessary; symmetric Sn-Sn02~Sn j u n c t i o n s , t h e r e f o r e , were used f o r the d e t e c t i o n experiments. E. E x c i t a t i o n s i n the Tunnel J u n c t i o n Charged P a r t i c l e Detector Now that i t has been e s t a b l i s h e d that a symmetric superconducting j u n c t i o n i s the most d e s i r a b l e c o n f i g u r a t i o n , i t i s necessary to consider the e x c i t a t i o n s produced i n the superconducting f i l m s by the charged p a r t i c l e as i t passes through. Throughout i t s t r a v e r s a l of one or both of the f i l m s comprising the j u n c t i o n , the charged p a r t i c l e loses energy p r i m a r i l y by the i o n i z a t i o n and e x c i t a t i o n of e l e c t r o n s a s s o c i a t e d w i t h the atoms along i t s path. I t seems reasonable (as discussed i n more d e t a i l i n chapter 7, s e c t i o n F) -9 t h a t w i t h i n a time the order of l e s s than 10 sec a f t e r the passage of the p a r t i c l e , the r e s u l t i n g e l e c t r o n and phonon e x c i t a t i o n s i n a slender c y l i n d e r c o - a x i a l w i t h the p a r t i c l e t r a c k have relaxed to a common l o c a l temperature which i s i n excess of the j u n c t i o n e q u i l i b r i u m (or helium bath) temperature. The aim of t h i s s e c t i o n i s to consider the q u a s i p a r t i c l e e x c i t a t i o n s which occur i n the superconductors a f t e r t h i s thermalizatiort has taken place. At t h i s point i t i s u s e f u l to r e c a l l the phonon d e t e c t i o n e x p e r i -ments described i n s e c t i o n B of t h i s chapter where the d e t e c t i o n process was described i n terms of the i n j e c t e d phonons d i s s o c i a t i n g Cooper p a i r s i n t o q u a s i p a r t i c l e s which could subsequently tunnel. The present s i t u a t i o n i s analogous. Here, the phonons i n excess of the thermal e q u i l i b r i u m number correspond to the i n j e c t e d phonons and, as. long as these excess phonons remain i n the energy range tuo £ 2A(T), they may generate q u a s i p a r t i c l e s v i a the breaking up of p a i r s . -63-From the microscopic p o i n t of view, when one considers an i n d i v i d u a l event i n which a p a i r i s broken up by a phonon, the r e s u l t i n g e x c i t a t i o n may be described as the c r e a t i o n of an e l e c t r o n - h o l e p a i r from a vacuum s t a t e c o n t a i n i n g the same number of coupled p a i r s before and a f t e r the (phonon-pair) i n t e r a c t i o n . Such a process i s p o s s i b l e because even though the p a i r broken up (say ( k t , -k+), w i t h kt going to k't) might have had k, k' > k^, the momentum s t a t e of an e x c i t a t i o n i s not s u f f i c i e n t , as i t i s i n a normal metal or superconductor, to uniquely determine i t s e l e c t r o n or hole nature. (In the Bogoliubov r e p r e s e n t a t i o n mentioned i n chapter 2 (equation 2-6) the q u a s i p a r t i c l e e x c i t a t i o n s are w r i t t e n e x p l i c i t l y as a coherent sum of amplitudes f o r an e l e c t r o n and hole.) Because of the coherence f a c t o r s v^ and u^, an i n d i v i d u a l e x c i t a t i o n process need not n e c e s s a r i l y conserve the p a r t i c l e number i n that i t d e f i n i t e l y creates an e l e c t r o n and d e f i n i t e l y creates a h o l e . On the average, however, p a r t i c l e number must be conserved so that the average d e n s i t y of e x c i t e d e l e c t r o n s and h o l e s , must be the same. Using the semiconductor model of a superconductor, which i s a thermal average not an i n d i v i d u a l e x c i t a t i o n event model, one may view the q u a s i p a r t i c l e e x c i t a t i o n s u l t i m a t e l y generated by the charged p a r t i c l e as equal numbers of e l e c t r o n s (above the gap) and holes (below the gap). These may subsequently t u n n e l , obeying Fermi's "Golden Rule" and the conservation of energy, as do the quiescent thermally e x c i t e d q u a s i p a r t i c l e s considered i n the d e r i v a t i o n of the tunnel current between two super-conductors given i n chapter 2 (equation 2-14). I t i s t h i s p i c t u r e which i s used i n the s i g n a l s i z e estimates to f o l l o w i n s e c t i o n F. Before l e a v i n g t h i s s e c t i o n , the c h a r a c t e r i s t i c time T w i t h which t h i s excess d e n s i t y of phonons and q u a s i p a r t i c l e e x c i t a t i o n s decays to zero should be considered. As mentioned b r i e f l y i n chapter 1 and more f u l l y i n chapter 7, x depends p r i m a r i l y on the r a t e at which the excess quasi-p a r t i c l e s tunnel and the ra t e at which phonons are l o s t from the energy range fico >, 2A(T); x was estimated (chapter 1) to l i e i n the range -8 -6 4 x 10 < x < 1.4x10 sec which includes the value of approximately 1.4 x 10 ^ sec deduced from t h i s experiment. (see chapter 7.) F. Small S i g n a l Equivalent C i r c u i t L o g i c a l l y , the next t o p i c to be discussed should be the expected s i g n a l s i z e but, to motivate and c l a r i f y such a d i s c u s s i o n , i t i s convenient to i n t e r j e c t at t h i s point (1) a general technique f o r o b t a i n i n g the s i g n a l -64-current and (2) an a n a l y s i s of the small s i g n a l equivalent c i r c u i t of the tunnel j u n c t i o n d e t e c t o r . 1. General Treatment of S i g n a l F o l l o w i n g a path o f t e n taken by e l e c t r i c a l engineers i n d e s c r i b i n g the c h a r a c t e r i s t i c s of an a c t i v e device l i k e a t r i o d e or t r a n -s i s t o r , one may consider f i g u r e 3-4 where a t y p i c a l f a m i l y of tunnel j u n c t i o n dc I-V curves taken at v a r i o u s temperatures T i s p l o t t e d . The load l i n e , which passes through the chosen b i a s i n g point Q — t h e c r i t e r i o n f o r choosing Q being that r = ( 9 V / 9 I ) T should be maximum as mentioned e a r l i e r — h a s a negative slope equal to the r e c i p r o c a l of the load r e s i s t a n c e seen by the tunnel j u n c t i o n . Using f i g u r e 3-4, one sees that as T i s increased above T q , the corresponding tunnel current i s uniquely determined. A general macroscopic method of determining the character-i s t i c s of the current pulse a r i s i n g from charged p a r t i c l e bombardment i s now evident. From the work of S p i e l et a l (1965) and C r i t t e n d e n (1968) i t seems that the manner i n which the heat energy d i f f u s e s away from the p a r t i c l e t r a c k i s adequately described by the c l a s s i c a l d i f f u s i o n equation (see chapter 7). Thus, i n p r i n c i p l e , one may c a l c u l a t e T(x, y, t ) where x and y l i e i n the plane of the j u n c t i o n and t i s the time elapsed since the "instantaneous" t h e r m a l i z a t i o n . For an isothermal j u n c t i o n at temperature T w i t h ( f o r the sake of s i m p l i c i t y ) a p e r f e c t l y uniform b a r r i e r of area A, the tunnel current d e n s i t y J(T) = I(T)/A where I(T) would be found from f i g u r e 3-4. I t f o l l o w s then that the s i g n a l tunneling current observed at time t i s given by I ( t ) = J ( T ( x , y, t ) ) dx dy A While s t r a i g h t f o r w a r d i n p r i n c i p l e , such an a n a l y s i s i s complicated i n p r a c t i c e because of problems i n s o l v i n g the d i f f u s i o n equation (see chapter 7) and by the f a c t that a r e a l b a r r i e r i s not p e r f e c t l y uniform making the tunnel current d e n s i t y a f u n c t i o n of x and y as well>as of T. C a l c u l a t i o n s along these l i n e s have been i n i t i a t e d at U.B.C. by Dr. B. L. White and Mr. G. May and w i l l not be pursued f u r t h e r i n t h i s t h e s i s . Lacking d e t a i l e d i n f o r m a t i o n therefore about the time dependence of T, i t w i l l be necessary to assume some time dependence of the -65-T > T. > T. -c 1 i - 1 Figure 3-4: T y p i c a l dc I-V Curves f o r Tunnel J u n c t i o n f o r Various Bath Temperatures T. < m i—i p q CO < W PQ c O •rH /-s u » c > R. 6 6e JO 6 i R_ A c t u a l C i r c u i t TheVenin Equivalent Norton Equivalent RB = RBIAS' r = < 9 V / 9 I )T' 6 6 = ( 3 V / 3 T - > T 6 T ' 6 1 ° (3I/3T) V6T RB > > r ' C C J u n c t i o n + C s t r a y Figure 3-5: Small S i g n a l Equivalent C i r c u i t f o r Tunnel J u n c t i o n -66-s i g n a l current i ( t ) — a s was done i n chapter 1—and estimate i t s magnitude from a s m a l l s i g n a l a n a l y s i s . 2. Small S i g n a l A n a l y s i s Consider f i g u r e 3-^ 4 again where the j u n c t i o n i s biased at Q which corresponds to some current I q = I ( V Q , T ). I f the average peak temperature excursion (6T) of the j u n c t i o n f o l l o w i n g bombardment by a charged p a r t i c l e i s small compared w i t h T , and w i t h the d i f f e r e n c e T - T , o C O the change i n current i s given approximately by the f i r s t two terms of the Taylor's s e r i e s expansion of I(V,T) 61 * ( 3 # V ) T o 6 V + ( 3 I / 3 T ) V o 6T +. .., eV Q < 2A(T) where i t i s assumed, i n consistency w i t h experimental evidence, that ( 3 I / 9 V ) T q and ( 3 I / 3 T ) V q are reasonably constant over a small range of 6V and 6T. An equivalent form may be w r i t t e n 6V = (3V/3I)_ 61 + (3V/3TV . 6T +. .. To Io, Thus, by Thevenin's theorem, the j u n c t i o n may be represented as a voltage generator 6e = ( 3 V / 3 T ) I q 6T i n s e r i e s w i t h an impedance r = (3V/3I)^, o r , by Norton's theorem, as a current generator 6 i = (3I/3T) V6T i n p a r a l l e l w i t h an impedance r = (3V/3I)^,. Both rep r e s e n t a t i o n s are shown i n f i g u r e 3-5. (Note that the current source form was used i n s e c t i o n D of t h i s chapter i n a n t i c i p a t i o n of t h i s r e s u l t . ) The importance of having both r and ( 3 I / 3 T ) V q as l a r g e as p o s s i b l e was s t r e s s e d i n s e c t i o n D. I t might be noted i n passing that ( 3 I / 3 T ) V q w i l l be l a r g e s t i n those j u n c t i o n s e x h i b i t i n g "pure" s i n g l e q u a s i p a r t i c l e t u n n e l i n g . Those units 1 having m e t a l l i c f i l a m e n t s or "bridges" j o i n i n g the f i l m s through imperfections i n the i n s u l a t i n g l a y e r w i l l pass a r e l a t i v e l y l a r g e current at f i n i t e v oltages which i s temperature i n s e n s i t i v e (see appendix B) making them useless as p o t e n t i a l detectors of charged p a r t i c l e s . In p r a c t i c e , the parameters (3I/3T)^ and r may be estimated experimentally from a s e r i e s of I-V c h a r a c t e r i s t i c s taken over a range of temperatures near T ; they may a l s o be estimated roughly from q u a s i p a r t i c l e -67-t u n n e l i n g theory as o u t l i n e d below. 3. D e r i v a t i o n Of Small S i g n a l Parameters The current i n a symmetrical j u n c t i o n ( c f . equation 2-14) i s , i n the n o t a t i o n of chapter 2, i = A | M | 1 n (E)n 2(E + eV)(f(E) - f ( E + eV)) dE (3-7) As was done i n s e c t i o n D of t h i s chapter, i t i s convenient to set 2 A | M | = G/e and, as w i l l be evident s h o r t l y , to use the approximation f ( E ) = (1 + ex p ( B E ) ) " 1 = exp(-BE). The range of v a l i d i t y of n(E) i s r e s t r i c t e d by equation 2-13 so t h a t , f o r eV < 2A(T), 3-7 may be r e - w r i t t e n as I - £ e A+eV E(E-eV)[(f(-E)-f(-E+eV)]dE ( E 2 - A 2 ( T ) ) * ( ( E - e V ) 2 - A 2 ( T ) ) * + E(E+eV)[(f(E)-f(E+ev)]dE ( E 2 - A 2 ( T ) ) 2 ( ( E + e V ) 2 - A 2 ( T ) ) * Thus, to a good approximation f o r Sn w i t h 0/ < T < 2K, the current i s I = G e ( e6 V - l ) E(E-V)e~3 EdE ,2 .2 2 .2, + d - e "3 V ) (E -A ) ((E -V ) -A*") A+V E(E+V)e" e EdE o o 1 o o l (E -A ) 2((E+V) -A ) 2 - ( I n t n + Int ) e 1 2 where, f o r convenience, i t i s understood that V-s-eV and A = A(T). S e t t i n g t = E-V-A and performing some elementary manipulations y i e l d s -BA .1 -BV-I n ^ = e (1-e ) (t+ A ) e "g t ( t 2 + 2 A t ) ^ 1- A t+V+A dt which, s i n c e q = t+V+A < 1, can be w r i t t e n -68-l n h - e - g A ( l - e - p V ) -Bt c i (t"+2At) 2 (t+A)e P dt 1 2 , 3 4 , . A s i m i l a r s u b s t i t u t i o n , t = E-A, i n I n t ^ r e v e a l s that I n t 2 = I n t ^ or -r 2G _ I = — Int., To o b t a i n a t r a c t a b l e form f o r I , i t i s necessary to terminate the s e r i e s a f t e r the f i r s t term which may only be done w i t h i n about 30% p r e c i s i o n by r e s t r i c t i n g V to be i n the range A < V < 2A. With t h i s approximation, the i n t e g r a l i s that of equation 3-2 making (with eV e x p l i c i t l y r e - i n s e r t e d ) I = (l-exp(-3eV))AK 1(BA) : , 0 < T < 2 K A < eV < 2 A (3-8) and r " 1 = ( 3 I / 9 V ) T = 2GB exp(~3eV)AK (8A) (3-9) To c a l c u l a t e ( 3 l / 3 T ) v , the i d e n t i t y | Z ( K X ( Z ) ) = - K Q ( Z ) - Z _ 1 K 1 ( Z ) i s used w i t h the r e s u l t V -2GAg eT K X(BA) ( B e V + l ) e - g e V - l -AK ( B A ) ( l - e " e e V ) o (3-10) where K i s a zero order modified Bessel f u n c t i o n of the second k i n d , o G. Estimate of the S i g n a l S i z e For purposes of a n a l y s i s , i t i s assumed that the current pulse superimposed on the ambient tunneling c u r r e n t i s of the form i ( t ) = i Q exp(-t/x) t * 0 (3-11) -69-where i i s p r o p o r t i o n a l to the magnitude of the excess q u a s i p a r t i c l e d e n s i t y and T (see s e c t i o n E of t h i s chapter) i s the c h a r a c t e r i s t i c time w i t h which t h i s d e n s i t y decays. At the outset i t must be emphasized that the a c t u a l time depen-dence i s probably much more complicated than the simple exponential form of equation 3-11. As pointed out i n the l a s t s e c t i o n , our t h e o r e t i c a l a n a l y s i s has not yet s u f f i c i e n t l y advanced to j u s t i f y the use of a more complex form and, as i t turns out i n chapter 7, the q u a l i t y of the data p r e s e n t l y a v a i l a b l e does not seem to warrant the use of such a form. I t must be remembered, however, that the form of i ( t ) chosen w i l l a f f e c t the value of i deduced from the experimental measurements (chapter 7) and w i l l u l t i m a t e l y a f f e c t , t h e r e f o r e the value found f o r w. For d e f i n i t e n e s s , alpha p a r t i c l e s of 5.1 MeV energy are assumed to be i n c i d e n t on the j u n c t i o n according to the geometry of f i g u r e 6-7. 1. Macroscopic or Phenomenological Approach As discussed p r e v i o u s l y i n t h i s chapter, the small s i g n a l approximation i s taken to apply at times s u f f i c i e n t l y long a f t e r the alpha p a r t i c l e t r a v e r s a l that a small volume of the j u n c t i o n (2XA') may be consid-ered to be at an average temperature T + 6T where <ST i s s u i t a b l y s m a l l . O (A' i s the area of the "hot" m a t e r i a l and X, t y p i c a l l y 2000 A, i s the f i l m t h i c k n e s s . ) Now the parameters of the preceding s e c t i o n were derived on the assumption that the e n t i r e j u n c t i o n i s at the same temperature and the -4 2 t o t a l e f f e c t i v e b a r r i e r area A ( t y p i c a l l y 4 x 10 cm ) i s passing current; i t seems reasonable then that the peak s i g n a l current may be w r i t t e n i o = ( 3 I / 9 T ) V 6 T - A l I f n (T) i s the thermal e q u i l i b r i u m d e n s i t y of q u a s i p a r t i c l e s (equation 3-2) o then the change i n the q u a s i p a r t i c l e d ensity '(in the volume 2XA') due to the momentary temperature r i s e 5T caused by the alpha p a r t i c l e i s 6n = 9 n o ( T ) 3T 2 N A V 6 T , _ J L .= _AE_ ( 3 _ 1 2 ) 2XA': wXA' -70-S o l v i n g 3-12 f o r 6T y i e l d s 4 T ^ ( S T 1 ATT (3-13) f 3 f ' 9  E9T V 3n o(T) wXA 1 = o The maximum energy l o s t by the alpha p a r t i c l e d i r e c t l y i n the t i n i s about 0.5 MeV which corresponds to an alpha p a r t i c l e i n c i d e n t on the j u n c t i o n at an angle of 80° with a path length i n the t i n of approximately 23,000 A ( c f . f i g u r e 3-3). C a l c u l a t i n g (3n o(T)/9T) from equation 3-2 and (3I/9T)^ from equation 3-10 at a temperature of 1.2 K f o r a j u n c t i o n having G = R = 0.1 U and using the value w(Sn) - .003 eV n as given i n t a b l e 1-1 y i e l d s an expected maximum current amplitude i = 15 uA • o which i s near the range of values found experimentally i n chapter 7. The corresponding temperature increase i s ST - 0.12 K. I t should be noted that these estimates are based on the assumption that the energy deposited by the alpha p a r t i c l e i n the substrate does not c o n t r i b u t e to the expected s i g n a l . I t turns out (see chapter 7) that t h i s assumption i s i n v a l i d ; hence A E ( e f f e c t i v e ) >AE = 0.5 MeV implying t h e r e f o r e that i i s underestimated. Furthermore, the values of (9I/9T) T T o V and (9n (T)/9T) used are constant and are v a l i d ( c f . f i g u r e 3-4) only i n the o l i m i t of sm a l l temperature changes. The manner i n which the a m p l i f i e r s e n s i t i v i t y was checked to ensure d e t e c t i o n of pulses w i t h t h i s magnitude and time dependence i s o u t l i n e d b r i e f l y i n chapter 5. [Note: an a l t e r n a t i v e method of estimating 6T i s to set ( c f . equation 3-12) 6T = AE/pc XA' P -3 where p i s the density and c the heat c a p a c i t y . Taking p = 7.29 g cm _c P _ i . and c (1.2 K) = 9 x 10 Jg~l(degree K) (Johnson, 1960) and the same P values f o r the other q u a n t i t i e s as before g i v e s - i - 24 uA which agrees reasonably w e l l w i t h that found above. (This l a t t e r value of i i s -71-probably an overestimate s i n c e c^ increases r a p i d l y w i t h i n c r e a s i n g temp-e r a t u r e ) . ] 2. " M i c r o s c o p i c " Approach The c o n s i d e r a t i o n s of t h i s paragraph provide some j u s t i f i c a -t i o n f o r the simple form of i ( t ) which has been assumed. As s t a t e d e a r l i e r , immediately f o l l o w i n g the passage of an alpha p a r t i c l e there e x i s t s a d e n s i t y N q of e x c i t e d q u a s i p a r t i c l e p a i r s which i s i n excess of n (T).. Decay of the excess d e n s i t y proceeds along two paths: (1) the q u a s i p a r t i c l e s tunnel through the i n s u l a t i n g l a y e r or (2) they recombine to form Cooper p a i r s and emit a phonon of energy "hed * 2A(T). A n a l y s i s i s g r e a t l y s i m p l i f i e d at t h i s point i f i t i s assumed that N q << n 0 ( T ) and that the recombination phonons may be ignored. As Rothwarf and Taylor (1967) s t a t e , however, t h i s approximation i s r a r e l y v a l i d and they have given an a n a l y s i s of the general, s i t u a t i o n assuming steady-state q u a s i p a r t i c l e i n j e c t i o n and uniform j u n c t i o n geometry. Subsequently, they a r r i v e at a p a i r of coupled equations d e s c r i b i n g the time dependence of N (the t o t a l number of q u a s i p a r t i c l e s ) and (the t o t a l number of phonons w i t h energy Tito > 2A). I d e a l l y , a s i m i l a r a n a l y s i s , modified to account f o r the t r a n s i e n t energy i n p u t , should be a p p l i e d to the tunnel j u n c t i o n p a r t i c l e detector but t h i s would r e q u i r e c a r e f u l treatment of the energy d i f f u s i o n problem t a k i n g i n t o account s p e c i f i c a l l y the j u n c t i o n boundaries and thermal impedances at those boundaries. Such a program w i l l not be undertaken here; i t i s mentioned at t h i s point only to i l l u s t r a t e the l i m i t a t i o n s of the a n a l y s i s to be given below. As a s t a r t i n g p o i n t , one i s i n t e r e s t e d i n the simplest p o s s i b l e model f o r the pulse generator to be i n s e r t e d i n f i g u r e 3-5. Thus, one ignores the time dependence of the r a t e W at which the q u a s i -p a r t i c l e s w i l l recombine and assumes that the tunneling r a t e i s constant and s u f f i c i e n t l y small w i t h respect to W that they are e f f e c t i v e l y inde-pendent of one another. The decay of the excess q u a s i p a r t i c l e density may then be viewed as sketched below. -72-I f R( N, n; t ) i s the j o i n t p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n that at time t , there are s t i l l N excess q u a s i p a r t i c l e p a i r s l e f t and n q u a s i p a r t i c l e p a i r s have tunneled, then R(N,n,.t + dt) = R(N,n;t)(1-NWdt) + R(N+l,n;t)(N+l) W Ddt + R(N+l,n-l;t)(N+l) W Tdt So l v i n g f - Vl,n<N+1> W R + \ +l,n-l< N + 1 ) WT " \nW y i e l d s <N(t)> = N q e " W t (3-14) <n(t)> = -2-1 (x-e-Wt) ( 3 . 1 5 ) W The current pulse due to the f r a c t i o n of the excess q u a s i -p a r t i c l e p a i r s which have tunneled i n time t i s thus i ( t ) = e d ^ t ) > = eN oW T exp(-Wt) ( 3 _ l f i ) where e i s the e l e c t r o n i c charge. [Note: the current i s not 2e d<n(t)>/dt because, at the usual b i a s i n g p o i n t eV - A, only one member of each q u a s i p a r t i c l e p a i r can tunnel to the other superconductor, due to the a c t i o n of the energy gaps.] S e t t i n g eN QW T = i Q y i e l d s i ( t ) = i exp(-t/-r) , t * 0 o which, of co u r s e , i s the form assumed e a r l i e r (equation 3-11) without -73-j u s t i f i c a t i o n . As i t should be, the qua n t i t y i j u s t introduced i s the same as that found by the phenomenological treatment (equation 3-13). This may be seen by s u b s t i t u t i n g f o r W v i a equation 3-17 to give • - G . M _ X o eN (0) wXA m which i s i d e n t i c a l to equation 3-13 f o r simple, approximate forms of n Q ( T ) (equation 3-2) and I(V = A/e,T) (equation 3-8). H. Tunneling P r o b a b i l i t y From equation 3-16, i t can be seen t h a t , i g n o r i n g the bandwidth l i m i t a t i o n s of r e a l a m p l i f i e r s , the maximum s i g n a l i i s obtained when the tun n e l i n g p r o b a b i l i t y per sec i s maximum. (N i s f i x e d by the choice of superconductor and the energy l o s t by a charged p a r t i c l e i n t r a v e r s i n g the tunnel j u n c t i o n ) . On the other hand, from the small s i g n a l equivalent c i r c u i t and equation 3-10, i t i s evident that the s i g n a l i = (8I/3T) 6T s v i s l a r g e s t f o r maximum j u n c t i o n conductance G. This s e c t i o n w i l l demonstrate the equivalence of these two r e -quirements and give an i n s i g h t i n t o the r e s t r i c t i o n s they place on the acceptable i n s u l a t i n g l a y e r t h i c k n e s s . A rough estimate of W may be made using a method suggested by Ginsberg (1962). For b i a s voltages eV >> 2A(T), the current should be I = e 2N (0) AXW V m 1 where N (0) i s the normal s t a t e d e n s i t y of f r e e e l e c t r o n s t a t e s per u n i t m volume near the Fermi surface and the other symbols have t h e i r usual meaning. In t h i s range of b i a s v o l t a g e , the d i f f e r e n t i a l r e s i s t a n c e i s very nearly constant and equal to the low temperature, normal s t a t e r e s i s t a n c e (R ) of the j u n c t i o n so that Wm = (e 2N (O)XAR ) - 1 = G(e 2N (O)XA)" 1 (3-17) T m n m -74-and, l i k e ( 8 I / 8 T ) V > WT i s p r o p o r t i o n a l to G. The number of f r e e e l e c t r o n s t a t e s per u n i t volume per u n i t energy range f o r both spins i s i/Z 1 _3 i N(e)de = 4ir(2m) e 2 h de = ke 2de (3-18) where m i s the e l e c t r o n mass and h i s Planck's constant, so that the average value of N(e) near the Fermi surface (e - e_) i n a range 6e « e r F i s N<feF> = 6 T e F+6e/2 i ke'de = ke 2 r e F-6c/2 (3-19) Now, the number of f r e e e l e c t r o n s N^ i s simply N f = N(e.)de = 2keJ /3 ' o (3-20) which g i v e s , upon combining equation 3-19 and 3-20, N m(0) = N(e p) = y N f / £ F (3-21) Equation 3-21 may then be evaluated by t a k i n g and(Wilson, 1965) N f = nN Ap/A t (3-22) e'F = (h^/8m)(3N f/Tr) 3 (3-23) where n = number of f r e e e l e c t r o n s per atom N^= Avogadro's number p = d e n s i t y A t= atomic weight For Sn, n = 1.1 (Wilson, 1965) so that -75-N = 4.05 x 1 0 2 2 e l s cm" 3 Ep = 4.4 eV (3-24) N (0) = 1.4 x 1 0 2 2 eV^cm" 3 m . o -4 2 T y p i c a l specimens have * = 2000 A, A = 0 . 2 m m x 0 . 2 m m = 4 x l 0 cm and R n = G _ 1 = 0.1ft y i e l d i n g WT = 5.6 x 1 0 5 s e c " 1 . So f a r , i t has been s t a t e d q u a l i t a t i v e l y that the maximum s i g n a l i Q i s great e s t f o r " l a r g e " . To place t h i s statement on a q u a n t i t a t i v e b a s i s , consider equations 3-15 and 3-16 which may be r e w r i t t e n as < n ( t ) > = N o w+w" [i-exp(-t(w T+w R))] R T i ( t ) = eN Q WT exp(-t(W T+W R)) r e s p e c t i v e l y . Assuming that the s t a t i s t i c a l f l u c t u a t i o n s i n the s i g n a l i go according to <n(t)> 2, i t i s imperative that <n(t)> be as la r g e as p o s s i b l e which means W should at l e a s t be comparable to W . (W i s the r a t e at T ... K K which the q u a s i p a r t i c l e p o p u l a t i o n i s reduced by processes other than t u n n e l i n g , see chapter 7.) Two p r a c t i c a l c o n s i d e r a t i o n s place an upper bound on W ; (1) minimum p r a c t i c a b l e thickness of i n s u l a t i n g l a y e r (a 1 7 - 1 or 2 mono-layer t h i c k i n s u l a t o r i s c o n s i s t e n t w i t h W = 7 x 10 sec ); 9 -1 (2) f i n i t e r i s e time of r e a l a m p l i f i e r s ( i f > 10 sec , the s i g n a l would decay before an a m p l i f i e r could respond). Un f o r t u n a t e l y , no f i r m t h e o r e t i c a l estimate of W i s a v a i l a b l e yet f o r tunnel j u n c t i o n s subjected to charged p a r t i c l e bombardment. For purposes of e s t a b l i s h i n g the orders of magnitude i n v o l v e d , however, the value of 2 x 10^ sec > W > 2 x 10"* sec \ as determined experimentally R by the steady s t a t e i n j e c t i o n of q u a s i p a r t i c l e s i n superconductors ( M i l l e r and Dayem, 1967 and Levine and Hsieh, 1968—see a l s o chapter 1) may be used. By i n s p e c t i o n of equation 3-17, i t i s c l e a r that f o r a given type of superconductor, WT may be increased by decreasing e i t h e r the j u n c t i o n -76-volume (XA) or the tunneling r e s i s t a n c e . The former p o s s i b i l i t y i s r u l e d out by the requirement t h a t , f o r d e t e c t i o n e f f i c i e n c y , the volume should be as l a r g e as p o s s i b l e . Junctions w i t h lower tunneling r e s i s t a n c e may be produced by decreasing the thickness of the i n s u l a t i n g l a y e r (see t a b l e 3-3) but only at the cost of g i v i n g r i s e to an unwanted dc Josephson current and o running the r i s k of the oxide being patchy (at 10 A i t i s only 2 or 3 monolayers t h i c k ) and/or p i e r c e d by m e t a l l i c f i l a m e n t s which become super-conducting s h o r t s . The f i r s t o b j e c t i o n i s not serious as a reasonably low magnetic f i e l d w i l l quench the supercurrent (see chapters 2 or 6); the second problem i s very serious and has proved to be a formidable obstacle to making a usable d e t e c t o r . Table 3-3, c a l c u l a t e d f o r a t y p i c a l j u n c t i o n of dimensions given above, i l l u s t r a t e s that t u n n e l i n g thicknesses (as derived from f i g u r e 2-3) o no greater than 10 A are p e r m i s s i b l e i f W i s to be comparable to W„. T R (Here, S i s the thickness of an i d e a l , uniformly t h i c k i n s u l a t i n g l a y e r ) S T(A) G - 1=R ft in -1 • W Tsec 6 8 x l 0 ~ 4 7.0xl0 7 8 1.6xl0~ 2 3 . 5 x l 0 6 10 0.27 2.1 x l 0 5 12 5.4 l.OxlO 4 Table 3-3: Tunneling P r o b a b i l i t y per sec f o r Several Tunneling Thicknesses. I . P r a c t i c a l Operation of Detector I t i s convenient at t h i s juncture to i n s e r t a b r i e f d e s c r i p t i o n of the manner i n which the superconducting tunnel j u n c t i o n would operate i n p r a c t i c e . The simplest method of bombarding a tunnel j u n c t i o n with alpha 249 p a r t i c l e s i s to mount an appropriate r a d i o a c t i v e source (Pu , 5.13 MeV alphas was the one chosen) i n s i d e 1 t h e helium dewar i n c l o s e p r o x i m i t y ' t o the j u n c t i o n (see f i g u r e s 4-1 and 4-3). Now the energy l o s s per u n i t path -77-l e n g t h (dE /dx) of a 5.1 MeV alpha p a r t i c l e i n l i q u i d helium i s roughly a -15 140 MeV/cm c a l c u l a t e d on the b a s i s of the stopping power e = 6.4 x 10 eV cm' _3 f o r helium (Whaling, 1958) and density p = 0.145 g cm f o r l i q u i d helium at 1.2 K (Wilks, 1967). As a s h u t t e r between the source and j u n c t i o n i s d e s i r a b l e , the minimum p r a c t i c a l s o u r c e - j u n c t i o n distance would be about 2 or 3 mm; hence, 5 MeV alpha p a r t i c l e s would be stopped i n the l i q u i d . Consequently, the specimen and source must be r a i s e d above the helium bath l e v e l , being maintained at 1.2 K by a t h i n f i l m of s u p e r f l u i d helium and by contact w i t h a copper c o l d f i n g e r whose lower end i s immersed i n the l i q u i d . At 1.2 K, the vapour pressure above the bath i s about 0.5 Torr which leads to a s p e c i f i c energy l o s s of approximately 25 keV/cm; the energy l o s t by 5 MeV alpha p a r t i c l e s i n t r a v e r s i n g t h i s attenuated helium atmosphere would the r e f o r e be a n e g l i g i b l e 0.5%. Of course, the alpha p a r t i c l e s must al s o pass through the s u p e r f l u i d f i l m which would cover —6 the j u n c t i o n but t h i s f i l m i s very t h i n , the order of 4 x 10 cm at a height of 0.5 cm above a 1.2 K helium bath (Wilks, 1967), and the c o r r e s -ponding energy l o s s i s only about 0.6 keV. 'For a l l p r a c t i c a l purposes t h e r e f o r e , the alpha p a r t i c l e s s t r i k i n g the' j u n c t i o n may be considered monoenergetic, having energy 5.1 MeV. Previous d i s c u s s i o n s have emphasized that the j u n c t i o n must be f a b r i c a t e d i n such a way as to have a high t u n n e l i n g conductance and th e r e f o r e l a r g e (91/91)^. Furthermore, i t has been observed that the operating temperature should be as low as p o s s i b l e and the operating b i a s v o l t a g e must be chosen such that r i s as l a r g e as p o s s i b l e . F i n a l l y , to suppress the unwanted dc Josephson supercurrent which j u n c t i o n s w i t h a s u f f i c i e n t l y t h i n i n s u l a t i n g l a y e r may e x h i b i t , i t might be necessary to supply a moderate magnetic f i e l d (<100 G) i n the plane of the j u n c t i o n . (A d e t a i l e d account of problems associated w i t h a t t a i n i n g these operating c o n d i t i o n s i n p r a c t i c e i s given i n section'A, chapter 6.) J . Noise This s e c t i o n o u t l i n e s very b r i e f l y the noise sources that should be considered i n d e s c r i b i n g the performance:of the tunnel j u n c t i o n as a d e t e c t o r . In the present experiment the magnitude of the j u n c t i o n noise i s not important f o r i t i s shown ( s e c t i o n C, chapter 6) that the dominant -78-noise source was the common base p r e a m p l i f i e r . As technology improves and lower noise p r e a m p l i f i e r s are designed—perhaps operating i n s i d e the helium dewar at 4 K — t h e noise a s s o c i a t e d w i t h the tunnel j u n c t i o n w i l l become more important and f o r that reason i s discussed at t h i s time. 1. Shot Noise on B i a s i n g Current (I) The shot n o i s e , which a r i s e s from the corpuscular nature of the c u r r e n t , i s given by i 2 - 2el6f s where 6f i s the bandwidth i n t e r v a l . 2. Johnson (Thermal) Noise For a tunnel j u n c t i o n i n thermal e q u i l i b r i u m , the Johnson or thermal noise i s given by the expression i 2 = 4kT«5f/r J o : where VQ i s the zero-voltage j u n c t i o n dynamic r e s i s t a n c e , k i s Boltzmann's constant and T i s the absolute temperature of the j u n c t i o n . Note: t h i s expression i s v a l i d only f o r an isothermal j u n c t i o n at zero ~2 b i a s v o l t a g e V; when V ^ 0 the noise i s given by i above. s 3. Generation-Recombination (GR) Noise GR noise i s due to f l u c t u a t i o n s i n the d e n s i t y of thermally e x c i t e d q u a s i p a r t i c l e s ' brought about by v a r i a t i o n s i n the generation and recombination r a t e s . 4. V a r i a t i o n s i n I n s u l a t i o n Thickness In any r e a l tunnel j u n c t i o n , the i n s u l a t i n g l a y e r thickness -79-i s not uniform but may be thought of as v a r y i n g s t o c h a s t i c a l l y (Hurych, 1966) over the j u n c t i o n area. As discussed i n s e c t i o n H the tunneling p r o b a b i l i t y depends s t r o n g l y on b a r r i e r thickness which i m p l i e s that the s t o c h a s t i c v a r i a t i o n i n i n s u l a t o r thickness w i l l be manifested as f l u c t u a t i o n s i n the ambient t u n n e l i n g c u r r e n t . C a l c u l a t i o n s of the magnitude of t h i s e f f e c t are i n progress at U.B.C. and w i l l be reported l a t e r . 5. F l u c t u a t i o n s i n S i g n a l F l u c t u a t i o n s i n the s i g n a l a r i s e p r i n c i p a l l y from the f o l l o w i n g sources, which are due to the same mechanisms which produce noise i n the ambient c u r r e n t : generation and recombination noise i n the current due to r a d i a t i o n - e x c i t e d q u a s i p a r t i c l e s , v a r i a t i o n i n oxide l a y e r t h i c k n e s s , and f l u c t u a t i o n s i n the number of q u a s i p a r t i c l e s o r i g i n a l l y e x c i t e d by the bombarding charged p a r t i c l e . K. Summary Beginning the chapter was a very b r i e f d e s c r i p t i o n of the con-v e n t i o n a l g a s - f i l l e d , s c i n t i l l a t i o n and semiconductor charged p a r t i c l e d e t e c t o r s . The use of s i n g l e s t r i p superconducting bolometers as r a d i a t i o n detectors was then o u t l i n e d followed by an account of the use of tunneling j u n c t i o n s as detectors of microwave photons and phonons. In d i s c u s s i n g the p r i n c i p l e s upon which the tunnel j u n c t i o n charged p a r t i c l e d etector i s based, i t was shown that the device having the best c h a r a c t e r i s t i c s was a Pb-Pb symmetric superconducting j u n c t i o n . (These u n i t s could not be used i n p r a c t i c e however, because of the d i f f i c u l t y i n f a b r i c a t i n g them w i t h s a t i s f a c t o r y c h a r a c t e r i s t i c s . ) The q u a s i p a r t i c l e e x c i t a t i o n s which are generated by the alpha p a r t i c l e i n the superconductors were considered next and i t was concluded t h a t , on the average, they could be regarded as equal numbers of e l e c t r o n s and holes appearing above and below the energy gap r e s p e c t i v e l y i n the semi-conductor model. Fo l l o w i n g a b r i e f d i s c u s s i o n of the general approach to analyzing the s i g n a l from the d e t e c t o r , a small s i g n a l equivalent c i r c u i t of the j u n c t i o n detector was found and expressions f o r the parameters (31/31)^ and r were derived from the tunneling current I = I _(V,T) between s i m i l a r -80-superconductors. The current pulse i ( t ) , which f o l l o w s the passage of a charged p a r t i c l e through the j u n c t i o n and i s superimposed on the thermal tunneling c u r r e n t , was assumed to be of the form i ( t ) = i exp(-t/-r) , t * 0 (3-11) This form of i ( t ) was found to be that given by a very simple model i n which the excess q u a s i p a r t i c l e density decayed v i a two decoupled and time independent c h a n n e l s — t u n n e l i n g and recombination. The s i z e of the s i g n a l current i was estimated on the b a s i s of t y p i c a l j u n c t i o n c h a r a c t e r i s t i c s and operating c o n d i t i o n s to be approx-imately 15 uA. That the a m p l i f i e r s e n s i t i v i t y was s u f f i c i e n t to detect pulses of that form and magnitude was checked experimentally. The importance and means of o b t a i n i n g a high tunnel p r o b a b i l i t y was then discussed on the b a s i s of a simple estimate of W due to T Ginsberg (1967) W - G(e 2N (O)XA)" 1 (3-17) i m A short d e s c r i p t i o n of the p r a c t i c a l operation of the j u n c t i o n was then given emphasizing the need to have the a c t u a l overlap area s l i g h t l y above the helium bath l e v e l and the need of a magnetic f i e l d i n the plane of the j u n c t i o n to b i a s out the dc Josephson c u r r e n t . F i n a l l y , a survey of noise s o u r c e s , i n the tunnel j u n c t i o n detector i s given. They are not considered i n depth as i t turns out experimentally that the p r e a m p l i f i e r was the p r i n c i p a l source of n o i s e . CHAPTER 4 . CRYOGENIC APPARATUS A. I n t r o d u c t i o n This chapter i s of t e c h n i c a l i n t e r e s t only and may be omitted by the reader u n i n t e r e s t e d i n experimental d e t a i l s . In essence, the cryogenic equipment c o n s i s t s of a double-walled g l a s s v e s s e l c o n t a i n i n g l i q u i d helium 4 — r e f e r r e d to h e r e a f t e r as the helium d e w a r — s i t t i n g i n s i d e a l a r g e r , s i m i l a r v e s s e l f i l l e d w i t h l i q u i d n i t r o g e n , r e f e r r e d to as the n i t r o g e n dewar (see f i g u r e 4-1). L i q u i d n i t r o g e n serves both as a pre-coolant and as a heat s h i e l d f o r the l i q u i d helium bath. The specimens are immersed i n the helium bath whose temperature can be v a r i e d from about 1.2 K to 4.2 K through r e g u l a t i o n of the vapour pressure above the l i q u i d . B. Dewars and Dewar Cap Both the n i t r o g e n and helium dewars, constructed by the departmental glassblower, are of conventional design except that a pyrex tee i s a f f i x e d to the top of the double-walled p o r t i o n of the helium dewar. The tee was n e c e s s i t a t e d by the f a i r l y unconventional manner i n which the dewars were i n s t a l l e d . They were mounted i n a f i r m l y anchored t a b l e , the top of the tee being about four fee t above f l o o r l e v e l , thereby^making i t p o s s i b l e to e a s i l y i n s t a l l or remove the specimens without d i s t u r b i n g the dewars. One port of the tee thus provided an opening f o r pumping on the helium dewar; the other port provided a r e a d i l y demountable vacuum coupling f o r the dewar cap. The brass dewar cap provides a vacuum t i g h t cover f o r the helium dewar w i t h p o r t s f o r the t r a n s f e r siphon, pressure sensing, e l e c t r i c a l feedthrough and, of course, the specimens. Because i t was necessary to use two d i f f e r e n t specimen h o l d e r s — d i s c u s s e d i n a l a t e r s e c t i o n — t h e cap was designed to accept e i t h e r one. -83-KEY TO FIGURE 4-1 Gl Marsh D i a l Gauge, Type 100 0-760 Torr G2 Edwards D i a l Gauge, Type C.G.3 0-20 Torr G3 Edwards Vacustat, Model 2E, 0-1 Torr, Lowest Reading S e n s i t i v i t y = 0.001 Torr G4 Marsh D i a l Gauge, (no type No.) 0-760 Torr G5 A s h c r o f t D i a l Gauge, Type 1850 0-760 Torr (vac), 0-380 Torr (press, gauge) 1. Sample mount. 2. E l e c t r i c a l l e a d feedthroughs. 3. Sample mount support and c o n t r o l tubes. 4. Dewar cap. 5. Pyrex Tee (3 x 3 x 2 i n . ) . 6. Helium Dewar Interspace pumpout p o r t . 7. Helium Dewar, g l a s s , 76 mm i . d . , 90 mm o.d., approx. 48 i n . long. 8. Nitrogen Dewar, g l a s s , 116 mm i . d . , 150 mm o.d., approx. 38 i n . long. 9. L i q u i d Nitrogen. 10. L i q u i d Helium. 11. Sample. 12. Helmholtz c o i l s . 13. He r e t u r n l i n e . 14. Welch Rotary Pump, Model 1402B, 5 cfm. 15. Stokes Pump, Model 49-10, 80 cfm. 16. Welch Rotary Pump, roughing purposes, Model 1405. 17. Mercury manostat (0-100 T o r r ) . 18. B o t t l e d dry N gas; f l u s h i n g purposes. 19. Transfer siphon p o r t . -84-C. Pumps The vacuum pumps used to reduce the vapour pressure of the l i q u i d helium are shown s c h e m a t i c a l l y i n f i g u r e 4-1. Temperatures of about 2 K could be obtained w i t h the Welch 1402B r o t a r y pump and about 1.2 K w i t h the Stokes, Model 49-10, Low Temperature l a b o r a t o r y "community" pump. D. Pressure-Temperature Measurement Temperatures are measured by observing the helium vapour pressure and r e l a t i n g i t to the absolute temperature w i t h a standard t a b l e . The vapour pressure-temperature r e l a t i o n s h i p of l i q u i d helium 4, the subject of extensive i n v e s t i g a t i o n , can be found i n many standard references, eg. Mendelssohn (1960). A mercury vacustat and two d i a l gauges, see f i g u r e 4-1, provide adequate coverage of the pressure range from 0.12 Torr (1.0 K) to 760 Torr (4.2 K). E. Sample Mounts In s p i t e of the great care taken i n t h e i r f a b r i c a t i o n , as described i n Appendix A, i t was conceded that probably no more than 25-50% of the tu n n e l i n g j u n c t i o n s prepared would have the s u i t a b l e c h a r a c t e r i s t i c s , as discussed i n an e a r l i e r chapter, f o r a s e n s i t i v e t e s t of t h e i r response to charged p a r t i c l e bombardment. For t h i s reason, two types of specimen mount are needed: one to check the I-V curves and magnetic f i e l d behaviour of a l l f i v e j u n c t i o n s at once, thereby d e t e c t i n g the good specimens; the other to. i n v e s t i g a t e the charged p a r t i c l e response of i n d i v i d u a l specimens. A unique f e a t u r e of the two mounts i s t h a t , i n order to change samples, only these pieces of apparatus need be d i s t u r b e d . The m u l t i p l e - j u n c t i o n sample holder i s shown i n f i g u r e 4-2. Permutation of the twelve e l e c t r i c a l leads connected to the substrate permits f o u r - t e r m i n a l t e s t i n g of any of the f i v e j u n c t i o n s . F i gure 4-3 shows the important features of the s i n g l e j u n c t i o n sample holder and alpha p a r t i c l e source support. P r o v i s i o n i s made f o r eight e l e c t r i c a l l e a d s , the glass-to-metal s e a l s being obtained by using the base of a discarded o c t a l e l e c t r o n i c tube. The vacuum t i g h t s l i d i n g s e a l permits r o t a t i o n of the c o n t r o l tube which a c t i v a t e s the shutter separating the r a d i o a c t i v e source from the tunnel j u n c t i o n . A small 2 Kft S p e c t r o l potentiometer, Model 60-1-1, serves p r i m a r i l y as a bearing f o r the con t r o l . t u b e but can be introduced as a c i r c u i t element i f r e q u i r e d . The shutter i s a t h i n piece of s t a i n l e s s s t e e l -85-15 P i n Cannon Plugs Glass Substrate w i t h Tunnel Junctions-12 P i n Kovar Seal (x2) © ! .Brass Flange 12 Leads (Advance, #38 AWG) 4" Dia. S/S Tube Tefl o n P l a t e (1/8" Thick) T e f l o n - I n s u l a t e d , Feedthroughs (xl2) .003" Dia. Au Wire Figure 4-2: dc Sample Holder -86-I I Thin Wall S/S Tubing ii Transmission Line =-#33 AWG Forme1 covered Cu wire "Q Dope Bead Transmission Line (see i n s e t ) Thin Wall S/S Tubing He l e v e l Thin Wall Cu Tubing S l i d i n g Seal O c t a l Plug S/S Con t r o l Tube (.125 x .006 i n . ) /S Support Tube (.250 x .006 in.) R a d i a t i o n S h i e l d Shutter P i v o t a - P a r t i c l e Source & Support Sample Cold Finger ( E l e c t r o l y t i c Tough P i t c h Copper) Figure 4-3: Pulse Test, Holder -87-to which a t h i n l a y e r of Pb was soldered to ensure the stoppage of 5 Mev alpha p a r t i c l e s and to make the s t a b l e s h utter p o s i t i o n the v e r t i c a l , or c l o s e d , one by lowering the s h u t t e r ' s centre of g r a v i t y below the p i v o t p o i n t . A source of 5.13 Mev alpha p a r t i c l e s (Pu 239) was prepared on a t h i n , smooth pie c e of s t a i n l e s s s t e e l (1/4" x 1/4") which was then soldered to the copper support as shown i n the diagram. The strength of the source was measured to be 1.1 x 10^ dpm i n t o 4ir s t e r a d i a n s or about 5 y C i . P i v o t i n g the support about a v e r t i c a l a x i s made i t p o s s i b l e to vary the angle at which the alpha p a r t i c l e s impinged on the sample. Because the source-sample d i s t a n c e i s roughly 1.0 cm—much greater than the range of alpha p a r t i c l e s i n l i q u i d helium (see chapter 3 ) — i t i s necessary that the helium bath l e v e l be kept below the j u n c t i o n when t e s t s are c a r r i e d out. On the other hand, the bath must be kept as c l o s e as p o s s i b l e to the j u n c t i o n f o r maximum c o o l i n g . A compromise between these c o n d i t i o n s was reached i n the f o l l o w i n g manner. The sample substrate was clamped to a copper bar of high thermal c o n d u c t i v i t y — e l e c t r o l y t i c tough p i t c h grade—most of which was immersed i n l i q u i d helium to serve as a c o l d f i n g e r . In a d d i t i o n , the four e l e c t r i c a l leads are passed around the bottom of the sample mount before connection to the sample, so that heat t r a n s f e r r e d down the e l e c t r i c a l leads and support tubes i s dumped d i r e c t l y i n t o the bath. Consequently, the j u n c t i o n may be somewhat above the bath l e v e l but i t s temperature should be but s l i g h t l y g r e a t e r than that of the bath because i t i s thermally anchored to the bath v i a the copper bar. S u p e r f l u i d flow over the j u n c t i o n serves as an a d d i t i o n a l c o o l i n g agent. The l e v e l of the helium bath f a l l s , of course, during an experiment so that a s l i d i n g vacuum-tight s e a l was d e s i g n e d — n o t shown on the s k e t c h — which, when interposed between the sample mount -flange and dewar cap, permitted the j u n c t i o n to be lowered i n step w i t h the bath : l e v e l . The s h i e l d e d p a r a l l e l p a i r t r a n s m i s s i o n l i n e f o r the voltage leads as shown i n the diagram i n s e r t , c o n s i s t s of two p a r a l l e l .007 i n . diam copper wires spaced an average d i s t a n c e of .06 i n . apart and kept c e n t r a l i n the s h i e l d by beads of Q-dope placed every few inches along i t s e n t i r e length of 1.62m. I t was not p r a c t i c a b l e to use the more obvious c o - a x i a l l i n e con-f i g u r a t i o n of a copper centre wire and s t a i n l e s s . s t e e l s h i e l d because the -88-r e s u l t i n g contact p o t e n t i a l would be larger than the bucking voltage a v a i l a b l e with the dc voltmeter monitoring the junction b i a s . Using a small diameter commercial cable, such as RG No. 174/U, was ruled out because of i t s r e l a t i v e l y large heat leak. A comparison of these three types of transmission l i n e i s given i n table 4-1. i • TYPE OF TRANSMISSION LINE ESTIMATED HEAT LEAKAGE WATTS *£/hr b o i l o f f of l i q u i d He CHARACTERISTIC IMPEDANCE (Q) Shielded p a r a l l e l p a i r (.125 i n . diam S. Steel tube, 2 x .0071 i n . Cu wire) 7.8xl0~ 3 .011 170 Co-axial (.125 i n . diam S. Steel tube, 1 x .0071 i n . Cu wire) 5.8xl0" 3 .008 150 Co-axial, RG 174/U 8 0 x l 0 - 3 • 115 50 Table 4-1: Comparison of Transmission Line C h a r a c t e r i s t i c s *White (p.198) and Hoare, et a l (p. 142) give the fi g u r e 1.43 £/hr/watt as the b o i l o f f rate of l i q u i d helium. -89-CHAPTER 5 ELECTRICAL MEASUREMENT TECHNIQUES A. dc Measurements 1. I n t r o d u c t i o n Before attempting to use any t h i n - f i l m j u n c t i o n to detect charged p a r t i c l e s , i t i s necessary as discussed i n Chapter 3 that i t s dc c h a r a c t e r i s t i c s be w e l l known. Of p a r t i c u l a r i n t e r e s t i s the dynamic r e s i s t a n c e , r = (BV/DI)^ 0<V<2A/e, and the magnetic f i e l d dependence of any supercurrent which the j u n c t i o n may pass. Examination of the dc I-V curves of various specimens thus serves a double purpose: to e l i m i n a t e q u i c k l y those j u n c t i o n s having undesirable c h a r a c t e r i s t i c s and to provide a b a s i s f o r d e c i d i n g optimal use of specimens having s u i t a b l e c h a r a c t e r i s t i c s . The major components of the dc c i r c u i t r y are, as shown i n f i g u r e 5-1, a b a t t e r y power supply, c u r r e n t - and volt-meters w i t h recorder outputs and an X-Y recorder. ' 2. Power Supply The power supply i s energized by a b a t t e r y because tunneling j u n c t i o n s , e s p e c i a l l y those e x h i b i t i n g the Josephson e f f e c t , are h i g h l y n o n - l i n e a r devices making i t imperative that the r i p p l e i n the b i a s i n g current be a b s o l u t e l y minimal. Potentiometer R l serves as a coarse current c o n t r o l ; potentiometer R2 serves as a voltage d i v i d e r such that only a f r a c t i o n of the noise generated i n the ten-turn potentiometer appears across the specimens. For some "l e a k y " specimens, the maximum output current (16 mA) from the r e g u l a r supply was i n s u f f i c i e n t to o b t a i n meaningful I-V curves. I t was to t e s t such j u n c t i o n s that the simple, high current (150 mA) a u x i l i a r y supply was employed. 3. S h i e l d i n g In p r a c t i c e , the voltage a p p l i e d to the samples i s very u l t i - s t r a n d Crvostat 3x12V c e l l s (#292) A u x i l i a r y Supply Figure 5-1: dc C i r c u i t r y (Schematic) -91-s m a l l — o n e or two m i l l i v o l t s — a n d comparable to that which might be induced by f l u c t u a t i n g s t r a y e l e c t r i c a l f i e l d s . Consequently, care was taken to encase the e n t i r e c i r c u i t r y i n s h i e l d i n g (see f i g u r e 5-1), the only exception being a one-foot s e c t i o n of patch-cords used to connect the power supply and voltmeter to the m u l t i - s t r a n d s h i e l d e d cable l e a d i n g to the c r y o s t a t . 4. D i s p l a y of I-V C h a r a c t e r i s t i c s A Moseley X-Y recorder, Model 2DR-2, was used to p l o t out the c u r r e n t - v o l t a g e curves. Because i t s maximum s e n s i t i v i t y was only 0.5 mV/in.— the maximum j u n c t i o n voltage of i n t e r e s t f o r a Sn specimen would be about 1.5 m V — i t was necessary to provide some a m p l i f i c a t i o n , i e . a Hewlett Packard 425A voltmeter w i t h recorder output, f o r a reasonable d i s p l a y . (see f i g u r e 5-1). The current was measured w i t h a K e i t h l e y , Model 600A, battery-operated electrometer which provided a voltage output s i g n a l f o r the Y a x i s of the recorder. A b a t t e r y powered ammeter was employed because passive meters having s u i t a b l y wide current ranges were i n s u f f i c i e n t l y s e n s i t i v e and reasonably p r i c e d line-powered meters were not-adequately i s o l a t e d from ground. Low frequency r i p p l e , mostly 60 Hz, present on the output of both the v o l t a g e and current meters introduced h y s t e r e s i s i n the curves traced out by the recorder. The amount of h y s t e r e s i s was reduced to n e g l i g i b l e p r o p o r t i o n s by p l a c i n g a low-pass f i l t e r , 7.5 Kft and 64uF, across the outputs of both meters but the i n t r o d u c t i o n of t h i s 0.5 sec time constant required that the curves be traced out q u i t e s l o w l y and uniformly. B. Helmholtz C o i l s f o r Magnetic B i a s i n g As pointed out i n Chapter 3, i t i s necessary to provide a magnetic f i e l d of up to 100 G i n the plane of the j u n c t i o n to suppress the Josephson t u n n e l i n g c u r r e n t . Helmholtz c o i l s were chosen as being the simplest and most economical means of producing a uniform, low l e v e l magnetic f i e l d . Space l i m i t a t i o n s r e q uired that the c o i l s b e 1placed outside the n i t r o g e n dewar s e t t i n g t h e r e f o r e a lower l i m i t of about 6.5 inches on the c o i l r a d i u s . Consequently, a p a i r of c o i l s were designed and constructed w i t h the f o l l o w i n g s p e c i f i c a t i o n s : . r-Power Supply Hewlett-Packard, Model 6268A, 40V-30A, operated i n constant voltage mode. -92-Magnetic Induction (G) Figure 5-2: C a l i b r a t i o n of Helmholtz C o i l s (Series Mode) -93-Maximum F i e l d 100 G, Series mode 200 G, P a r a l l e l mode Power D i s s i p a t i o n 330 watts, s e r i e s mode 1150 watts, p a r a l l e l mode Number of Turns 298 per c o i l C o i l Diameter 14 i n . i . d . , 17 in,,o.d. C o i l Width 1.31 i n . C o i l Form Aluminum Wire 13 gauge, heavy Formel covered copper. The power supply was operated i n the constant v o l t a g e r a t h e r than the constant current mode because the quoted r i p p l e and noise (luV rms) was much l e s s i n the former. The c o i l s were wound i n such a way that current could be passed through a l l 298 turns connected i n s e r i e s or two 149 turn superimposed c o i l s connected i n p a r a l l e l (see i n s e t of f i g u r e 5-2). In t h i s way, p r o v i s i o n was made f o r f i e l d s up to almost 200 G, without having to use a higher v o l t a g e supply. C a l i b r a t i o n of the centre f i e l d r e g i o n , c a r r i e d out w i t h a B e l l , Model 240, Incremental Gaussmeter, showed the c o i l s to meet design s p e c i f i -c a t i o n s . Figure 5-2 contains the c a l i b r a t i o n curve f o r the c o i l s connected f o r s e r i e s o p e r a t i o n . C. Pulse D e t e c t i o n E l e c t r o n i c s 1. B i a s i n g and Pulse C i r c u i t r y J unctions to be used f o r alpha p a r t i c l e d e t e c t i o n were mounted on the s o - c a l l e d "pulse t e s t " sample holder described i n f i g u r e 4-3; the c i r c u i t r y r e q u i r e d to provide dc b i a s and connection to the pulse a m p l i f i e r i s shown i n f i g u r e 5-3. The f i l t e r box serves to i s o l a t e the dc a m p l i f i e r s from unwanted ac s i g n a l s and provide a low impedance path to the pulse a m p l i f i e r . D i s c u s s i o n of the e l e c t r i c a l p r o p e r t i e s of the sh i e l d e d p a r a l l e l p a i r t r a n s m i s s i o n i s postponed u n t i l s e c t i o n E of t h i s chapter. 2. P r e a m p l i f i e r Design Figure 5-4 i s a schematic of the low n o i s e , current s e n s i t i v e common base (CB) p r e a m p l i f i e r , w i t h a common c o l l e c t o r second stage s e r v i n g For d e t a i l s of t h i s Dotted Region, See F i g . 5-1 F i l t e r Box C r y o s t a t - P u l s e Holder (Shielded) 1 Specimen Shielded P a r a l l e l P a i r Transmission Line To Main A m p l i f i e r s I J r = d . I Figure 5-3: Schematic of J u n c t i o n B i a s i n g and Pulse Detection C i r c u i t 100 470 Input d) 1 h 1.0 CB t70 16G2 .01 H l - o — * dro.i <j> 3.3K0 16G2 1.0 27Kft ;56 ) - 6V CE Q) i i 220 Output -O4" 6V Note: Capacitances i n uF Resistances i n Q unless otherwise noted. Figure 5-4: P r e a m p l i f i e r Schematic -96-as impedance transformer. (The common emitter (CE) p r e a m p l i f i e r (Z. = 400 Q) i n shown i n the diagram was constructed on the same c h a s s i s to provide f o r v o l t a g e s e n s i t i v e a m p l i f i c a t i o n should the need have a r i s e n . ) The input impedance (Z. = R. + j X . ) of the CB p r e a m p l i f i e r , determined d i r e c t l y r m i n J i n r r > J w i t h a Hewlett-Packard RF Vector Impedance Meter (Model 4815A), i s p l o t t e d as a f u n c t i o n of frequency i n f i g u r e 5-5. An independent check of the input impedance of the p r e a m p l i f i e r and the p r e a m p l i f i e r - f i l t e r box combination was made (see f i g u r e 5-5) w i t h an o s c i l l o s c o p e and s i g n a l generator by measuring the phase and amplitude r e l a t i o n s between input current and v o l t a g e . Some of the i n d u c t i v e reactance may be a t t r i b u t e d to the l y F c o u p l i n g c a p a c i t o r s i n the f i l t e r box and p r e a m p l i f i e r which, when checked s e p a r a t e l y , were found to look l i k e l u F i n s e r i e s w i t h 0.5uH, having a s e l f r e s o n a t i n g frequency of about 0.9 MHz. The 10-90% r i s e t i m e ( T - ) was determined by observing the response of the a m p l i f i e r to an input pulse r i s i n g i n l e s s than 1 nsec. I t was found that T d = 15 nsec which i s c o n s i s t e n t w i t h the measured upper 3db frequency of 28 MHz. 3. A n c i l l a r y E l e c t r o n i c s Apart from the p r e a m p l i f i e r s and veto d i s c r i m i n a t o r , the remaining a m p l i f y i n g and pulse handling devices used were of commercial manufacture; f i g u r e 5-6 i s a block diagram i l l u s t r a t i n g the arrangement of the v a r i o u s instruments. Table 5-1 l i s t s the i n d i v i d u a l u n i t s . A check of the a m p l i f i e r system s e n s i t i v i t y was performed by using i t to amplify known amplitude pulses developed by a pulse generator i n a lumped constant model of the small s i g n a l e quivalent c i r c u i t of the specimen. The system was considered to be s u f f i c i e n t l y s e n s i t i v e when the s o - c a l l e d "minimum d e t e c t a b l e " input p u l s e — t h a t pulse whose corresponding output s i g n a l was equal to 3 times the rms noise v o l t a g e — w a s comparable to the a n t i c i p a t e d pulses (see s e c t i o n G, Chapter 3) o r i g i n a t i n g i n the tunnel j u n c t i o n due to alpha p a r t i c l e bombardment. D. In t e r f e r e n c e Vetoing System Such a high gain a m p l i f i e r system i s , u n f o r t u n a t e l y , h i g h l y s e n s i t i v e to electro-magnetic i n t e r f e r e n c e . In s p i t e of c a r e f u l s h i e l d i n g , spurious s i g n a l s were detected which were found to o r i g i n a t e i n such sources as f l u o r e s c e n t l i g h t s , e l e c t r i c motors and the Helmholtz c o i l power supply. -98-B u w C O 0 0 c • H > O C OJ 0) <4-l u 0 J Bias Current — W W } R I n t e r -ference Pick-up Antennae CE Preamp (F i g . 5-4) Amp (A- 2) Amp. (B-2) Amp. (D) Veto D i s c r i m i n -a t o r (K) < Specimen dc VTVM 0 or d e t a i l s above t h ine , see f i g u r e s 5-and 4. Pulse Height Analyzer (E) Wideband VTVM ' (F) Pulse Gate D i s c r i m i n -ator S c a l e r (G) D i s c . Output Scaler (H) F i l t e r Box CB Preamp. 1 Amp. (A-D Amp. (B-1) Amp & Bandpass F i l t e r (C) Del (500 n ay sec) O s c i l l o -scope (I) 0 J U CO >% C O 6 0 C • H •4-1 (X 6 < B 6 0 • H C O ^Camera Figure 5-6: E l e c t r o n i c s Arrangement; E l e c t r o n i c U n i t s are l i s t e d i n Table 5-1 -99-Th e amplitude and number of these unwanted s i g n a l s were such that e x t r a c t i n g i n f o r m a t i o n concerning the l e g i t i m a t e pulses coming from the j u n c t i o n detector by d i s c r i m i n a t o r l e v e l s e t t i n g or s t a t i s t i c a l a n a l y s i s TABLE 5-1 E l e c t r o n i c U n i t s Used on the Experiment A - LeCroy Research Systems, Dual Linear A m p l i f i e r , Model 107F B - Hewlett-Packard, Linear A m p l i f i e r , Model 462A, (2) C - Ortec, L i n e a r A m p l i f i e r , Model 410 D - Hewlett-Packard, L i n e a r A m p l i f i e r , Model 460A E - Nuclear-Data, Pulse Height Analyzer, Model 101 (Note: This u n i t was out of order when the pulse data were a c t u a l l y acquired from specimen.) F - K e i t h l e y , Wideband VTVM, Model 120 G - Ortec, D i s c r i m i n a t o r - S e a l e r , Model 429 H - Ortec, S c a l e r , Model 430 1 ~ T e k t r o n i x , O s c i l l o s c o p e , Model 454 J - Ortec, Pulse S t r e t c h e r , Model 411 K - U.B.C., In t e r f e r e n c e Veto D i s c r i m i n a t o r , C i r c u i t Diagram, f i g . 5-7 L - Datapulse, Pulse Generator, Model 106A, (not shown on f i g . 5-6) was not f e a s i b l e (see a l s o the b r i e f d i s c u s s i o n of the Veto system i n Chapter 6 ) . Consequently an " I n t e r f e r e n c e Vetoing" system was devised (see f i g u r e 5-6) which operated i n p a r a l l e l w i t h the main a m p l i f i e r s y s t e m — w i t h appropriate antennae to p i c k up the same i n t e r f e r e n c e as the main system—and put out g a t i n g pulses to paralyze the s c a l e r and pulse height analyzer whenever the i n t e r f e r e n c e s i g n a l exceeded a threshold v a l u e . (The threshold l e v e l i s conveniently set by stopping the alpha p a r t i c l e s before they reach the j u n c t i o n by lowering i t below the helium bath surface and a d j u s t i n g the d i s c r i m i n a t o r u n t i l "zero" counts are observed on the gated s c a l e r . ) Figure 5-7 i s a schematic of the veto d i s c r i m i n a t o r u n i t designated by "K" i n f i g u r e 5-6. E. E l e c t r i c a l P r o p e r t i e s of Transmission L i n e 1. I n t r o d u c t i o n A b r i e f p h y s i c a l d e s c r i p t i o n of the so c a l l e d shielded p a r a l l e l p a i r t r a n s m i s s i o n l i n e used i n t h i s experiment was given i n chapter 4. This s e c t i o n summarizes the e l e c t r i c a l c h a r a c t e r i s t i c s of the 754A 914 -°+12V o f -0+5.2V •220ft iMC408 iMC408 iMC408 1N100 100 pF-»-• 2N2925 13 2N1304 2 1N100 Ortec Veto +4.8 1 220ft 3.3Kft lKft ND 101 Veto lOKftC i l K f t »S>»U -• •-2N1305 -0-6V 2N1304 Figure 5-7: Schematic of Veto D i s c r i m i n a t o r -101-l i n e needed f o r the noise and pulse shape a n a l y s i s of chapter 7 and describes how they were determined. To a reasonable approximation, the l i n e (assumed to be l i n e a r ) may be represented by lumped constants f o r i t s length (£ =1.6m) i s much l e s s than the minimum wavelength of 85m corresponding to the upper 3db-down frequency of the a m p l i f i e r system (3.5 MHz). The impedance Z. seen l o o k i n g i n t o such a l i n e when i t i s terminated i n Z = R + i X i s I b t t J t simply Z i = R^ + R + j(X^+ X f c) where Z^ = R^ + j X ^ i s the e f f e c t i v e s e r i e s impedance of the l i n e . The f o l l o w i n g two paragraphs describe the measurement of R. and X. and the establishment of Z. f o r p r a c t i c a l t r a n s m i s s i o n l i n e - l o a d c o n f i g u r a t i o n s . Paragraph 5 i s an i n v e s t i g a t i o n made to check the consistency between those parameters measured and those which may be estimated approximately from theory. 2. E f f e c t i v e Line Impedance Although Z may have been estimated from theory, l i t t l e JO confidence could be placed i n the r e s u l t s because of the inhomogenity of the l i n e (see paragraph 5) and the temperature range i t spanned. (One end was immersed i n 1.2 K l i q u i d helium w h i l e the other end was thermally anchored at room temperature 295 K.) Consequently, R^ and X^ were determined experimentally over the a m p l i f i e r passband by s h o r t i n g the "detector end" of the tra n s m i s s i o n l i n e (making R = X = 0 ) and measuring Z. = Z =R +jX t U 1 Jo J6 Xr w i t h a Hewlett-Packard Model 4815A RF Vector Impedance Meter (Z meter) f o r three d i f f e r e n t temperature c o n d i t i o n s , namely: complete l i n e at room temperature (295 K), one end at 295 K w i t h the remaining 90% cooled to 77 K w i t h helium gas, one end at 295 K with the other at 1.2 K as i n the a c t u a l experiment. The r e s u l t s are p l o t t e d i n f i g u r e 5-8 where the e r r o r bars shown are t y p i c a l and a r i s e from the accuracy l i m i t s of the Z meter. (For a reading of Z = |z| exp ( j <j>), the accuracy l i m i t s quoted by the manufacturer are AZ = ± 4% F.S.D. and 6 = Aq> = + 3° so that R = | Z | cos a>(l + AZ/ | Z | + 6tan <f>) and X = |z| s i n 4> (1 + Az/ | Z j ± Scot<f>).) (a) E f f e c t i v e Resistance The e f f e c t i v e l i n e r e s i s t a n c e i s c l e a r l y both temperature and frequency dependent. A l e a s t squares f i t of R = R + R n * f to the data -102-i n f i g u r e 5-8 y i e l d s the f o l l o w i n g v a l u e s . (For f u t u r e reference, when the l i n e i s s a i d to be at temperature T, i t i s to be understood that one end i s at 295 K and the other at T; eg. R o r i(77 K) = 1.73 ±.3452 ). Temperature Range (K) ho™ R £ 1 ( f t MHz 1 ) 295-295 295-77 295-1.2 2.47 + .49 1.73 + .34 1.60 + .32 .875 + 0.17 .534 + .10 .557 + .11 (b) E f f e c t i v e Inductance Wi t h i n experimental accuracy, the r e a c t i v e component of i s temperature independent and purely i n d u c t i v e so that i t may be w r i t t e n = 2 T T f L ^ where i s the e f f e c t i v e inductance of the l i n e over the bandwidth of i n t e r e s t . S e t t i n g 2 TTL^ equal to the slope of the best s t r a i g h t l i n e through the experimental p o i n t s f o r a l l three temperatures y i e l d s = 1.63 ± .13'uH This value of agrees c l o s e l y w i t h the value of 1.6uH obtained by t e r m i n a t i n g the l i n e i n a c a l i b r a t e d , (925 pF) high frequency c a p a c i t o r and 2 -1 f i n d i n g the resonant frequency a> = (LC) 3. Line Input Impedance w i t h P r a c t i c a l Loads On the b a s i s of the preceding r e s u l t s , i t may be s a i d t h a t , f o r the frequencies used i n t h i s experiment, the transmission l i n e terminated i n Z- may be regarded as a two-terminal network whose input impedance i s Z^ = R + j X where, f o r temperature T R = R T + R £ Q ( T ) + R u ( T ) - 0 ) / 2 T T (5-1) X - X t + j U L 4 . I f equation 5-1 i s to be used to describe the input impedance l o o k i n g i n t o -103-i r—\ a co u c CO 4J W • H W OJ R„ = R„^ + R n,*f A £0 £1 short Trans-mission Line Temperature (T) V^X£ ft T » 1 a X 4 r i r-Frequency (MHz) (T) X 295 K O 77 K £ 1.2 K X I C J OJ a c cd o co <u 35 30 25 20 15 10 L X £ " 2 ^ f L £ 2! ft 21 6 L * _L X 295 K O 77 K A l - 2 K 0 1 "2 ' H "" Frequency (MHz) Figure 5-8: Resistance & Reactance of Transmission Line -104-e l t h e r end of the l i n e , i t must be assumed that the e f f e c t i v e l i n e impedance = R^(T) + J w L£ i s t n e same whether determined from the warm or " c o l d " end. For T = 295 K, t h i s assumption was v e r i f i e d experimentally but i t could not be confirmed, of course, at T = 77 or 1.2 K. Two s p e c i f i c t r a n s m i s s i o n l i n e - l o a d c o n f i g u r a t i o n s are of i n t e r e s t to the i n t e r p r e t a t i o n of the experimental r e s u l t s . For pulse shape a n a l y s i s , the d e s i r e d impedance i s that seen by the 1.2 K j u n c t i o n as i t looks i n t o the c o l d end of the transmission l i n e when i t i s terminated w i t h the p r e a m p l i f i e r at 295 K ( s i m i l a r l y to i n s e t , f i g u r e 5-9); t h i s impedance w i l l be denoted by Z. n , = Z. . For noise c a l c u l a t i o n s , the r i , c o l d xc ' re q u i r e d impedance i s that seen by the p r e a m p l i f i e r as i t looks i n t o the warm end of the tra n s m i s s i o n l i n e when i t i s terminated w i t h a c o l d tunnel j u n c t i o n (see f i g u r e 7-1); t h i s impedance w i l l be denoted by Z. = Z. . J x,warm IW To t e s t the accuracy of equation 5-1 i n p r e d i c t i n g Z^c and Z. f o r the a c t u a l j u n c t i o n , s t u d i e s were made (at 295 K) of the l i n e xw terminated w i t h the p r e a m p l i f i e r and w i t h the l i n e terminated i n a lumped constant model of a tunnel j u n c t i o n . To i l l u s t r a t e these t e s t s , the r e s u l t s w i t h the p r e a m p l i f i e r as load are given below. (a) Z. f o r Line w i t h P r e a m p l i f i e r as Load x c  The p r e a m p l i f i e r input impedance Z. = R. + j X . was xn xn xn determined independently (see f i g u r e 5-5) so that by equation 5-1, the pr e d i c t e d impedance l o o k i n g i n t o the l i n e i s Z. = R. + R„(295 K,o>) + j ( X . + ooL.) (5-2) x xn I J xn V This may be compared to Z^ = R' + j X ' obtained w i t h the Z meter by connecting the p r e a m p l i f i e r to the l i n e as shown i n the i n s e t of f i g u r e 5-9. I f the model of equation 5-1 i s v a l i d i t i s expected that Z^ = Z\ The r e s u l t s are given i n f i g u r e 5-9. C l e a r l y , the measured reactance X' agrees w e l l w i t h that obtained from equation 5-2. The agreement between the corresponding r e s i s t i v e components of the impedance i s not as good but may be made c o n s i s t e n t by using the s l i g h t l y smaller average value of R. = 7.5ft instead of 7.9ft as determined from f i g u r e 5-5. 5 i n (Such a change i s w i t h i n the accuracy l i m i t s s t i p u l a t e d by the Z meter -105-Z' Trans r — j l ! Line i J i 1 ^ F i l t P. A. X+R' (Measured) er Box Z' = R' + j X ' 01 C J C to 4-1 CO • H CO OJ Pd 14 13 12 11 10 E x p t ' l : R. + Attenuation l n (R. = 7.9 ft) i n E x p t ' l : R. + i n A ttenuation (R. = 7.5 n) D i r e c t Measurement (R') -A" R. (See eq'n 5-3) I C -L 4 Freq. (MHz) 35 30 20 A.-*X' (Meas.) O -»X=X. +X„ i n SL X. from F i g . 5-5 i n X £ from F i g . 5-8 0 ) o c cfl 4-1 ( J fS OJ Pi 10 4 Freq. (MHz) Figure 5-9: Impedance of Transmission Line w i t h P r e a m p l i f i e r as Load -106-manufacturer.) (b) Z± f o r "Cold" Transmission l i n e + P r e a m p l i f i e r On the b a s i s of the Z ^ symmetry assumption s t a t e d e a r l i e r and the agreement between measurement and the model of the l i n e represented by equation 5-1, Z^ may be w r i t t e n Z . = R. + R„.(1.2 K,u>) + jtoL = R. + j X (5-3) xc i n £ J c i c J c where R. = 7.5ft and L = L. + L = 1.69yH. For reference, R. i s m c £ preamp. ' i c p l o t t e d i n f i g u r e 5-9. (c) Z ^ f o r Transmission l i n e + Tunnel J u n c t i o n By the same reasoning as given above, Z^ may be w r i t t e n Z i w = R j + V1*2 ' K' w ) + j ( X j " H j j L£ ) ( 5 " 4 ) where Z.. = R.. + jX^ i s the tunnel j u n c t i o n output impedance. 4. C h a r a c t e r i s t i c Impedance The e f f e c t i v e l i n e r e s i s t a n c e (R ) and inductance (L ) which have been deduced are a l l that i s r e q u i r e d to represent the l i n e i n the lumped constant s e r i e s impedance approximation that has been chosen. To a i d i n understanding the l i n e and to ensure that the measured parameters f o r the l i n e are s e l f - c o n s i s t e n t , the c h a r a c t e r i s t i c impedance Z of the o l i n e was a l s o determined. Z q was found by terminating the transmission l i n e i n a purely r e s i s t i v e l o a d R^ and a d j u s t i n g R^ u n t i l the input impedance of the l i n e ( Z ^ ) as measured by the Z meter e x h i b i t e d no r e a c t i v e component. (Note: "purely r e s i s t i v e " and "no r e a c t i v e component" mean that w i t h i n the + i£° e m p i r i c a l l y determined phase angle s e n s i t i v i t y of the instrument near zero, the observed phase angle i n the two impedances was zero.) The value of Z q thus deduced was Z = 168 + 17ft o -where the e r r o r s a r i s e from the accuracy l i m i t s on the phase angle (as j u s t -107-stated) and on the magnitude (as given i n paragraph 2) f o r both and Z.. 1 5. Consistency Check of Measured Values w i t h Theory The preceding paragraphs have shown that a l l the parameters needed to c h a r a c t e r i z e the l i n e i n the a n a l y s i s of succeeding chapters may be determined experimentally. No f a i t h could be placed i n these q u a n t i t i e s c a l c u l a t e d from theory because the l i n e operated over a s i g n i f i c a n t range of temperatures and bore but f a i n t resemblance to the i d e a l , simple c o n f i g u r a -t i o n s f o r which c a l c u l a t i o n s are r e a d i l y made (see eg. B l e i l , 1957). This s e c t i o n f i r s t d e s c r i bes the very non-ideal c o n s t r u c t i o n of the l i n e and then compares the measured parameters w i t h those estimated f o r the s h i e l d e d p a r a l l e l p a i r (spp) t r a n s m i s s i o n l i n e — t h e geometry which most c l o s e l y approximates the a c t u a l l i n e . U n l i k e the i d e a l spp l i n e , the experimental l i n e was not operated i n balanced mode f o r both the " s h i e l d " and one of the wires were grounded. Being homemade, i t had a non-uniform s t r u c t u r e . For example, because of space l i m i t a t i o n s , the " s h i e l d " was only 2/3 as long as the center conductors; the two wires were somewhat unevenly spaced both w i t h respect to each other and to the " s h i e l d " ; three d i e l e c t r i c s were present: (1) a t h i n l a y e r (= 0.001 i n . ) of Formel v a r n i s h which covered the w i r e s , (2) q u a s i -continuous lumps of Q-dope, which occupied perhaps 20-40% of the volume i n s i d e the " s h i e l d " , (the Q-dope was used to space the wires) and (3) room a i r or helium which f i l l e d the remaining volume. Table 5-2 summarizes the comparison between the experimentally and t h e o r e t i c a l l y estimated parameters. Considering the inhomogeneous nature of the l i n e , the agreement i s q u i t e reasonable. 6. Propagation V e l o c i t y From the foregoing d i s c u s s i o n , i t i s evident that a reasonable r e p r e s e n t a t i o n of the l i n e has been obtained i n that impedances measured w i t h t h i s model are s e l f - c o n s i s t e n t and the parameters of the model agree f a i r l y w e l l w i t h those estimated from theory. However, one aspect of the t r a n s m i s s i o n l i n e remains which i s not yet completely understood but which, f o r t u n a t e l y , i s not e s s e n t i a l to the a n a l y s i s of l a t e r c h a p t e r s — t h e propagation v e l o c i t y u = (LC) 2 = Z /L - (.56 ±.l)c seems to -108-be i n o r d i n a t e l y low. This value of u = c(K'K') 2 i m p l i e s that K'K* = 3.2 „ m r m - .9 where K' = e'/e i s the e f f e c t i v e d i e l e c t r i c constant and K' = u'/u i s the o m o e f f e c t i v e r e l a t i v e p e r m e a b i l i t y . I t i s d i f f i c u l t to perceive how such Parameter Experiment Theory (Model) Resistance (R, 3 MHz) Inductance (L) C h a r a c t e r i s t i c Impedance Z o 3.1 ± .6ft/m * 1.01 ± .08pH/m * 168 ± 17ft 2.1ft/m (spp) 1.3uH/m (pp) t 190ft (spp) § Table 5-2: Comparison of Experimental and T h e o r e t i c a l Transmission Line Parameters. * The p h y s i c a l length of the l i n e i s 1.62 m. t (pp) denotes p a r a l l e l p a i r — n o estimate of L i s a v a i l a b l e f o r the spp. § This value of Z depends on the value of e f f e c t i v e d i e l e c t r i c constant and e f ? e c t i v e r e l a t i v e p e r m e a b i l i t y assumed f o r the l i n e — s e e paragraph 6. a l a r g e product obtains i n view of the f a c t that the major d i e l e c t r i c i n the l i n e , by volume, i s a i r and there are no obviously magnetic m a t e r i a l s present. Consider f i r s t the e f f e c t i v e d i e l e c t r i c constant. In a d d i t i o n to a i r , the other d i e l e c t r i c s present i n the l i n e were Formel (a p o l y v i n y l acetate r e s i n ) w i t h K = 2.92 (von H i p p e l , 1954) and Q-dope (mostly polystyrene) w i t h K = 2.52-2.65 as stat e d by the manufacturer (GC E l e c t r o n i c s , Rockford, I l l i n o i s ) and checked experimentally. The frequency s t a b i l i t y of both d i e l e c t r i c s i s i l l u s t r a t e d by the f a c t that K(polystyrene) i s e s s e n t i a l l y independent of frequency over the range 1 - 3000 MHz and K (Formel) decreases by only 5% over the same range (von H i p p e l , 1954). Although the Formel and Q-dope occupy only a r e l a t i v e l y s m a l l f r a c t i o n of the volume i n s i d e the " s h i e l d " , t h e i r c o n t r i b u t i o n to the e f f e c t i v e d i e l e c t r i c constant i s l a r g e r than t h e i r c o n t r i b u t i o n to the volume. This a r i s e s from the f a c t that these substances are l o c a t e d c l o s e -109-to the wire surfaces where the e l e c t r i c f i e l d , and t h e r e f o r e the stored energy d e n s i t y , i s l a r g e s t . (To i l l u s t r a t e t h i s p o i n t , consider a l i n e w i t h simple c o - a x i a l geometry w i t h r a d i i a(wire) and c ( s h i e l d ) and a d i e l e c t r i c K surrounding the wire out to a r a d i u s b such that f o r a<r$b, K>1 and f o r b<r£c, K = 1. The e f f e c t i v e d i e l e c t r i c constant of such an arrangement i s K' = l n ( c / a ) ( K _ 1 ln(b/a) + l n ( c / b ) ) _ 1 ; s e t t i n g a = .0035 i n . , b = .03 i n . , c = .06 i n . and K = 2.6, as a very rough approximation to the s i t u a t i o n i n the a c t u a l t r a n s m i s s i o n l i n e , y i e l d s K' =1.86 which d i f f e r s s i g n i f i c a n t l y from the value K'(volume) = 1.37 c a l c u l a t e d on the assumption that each d i e l e c t r i c c o n t r i b u t e s i n p r o p o r t i o n to i t s r e l a t i v e volume.) I t might be expected that t h i s e f f e c t would be even more pronounced i n the twin wire l i n e because the e l e c t r i c f i e l d l i n e s would tend to be concentrated i n the Q-dope f i l l e d r e g i on keeping the wires apart. Thus, an e f f e c t i v e d i e l e c t r i c constant as l a r g e as 2.3 (the l o w e r . l i m i t on K'K^) i s perhaps conceivable. There seems to be no way i n which the transmission l i n e could e x h i b i t an e f f e c t i v e r e l a t i v e p e r m e a b i l i t y s i g n i f i c a n t l y greater than one. I t i s p o s s i b l e that the r e l a t i v e p e r m e a b i l i t y of the type 304 s t a i n l e s s s t e e l used i n the s h i e l d was greater than one—due to c o l d working e f f e c t s and the i r r e v e r s i b l e p a r t i a l transformation from the a u s t e n i t i c to the m a r t e n s i t i c phase which takes place i n t h i s type of s t a i n l e s s s t e e l as i t i s repeatedly cooled to helium temperatures (Reed and M i k e s e l l , 1960)—but because of i t s r e l a t i v e l y small volume, which i s f u r t h e r e s s e n t i a l l y reduced by the s k i n e f f e c t , the c o n t r i b u t i o n of the " s h i e l d " to the o v e r a l l p e r m e a b i l i t y of the l i n e i s expected to be n e g l i g i b l e . (Because i t was not p o s s i b l e to t e s t K f o r the tubing used as the " s h i e l d " without destroying the l i n e , a piece of s i m i l a r " o f f the s h e l f " tubing (not known to have been cooled to helium temperatures) was checked but i t gave no evidence of having K > 1.) m From these c o n s i d e r a t i o n s , i t appears that the propagation v e l o c i t y deduced from Z q and L i s not c o n v i n c i n g l y c o n s i s t e n t w i t h reasonable values of the e f f e c t i v e d i e l e c t r i c constant and r e l a t i v e p e r m e a b i l i t y but that the maximum value of u allowed by t h e - e r r o r s does approach a value which i s c o n s i s t e n t w i t h theory. This problem was not pursued f u r t h e r as i t was not considered c e n t r a l to the purpose of the experiment. To s a t i s f a c t o r i l y r e s o l v e the anomaly would r e q u i r e a c a r e f u l study of the a c t u a l l i n e which, because of i t s unwieldy geometry and horrendous mixture of d i e l e c t r i c i n t e r f a c e s , presents a formidable challenge. -110-F. Summary The apparatus f o r t r a c i n g out the dc I-V curves, magnetically b i a s i n g the specimens and observing the pulses output from the detector has been described. Q u a n t i t i e s of p a r t i c u l a r i n t e r e s t to subsequent chapters are: 1. P r e a m p l i f i e r + F i l t e r Input Impedance (Z!^ ) Z'. = R! + jx! where, i n the a m p l i f i e r bandpass, i n i n i n ' r r » R! =' 7.9ft = R. and X! = coL - (toC)" 1 (5-5) i n m m w i t h L = 0.23uH and C = 0.5yF. 2. Transmission Line + Load Equivalent C i r c u i t For the transmission l i n e at. temperature T terminated i n Z = Rfc + jXt» the e f f e c t i v e input impedance i s approximately Z^ = R + j X where R = R t + R^co.T) X - X t + j o ^ w i t h co being the angular frequency, L^ = 1.63uH the e f f e c t i v e l i n e inductance and R^ = R^n + R ^ "10/211 the frequency and temperature dependent e f f e c t i v e l i n e r e s i s t a n c e described i n s e c t i o n E, paragraph 2. -111-CHAPTER 6 RESULTS A. dc C h a r a c t e r i s t i c s i As mentioned i n e a r l i e r chapters, i t was necessary to obtain the dc I-V c h a r a c t e r i s t i c s of each j u n c t i o n at helium temperatures to determine i t s s u i t a b i l i t y as a de t e c t o r . The q u a n t i t i e s of p a r t i c u l a r i n t e r e s t , as o u t l i n e d i n Chapter 3, were the dynamic r e s i s t a n c e r = (3V/8I)^, and the degree to which the supercurrent could be suppressed by the a p p l i c a -t i o n of a magnetic f i e l d . This l a s t mentioned t e s t p a r t i c u l a r l y d i s c r i m i n a t e s against " l e a k y " j u n c t i o n s — t h o s e w i t h e l e c t r i c a l shorts through the i n s u l a t i n g l a y e r — a s the r e s u l t a n t supercurrent i s very much l e s s s e n s i t i v e to a magnetic f i e l d than i s the dc Josephson supercurrent. "Leaky" j u n c t i o n s and t h e i r c h a r a c t e r i s t i c s are described b r i e f l y i n Appendix B. 1. Temperature Dependence of Tunneling Current Figure 6-1 i l l u s t r a t e s the v a r i a t i o n i n the dc I-V character-i s t i c w i t h change i n temperature f o r a so c a l l e d "good" Sn-Sn j u n c t i o n . The marked decrease i n current w i t h decreasing temperature and the f a c t that a f i e l d of 100 G has c l e a r l y suppressed the zero-voltage current i n d i c a t e that the observed current flow i s that of t u n n e l i n g s i n g l e q u a s i p a r t i c l e s . The d i s t i n c t i o n between t h i s I-V c h a r a c t e r i s t i c and one obtained from a t y p i c a l leaky j u n c t i o n , as shown i n Appendix B, i s thus r e a d i l y apparent. In a d d i t i o n , i t may be seen that the slope ( 9 I / 3 V ) T of the I-V curve decreases as the temperature i s lowered. This p o i n t s out the need to operate the j u n c t i o n at the lowest convenient temperature so that the dynamic r e s i s t a n c e r may be as l a r g e as p o s s i b l e to maximize the s i g n a l to noise r a t i o i n accord w i t h the theory o u t l i n e d i n Chapter 3. 2. Magnetic F i e l d Dependence of Tunneling Current The observed v a r i a t i o n of the j u n c t i o n current w i t h a magnetic -112-0 0.2 0.4 0.6 0.8 1.0 1.2 V(mV) Figure 6-1: I-V C h a r a c t e r i s t i c s , Sn-SnO -Sn Tunnel J u n c t i o n , B=100 G -113-f l e l d p a r a l l e l to the plane of the j u n c t i o n i s shown i n the composite X-Y recorder t r a c i n g s of f i g u r e 6-2. (In p r a c t i c e , p l o t s of t h i s s o r t were more u s e f u l d i a g n o s t i c a l l y than the conventional I ( c r i t ) - B or Fraunhoffer type p l o t shown i n f i g u r e 2-8 (b)). In p a r t i c u l a r , three features are evident: (a) Magnitude of the Supercurrent (B=0) For a given j u n c t i o n at a f i x e d temperature, the observed maximum zero-voltage current ( I £ ) was found to vary from t r a c i n g to t r a c i n g and to be only a small f r a c t i o n of the t h e o r e t i c a l maximum supercurrent (see equation 2-16 ) given by I = J (T)-A=TrA(0)/2eR cm o n The observations are summarized i n t a b l e 6-1. SPECIMEN THEORETICAL ''"cm AVERAGE OBSERVED I c m Icm 0 B S -Icm T H Y ' H-4 3mA ,.04±.02mA .013 H-5 4.5mA .045±.005mA .01 J-5 11.2mA .35±.15mA .031 Table 6-1: Comparison of Experimental & T h e o r e t i c a l Maximum Supercurrent. That the observed 1^ i s almost two orders of magnitude s m a l l e r than the t h e o r e t i c a l maximum f o r zero magnetic f i e l d may be a t t r i b u t e d to the f a c t that no attempt was made to magnetically s h i e l d the specimens. Workers studying the dc Josephson e f f e c t and d e t e c t i n g 1^ as high as 50-90% of I ( t h e o r e t i c a l ) , report ( J a k l e v i c e t a l , 1965) f i n d i n g i t necessary to c o o l t h e i r j u n c t i o n s w h i l e s h i e l d e d from the earth's magnetic f i e l d . Otherwise, f l u x was apparently trapped i n the f i l m s or j u n c t i o n s which se v e r e l y attenuated the maximum Josephson c u r r e n t . In a d d i t i o n , the c o n d i t i o n "B- = 0" i n the present experiment i n d i c a t e s only that the f i e l d due to the Helmholtz c o i l s c a n c e l l e d out the component of the earth's f i e l d l y i n g i n the j u n c t i o n plane (see f i g u r e 5-2). Most l i k e l y , t h e r e f o r e , at "B" = 0" s t r a y magnetic f i e l d s were present which, as -115-discussed i n Chapter 2, need only be of small magnitude to have a profound e f f e c t on I . c (b) Magnetic F i e l d Dependence of I As o u t l i n e d i n chapter 2, I should decrease w i t h i n c r e a s i n g magnetic f i e l d according to sinCrcXB) w n e r e B i s the f i e l d T T A J J magnitude and X i s a term c h a r a c t e r i s t i c of a given j u n c t i o n and t y p i c a l l y of the order 1.5 G _ 1. (For B=70 G, I $ I /IOOTT) . Q u a l i t a t i v e agreement c cm w i t h theory i s c l e a r l y evident i n f i g u r e 6-2 which i n d i c a t e s that the "good" j u n c t i o n s do indeed e x h i b i t the dc Josephson e f f e c t . The important p o i n t , as f a r as the present experiment i s concerned, i s that the zero-voltage supercurrent can be e f f e c t i v e l y suppressed by the a p p l i c a t i o n of a magnetic f i e l d which i s much smaller than the c r i t i c a l f i e l d of the super-conducting f i l m s . (See paragraph 5 of t h i s ' s e c t i o n . ) (c) V a r i a t i o n of 3V/3I w i t h Magnetic F i e l d For a given magnetic f i e l d B and temperature T, there i s a p o i n t (V , I ) at which r = 3V/31 (eV$2A) i s a maximum. I t i s r m m evident i n f i g u r e 6-2 that r = (3V/3I), 7 ,,T v a r i e s somewhat w i t h magnetic m vnij Im f i e l d , decreasing sharply at about 70 G as. shown i n f i g u r e 6-3. This decrease i n r , the consequence of an i n c r e a s i n g l y s i g n i f i c a n t r eduction m i n the superconducting energy gap width e f f e c t e d by the magnetic f i e l d , i s s i m i l a r to the decrease i n r ^ brought about by i n c r e a s i n g the temperature— thereby reducing the gap—as i s evident i n f i g u r e 6-1. (d) Optimum B i a s i n g Point The c r i t e r i a f o r and means of determining the optimum operating p o i n t are now c l e a r . From a s e r i e s of curves l i k e those i n f i g u r e 6-2, taken at the lowest a c c e s s i b l e temperature, the values of I , V and B are found at which i s maximum and the supercurrent i s minimum. For example, had specimen H-4 not degenerated w i t h thermal c y c l i n g — t o be discussed i n paragraph 4 of t h i s s e c t i o n — t h e b i a s i n g point chosen would have been I = 9.5uA, V = 0.22mV and B = 70 G where r = 100ft at 1.2 K. m 3. Determination of Dynamic Resistance (3V/3I) as a Function of Voltage No p r o v i s i o n was made f o r o b t a i n i n g 3V/3I d i r e c t l y on an X-Y p l o t . For purposes of s e l e c t i n g the operating point i t was s u f f i c i e n t l y accurate to draw tangents to the I-V curves and determine t h e i r slope. I t -116-e *6 100 C N « 50 0 Specimen H-4 . T = 1.2 K -o-X X X X X 0 20 40 60 80 100 B(G) Fipnrp 6-3: V a r i a t i o n of maximum dynamic r e s i s t a n c e w i t h magnetic f i e l d i d = (av/a^ n T = 1.2 K B = 30 G X X .2 .3 Ju n c t i o n Voltage (mV) .5 .6 Figure 6-4: Dynamic r e s i s t a n c e vs voltage f o r specimen J-5 -117-turned out, however, that i n order to understand the manner i n which the noise v a r i e d w i t h b i a s v o l t a g e — s e e s e c t i o n D of t h i s c h a p t e r — a more p r e c i s e e v a l u a t i o n of r = 3V/9I was necessary f o r sample J-5, the sample w i t h which the alpha p a r t i c l e s were detected. 4 A f o u r t h order polynomial I = E a nV was l e a s t squares f i t t e d to each of two experimental curves n ° (0$V$.8 mV) obtained at T = 1.2 K and B = 30 G f o r t h i s specimen. For each curve, r = (9 I / 9 V ) " 1 was c a l c u l a t e d w i t h the mean value being shown i n f i g u r e 6-4. 4. D e t e r i o r a t i o n of Specimens a f t e r Thermal C y c l i n g Because of the high percentage of f a u l t y j u n c t i o n s , the t e s t i n g apparatus was designed (see Chapter 4) so that f i v e specimens could be "dc" te s t e d at one time i n the so c a l l e d "dc hol d e r " to determine the specimen w i t h the most d e s i r a b l e c h a r a c t e r i s t i c s . A f t e r warming slowly (24-36 hours) up to room temperature i n a helium atmosphere, the specimens were removed from the "dc ho l d e r " , the best specimen was separated from the others and mounted i n the "ac"or "pulse t e s t h o l d e r " (see f i g u r e 4-2 and 4-3). This e x t r a handling of the specimens was n e c e s s i t a t e d by the f a c t that the "pulse t e s t h o l d e r " , which was equipped w i t h the alpha source and s h u t t e r , was o r i g i n a l l y designed to take only one 1 cm x 2 cm substrate w i t h i t s four e l e c t r i c a l l e a d s . I t was discovered—when Sn specimens were f i n a l l y made having s a t i s f a c t o r y dc c h a r a c t e r i s t i c s — t h a t upon being warmed up to room temperature, i n s t a l l e d i n the "pulse t e s t h o l d e r " , and cooled again to helium temperatures, a specimen s u f f e r e d c o n s i d e r a b l e , apparently i r r e v e r -s i b l e , changes i n i t s t u n n e l i n g p r o p e r t i e s rendering i t useless as a d e t e c t o r . Of the 6 samples f o r which data are a v a i l a b l e , the general symptoms were a decrease i n the maximum dynamic r e s i s t a n c e , l i t t l e or no temperature dependence of the current ( u n l i k e f i g u r e 6-1) and reduced s e n s i t i v i t y of the supercurrent to a magnetic f i e l d ( u n l i k e f i g u r e 6-2). Table 6-2 l i s t s the p r o p e r t i e s of one such j u n c t i o n (H-4) which was cycled i n the above manner. These " a f t e r c y c l i n g " c h a r a c t e r i s t i c s are very s i m i l a r to those of the "l e a k y " specimens discussed i n Appendix B which i m p l i e s that during the c y l i n g procedure, minute m e t a l l i c f i l a m e n t s or superconducting shorts had developed through the i n s u l a t i n g oxide. The f a c t that these -118-f i l a m e n t s produced no s i g n i f i c a n t change i n the 4.2 K normal s t a t e j u n c t i o n r e s i s t a n c e may be explained as f o l l o w s . I f R i s the normal s t a t e r e s i s t a n c e n SPECIMEN H-4 J-5 Property Before C y c l i n g A f t e r Before C y c l i n g A f t e r R (4.2 K) n I (B) c r =3V/3I m Temp. Dependence B Dependence .286ft .0003mA(100G) 64ft F i t s Theory I quenchable .289ft .lmA(lOOG) 4ft I n s e n s i t i v e to Temp. I not c quenchable .077ft .001mA(40G) 70ft Not Measured I quenched .076ft .01mA(40G) 9.3ft Reasonable I quenched Table 6-2: Specimen C h a r a c t e r i s t i c s Before and A f t e r Thermal C y c l i n g . before and R ' i s the normal r e s i s t a n c e a f t e r c y c l i n g , then 1 1 + 1 R ' R ' Rr . - i n n f i i where R,., i s the r e s i s t a n c e of the filaments when i n t h e i r normal s t a t e , f i i Rough estimates of R J . Q obtained by assuming that the current c a r r i e d by the f i l a m e n t s If.Q i s given by I . . . = I(V , T , a f t e r c y c l i n g ) - I(V , T , before c y c l i n g ) = f i l o ' o ' 6 < o ' o ' J & R f i l i n d i c a t e R,..., = 3-6ft which means t h a t , w i t h i n experimental e r r o r , R '=R . f i l n n In an attempt to prevent d e t e r i o r a t i o n , a new method was devised f o r c y c l i n g sample J-5 which was s u c c e s s f u l i n s o f a r as the ensuing changes i n the p r o p e r t i e s of the j u n c t i o n were not so severe as to prevent -119-i t s being s u c c e s s f u l l y used as a d e t e c t o r . This new technique co n s i s t e d of warming the specimens up r a t h e r r a p i d l y (about 1 1 / 2 hours) from 4.2 K by sl o w l y r a i s i n g them out of the helium bath i n t o a h e l i u m - f i l l e d p l a s t i c bag kept at room temperature. Specimen J-5 was then i n s t a l l e d i n the "pulse h o l d e r " and the reverse procedure c a r r i e d out. In t h i s way the t u r n around time was reduced by an order of magnitude and the time of exposure to "high" temperatures minimized. The mechanism by which the m e t a l l i c f i l a m e n t s appear i s not c l e a r . I t i s i n t e r e s t i n g to note that f o l l o w i n g the i n i t i a l thermal c y c l i n g , specimen J-5 was kept at a temperature of 4.2 K or l e s s f o r some 40 hours w i t h no change i n performance which would seem to exonerate the t e s t i n g and operating procedures. Great care was always taken to avoid the condensation of water on any j u n c t i o n as the d e l e t e r i o u s e f f e c t of water on t h i n f i l m s i s w e l l known. Three p o s s i b l e causes of the f i l a m e n t appearance which cannot be r u l e d out however on the b a s i s of accumulated evidence are: (1) S t r a i n i n the j u n c t i o n r egion caused by d i f f e r e n t i a l c o n t r a c t i o n of the f i l m s and s u b s t r a t e , (2) Contamination by water vapour during the re-mounting time, (3) Contamination by vapour from remnants of solder f l u x l e f t on the "pulse h o l d e r " e l e c t r i c a l t e r m i n a l s . The obvious s o l u t i o n to the d e t e r i o r a t i o n problem i n f u t u r e work i s to avoid thermal c y c l i n g by b u i l d i n g a specimen holder i n which s e v e r a l specimens can be tested at once and the best ones s e l e c t e d f o r pulse measurements without ever being removed from the helium bath. 5. Magnetic F i e l d Dependence of Energy Gap The observed magnetic f i e l d dependence of the superconducting energy gap i s shown i a f i g u r e 6-5. I f the value of the gap i s chosen by e x t r a p o l a t i o n to the v o l t a g e a x i s , one f i n d s 2A(B = 100) = 1.05 meV = 2A(B = 0 ) 1.11 meV which i s c o n s i s t e n t w i t h the f i n d i n g s of other workers (Douglass & F a l i c o v , 1964) f o r f i l m s at these temperatures i n that t h e i r e m p i r i c a l r e l a t i o n -120-Figure 6-5: Magnetic F i e l d Dependence of Energy Gap -121-(equatlon 2-7) y i e l d s A(H) A(0) = 1 -' H 1 2 1 — fioo' cf s x — 382 = .93 Here, H = 382 Oe i s the c r i t i c a l f i e l d f o r a t h i n Sn f i l m at temperature T c a l c u l a t e d from (Meservey & Douglass, 1964) H (T) = H (0)(1 - (T/T ) 2 ) ( 1 - (2X/d)tanh(d/2X))~i ct c c w i t h T = 1.2 K T c = 3.72 K ( C r i t i c a l temperature) H (0) = 306 Oe (Bulk C r i t i c a l f i e l d at I = 0 K) c X = 500 A (Penetration Depth) d = 2000 A ( F i l m Thickness) The magnetic f i e l d used w i t h the detector j u n c t i o n (J-5) was only 30 G so that the decrease i n the energy gap due to magnetic b i a s i n g may be taken to be l e s s than 5% and may be s a f e l y neglected i n f u r t h e r a n a l y s i s . B. Observation of Pulses from Alpha P a r t i c l e s F o l l o w i n g the dc t e s t s and re-mounting procedure described i n the previous s e c t i o n , sample J-5 was cooled to 1.2 K and the optimum operating p o i n t determined as discussed p r e v i o u s l y . (see f i g u r e 6-6) 1. Counting of Pulses With the e l e c t r o n i c s set up as i n f i g u r e 5-6, the bandwidth having been adjusted f o r equal r i s e and f a l l times of 100 nsec, counts were observed on the d i s c r i m i n a t o r - s e a l e r . That the pulses were produced by bombardment of the j u n c t i o n area i t s e l f was evident from the observation that the pulses could be c o n t r o l l e d by c l o s i n g the mechanical shutter (shown i n f i g u r e 4-3) or by lowering the substrate u n t i l the overlap area was^just below the l e v e l of the helium bath. -122-I (mA) o l , 1 ; » I L L 0 0.2 0.4 0.6 0.8 1.0 V(mV) Figure 6-6: Detector B i a s i n g Conditions -123-(a) Count Rate The observed count r a t e , w i t h s h u t t e r open and the d i s c r i m i n a t o r set at 5 times the rms noise output from the a m p l i f i e r s (33 mV f o r t h i s bandwidth), was 222 + 3 counts per minute (cpm). This agrees reasonably w e l l w i t h the p a r t i c l e f l u x at the j u n c t i o n p r e d i c t e d from the source str e n g t h and junc t i o n - s o u r c e geometry i l l u s t r a t e d i n f i g u r e 6-7. I f N = 1.1 x 10^ i s the p a r t i c l e f l u x from the source then N., the number of alpha p a r t i c l e s s t r i k i n g the j u n c t i o n per minute i s J 2 ! N. = N A COS(0)/4TT£ =198. 3 o A reduced number of counts, about 65 cpm, was also observed when the shutter was i n the " c l o s e d " p o s i t i o n — a d i s t u r b i n g r e s u l t which f o r t u n a t e l y has a ready explanation. In order to maximize the path l e n g t h of the alpha p a r t i c l e s through the j u n c t i o n , the source was l o c a t e d to one side of the s u b s t r a t e (see f i g u r e 6-7) so that the s h u t t e r , which moved i n the plane of the j u n c t i o n , when."closed" or pressed against the source l e f t about 35% of the source s t i l l exposed to the j u n c t i o n . The p r e d i c t e d count r a t e w i t h the s h u t t e r " c l o s e d " would thus be .35 N.. = 69 cpm which i s c o n s i s t e n t w i t h the observed r a t e of: 65 cpm. Some contamination was detected on the s h u t t e r but the number of alpha p a r t i c l e s coming from -4 t h i s course were a n e g l i g i b l y s m all f r a c t i o n , roughly 5 x 10 , of those coming from the main source. (b) Influence of Veto System Chapter 5 contains a d e s c r i p t i o n of an i n t e r f e r e n c e -v e t o i n g system which was designed to paralyze the d i s c r i m i n a t o r - s e a l e r whenever b u r s t s of electro-magnetic i n t e r f e r e n c e were present thereby preventing these being counted as meaningful pulse s . Thus, only those pulses from the detector a m p l i f y i n g system were"counted whose amplitude exceeded the s c a l e r d i s c r i m i n a t o r l e v e l (V,. •) and whose a r r i v a l i n time d i s c was a n t i - c o i n c i d e n t w i t h an i n t e r f e r e n c e pulse. A l l other p u l s e s , whose amplitude exceeded V,. , were counted on another s c a l e r — s e e f i g u r e 5-6— c d i s c ° and the count ra t e f o r these ranged from 13,000 to 18,000 cpm. C l e a r l y , w i t h such a high i n t e r f e r e n c e count l e v e l , the small count r a t e of 200 cpm due to l e g i t i m a t e pulses from the detector i t s e l f would have been undetectable without an e f f i c i e n t v e t o i n g system -124-- 1.1 cm d,= 51 + 1° © 2 = 79 + 1° F i g . 6-7: J u n c t i o n - S o u r c e Geometry ( S h u t t e r not shown f o r c l a r i t y ) F i g . 6-8: P u l s e s observed a t optimum bandwidth; V e r t i c a l - 1 . 7 5 ^ i A / d i v . ; H o r i z . - 200 n s e c / d i v . ( V e r t i c a l scale c a l i b r a t e d f o r a square input pulse as described i n s e c t i o n B.2 (p.125) .) -125-Even with such a relatively high interference pulse count rate, the corresponding "dead time" or percentage of time during which the discriminator-sealer was paralyzed and unable to accept pulses from the detector was negligibly small, being less than about 0.5%. 2. Pulse Characteristics While i t i s interesting and satisfying to observe that the count rate agrees with the predicted alpha particle flux at the junction, the key to understanding the physical processes taking place in the detector l i e s in the interpretation of the characteristics of the actual pulses. The pulses were recorded photographically on a wide-band oscilloscope (2.4 nsec rise time) which was connected to the amplifier and vetoing systems as shown in figure 5-6. Optimization of the amplifier system bandwidth (x. ^  andx,.,., "out" on Ortec amplifier) led to the obser-lnt dif f vation of pulses (shown in figure 6-8) whose amplitudes were up to 19 times the rms noise level. The upper and lower 3-db points of the amplifier system at this bandwidth setting were later measured to be 0.1 MHz and 3.5 MHz respectively which is consistent with the observed response (100 nsec rise time and 1.7 psec f a l l time) of the system to square pulses injected from a pulse generator. To determine i t s current sensitivity, the amplifier system was used to measure pulses of known amplitude produced by a pulse generator in a lumped constant model of the junction small signal equivalent circuit (see figures 6-9 and 3-5) V. V (I ) o i S _ = ES _ o step 3 9 -1 (attenuation) (44ft) For V = 110 mV and attenuation of 50 db, (I ) ^ = 8pA which corresponds pg o step r to an output voltage of ^ o u^. = 2.3 V making the amplifier system sensitivity S = 1V/3.5 pA (6-1) The signal to noise ratio was a maximum for the operating point shown in figure 6-6 at which r = (9V/8I) was maximum. Reducing r -126-I F ( i ) v o step Pulse Generator Attenuator 39Q. Pulse Generator Attenuator V. 10fi; T . Pulse Generator Attenuator Lumped constant Model of specimen (C = 3500 pF) Preamp. Figure 6-9: Current S e n s i t i v i t y C a l i b r a t i o n a. (\ A Pulse Amplitude r a t i o as measured from pulse photographs — Theory V "\_ (r = r .) J out 1 V Cr = r ) out o 'Normalization point a i load out 10 r = (3V/3I) Tn Figure 6-1Q: Pulse Amplitude vs Dynamic Resistance -127-by s h i f t i n g the b i a s voltage above or below 0.3 mV caused both the noise l e v e l to i n c r e a s e , as shown i n f i g u r e 6-12, and the pulse amplitude to decrease as recorded by a s e r i e s of photographs taken at various b i a s v o l t a g e s . This v a r i a t i o n of pulse amplitude w i t h dynamic r e s i s t a n c e i s p l o t t e d i n f i g u r e 6-10 along w i t h the t h e o r e t i c a l v a r i a t i o n c a l c u l a t e d on the b a s i s of the s m a l l s i g n a l equivalent c i r c u i t of the j u n c t i o n depicted i n the i n s e t . ( c f . Figure 3-5). I f r = ( 8 V / 9 I ) T and R = 10.3ft i s the input impedance of the t r a n s m i s s i o n l i n e - p r e a m p l i f i e r system (see Chapter 5) then i s r -61 = 1+r a n d R + r r . o # 1 r * R + r . o 1 which i s c o n s i s t e n t w i t h the measured va l u e s . 3. Pulse Height Spectrum Figure 6-11 i s a histogram showing the number of times pulses of a given amplitude were observed p l o t t e d against pulse amplitude. As no multi-channel pulse height analyzer was a v a i l a b l e , t h i s pulse height spectrum was obtained by examining a s e r i e s of pulse photographs ( s i m i l a r to that of f i g u r e 6-8) and simply counting the number of pulses f a l l i n g w i t h i n the ranges denoted by the small d i v i s i o n l i n e s of the o s c i l l o s c o p e g r a t i c u l e . ( A l l photographs so analyzed were taken under i d e n t i c a l a m p l i f i e r c o n d i t i o n s w i t h the j u n c t i o n biased at the optimum operating point.) The sharp c u t - o f f at the low amplitude end of the spectrum denotes the pulse t r i g g e r i n g l e v e l . At f i r s t glance, t h i s wide range i n pulse amplitudes i s s u r p r i s i n g f o r the alpha p a r t i c l e s s t r i k i n g the j u n c t i o n are e s s e n t i a l l y monoenergetic (5.10-5.15 MeV). I t turns out, as discussed i n more d e t a i l i n Chapter 7, that the amplitude v a r i a t i o n may be a t t r i b u t e d to two causes: (1) the energy l o s t by the p a r t i c l e s i n passing through the f i l m s v a r i e d from 0.5 to 0.14 MeV depending on the angle of incidence ( f i g u r e 6-7); (2) the j u n c t i o n i s thermally coupled to the substrate and some of the energy expended i n the substrate as the p a r t i c l e comes to r e s t d i f f u s e s back to V „(r = r.) out i _ V . (r = r ) out o -128-w + w 00 c u CO 01 CO I—I 3 p-l 20 o z 10 j - 1 Pulse Amplitude (E) ( O s c i l l o g r a p h d i v i s i o n s ) Figure 6-11: Pulse Height Spectrum -129-th e j u n c t i o n i n s u f f i c i e n t time to c o n t r i b u t e to the pulse. C. Noise 1. Observed Noise The noise output from the d e t e c t o r - a m p l i f i e r system was measured w i t h a K e i t h l e y Model 120 wideband (10 Hz-100 MHz) v o l t m e t e r — c a l i b r a t e d to read the rms value of a t r u e sine wave—connected as shown i n f i g u r e 5-6. ( S t r i c t l y speaking, because the meter i s c a l i b r a t e d to read the rms of s i n u s o i d a l , not random, s i g n a l s , noise voltages i n d i c a t e d by the meter should be m u l t i p l i e d by 1.13 to o b t a i n t h e i r t r u e rms value ( P a r t r i d g e , 1958).) Of s p e c i a l i n t e r e s t was the observed v a r i a t i o n of noise w i t h j u n c t i o n dynamic r e s i s t a n c e p l o t t e d i n f i g u r e 6-12. (These data w i l l be considered again i n S e c t i o n D f o r purposes of e s t i m a t i n g the j u n c t i o n capacitance^ Noise readings were a l s o taken w i t h a number of c a p a c i t o r s (C ) connected i n p a r a l l e l to the j u n c t i o n . As i t was not p o s s i b l e during the course of the run to place them i n c l o s e p r o x i m i t y to the j u n c t i o n , they were connected at room temperature i n the " f i l t e r b ox"—see f i g u r e 5-3. U n f o r t u n a t e l y , only the nominal values of these c a p a c i t o r s are known as the a c t u a l components used were a c c i d e n t a l l y m i s l a i d before they could be c a l i b r a t e d . Other c a p a c i t o r s of s i m i l a r manufacture were, by measurement, found to be of t y p i c a l l y ± 20% p r e c i s i o n at 1 kHz but s e l f - r e s o n a t i n g at about 9 MHz. This means that i n the a m p l i f i e r bandpass (0.1 to 3.5 MHz) the a c t u a l c a p a c i t o r s used i n the t e s t would probably e x h i b i t a reactance corresponding to much l e s s than the nominal capacitance which makes the noise measurements taken as a f u n c t i o n of C of l i t t l e value other than a ext c o nsistency check (see f i g u r e 6-13). 2. O r i g i n of Noise This s e c t i o n w i l l show that the dominant source of noise i n the experiment was the common base p r e a m p l i f i e r and not the superconducting tunnel j u n c t i o n p a r t i c l e d e t e c t o r . (a) Representation of Detector as Noise Generator The equivalent c i r c u i t i n c l u d i n g noise generators f o r the d e t e c t o r as biased and connected to the a m p l i f y i n g system (see f i g u r e s 5-3 and 5-6) i s given i n f i g u r e 6-14(a) where -130-140 130 E ' I 120 a) 110 w • H O c 100' \ \ \ X \ \ \ _L r = ( 8 V / 3 I ) T ft F i g u r p 6-17: Output Noise vs j u n c t i o n Dynamic Resistance C O 0) w • H O C 300 200 100 X ± .01 Figure 6-13: ,02 .03 C(Ext) uF(Nominal) .04 .05 Output Noise vs Ex t e r n a l Capacitance -131-T = 1.2 K T = 295 K I 1 2 1 Kt) © "b ©K GX To Preamp. r = ( 3 V / S l ) T , = Bias Resistance, I(t) = I q exp^-t/x) = | s i g n a l current i i 1 Figure 6-14 (a): Detector Equivaient c i r c u i t with Noise Generators © .2 . 2 , . 2 1 = \ + XB 1 2 .2 2 v, = l r Figure 6-14 (b): Detector noise Generator equivalent C i r c u i t s r , v i e e input e b' Lg>J r , = emitter r e s i s t a n c e , r^, = base r e s i s t a n c e , r c , = c o l l e c t o r e r e s i s t a n c e Figure 6-14 ( c ) : CB P r e a m p l i f i e r with Noise -132-2 2 i ^ a Johnson noise i n b i a s r e s i s t a n c e (R^)=IR^=4kTB/R^ 2 ~~2 i , = shot noise on b i a s current (I) = I = 2eIB D B w i t h T the room temperature (295 K), k Boltzmann's constant and B the system bandwidth. In p r a c t i c e , = 20 Kft>>r, hence the equivalent c i r c u i t may be reduced to the forms shown i n f i g u r e 6-14(b), where, being u n c o r r e l a t e d , the noise currents add q u a d r a t i c a l l y . Thus, over the bandwidth B, 2 2 v d = r 4kTB + 2eIB = 4kTBr r 2 e l r 4kT (6-2) 4kTBr (M) At the operating p o i n t I -2 makes M - 1.35 x 10 70 uA which, along w i t h the other values c i t e d , (b) Equivalent C i r c u i t f o r CB T r a n s i s t o r w i t h Noise The "low frequency" equivalent c i r c u i t f o r a t r a n s i s t o r o p e rating i n common base (CB) mode w i t h white noise generators as given by Woll and Herscher (1962) i s shown i n f i g u r e 6-14(c). In t h i s case, the noise generators are: 2 V T2 2 e e " 2 e I E B r e ' 2kTBr' e kT e l . V 4kTBr' b (6-3) and v 2 = 2 e r , a ( l - a ) r 2 B = 2 k T a ( l - a ) r 2 B/r , c E c c e 2 a where I i s the shot noise on the emitter current I„ and H„„= ^  = 40 e E FE 1-a i s the measured common emitter forward current g a i n . (The "low frequency" r e p r e s e n t a t i o n i s used as the upper frequency of the a m p l i f y i n g system -133-bandpass i s about 3.5 MHz whereas the upper c u t o f f frequency f ^ f o r the 16G2 t r a n s i s t o r used i n the p r e a m p l i f i e r i s quoted by the manufacturer as 1.1 GHz.) (c) C a l c u l a t i o n of Noise Figure To appreciate the r e l a t i v e importance of the detector and p r e a m p l i f i e r as noise sources, i t i s convenient to de f i n e the f o l l o w i n g n o i s e f i g u r e F = t o t a l output noise power output noise power due to noise of detector From the c i r c u i t f o r the p r e a m p l i f i e r and detector shown i n f i g u r e 6-15 i t i s evident that the noise power i s p r o p o r t i o n a l to the 2 mean square v o l t a g e e Q appearing across R^. Taking the open loop case f o r s i m p l i c i t y ( i e . R^ = °°), one f i n d s ; d e' b' ., I S i = (6-4) (X + r e , + r v . ) and •• - • e =-v, + v + i (r, . + ar ) o b c e b' c S u b s t i t u t i n g i i n t o e^, c o l l e c t i n g terms and ta k i n g the mean square value of both sides y i e l d s 2 2 71 , ~1 , ~1 . . ~2 .••• . .2 2 a r c ( v d + V b + V e ' } + V c <r.-+ r e ' + ' b ^ e o = — (6-5) ( r + r e ' + r b ' } where i t has been assumed that the noise generators are u n c o r r e l a t e d , ar >>r, , and ar >>r . + r . c b c e' The output noise power due to the noise of the detector —2 ~2 "2 i s found by s e t t i n g v>j» = v e « = v c - 0 i n 6-5 which gives ~2 , ~2 ~2 . . , ,2 V b ' + Ve» V c ( r + r e ' + r b ' } F = 1 + — — + *T 2 2 2 v, a r v, d c d -134-o i i e Figure 6-15: Equivalent c i r c u i t f o r Detector and A m p l i f i e r w i t h noise a i v = v or V 8 s Figure 6-16: Equivalent c i r c u i t f o r S i g n a l to Noise Ratio estimate -135-Using equations 6-2 and 6-3 makes .2 F " 1 = rM (r + r , + r, , ) ' V + ^ e ' + e* b 1 2 r e ' H F E (6-6) The t r a n s i s t o r b i a s current ( I £ ) was 10 mA, so that by 6-3 r g l = 2.6 ft. A n g e l l (1967) gives the input impedance of a CB t r a n s i s t o r as R i n = r g l + r b , ( l - a ) making r ^ , = 216 ft using the measured (see chapter 5) input impedance of approximately 8 ft. P u t t i n g these values i n t o equation 6-6 y i e l d s F = 3840 c l e a r l y i d e n t i f y i n g the p r e a m p l i f i e r as the major source of n o i s e . (d) Equivalent Input Noise (for C i r c u i t Used) I t i s convenient to represent the p r e a m p l i f i e r noise generators by an equivalent voltage generator (v ) across the j u n c t i o n 8 2 2 dynamic r e s i s t a n c e . (see f i g u r e 6-14(b) and 6-15). Thus F = v /v, or g d v = (4kTBrMF)* = 5.2 yV(rms), r = 9.3ft (e) S i g n a l to Noise R a t i o Estimate From the pulse photograph ( f i g u r e 6-8), the maximum s i g n a l i s seen to correspond to about 8.1 uA which can be represented by a v o l t a g e generator V = 8.1 uA • r = 75 uV across the j u n c t i o n dynamic r e s i s t a n c e . (see f i g u r e s 6-14(b) and 6-16). The two generators v and V g s are thus l o c a t e d at the same poi n t i n the d e t e c t o r - p r e a m p l i f i e r equivalent c i r c u i t so that the q u a s i - t h e o r e t i c a l s i g n a l to noise r a t i o (S/N) i s simply S/N = V /v = 14.5 i n reasonable agreement with the measured value of 19. s g ( f ) Noise w i t h Lumped Constant Models of Specimens To confirm experimentally that the observed output noise depends e s s e n t i a l l y only on the source impedance seen by the pre-a m p l i f i e r and not on noise sources a s s o c i a t e d w i t h the d e t e c t o r , a lumped constant model of a tunnel j u n c t i o n , c o n s i s t i n g of a 9.6ft r e s i s t o r i n p a r a l l e l w i t h 3000 pF, was s u b s t i t u t e d f o r the- a c t u a l specimen. The r e s u l t a n t n o i s e , measured w i t h the a m p l i f i e r gains unchanged from measurements w i t h the a c t u a l specimen, was indeed equal to that obtained w i t h the operating j u n c t i o n . To some extent, t h i s p r e c i s e agreement i s f o r t u i t o u s because the output noise was observed (see f i g u r e 6-12,13) to vary w i t h j u n c t i o n -136-r e s i s t a n c e and p a r a l l e l capacitance and yet the lumped constant components of 9.6ft and 3000 pFs. were only approximations to the a c t u a l values. Nonetheless, because the absence of the detector has not s i g n i f i c a n t l y a l t e r e d the n o i s e , i t i s reasonable to conclude that the p r e a m p l i f i e r was i n f a c t the dominant noise source. D. Determination of J u n c t i o n Capacitance For mathematical s i m p l i c i t y the presence of C j was ignored i n the noise i n v e s t i g a t i o n s of the previous s e c t i o n . The experiments mentioned above i n which the noise was measured w i t h a r e s i s t o r and a c a p a c i t o r s u b s t i t u t e d f o r the r e a l specimen j u s t i f i e d t h i s s i m p l i f i e d approach i n that they were c o n s i s t e n t w i t h the c o n c l u s i o n (about the a m p l i f i e r being the main noise source) a r r i v e d at t h e o r e t i c a l l y w i t h Cj = 0. In order to proceed w i t h the a n a l y s i s of the r e s u l t s c a r r i e d out i n chapter 7, however, a reasonably c l o s e estimate of C i s r e q u i r e d . The purpose of t h i s s e c t i o n i s to s a t i s f y t h i s requirement, f i r s t by c a l c u l a t i n g Cj from a p a r a l l e l p l a t e model and second by a d e t a i l e d a n a l y s i s of the noise measurements t a k i n g the e f f e c t of Cj i n t o account. I t turns out that the range of values found by the two approaches are reasonably compatible. 1. P a r a l l e l P l a t e Model A rough estimate of the j u n c t i o n capacitance may be obtained by regarding the device as a simple p a r a l l e l p l a t e c a p a c i t o r having t h i n Sn f i l m e l e c t r o d e s and a d i e l e c t r i c of Sn0 2. (a) D i e l e c t r i c Constant of SnO^ Van Daal (1968) gives the s t a t i c or zero frequency d i e l e c t r i c constant (K = £/eQ) °f bulk SnO^ as l y i n g between 9 and 14 depending on the c r y s t a l o r i e n t a t i o n . (Other workers, eg. Coon and F i s k e (1965), have used K = 5 without q u a l i f i c a t i o n — p r e s u m a b l y t h i s i s an approximation to the o p t i c a l d i e l e c t r i c constant,, (which i s i n a p p r o p r i a t e here) quoted by some authors, eg. Summitt and B o r r e l l i (1965) and A r a i (1960), as being approximately 4.) I t i s not r e a d i l y apparent that the d i e l e c t r i c constant f o r bulk m a t e r i a l may be meaningfully c a r r i e d over to very t h i n l a y e r s — a s i n a tunnel j u n c t i o n — a l t h o u g h there i s some work by Kohn (1958) and Mead (1961) which would seem to; j u s t i f y such an e x t r a p o l a t i o n . -137-(b) I n s u l a t o r Thickness The t h i c k n e s s of the i n s u l a t i n g l a y e r i s estimated from the measured low temperature 4.2 K, low v o l t a g e , normal s t a t e t u n n e l i n g r e s i s t a n c e of the j u n c t i o n , R = 0.077 ± .001ft. To d e r i v e the n e f f e c t i v e t u n n e l i n g thickness (S T) from f i g u r e 2-3, i t i s necessary to know, i n a d d i t i o n to K, the mean b a r r i e r height of the i n s u l a t o r d> and the -4 2 ° normalized r e s i s t a n c e a = R n*A where A = 7 x 10 cm i s the j u n c t i o n over-l a p area. The magnitude of the forbidden energy band i n Sn02 (E ) i s given by A r a i (1960) and Summitt et a l (1964) as 4 eV. Assuming gap that the Fermi surface l i e s w i t h i n the band gap, <j>Q must be i n the range 0 < d> < E . A p r e c i s e value of a) cannot be s p e c i f i e d because the o gap o r presence of i m p u r i t i e s , non-stoichiometry and d e f e c t s i n the i n s u l a t o r make the p o s i t i o n of the Fermi surface i n the band gap u n c e r t a i n . -3 2 Taking o = 5.4 x 10 ft mm , 1 < <J> < 4 eV ( f o r example), O and 9 $ K $ 14 y i e l d s S^, l y i n g between 7 and 14 A ( c f . f i g u r e 2-3). (Note: S^ i s the t h i c k n e s s of an i d e a l , u n i f o r m l y t h i c k i n s u l a t i n g l a y e r of area A whose t u n n e l i n g r e s i s t a n c e i s R^ and i s not t o be confused w i t h some average th i c k n e s s <S> A = i n s u l a t o r t h i c k n e s s at x,y.) ^ S(x,y) dx dy where S(x,y) i s the a c t u a l Chow (1963) and Hurych (1966) po i n t o u t , however, that the e f f e c t i v e t h i c k n e s s (S c) to be used f o r capacitance c a l c u l a t i o n s i s 1.5 to 3 times g r e a t e r than S T. This d i f f e r e n c e i s a t t r i b u t a b l e to the f a c t that the thickness of the i n s u l a t i n g l a y e r i s not constant but v a r i e s s t o c h a s t i c a l l y from p o i n t to p o i n t . Such n o n - u n i f o r m i t i e s w i l l s t r o n g l y a f f e c t the tunnel c u r r e n t — b e c a u s e of i t s exponential dependence upon the t h i c k n e s s — b u t only weakly a f f e c t the capacitance. For 9 $ K ^ 1 4 amd 10 $ S $ 42 A, the c a l c u l a t e d value c , of C j i s i n the range 1300 S Cj £ 8OO0 pF. Since ^ < 1 eV i m p l i e s S > 42 A and C T < 1300 pF, a l l t h a t can be concluded i s t h a t f o r c J 0 < d> < 4 eV ! o 0 < Cj $ 8i000 pF -138-A more d i r e c t and more p r e c i s e estimate was obtained from the noise measurements as described i n the next s e c t i o n . 2. Noise Measurements In s e c t i o n C of t h i s chapter, the noise output from the j u n c t i o n - a m p l i f i e r system (V ) was discussed at some len g t h and was found to be a f u n c t i o n of the impedance i n the p r e a m p l i f i e r input loop e x e m p l i f i e d by the observed v a r i a t i o n of noise w i t h r (the j u n c t i o n dynamic r e s i s t a n c e ) f o r unknown constant j u n c t i o n capacitance Cj ( a c t u a l ) = C j a ; t h i s q u a n t i t y w i l l be denoted as ^ ( r , C j ^ , Meas.). From the small s i g n a l equivalent c i r c u i t of the d e t e c t o r - a m p l i f i e r system, the output noise may be c a l -c u l a t e d as a f u n c t i o n of r and C^; t h i s q u a n t i t y w i l l be denoted as V ( r , C , Thy.). The purpose of t h i s s e c t i o n i s to estimate the j u n c t i o n XI J capacitance by determining the range of C f o r which V ( r , C , Thy.) i s c o n s i s t e n t w i t h the experimentally measured noise v n ( r > Gj a» Meas.). The reader u n i n t e r e s t e d i n the f o l l o w i n g mathematical d e t a i l s may wish to s k i p on to s e c t i o n E where the acceptable range of C. values obtained from the noise measurements, along w i t h the other p r i n c i p a l r e s u l t s from t h i s chapter, are summarized. (a) C a l c u l a t i o n of V ^ r , C j , Thy.) Figure 6-17 shows the equivalent c i r c u i t of the tunnel j u n c t i o n — t r a n s m i s s i o n — l i n e — p r e a m p l i f i e r system (see f i g u r e s 3-5, 6-14 and 6-16) w i t h the p r e a m p l i f i e r noise generators v e , , v^, and v c , (equation 6-3) i n p l a c e . In keeping w i t h the conclusions of s e c t i o n C, the j u n c t i o n noise generators have been neglected. Assuming rc>>R^+r^,, i t may be shown that the mean square noise voltage across (see f i g u r e 6-15) i s given by 2 e = R 2 a 2.4kTB ° ( r + r e , + r b , ( l - a ) ) 2 [ ( r + r e , + r b , ) FE re' To extend t h i s r e s u l t to the c i r c u i t of f i g u r e 6-17, one may r e c a l l equation 5-4 and r e p l a c e r by the transmission l i n e - d e t e c t o r impedance Z = Z ( r , C ) = R + j X where -139-l.OtlF. l.OuF I I I L _ J u n c t i o n Transmission F i l t e r P r e a m p l i f i e r Line Box Figure 6-17: Equivalent C i r c u i t of J u n c t i o n - P r e a m p l i f i e r System • with Noise Sources only (no alpha pulses) -- 1 4 0 -R - + R (1.2 K,co). 1 + (corC ) Z % - r ^ u C j r ) X = j + ju L ; L = 1.63 uH. 1 + (torCj) Since 1 Kft > 470ft >> r g I + r b , ( l - a ) , the f i l t e r box-CB t r a n s i s t o r combination may be replaced by a s e r i e s impedance Z.' = R! + jX'. where ( c f . f i g u r e 5-5 and equation 5-5) i n i n i n n R' . = 7.9ft = r , + (1 - c t ) r u l m e' b' X ' i n = j ( a ) L " 1 / a ) C ) = ^ " X c w i t h L = 0.23 uH. and C = 0.5 uF. (Noise generators corresponding to the Johnson noise i n the p a r a l l e l r e s i s t o r s Rc and R_ i n f i g u r e 6-17 are ignored t D as t h e i r c o n t r i b u t i o n to the net output noise v o l t a g e may r e a d i l y be shown to be n e g l i g i b l e . ) For detector impedances Z of i n t e r e s t , r^,>>re,+|Z|, which means that over a bandwidth df e o d f = 4 ° 2 * k T d f < V + * re» + rb? / HFE%»> ( Z + Z i n ) _ 2 K df (Z + Z! ) - 2 , Izl « r, , i n b (6-7) The magnitude of the t o t a l noise v o l t a g e f o r p a r t i c u l a r values ( r ' , Cj') of the j u n c t i o n equivalent c i r c u i t parameters i s therefore -141-e df = K o . |2 df/| Z ( r ' , C.') + Z! | «J xn. where the l i m i t s of i n t e g r a t i o n are the a m p l i f y i n g system 3db-down p o i n t s , f = 0.1 MHz and f 2 = 3.5 MHz. may form the noise voltage r a t i o , . MHz. Now V ( r , C T, Thy.) °= e ( r , C T) so that one 1 2 n J y o J Y(Thy.) = V ( r , C , Thy.) n J V N ( r \ C j ' , Thy.) e;;•<*. CJ> (6-8) With C j ' = Cj and r ' = 9.3P., y(Thy.) was evaluated by Simpsons' r u l e i n t e g r a t i o n f o r an appropriate range of values f o r r , Cj thereby generating the f a m i l y of curves shown i n f i g u r e 6-18. (b) Computation of ^ ( r , Cj , Meas.) From equation 6-7 i t may be seen that the t h e o r e t i c a l output noise v o l t a g e depends upon the p r e a m p l i f i e r input loop impedance Z + Z.' . The measured output noise V (Meas.), however, includes c o n t r i b u -m n t i o n s from the p o s t a m p l i f i e r s whose noise i s i n s e n s i t i v e to Z + Z.' so that i n i t i s convenient to w r i t e (6-9) V (Meas.) = [ V 2 ( Z + Z.') + V 2 ] 2 n m o Here V(Z + Z^) i s the component of output noise v o l t a g e depending on Z + and V q i s the "open input loop" output noise v o l t a g e . The q u a n t i t y of i n t e r e s t f o r comparison to theory i s t h e r e f o r e V ( r , C T , Meas.) = [V 2 (Meas.) - V 2 ] 2 n Ja n o (6-10) where, by experiment, V ' = 36 mV. For comparison to y(T\iy.)t the r a t i o y(Meas.) = V ( r , C T , Meas.) n J a ?  V (9.3, C T , Meas.) n ' J a ' i s computed from the data of f i g u r e 6-12 and equation 6-9 and p l o t t e d i n f i g u r e 6-18. (The c i r c l e s around the v a r i o u s y(Meas.) p o i n t s are a measure -142-Figure 6-18: T h e o r e t i c a l and Measured P r e a m p l i f i e r Noise Output vs Dynamic Resistance of J u n c t i o n -143-of the experimental u n c e r t a i n t y . ) (c) "Best" value of Cj Ins p e c t i o n of f i g u r e 6-18 shows reasonable agreement of theory to experiment f o r a j u n c t i o n capacitance 1500 $ C j $ 4500 pF which l i e s i n s i d e the range of values p r e d i c t e d from the p a r a l l e l p l a t e model i n s e c t i o n 1 of t h i s chapter. (d) Consistency Check Wi t h i n experimental e r r o r , t h i s value of the j u n c t i o n capacitance agrees w i t h that obtained g r a p h i c a l l y from s e t t i n g (r = 0 , Cj = 0, Thy.) V r (transmission l i n e shorted, Meas.) ^ ^ V (r = 9.3, C T, Thy.) V (r =9.3, C T , Meas.) XI J XI J 3 . The l e f t hand s i d e of 6-11 was computed s i m i l a r l y to equation 6-8; the r i g h t hand s i d e i s the r a t i o of the output noise measured w i t h the transmission l i n e terminated w i t h a dc short (at 1.2 K) to the noise measured w i t h the t r a n s m i s s i o n l i n e terminated w i t h the operating tunnel j u n c t i o n . E. Summary This chapter has described i n some d e t a i l the r e s u l t s obtained i n the course of d e t e c t i n g alpha p a r t i c l e s w i t h tunnel j u n c t i o n s . The s i g n i f i c a n t p o i n t s are l i s t e d below: (1) The dc I-V c h a r a c t e r i s t i c s were c o n s i s t e n t w i t h theory and the r e s u l t s of other workers. From these c h a r a c t e r i s t i c s , the maximum r = ( 9 V / 3 I ) T was found to be 9.3ft. (2) With the j u n c t i o n biased at i t s optimum operating p o i n t , which was found, as p r e d i c t e d t h e o r e t i c a l l y , to correspond to those values of I , V and B f o r which r was a maximum and the supercurrent was minimum, pulses were detected whose count r a t e agreed w i t h that p r e d i c t e d from the source s t r e n g t h and the j u n c t i o n geometry. This observation, together w i t h the f a c t that the pulses could be turned o f f by i n t e r p o s i n g a mechanical s h u t t e r between the source and j u n c t i o n , l e d to the con c l u s i o n that the -Im-pulses were produced by alpha p a r t i c l e bombardment of the j u n c t i o n i t s e l f . (3) The pulses observed, at optimum a m p l i f i e r bandwidth ( x . = 100 nsec, xr = 1.7 psec), had amplitudes up to 19 times the r i s e f a l l r r preamplifier-dominated rms noise l e v e l of roughly 100 keV r e f e r r e d to the i n p u t . (4) From noise measurements, the j u n c t i o n capacitance Cj was found to l i e i n the range 1500 S Cj $ 4500 pF which i s c o n s i s t e n t w i t h the range of values p r e d i c t e d from a p a r a l l e l p l a t e model. (5) Of minor s i g n i f i c a n c e to the reader but of major concern to the experimenter was the observed tendency of tu n n e l i n g j u n c t i o n s to d e t e r i o r a t e upon thermal c y c l i n g . -145-CHAPTER 7 ANALYSIS OF RESULTS A. I n t r o d u c t i o n As discussed i n Chapter 1, the qu a n t i t y of most i n t e r e s t to be obtained from the present experiment i s w — t h e average energy l o s s by a charged p a r t i c l e to e x c i t e a q u a s i p a r t i c l e p a i r — f o r comparison w i t h the corresponding q u a n t i t y i n other types of detector and, eventually, f o r comparison w i t h theory when i t i s s u f f i c i e n t l y w e l l developed. To determine w from the measured pulse response of the superconducting tunnel j u n c t i o n , i t i s necessary to know the form of the current pulse i ( t ) which i s super-imposed on the thermal e q u i l i b r i u m t u n n e l i n g current f o l l o w i n g each alpha p a r t i c l e bombardment. Once i ( t ) i s known, the number N q of q u a s i p a r t i c l e s created by the alpha p a r t i c l e having l o s t energy AE i n the j u n c t i o n may be estimated, g i v i n g w = A E / N Q . In t h i s experiment, the r e s u l t s were not s u f f i c i e n t l y p r e c i s e to determine d i r e c t l y the form of i ( t ) . Therefore, i t i s assumed that the current pulses have the form (equation 3-11) i ( t ) = i e x p ( - t / t ) , ( t £ 0) where the peak amplitude i i s p r o p o r t i o n a l to the magnitude of the excess d e n s i t y of q u a s i p a r t i c l e s generated by the passage of the alpha p a r t i c l e and T i s a c h a r a c t e r i s t i c time during which t h i s excess d e n s i t y decays to zero. Admittedly t h i s model has no d i r e c t t h e o r e t i c a l b a s i s , being assumed f o r the sake of convenience i n a n a l y s i s , but our t h e o r e t i c a l a n a l y s i s i s not yet s u f f i c i e n t l y developed to j u s t i f y a more complex time dependence. (A t h e o r e t i c a l a n a l y s i s of i ( t ) based on c l a s s i c a l heat d i f f u s i o n theory i s being c a r r i e d out by Mr. George May of t h i s l a b o r a t o r y . ) The purpose of t h i s chapter i s to estimate w by deducing the parameters x and i from the a m p l i f i e r output pulse data. In order to deduce these parameters and t h e i r e r r o r s , t h e e f f e c t s of the t r a n s f e r f u n c t i o n of the j u n c t i o n - t r a n s m i s s i o n l i n e - a m p l i f i e r system must be taken i n t o account -146-( s e c t i o n B ) , and the e f f e c t s of noise i n the system must be allowed f o r ( s e c t i o n s C and D). Having determined x and i Q , one may e a s i l y f i n d the t o t a l amount of charge which a c t u a l l y tunneled. An estimate of AE then makes i t p o s s i b l e to give ( s e c t i o n E) a f i g u r e f o r w(exp), the average energy l o s s per t u n n e l i n g q u a s i p a r t i c l e , and w the average energy l o s s per q u a s i p a r t i c l e p a i r e x c i t e d . A d i s c u s s i o n ( s e c t i o n F) of problems p e r t a i n i n g to a rigorous c a l c u l a t i o n of AE concludes the chapter. B. D e r i v a t i o n of Detector-Transmission L i n e - A m p l i f i e r Transfer Function A fundamental concept of l i n e a r systems a n a l y s i s (see eg. Bohn, 1963) i s the system t r a n s f e r f u n c t i o n H(s) defined as H(s) = v o ( t ) / v ± ( t ) where v ^ ( t ) and v Q ( t ) are the input and output s i g n a l s r e s p e c t i v e l y , s=jui i s the complex frequency and t i s the time. The aim of t h i s s e c t i o n i s to d e r i v e H(s) from the known c h a r a c t e r i s t i c s of the d e t e c t o r — t r a n s m i s s i o n l i n e - a m p l i f i e r system and f i n d V Q ( t ) using an assumed form of v ^ ( t ) . 1. Small S i g n a l Equivalent C i r c u i t Figure 7-1 (a) i s a schematic of the pulse a m p l i f y i n g network and f i g u r e 7-1 (b) i s the corresponding' small s i g n a l equivalent c i r c u i t where the a m p l i f i e r i s now assumed n o i s e l e s s . Moving from l e f t to r i g h t , v_^(t) i s the input v o l t a g e s i g n a l obtained from the assumed form of i ( t ) and a p p l i c a t i o n of Thevenin's theorem to the c i r c u i t to the l e f t of P-P i n f i g u r e 7-1(a); thus v . ( t ) = i ( t ) Z, = i exp(-t/x) r (7-1) 1 d o —;—: — 1 + JtorCj I n c o r p o r a t i n g the r e s u l t s of chapter 5, (equation 5-3), the e f f e c t i v e input impedance of the " c o l d " transmission l i n e - f i l t e r box-p r e a m p l i f i e r may be represented approximately as a s e r i e s inductance L and r e s i s t a n c e R where L = 1.69 uH and R = 10.3ft (From f i g u r e 5-9, R = R^c i s seen to range from 9.5ft to 11.6ft over the system bandwidth but, i n order to keep the t r a n s f e r f u n c t i o n i n a t r a c t a b l e form, the mid-band value of R=10.3ft -148-has been chosen as a s u i t a b l e average.) The t r a n s f e r f u n c t i o n of t h i s part of the network i s then Y(s) - i e ( t ) / v . ( t ) (7-2) The CB t r a n s i s t o r p r e a m p l i f i e r transforms the input s i g n a l current i e ( t ) a t l o w impedance l e v e l i n t o output s i g n a l a i g at high impedance l e v e l such that v ' ( t ) = «i e(t)R L = i e ( t ) R ' (7-3) where R' i s taken to be frequency independent. I t was pointed out i n chapter 6 that the optimum a m p l i f i e r system bandwidth f o r pulse d e t e c t i o n corresponded to a 10-90% r i s e time of 100 nsec. To take t h i s time constant i n t o account, an i n t e g r a t i n g c i r c u i t i s i n c l u d e d where the a m p l i f i e r r i s e time x = R C . For t h i s p o r t i o n of the a o o c i r c u i t W(s) = v " ( t ) / v ' ( t ) (7-4) and, r e p r e s e n t i n g the gain of the a m p l i f y i n g system by G, v Q ( t ) = G v " ( t ) (7-5) The d e s i r e d t r a n s f e r f u n c t i o n H(s) may thus be obtained from equations 7-2 to 7-5 v ( t ) V (s) H(s) = ^ _ = GR' W(s) Y(s) = (7-6) where, by d e f i n i t i o n (see, eg. Bohn, 1963), V(s) i s the Laplace transform of v ( t ) w r i t t e n V(s) = jT[v ( t ) ] = v ( t ) e x p ( - s t ) d t o -149-Thus, V (s) = GR'V, (s) W (s) Y (s) o i and the d e s i r e d q u a n t i t y V Q ( t ) may be found by t a k i n g the inv e r s e transform v ( t ) = " ^ " V (s)]' A l l that remains now i s to determine V ^ ( s ) , W(s) and Y(s) from f i g u r e 7-1(b), 2. Laplace Transform Representation (a) V.(s) W r i t i n g the impedances i n terms of t h e i r transforms and a p p l y i n g the d e f i n i t i o n of V(s) to equation 7-1 y i e l d s v ( s ) J» . J L . J L (7-7) i v ' Cj s+a s+d where a = x \ d = (rC.) 3 (b) W(s) Insp e c t i o n of the i n t e g r a t i n g c i r c u i t shows W< S> " C T ' R A/C s • i+b ( 7 ' 8 ) o o o w i t h b = (x ) _ 1 = (R C a o o (c) Y(s) Somewhat more complicated a l g e b r a i c a l l y than the other terms, Y(s) i s given by -150-Y(s) = [R + Ls + r / ( l + s r C j ) ] " 1 = (s + d ) / L ( s 2 + a ns + a ) 1 o where a = (R + Ld)/L, a = (RC.d + 1)/LC T. 1 o j J (7-9) Combining equations 7-7 to 7-9 gives GR'bi V q ( S ) = 2. • -=— • 2 1 (7-10) C TL s+a s+b s + a, s + a J 1 o Inverse Transform To o b t a i n the inverse transform of 7-10 i t i s convenient = constant = A, wl f a c t o r the quadratic term such that t o set GR'bi o/CjL here A i s independent of time, and to V (s) o (s+a) (s+b) (s+a 1)(s+a 2) » 3 1 * 4 a o < O R C J « CJO> (s+a) (s+b) 2 2 ' (s+ct) + y a l * 4 a o ( O R C J > So) Both forms of V (s) must be considered as Crn, the value of C T which o J U J s a t i s f i e s a„ = 4 a , i s about 3300 pF which l i e s r i g h t i n the middle of 1 o the accepted range of values f o r Cj as determined from noise measurements i n chapter 6. The in v e r s e transforms, obtained from standard Laplace -151-Transform Tables [eg. E r d e l y i (1954) and McCollum and Brown (1965)], are as f o l l o w s : V o ^ _ _ exp (-at) exp(-bt)  A (a-b) ( a - a ^ (a-ct 2) (a-b) (b-c^) (b-«2) e x p ( - a 1 t ) e x p ( - a 2 t ) Cj * C J Q ^ al~ a2^ a~ al^ b"°'l^ (a-a 2) (b-a 2) ( a 1 - a 2 ) t * 0 V Q ( t ) exp (-bt) exp (-at)  ( a - b ) [ ( a - b ) 2 + Y 2 ] ( a - b ) [ ( a - a ) 2 + y2} Y [ ( a - b ) 2 + Y V [ ( a - a ) 2 + Y 2 r t _ - - l Y <_ - i a-b where o> = tan — 1— - tan a-a Y a = f l = R + Ld 2 2L Y = ( a o - 4 A ] V A = GR'bi /C L o J [7-11(a)] exp(-at)cos(Yt + j>) , C 5 C J ' JO t Z 0 [ 7 - l l ( b ) ] -152-4. J u s t i f i c a t i o n of Approximate Transmission Line Treatment In chapter 5, the j u s t i f i c a t i o n f o r r e p r e s e n t i n g the input impedance of the transmission l i n e terminated w i t h the p r e a m p l i f i e r as R^c + J X ^ c (equation 5-3), was based on the agreement between t h i s model and the measured impedance. To check the v a l i d i t y of t h i s approximation i n d e s c r i b i n g the impulse response of the system, square pulses from a pulse generator were developed i n a lumped constant model of the specimen and the output waveform from the p r e a m p l i f i e r was photographed on an o s c i l l o s c o p e f o r comparison to the t h e o r e t i c a l waveform. (The arrangement was s i m i l a r to f i g u r e 6-9 except that four d i f f e r e n t capacitances (C) were employed i n separate t e s t s . ) The t h e o r e t i c a l waveform was c a l c u l a t e d i n the manner described above using the average room temperature r e s i s t a n c e and i n d u c t i v e components of the l i n e (R = 13 P., L = 1.69uH, see f i g u r e 5-9). In a l l four cases, the agreement between the t h e o r e t i c a l and experimental output waveforms was s u f f i c i e n t l y good (and s u f f i c i e n t l y s e n s i t i v e to v a r i a t i o n s i n R or L) that t r e a t i n g the l i n e and i t s load by the more p r e c i s e , but a l g e b r a i c a l l y complicated s u c c e s s i v e - r e f l e c t i o n s technique (Bohn, 1963) seemed unwarranted. C. Determination of Current Pulse Parameter T The object of t h i s s e c t i o n i s to estimate T from a l e a s t squares f i t of the t h e o r e t i c a l pulse shape (equations 7-11) to the pulse shape observed on photographs (see eg. Figure 6-8). This s t a t i s t i c a l approach was made necessary by the noise superimposed on the s i g n a l pulses. Thus, f o l l o w i n g Orear (1958), i f x^ are the t h e o r e t i c a l values c h a r a c t e r i z i n g the pulse shape and x^ are the corresponding measured v a l u e s , assumed to be Gaussian d i s t r i b u t e d w i t h standard d e v i a t i o n a •; the l e a s t squares f u n c t i o n to be minimized i s N M* = t (x. - x . ) 2 / o . 2 (7-12) ... 1 X 1 i = l 2 I t turns out (Orear, 1958) that M* i s merely the x d i s t r i b u t i o n of (N-n) degrees of freedom where n i s the number of parameters solved f o r and N i s the number of experimental p o i n t s . The probable value of M* i s M* = N-n w i t h the standard d e v i a t i o n AM* = [ 2 ( N - n ) ] z . -153-( l ) E x t r a c t i o n of the x^ and o\ from Photographs Consider the t y p i c a l pulse photograph i n f i g u r e 6-8. Let t = t - 250 nsec be the time at which a given pulse reaches i t s maximum amplitude, then the pulse amplitude a ( t ) at N other times ( t ^ ) , f o r 0 $ t ^ £ 2 t Q , may be measured d i r e c t l y from the photograph ( f o r the same pulse) to form the r a t i o y i = a ( t i ) / a ( t o ) ' 1 = 1 " - ' « N Because the a m p l i f y i n g system i s l i n e a r , the pulse shape (as c h a r a c t e r i z e d by y^) depends only on the time constants of the system and not on the absolute a m p l i t u d e — p r o v i d e d of course that the amplitude i s s u f f i c i e n t l y greater than the rms noise v o l t a g e . Consequently, maximum amplitude pulses on 5 d i f f e r e n t photographs were chosen to be analyzed i n t h i s f a s h i o n to a r r i v e at mean values x. = - 7 T y.. , 1 = 1,...,7 i 5 ... i i J=l w i t h variance (Evans, 1955) 2 1 ? i — v 2 . a. = 7 - ) ( y . . - x . ) , i = l , . . . , 7 I 4 . , i i i J=l where y.. i s the measured r a t i o a ( t . ) / a ( t ) on the j t h pulse. 'IJ l o J v The remaining q u a n t i t i e s needed to c a l c u l a t e M* are the corresponding t h e o r e t i c a l r a t i o s c a l c u l a t e d from equations 7-11, x i = v o < t i ) / v o ( t o ) ' 1 = 1 7 which, as i t should be, i s independent of i Q . At t h i s j u n c t u r e , a word i s i n order concerning the p r e c i s i o n w i t h which the times t . are known. In p r a c t i c e , since the zero of time f o r l a p a r t i c u l a r pulse i s not c l e a r l y d i s c e r n i b l e from the photograph, the t ^ must be measured w i t h respect to t Q , which can only be estimated from photographs to l i e w i t h i n a range of values t Q + A t Q . Hence, -154-t = t + t ' ± At + At! where - t 4 t . t and At! i s the e r r o r i o i o i 0 1 0 1 i n v o l v e d i n marking o f f time i n t e r v a l s t ' To take t h i s u n c e r t a i n t y i n t o X • account, M* must be c a l c u l a t e d over a range of t . o (2) M* C a l c u l a t i o n I n s p e c t i o n of equations 7-11 r e v e a l s that x i s the only unknown v a r i a b l e i n the x^ but that the value of x which minimizes M* w i l l depend upon the p a r t i c u l a r values chosen f o r C T and t . I t was shown i n J o s e c t i o n B that 1500 * Cj S? 4500 pF and t Q may be estimated from f i g u r e 6-8, by e x t r a p o l a t i o n of the maximum amplitude pulse to the ac ground l e v e l , to l i e i n the range 245 $ t $ 265 nsec o A computer program was w r i t t e n to evaluate equations 7-11 over the s t i p u l a t e d ranges of C and t y i e l d i n g the t y p i c a l p l o t of f i g u r e J o 7-2. As a measure of the u n c e r t a i n t y i n the s o - c a l l e d most probable x — i . e . t h a t value which minimizes M* f o r a p a r t i c u l a r C T and t — i t i s convenient, J o although a r b i t r a r y , to accept a l l x which f a l l w i t h i n a 90% confidence l e v e l . In t h i s context "90% confidence l e v e l " means that i f the experiment were repeated under e x a c t l y the same c o n d i t i o n s , x = x ( C j , t Q ) i s such that the p r o b a b i l i t y i s 90% that the newly measured x. would give M* £ Q where Q=10.6 2 1 i s the value f o r Q f o r a x d i s t r i b u t i o n w i t h N-n = 7-1 = 6 degrees of freedom 2 (see eg. x t a b l e s , Handbook of Physics and Chemistry). Thus, f o r example (see f i g u r e 7-2) a 90% confidence l i m i t on x = *x (C = 3000 pF, t = 260 nsec) J o gives 123 £ x $ 167 nsec w i t h a most probable value of roughly 142 nsec. Figure 7-3 summarizes t h e . r e s u l t s of the M* c a l c u l a t i o n s . The region shown i n heavy b l a c k o u t l i n e i s the 790% confidence volume" i n Cj, t Q , x space compatible w i t h the u n c e r t a i n t y i n t Q and Cj described above; the surface of most probable values of x l i e s roughly midway between the upper and lower extremes. C l e a r l y , i t i s not meaningful to speak of a unique x -155--157-as g i v i n g the best f i t of the t h e o r e t i c a l to measured pulse shapes. For purposes of e s t i m a t i n g i Q , however, i t i s convenient to take a c e n t r a l value (shown i n f i g u r e 7-3) of T = T(3000 pF, 255 nsec) - 138 nsec. (3) V a l i d i t y of the Assumed Input Current Pulse The values of x chosen by the l e a s t squares f i t are based on the assumption that the current pulse i ( t ) = i exp (-t/x ) and the measured t r a n s f e r f u n c t i o n c o n s t i t u t e a s u f f i c i e n t b a s i s from which to c a l c u l a t e the form of the observed pulse. Some informat i o n about the v a l i d i t y of t h i s assumption may a l s o be obtained from the M* c a l c u l a t i o n s . In p a r t i c u l a r , i f ( C j , t ^ , x') minimizes M* then M* (C' t ' , x*) should l i e i n the range M* ±AM* = 6 ± 3.5 i f the t h e o r e t i c a l J o pulse shape i s adequate (Braddick, 1966) whereas M*(C', t ' , x') < 2.5 would J o imply x i s not determined by the measurement and M*(C', t ' , x') > 9.5 J o would imply more parameters should have been used. The p l o t of f i g u r e 7-4 shows that the c a l c u l a t e d M* (minimum) do indeed f a l l g e n e r a l l y w i t h i n the d e s i r e d range i n d i c a t i n g not only that x i s meaningful but that a higher order parameter form of i ( t ) i s not j u s t i f i e d w i t h the present data. D. Estimate of Current Pulse Amplitude ( i Q ) A r e s u l t which w i l l be needed l a t e r i s the current pulse amplitude i corresponding (for reasons to be discussed i n s e c t i o n E f o l l o w i n g ) to the maximum amplitude observed pulses. The method of obta i n i n g t h i s parameter from the photographed pulses ( f i g u r e 6-8) and the a m p l i f i e r system s e n s i t i v i t y c a l i b r a t i o n i s o u t l i n e d below. Equations 7-11 cont a i n i i m p l i c i t l y i n that GR' i b v (t) A = 2. = -2 :•: CjL f ( t ) where v (t ) i s the peak pulse amplitude measured p h o t o g r a p h i c a l l y from o o the o s c i l l o s c o p e ( i n output v o l t s V(output)) at time t Q and f ( t ) i s the r i g h t hand side of equations 7-11 evaluated at t = t . A l l that remains i s to f i n d the a m p l i f i e r system t r a n s f e r constant GR'. This q u a n t i t y i s determined, as shown i n the next two paragraphs, from the a m p l i f i e r s e n s i t i v i t y c a l i b r a t i o n described i n chapter 6 where very -158-6 3 e • H c • H 16 14 12k 100 120 O 1500 pF • 2000 X 3000 © 4000 & 4500 Subscripts 1-5 i n d i c a t e t = 245-o 265 nsec r e s p e c t i v e l y 180 140 160 i ( n s e c ) Figure 7-4: M*(minimum') v/s Re l a x a t i o n time ( T ) 200 - 1 5 9 -f a s t r i s i n g (step) current pulses of known amplitude were i n j e c t e d i n t o a lumped constant model of the small s i g n a l e q u i v a l e n t c i r c u i t of the specimen (see f i g u r e 6 - 9 ) and the output pulse v o c ( t ) was recorded p h o t o g r a p h i c a l l y on an o s c i l l o s c o p e . (The gain and bandwidth s e t t i n g s of the a m p l i f i e r s were of course unchanged from the s i g n a l pulse measurements.) 1. Transfer Function f o r Input Current Step Pulse The f a s t r i s i n g input current pulse may be represented approximately by i c ( t ) = i ^ 1 ^ 1 - ) where u( t ) = 0 , t < 0 k , t = 0 1 , t > 0 and i i s the measured maximum amplitude input c a l i b r a t i n g c u r r e n t . oc . , S u b s t i t u t i n g i - c ( t ) i n t o equation 7 - 1 and transforming gives T T / \ 1 o c 1 1 V i s ) = • — • i c v ' C T s s+d Jc which i s the same as equation 7 - 7 w i t h a = 0 . ( C J c = 3 5 0 0 pF i s the "dummy" specimen value of Cj.) The forms of both W(s) and Y(s) are unchanged from the d i s c u s s i o n i n s e c t i o n B so that the appropriate t r a n s f e r f u n c t i o n i s given by 7 - 1 1 ( b ) — s i n c e C J c > C J Q — w i t h a = 0 . Thus V o c ( t ) = _ exp(-bt) 1 B b [ ( a - b ) 2 + Y 2 ] b ( a 2 + y 2 ) exp(-ctt) cos (yt + <ft) Yt(a-b) Z + y J 2 (a + Y V ( 7 - 1 3 ) where and , . —1 Y ^ - 1 a-b A = tan -1- - tan a Y GR'i b B = r T -160-Q u a n t i t i e s a and y, as before, are given i n terms of a^, a^ and d which must be s u i t a b l y r e d e f i n e d f o r the lumped constant model values of C = C T , r = r = 10 Jl and R = R = 12 Q. (R i s the midband room temperature J J c c c c value of the t r a n s m i s s i o n l i n e - p r e a m p l i f i e r s e r i e s r e s i s t a n c e ; L and b are unchanged.) 2. E v a l u a t i o n Of GR' From s e c t i o n B, b = 10 sec and the values of a and y 7 - 1 7 - 1 p e r t i n e n t to 7-13 are about 1.8 x 10 sec and 0.74 x 10 sec r e s p e c t i v e l y . These v a l u e s , which make the denominators of a l l three terms of the equation the same order of magnitude, combined w i t h the f a c t that the times of i n t e r e s t f o r measured v ( t ) are the order of 1 usee, d r a s t i c a l l y —6 s i m p l i f y equation 7-13. Hence v (t - 10 sec)-* v (t = °°) or n oc oc v (t ) oc 0 0 2 2 b(a + Y ) b a Q Remembering the d e f i n i t i o n s of B and a Q given e a r l i e r y i e l d s v (t ) R + r GR' = , _ oc °° c c i r oc c From chapter 6, v (t ) / i = 1 V/3.5 uA so that oc 0 0 oc , .11 . 22 _ - , 0 Q V(output) GR' = YJ ^ • — = 0.628 uA(input) 3. E v a l u a t i o n of i o From the d e f i n i t i o n of A given e a r l i e r , the s i g n a l current pulse amplitude may now be w r i t t e n i n terms of known q u a n t i t i e s : v (t ) C. L o o J 1 = ° f ( t ) GR'b o Table 7-1 contains the values of i obtained f o r the maximum amplitude pulse o -161-of f i g u r e 6-8 (corresponding to v (t ) - 2.35 V(output)) and f ( t ) and C o o o J evaluated f o r extreme, values of C , t , T c o n s i s t e n t w i t h f i g u r e 7-3. J O CjfeF) t (nsec) o x (nsec) i o ( y A ) 1500 245 153 21.2 1500 245 120 24.8 2600 265 160 20.6 3000 265 171 19.7 3500 245 114 24.9 3900 249 110 25.5 4000 265 169 19.6 4500 250 105 26.2 4500 265 160 20.3 *3000 255 138 22.3 Table 7-1: Estimate of i f o r maximum- Amplitude Pulse. * C e n t r a l values of C T, t , T J o The u n c e r t a i n t y i n C T, t and x i s seen to lead to a J o spread i n i so that f o r the maximum amplitude observed output pulse i n ques t i o n , i may be s a i d to l i e i n the 90% confidence i n t e r v a l 20 £ i $ 26uA w i t h a most probable value of 22pA. E. Energy Loss per Q u a s i p a r t i c l e 1. Number of Q u a s i p a r t i c l e s Produced : Once i i s known, the t o t a l number N of e x c i t e d q u a s i -o o p a r t i c l e p a i r s produced by the alpha p a r t i c l e may be found by r e c a l l i n g the expression <n(t)>, the expected number of q u a s i p a r t i c l e s tunneling i n time t , from equation 3-15. S e t t i n g oo i exp (-t/x)dt (7-14) <n( oo )> = w gives N = i /WTe o o T (7-15) -162-where i ( t ) = i exp (-t/x) i s the form assumed f o r the current pulse superimposed on the ambient tunneling current. Before going on to evaluate 7-15 i t i s necessary to estimate W and discuss the s i g n i f i c a n c e of W = W-W . i R T (a) Tunneling P r o b a b i l i t y From equation 3-17, WT = G(e 2 -N (0)^ X A ) _ 1 where G i s the low temperature, normal s t a t e conductance of the specimen, e i s the e l e c t r o n i c charge, ^ ( 0 ) i s the normal s t a t e d e n s i t y of f r e e e l e c t r o n s t a t e s per u n i t volume near the Fermi s u r f a c e , X i s the f i l m t hickness and A the j u n c t i o n overlap area. For the sample (J-5) w i t h which t h i s a n a l y s i s i s concerned, G = (R n (4.2 K ) ) " 1 = (.077ft ) _ 1 X = 2030 ± 280 A (7-16) -4 2 A = 7.1 x 10 cm Using equations 3-18 to 3-23, w i t h the number of f r e e e l e c t r o n s per atom n = 1.1 as derived from anomalous s k i n e f f e c t measurements (Wilson, 1965), 22 3 one f i n d s N f = 4.05 x 10 els/cm , E (Sn) = .4.4 eV making N (0) = 1.4 x 1 0 2 2 eV 1 cm" 3 (7-17) m The value f i n a l l y obtained f o r the tunneling p r o b a b i l i t y i s t h e r e f o r e WT = 4.1 x 1 0 5 s e c " 1 (7-18) (b) Recombination Process The p o p u l a t i o n of excess q u a s i p a r t i c l e s generated by the alpha p a r t i c l e i s reduced as these q u a s i p a r t i c l e s recombine w i t h others to form Cooper p a i r s and r e j o i n the s u p e r f l u i d (see chapter 2). This recombination takes place p r i m a r i l y by emission of phonons of energy -163-ntoi 2 A (T) r a t h e r than photons, as the r a t e of the former exceeds that of the l a t t e r by a f a c t o r of about 10^ (Rothwarf and Cohen, 1963). Subsequently these phonons, i f not l o s t from the energy range tuo£ 2A(T) by d i f f u s i o n to the s u b s t r a t e (Jones and Pennebaker, 1963) or the s u p e r f l u i d helium (Wilks, 1967), may create a d d i t i o n a l q u a s i p a r t i c l e s by Cooper p a i r e x c i t a t i o n so that e v e n t u a l l y a dynamic e q u i l i b r i u m i s r e s t o r e d i n which p a i r s are c o n t i n u a l l y being d i s s o c i a t e d by thermal phonons of energy tuo£ 2A(T) and q u a s i p a r t i c l e s are c o n t i n u a l l y recombining i n t o p a i r s v i a the emission of phonons. Rothwarf and Taylor (1967) have shown that f o r quas i p a r t -i c l e s i n j e c t e d i n t o a superconducting f i l m at a constant r a t e , the observed recombination r a t e W' i s dominated by the r a t e W = x at which phonons R 7 y y _x are l o s t from the range tiu> 2A(T) and not by the r a t e W = x at which R R i n d i v i d u a l q u a s i p a r t i c l e s recombine i n t o Cooper p a i r s . From these c o n s i d e r a t i o n s , i t i s apparent that the recombination r a t e W' deduced from the observed c h a r a c t e r i s t i c decay rat e -1 W = WT + WR (WR, W ) = x i s not simply that c a l c u l a t e d from the theory of q u a s i p a r t i c l e recombination (W , see eg. Woo and Abrahams, 1968) but i s R a l s o a f u n c t i o n of the phonon l o s s processes which go according to W^ . (c) E f f e c t of the Energy Gap Parameter I t should perhaps be noted that as the temperature d i s t r i b u t i o n i n the tunnel j u n c t i o n s r e l a x e s back to the e q u i l i b r i u m s i t u a t i o n , the magnitude of the energy gap 2A(T) increases and w i l l a f f e c t the manner i n which the excess q u a s i p a r t i c l e population decays. Some a p p r e c i a t i o n f o r the i n f l u e n c e of A(T) may be. gained from the f o l l o w i n g approximate c a l c u l a t i o n s . For a superconductor at temperature T, the thermal e q u i l i b r i u m d e n s i t y of q u a s i p a r t i c l e s n(T) i s given by equation 3-2: n(T) = N (0) A(T) K... ($A(T)), T < T (3-2) m 1 c « N (0) A(T) exp^ACO/k^T) (3- 3) m 15 (Here, N (0) i s the normal s t a t e d e n s i t y of f r e e e l e c t r o n s t a t e s per u n i t -164-volume near the Fermi surface (E = 0 ) , A(T) i s the energy gap parameter and kg i s Boltzmann's constant. Fpr convenience K ^ ( x ) , the f i r s t order modified Bessel f u n c t i o n of the second k i n d , i s approximated as ( c f . p.55) K^(x) = oc(x) exp (-x) = exp(-x) as o(x) v a r i e s slowly w i t h x and i s the order of one.) Thus, as the temperature i n the tunnel j u n c t i o n decreases to the e q u i l i b r i u m l e v e l (T ), the change i n the excess q u a s i p a r t i c l e d e n s i t y 6n = n(T) - n ( T Q ) , i g n o r i n g the recombination phonons, would be given by ( f o r T < T < T ) o c 3(6n) 9T N (0) exp m A(T) k B T 1- A(T) V 9A(T) A 2(T) 9 T k T 2 B From chapter 2, i t may be r e c a l l e d that f o r T < { T , 2A(T)=3.5 k„T c B e so that i n t h i s range of temperatures, 3(6n)/9T would be governed by a Boltzmann f a c t o r exp (-1.75 TjT) s i m i l a r to that governing the den s i t y of e x c i t e d c a r r i e r s i n a semiconductor. Near T .however, A(T) v a r i e s as i c (1 - T/T^) 2 r e s u l t i n g i n considerable departure from the s t r a i g h t f o r w a r d e x p o n e n t i a l decay. (d) E v a l u a t i o n of N o The.number of q u a s i p a r t i c l e p a i r s i n i t i a l l y created by the passage of an alpha p a r t i c l e may now be c a l c u l a t e d from equation 7-15 and the estimate that = 4.1 x 10^ sec 1 (equation 7-18). I t was shown i n s e c t i o n D that the most probable value of i corresponding to a maximum amplitude observed pulse was 22yA; t h e r e f o r e ' N = i /W^ e = 3.4 x 10 o o T (7-19) 2. Energy Loss Corresponding to Maximum Amplitude Pulse The remaining q u a n t i t y needed to c a l c u l a t e w = A E / N q i s A E , the t o t a l amount of energy deposited i n the superconducting tunnel j u n c t i o n by the alpha p a r t i c l e . I d e a l l y , f o r purposes of nuclear spectroscopy, A E should be equal to the t o t a l alpha p a r t i c l e energy E^ = 5.1 MeV but t h i s c o n d i t i o n could not be achieved w i t h the t h i n f i l m j u n c t i o n s used. -165-The alpha p a r t i c l e path length through the f i l m of t o t a l t h ickness 2X i s x = 2A /cos6 where 6 i s the angle of incidence. From f i g u r e 6-7 the most oblique angle of incidence was 80° which corresponds -4 to a path length of 2.3 x 10 cm. Thus, w i t h the use of f i g u r e 3-3, the maximum amount of energy l o s t d i r e c t l y AE ( d i r e c t ) i n the t i n f i l m s i s seen to be only about 0.5 MeV. The remainder of E^, which was deposited i n the glass s u b s t r a t e as the alpha p a r t i c l e comes to r e s t , may not however, be neglected i n determining w. No attempt was made to thermally decouple the f i l m s from the sub s t r a t e w i t h the r e s u l t t h a t , perhaps as much as one h a l f of the energy deposited i n the glass may have d i f f u s e d back i n t o the t i n s u f f i c i e n t l y r a p i d l y to c o n t r i b u t e to the pulse. For purposes of the present c a l c u l a t i o n t h e r e f o r e , AE = A E ( d i r e c t ) + h ( E a - A E ( d i r e c t ) ) (7-20) where 0 $ h $ 0.5 making AE $ 2.75 MeV. The r e l a t i o n of equation 7-20 to the observed pulse spectrum ( f i g . 6-11) i s considered i n s e c t i o n F of t h i s chapter. 3. Estimate of w Taking AE £ 2.75 MeV as the t o t a l energy deposited i n the j u n c t i o n corresponding to the maximum amplitude pulse and the r e s u l t f o r N q from equation 7-19 gives an upper l i m i t w ( t i n ) * ~ = 8.2 x 10~ 3eV/q.p. p a i r (7-21) o which i s roughly 7.5 times the value of the energy gap i n superconducting t i n at 1.2 K (see t a b l e 3-2). P u r e l y a phenomenological q u a n t i t y , w i s the average amount of energy which must be deposited i n a superconductor to create an e x c i t e d q u a s i p a r t i c l e p a i r ; i t i s not to be confused w i t h E = 2A(T) derived from microscopic theory which i s the minimum amount of g energy r e q u i r e d to break up a Cooper p a i r at temperature T. Another q u a n t i t y of i n t e r e s t . f o r comparing the energy r e s o l u t i o n of the tunnel j u n c t i o n detector to conventional semiconductor -166-spectrometers i s w(exp), the energy l o s s per q u a s i p a r t i c l e which a c t u a l l y  tunneled. Thus, from equation 7-15 w(exp) $ = — • — - .145 eV (7-22) <n(°°)> N Wm  v ' o T F. Energy Loss and D i f f u s i o n This s e c t i o n o u t l i n e s some co n s i d e r a t i o n s r e l e v a n t to the c a l c u l a t i o n of the manner i n which the energy l o s t by the alpha p a r t i c l e d i f f u s e s through the j u n c t i o n and substrate g i v i n g r i s e to a heat pulse. 1. Energy Loss Processes The primary means by which the alpha p a r t i c l e loses energy i s by the i o n i z a t i o n and e x c i t a t i o n of the e l e c t r o n s a s s o c i a t e d w i t h the atoms along i t s path. (In metals, semiconductors and i n s u l a t o r s , the p a r t i c l e gives up only about 0.1% of i t s energy d i r e c t l y to the l a t t i c e — c r e a t i n g phonons and d i s l o c a t i o n s — w i t h the remainder going to the e l e c t r o n s ( S e i t z and Koehler, 1956).) As the energy t r a n s f e r at each i o n i z i n g c o l l i s i o n i s g e n e r a l l y a small f r a c t i o n (approximately the order of 10 ) of the p a r t i c l e energy (Dearnaley and Northrop, 1966), the d e f l e c t i o n s from a s t r a i g h t l i n e path are s l i g h t . The p a r t i c l e comes to r e s t i n about -12 -11 10 to 10 sec w i t h roughly 90% of i t s path (see f i g u r e 7-5) being i n the g l a s s s u b s t r a t e making i t necessary the r e f o r e to consider the energy l o s s processes i n both metals and i n s u l a t o r s . In a metal, the primary e l e c t r o n e x c i t a t i o n s are s u f f i c i e n t l y e n e r g e t i c that the usual argument (Jones, 1956) s t a t i n g that e l e c t r o n -e l e c t r o n i n t e r a c t i o n s are n e g l i g i b l e compared w i t h electron-phonon i n t e r -a c t i o n s does not h o l d ; f i r s t because the It space r e s t r i c t i o n s on s c a t t e r i n g due to the e xistence of the Fermi sphere are g r e a t l y reduced and second because the phonon d e n s i t y i s small as a r e s u l t of the f a c t o r of 1000 i n r e l a t i v e e x c i t a t i o n p r o b a b i l i t i e s r e f e r r e d to above. (The argument given by Jones depends on there being a s t a t e of thermal e q u i l i b r i u m i n which energy i s shared e q u a l l y between e l e c t r o n and l a t t i c e degrees of freedom.) Therefore the coupling between e l e c t r o n and phonon e x c i t a t i o n s i s much weaker than e i t h e r the e l e c t r o n - e l e c t r o n or phonon-phonon coup l i n g s , so that the two types of e x c i t a t i o n may be considered to thermalize -167-Figure 7-5: P a r t i c l e Track Geometry Figure 7-6: A E ( d i r e c t ) (MeV) T h e o r e t i c a l Pulse Height Spectrum, h=0 (Un-normalized) -168--12 -11 i n d i v i d u a l l y i n times of the order of 10 - 10 sec (Dearnaley and Northrop, 1966) w i t h the e l e c t r o n temperature approximately 200 times greater than the l a t t i c e temperature. F i n a l l y , the weak electron-phonon co u p l i n g causes the two types of e x c i t a t i o n to r e l a x to a common -9 temperature i n about 10 sec ( S e i t z and Koehler, 1956). In an i n s u l a t o r l i k e g l a s s , the e l e c t r o n s may not be considered f r e e . In t h i s case the time r e q u i r e d f o r the e l e c t r o n s and l a t t i c e to r e l a x -12 to some common temperature i s probably the order of 10 sec (as estimated f o r semiconductors by Dearnaley and Northrop, 1966). I t seems reasonable to assume th e r e f o r e that w i t h i n a time the -9 order of 10 sec i t i s meaningful to speak of a l o c a l e l e c t r o n - l a t t i c e temperature, i n a narrow c y l i n d e r c o - a x i a l w i t h the p a r t i c l e t r a c k , which i s i n excess of the j u n c t i o n e q u i l i b r i u m temperature. The next paragraph considers the manner i n which t h i s heat energy d i f f u s e s outward from the t r a c k . 2. Energy Transfer The heat energy i s subsequently transported away from the p a r t i c l e t r a c k by e l e c t r o n and phonon e x c i t a t i o n s i n a way which may be described by c l a s s i c a l heat d i f f u s i o n theory ( C r i t t e n d e n , 1968). A complete a n a l y t i c s o l u t i o n of the d i f f u s i o n equation pc(9T/8t) = v>KVT where p i s the d e n s i t y , c the s p e c i f i c heat and K the thermal c o n d u c t i v i t y > i s precluded by the f o l l o w i n g c o m p l i c a t i n g f a c t o r s : (a) Temperature Dependence of c and K Because of the wide range of temperatures o c c u r r i n g i n the v i c i n i t y of the t r a c k , the temperature dependence of the s p e c i f i c heat and thermal c o n d u c t i v i t y may not be s a f e l y neglected. (For example, published data f o r glass (Johnson, 1960) i n d i c a t e a v a r i a t i o n i n K and c of 2 and 5 orders of magnitude r e s p e c t i v e l y over the temperature range from 1 K to 300 K.) With c = c(T) and K = K(T), however, the d i f f u s i o n equation i s n o n - l i n e a r making numerical methods imperative. (b) Track Geometry The d i s t a n c e which the alpha p a r t i c l e penetrates the glass substrate i s given by R where R s a t i s f i e s -169-fR AE* = E - AE ( d i r e c t ) = a a a ft) dx. Here, x i s the d i s t a n c e measured along the t r a c k from the t i n - g l a s s i n t e r f a c e , E^ i s the i n i t i a l alpha p a r t i c l e energy (5.1 MeV) and A E ( d i r e c t ) i s the energy l o s t by the alpha p a r t i c l e i n the t i n f i l m s . An approximate estimate of R may be obtained by tak i n g (dE^ /dx) = constant i n glass so that R = AE^ /OE^ /3x). The r a t e of alpha p a r t i c l e energy l o s s (8E /8x), estimated from the atomic stopping cross s e c t i o n s of the a 3 c o n s t i t u e n t s of soda lime g l a s s , i s found to be roughly 1.6 x 10 MeV/cm. Now AE^ depends only s l i g h t l y on the angle at which the alpha p a r t i c l e s s t r i k e the j u n c t i o n (see f i g u r e 6-7) w i t h the r e s u l t that R ranges from -3 2.9 to 3.1 x 10 cm which, f o r a l l p r a c t i c a l purposes, may be taken as -3 R = 3 x 10 cm independent of angle of i n c i d e n c e . Consequently, as sketched i n f i g u r e 7-5, about 90% of the t r a c k e x i s t s i n the s u b s t r a t e . (c) Boundary Conditions With reference to f i g u r e 7^5, i t i s c l e a r that the heat 1 f l o w across two i n t e r f a c e s should be considered. Although the a c o u s t i c impedances of t i n and glass are q u i t e c l o s e (American I n s t i t u t e of Physics Handbook), the r e f l e c t i o n c o - e f f i c i e n t f o r phonons at the t i n - g l a s s i n t e r f a c e i s s u f f i c i e n t l y l a r g e that a f i n i t e thermal impedance e x i s t s . L i t t l e (1959) has c a l c u l a t e d that at low temperatures the heat flow between pyrex and copper i s dQ/dt = 3.7 x l O 5 ^ 4 - T 2 4 ) erg s e c - 1 c m " 2 which, i n the l i m i t of small temperature d i f f e r e n c e s becomes 3 -2 dQ/dt = 1.5 T AT W cm where AT i s the temperature d i s c o n t i n u i t y at the -3 2 -1 i n t e r f a c e . This corresponds then to a thermal impedance Z^ = .67 T K cm W which i s presumably not f a r d i f f e r e n t from a tin-soda lime g l a s s boundary. The s i t u a t i o n at the t i n - s u p e r f l u i d helium (He**) i n t e r f a c e i s more d i f f i c u l t to handle t h e o r e t i c a l l y as the observed thermal impedance cannot be accounted f o r by c o n s i d e r a t i o n of the l a r g e a c o u s t i c a l mismatch alone. Other f a c t o r s which have been suggested ( L i t t l e , 1961) are the coupling of the conduction e l e c t r o n s i n the metal to the t o t a l l y -170-r e f l e c t e d phonons of the l i q u i d and the adsorption of a t h i n l a y e r of helium atoms on the metal i n t e r f a c e . This l a y e r of s o l i d helium would provide an a c o u s t i c match between the l i q u i d and metal because of i t s intermediate a c o u s t i c impedance. Experimental measurements (Gittleman and Bozowski, 1962) f o r the normal t i n - l i q u i d helium i n t e r f a c e give -3 2 - 1 Z T = 5.48 T K cm W . The value of Z^ f o r a superconducting t i n - l i q u i d helium i n t e r f a c e was some 10.6% greater. From t h i s b r i e f d i s c u s s i o n , i t i s apparent that the problem of the heat pulse d i f f u s i o n through the t i n - g l a s s system i s s u f f i c i e n t l y complex to merit d e t a i l e d c o n s i d e r a t i o n and, as mentioned e a r l i e r , such work i s being continued i n t h i s l a b o r a t o r y . 3. Evidence f o r Heat C o n t r i b u t i o n from Substrate The observed pulse height spectrum i n f i g u r e 6-11 contains some inf o r m a t i o n concerning the heat fed back i n t o the j u n c t i o n from the s u b s t r a t e which w i l l now be considered. I t was s t a t e d e a r l i e r that the t o t a l energy c o n t r i b u t i n g to the pulse should be of the form AE * AE (direct)+h(E - A E ( d i r e c t ) ) (7-20) * a where h i s an experimental parameter assumed to l i e i n the range 0 £ h £ 0.5. The upper l i m i t of h = 0.5 was i n f e r r e d from a rough c a l c u l a t i o n which shows that the minimum time required f o r phonons, w i t h t h e i r i n i t i a l v e l o c i t y components d i r e c t e d away from the g l a s s - t i n i n t e r f a c e , to be p e r f e c t l y r e f l e c t e d from the f a r s i d e of the glass and —3 3 —1 —6 r e t u r n to the i n t e r f a c e i s t = 2s/v = 2x10 m/2xl0 m sec = 10 sec. P (Here, v i s the propagation v e l o c i t y (Klemens, 1951) and the phonon mean fr e e path f o r glass at low temperatures, which i s known to be "long" (Ziman, 1965 and K i t t e l , 1956), i s assumed to be of the order of the s u b s t r a t e thickness s.) From f i g u r e 6-8, i t can be seen that the a c t u a l —6 heat pulse must be l e s s than 0.5 x 10 sec i n width so i t i s not expected that the " r e f l e c t e d " heat pu l s e , corresponding to roughly one-half of the o r i g i n a l energy i n p u t , w i l l c o n t r i b u t e s i g n i f i c a n t l y to the observed pulse. -171-To see the e f f e c t of the heat c o n t r i b u t i o n from the s u b s t r a t e , i t i s convenient to consider two l i m i t i n g cases: h = 0 and h = 0.5. I f h = 0, which i m p l i e s there i s no feedback of heat energy from the s u b s t r a t e , the pulse height spectrum should be of the form sketched i n f i g u r e 7-6, f o r the energy A E ( d i r e c t ) deposited i n the f i l m i s i n v e r s e l y p r o p o r t i o n a l to cos 6 where 6 i s the angle of incidence. Contrary to what was observed ( f i g u r e 6-11), t h i s spectrum p r e d i c t s that lower energy pulses are very much more probable (up to 2 order of magnitude) than higher energy ones. I t must be concluded therefore that heat d i f f u s i n g back from the s u b s t r a t e , the amount of which i s l e s s s e n s i t i v e to 9, was r e s p o n s i b l e to a l a r g e extent f o r the observed pulse s . Thus h = 0 may be r u l e d out. On the other hand i f h = 0.5 and i s independent of 0 so that the j u n c t i o n acts mainly as a thermometer s i t t i n g on top of the g l a s s , the p r e d i c t e d pulse height spectrum would be a " s p i k e " centered around the 2.3 MeV po i n t on the energy a x i s — f a r removed from the 0.5 MeV pulses due to A E ( d i r e c t ) of f i g u r e 7-6. Such a d i s t r i b u t i o n i s incompatible w i t h the observed spectrum which the r e f o r e r u l e s out the simple case of h = 0.5 f o r a l l 9. I n t u i t i v e l y , i t i s expected that h w i l l depend upon 0, perhaps as the p r o j e c t i o n ( s i n 0) of the t r a c k on the g l a s s - t i n i n t e r f a c e so that from these simple c o n s i d e r a t i o n s , h = h (0) and 0 < h (0) $ 0.5. I t might be argued that the minimum value of h would be the one y i e l d i n g AE such that w = AE/N Q = 2A(T) where A(T) i s the energy gap parameter at the bath temperature T. This c r i t e r i o n i s d i f f i c u l t to j u s t i f y at present as a d e t a i l e d theory r e l a t i n g secondary i o n i z a t i o n and microplasma phenomena i n superconductors to an energy gap model i s not yet a v a i l a b l e . Although more s o p h i s t i c a t e d a n a l y s i s might be used to e s t a b l i s h narrower l i m i t s on h, i t was f e l t such e f f o r t s were u n j u s t i f i e d because of the p a u c i t y of data p r e s e n t l y a v a i l a b l e . In a d d i t i o n , t h i s problem of heat feedback w i l l be obviated when the experiment i s repeated i n t h i s l a b o r a t o r y by Dr. B. White and Mr. G. May with a phonon b a r r i e r between the su b s t r a t e and tunnel j u n c t i o n . -172-G. Summary For convenience, a resume''of the important r e s u l t s of t h i s chapter i s given below. I t was shown that the input impedance of the transmission l i n e -p r e a m p l i f i e r system could be w r i t t e n as Z = R + jcoL where R = 10.3ft , L = 1.69uH and OJ i s the angular frequency. From the small s i g n a l equivalent c i r c u i t of the complete system, the output p u l s e , corresponding to an input current pulse of the assumed form i ( t ) = i e x p ( - t / t ) was derived using Laplace Transform techniques. By least-squares f i t t i n g the t h e o r e t i c a l output pulse shape to that observed on photographs, the value of T was deduced to be x = 138 ± 33 nsec where the e r r o r s are those corresponding to a 90% confidence l i m i t . Knowing the a m p l i f i e r system t r a n s f e r constant GR' from a s e n s i t i v i t y c a l i b r a t i o n , one could then c a l c u l a t e 20 $ i $ 26uA f o r maximum amplitude p u l s e s . The t o t a l number of q u a s i p a r t i c l e p a i r s produced by the alpha p a r t i c l e N was found to be o N o = W ( 7 _ 1 5 ) where = 4.1 x 10^ sec 1 i s the estimated tunneling p r o b a b i l i t y per excess q u a s i p a r t i c l e per second and e i s the e l e c t r o n i c charge. Taking i n t o account that up to one-half of the energy deposited i n the substrate by the alpha p a r t i c l e might have c o n t r i b u t e d to the observed p u l s e s , the t o t a l energy corresponding to the maximum amplitude pulses was estimated to be A E $ 2.75 MeV. The average energy l o s s per q u a s i p a r t i c l e p a i r i s then w = A E / N q $ 8.2 x 10~ eV/q.p.pair (7-21) -173-Another q u a n t i t y of i n t e r e s t , the average energy l o s s per q u a s i p a r t i c l e ( q u a s i e l e c t r o n or quasihole) which a c t u a l l y tunneled w(exp), was found to be w(exp) $ 0.145 eV (7-22) Concluding the chapter i s a short d i s c u s s i o n of problems concerning d e t a i l e d c a l c u l a t i o n of the way i n which the heat energy deposited along the alpha p a r t i c l e t r a c k d i f f u s e s through the g l a s s - t i n - l i q u i d helium system. -174-CHAPTER 8 CONCLUSIONS This chapter summarizes the important f i n d i n g s from the present work ( s e c t i o n A) and attempts to place them i n p e r s p e c t i v e through e v a l u a t i o n of the superconducting t h i n f i l m tunnel j u n c t i o n as a nuclear spectrometer ( s e c t i o n B). S e c t i o n C o u t l i n e s the d i r e c t i o n i n which f u t u r e work i n t h i s area should proceed to improve both the understanding and performance of t h i s device. A. Major Results The experiment has demonstrated that the superconducting t h i n f i l m tunnel j u n c t i o n may be used as a detector of charged p a r t i c l e s . In p a r t i c u l a r , i t has shown that the pulse of t u n n e l i n g current r e s u l t i n g from the bombardment of the j u n c t i o n by 5.1 MeV alpha p a r t i c l e s was of s u f f i c i e n t amplitude and d u r a t i o n to be r e a d i l y observable. F a b r i c a t i o n refinements r e q u i r e d to improve the tunnel j u n c t i o n c h a r a c t e r i s t i c s and so surpass the measured s i g n a l to noise r a t i o of 19 are, as o u t l i n e d i n s e c t i o n C, w i t h i n the scope of present technology. -3 An upper l i m i t of 8.2 x 10 eV has been placed on w(Sn), the average energy l o s s by a charged p a r t i c l e r e q u ired to e x c i t e a q u a s i p a r t i c l e p a i r i n superconducting t i n at 1.2 K. This phenomenological q u a n t i t y , which h i t h e r t o had not been measured i n superconductors, i s of i n t e r e s t p r i m a r i l y to nuclear s p e c t r o s c o p i s t s f o r the comparing of one d e t e c t i n g medium w i t h another (see s e c t i o n B). This value of w(Sn) i s approximately 7.5 times the energy gap (2A) i n superconducting t i n at 1.2 K which may be compared w i t h the average energy l o s s per e l e c t r o n - h o l e p a i r created i n a semiconductor (eg. s i l i c o n ) where w(Si) =3.6 eV which i s 3.3 times the energy gap i n s i l i c o n . (Dearnaley and Northrop, 1966) -175-B. E v a l u a t i o n of the Device as a Nuclear Spectrometer 1. Energy R e s o l u t i o n A convenient d e f i n i t i o n of s t a t i s t i c a l l y l i m i t e d energy r e s o l u t i o n R i s (Dearnaley and Northrop, 1966) R = a/q = N_2" = (w/E)2" _ x where q = Ne i s the s i g n a l charge c o l l e c t e d , a/e = N 2 i s the standard d e v i a t i o n i n N, (the number of c a r r i e r s produced by the charged p a r t i c l e ) , w i s the mean energy r e q u i r e d to generate a c a r r i e r , E i s the p a r t i c l e energy and e i s the e l e c t r o n i c charge. To compare the tunnel j u n c t i o n detector to another spec-trometer the value of w to be used i s w(exp) = 0.145 eV where w(exp) i s the maximum value of the energy l o s s per r q u a s i p a r t i c l e which a c t u a l l y  tunneled (see s e c t i o n E, chapter 7). Thus, f o r the same p a r t i c l e , i t may be i n f e r r e d that the s t a t i s t i c a l l y l i m i t e d energy r e s o l u t i o n of the i present superconducting tunnel j u n c t i o n i s a f a c t o r (0.145/3.6) 2 = 0.20 smaller than that obtained w i t h a s i l i c o n semiconductor d e t e c t o r , assuming the same Fano f a c t o r f o r both m a t e r i a l s . I t i s conceivable, w i t h forseeable improvements i n j u n c t i o n f a b r i c a t i o n technology, that t h i s r e s o l u t i o n f a c t o r could be improved to i converge perhaps on the approximate l i m i t t E g a P ( s u p e r ) / E g a p ( s e m i ) ] 2 = 0.03. For example, the t u n n e l i n g p r o b a b i l i t y may be improved by decreasing the normal s t a t e j u n c t i o n r e s i s t a n c e by the use of i n s u l a t o r s w i t h smaller d i e l e c t r i c constants and energy gaps than SnO^. (The normal s t a t e j u n c t i o n r e s i s t a n c e may al s o be decreased, of course,, by minimizing the i n s u l a t o r o t h i c k n e s s , but a p r a c t i c a l lower l i m i t of about 10 A precludes s i g n i f i c a n t advances from t h i s approach.) Furthermore, the recombination p r o b a b i l i t y W — d i s c u s s e d i n chapter 7 — c o u l d be decreased by surrounding the j u n c t i o n R w i t h a phonon i n s u l a t o r , such as a c r o s s l i n k e d polymer (see s e c t i o n C), to reduce the l o s s of recombination phonons to the substrate and super-f l u i d helium f i l m . The a c t u a l , r a t h e r than the s t a t i s t i c a l , l i m i t of energy r e s o l u t i o n i s determined by other f a c t o r s beside w. As discussed i n -176-chapter 3, s t o c h a s t i c v a r i a t i o n s i n i n s u l a t o r thickness across the j u n c t i o n overlap area imply that i s not uniform across the j u n c t i o n and f l u c t u a t i o n s w i l l occur i n the number of q u a s i p a r t i c l e s which tunnel f o r a given input of energy. At present, the p r e a m p l i f i e r noise dominates j u n c t i o n noise and degrades r e s o l u t i o n but the f a b r i c a t i o n of j u n c t i o n s w i t h higher output impedances w i l l improve t h i s s i t u a t i o n . 2. L i n e a r i t y ' Another p o i n t of concern i s the dependence of w upon the type of p a r t i c l e , i t s energy (E) and the geometry of the j u n c t i o n . In the i d e a l spectrometer, w(E) would be a constant k so that the s i g n a l charge c o l l e c t e d q = AE/k i s l i n e a r l y r e l a t e d to the energy AE which a p a r t i c l e of a r b i t r a r y i n i t i a l energy l o s e s i n the system. This s i t u a t i o n obtains approximately, f o r l i g h t l y i o n i z i n g r a d i a t i o n l i k e protons and alpha p a r t i c l e s , i n p r a c t i c a l c o n f i g u r a t i o n s of s e v e r a l gas and semi-conductor detectors (see eg. Dearnaley and Northrop, 1966). The reasons f o r t h i s are that the i o n i z a t i o n p o t e n t i a l i n the gas and the energy gap i n the semiconductor remain e f f e c t i v e l y constant as the primary p a r t i c l e slows down and that the counters are s u f f i c i e n t l y l a r ge that the i n t e r a c t i o n of t h e i r boundaries w i t h processes t a k i n g place near the p a r t i c l e t r a c k may be s a f e l y ignored. As discussed below, n e i t h e r c o n d i t i o n holds i n the superconducting tunnel j u n c t i o n d e t e c t o r . The energy gap i n the superconductor i s , of course, temperature dependent such that i t vanishes i n the hot region near the p a r t i c l e t r a c k center, f i n a l l y reappearing and i n c r e a s i n g i n magnitude as the temperature of the superconducting f i l m s r e l a x e s back to that of the bath. In a sense, the s i t u a t i o n i s analogous to that i n semi-conductor detectors when the d i s o r d e r l e f t i n the wake of a densely i o n i z i n g f i s s i o n fragment permanently a l t e r s the band gap r e s u l t i n g i n a departure from l i n e a r i t y o f t e n r e f e r r e d to as the " f i s s i o n d e f ect". In chapter 7, a t t e n t i o n was drawn to the f a c t that the r a t e at which phonons escaped from the tunnel j u n c t i o n played an important r o l e i n determining the temperature r e l a x a t i o n time T and consequently w. I t would seem th e r e f o r e that c o n d i t i o n s at the boundaries of the tunnel j u n c t i o n detector and the l o c a t i o n of the p a r t i c l e t r a c k s w i l l a f f e c t the measured value of w. -177-C l e a r l y , considerably more t h e o r e t i c a l and experimental evidence i s r e q u i r e d to e s t a b l i s h the v a r i a t i o n of w w i t h p a r t i c l e type and energy i n superconducting tunnel j u n c t i o n s having a p r a c t i c a b l e geometry. 3. Stopping E f f i c i e n c y I f the j u n c t i o n i s to be u s e f u l as a spectrometer, i t should be s u f f i c i e n t l y massive to stop i o n i z i n g r a d i a t i o n so that a l l i t s energy i s deposited i n the j u n c t i o n . The stopping e f f i c i e n c y of a s i n g l e tunnel j u n c t i o n i s l i m i t e d by i t s mass pXA which, f o r the present specimen, —7 8 i s approximately 2 x 10 g — a f a c t o r 10 smaller than f o r contemporary semiconductor d e t e c t o r s . The area A cannot be made much l a r g e r than at present because the j u n c t i o n capacitance would become s u f f i c i e n t l y l a r g e to prevent the current pulse from being observed; X could be increased but only at the expense of decreasing (equation 3-17), the f r a c t i o n of e x c i t e d q u a s i p a r t i c l e s which tunnel and the speed of response of the detector system; p could be increased, but only some 55%, by using lead i n s t e a d of t i n . I t i s conceivable however, w i t h the present r a p i d growth of m i c r o - e l e c t r o n i c and j u n c t i o n f a b r i c a t i o n techniques, (Schroen, 1968) that i t may be p o s s i b l e to overcome t h i s handicap of t h i n f i l m j u n c t i o n s by manufacturing them i n three-dimensional a r r a y s . 4. Advantages Over Conventional Spectrometers In p r i n c i p l e , the superconducting tunnel j u n c t i o n should e x h i b i t s u p e r i o r energy, time and s p a t i a l r e s o l u t i o n over conventional semiconductor spectrometers i n the measurement of nuclear and X r a d i a t i o n o which can be stopped i n , say, 10,000 A of l e a d . The r e s u l t s f o r the t i n j u n c t i o n reported i n t h i s t h e s i s i n d i c a t e the f e a s i b i l i t y of a t t a i n i n g t h i s goal w i t h a reasonable amount of f u r t h e r developmental e f f o r t , (see s e c t i o n C) As mentioned i n paragraph 2, stopping e f f i c i e n c y might be increased so to be comparable to semiconductors, by f a b r i c a t i n g the j u n c t i o n i n three-dimensional a r r a y s . I t remains to be c a l c u l a t e d , however, how s e r i o u s l y the f l u c t u a t i o n s i n i n s u l a t o r thickness across the many j u n c t i o n s r e q u i r e d to stop an energetic charged p a r t i c l e would d e t e r i o r a t e the o v e r a l l energy resolution.---178-5. Disadvantages f o r Operation as a Spectrometer An obvious shortcoming i s the operating temperature of 1.2 K which n e c e s s i t a t e s a l i q u i d helium c r y o s t a t . Even i n l a b o r a t o r i e s where a supply of helium i s a v a i l a b l e , the bulk and complexity of a cryogenic system would probably deter a l l but the most s t r o n g l y motivated p o t e n t i a l users. To place a superconducting detector on l i n e w i t h an accelerator-produced beam i n v o l v e s s t i l l other d i f f i c u l t i e s . Because of the t i n y j u n c t i o n area, the s o l i d angle subtended by the j u n c t i o n s to an e x t e r n a l source i s very small indeed. Good s p a t i a l r e s o l u t i o n i s c e r t a i n l y p o s s i b l e but only at the expense of d e t e c t i o n e f f i c i e n c y . I n t r o d u c i n g the r a d i a t i o n i n t o the c r y o s t a t i n such a way as to minimize energy s t r a g g l i n g and i n t e n s i t y a t t e n u a t i o n could w e l l prove to be a c h a l l e n g i n g cryogenic and nuclear engineering problem. Too high a p a r t i c l e f l u x would tend to overheat the j u n c t i o n and reduce i t s s e n s i t i v i t y i f not render i t i n o p e r a t i v e . A f u r t h e r hindrance to r o u t i n e use i s the tendency of t h i n f i l m tunnel j u n c t i o n s to change t h e i r p r o p e r t i e s upon storage or thermal c y c l i n g ( c f . Chapter 6). Workers i n the area of development of tunnel j u n c t i o n arrays as computer elements (eg. Schroen, 1968) appear to have overcome t h i s d i f f i c u l t y , however, by c a r e f u l l y d e f i n i n g the j u n c t i o n area w i t h p h o t o - r e s i s t and growing the oxide l a y e r w i t h an oxygen glow discharge. Although none seems insuperable, these obstacles w i l l no doubt impede the development of a p r a c t i c a b l e superconducting spectrometer. C. Future Work 1. Use of Junctions w i t h Larger Dynamic Resistance (r = 3V/3I) The s i g n a l to noise r a t i o of a j u n c t i o n - c u r r e n t s e n s i t i v e p r e a m p l i f i e r system b e n e f i t s doubly from the use of j u n c t i o n s w i t h reasonably l a r g e (500-1000°-) output' impedances, f o r not only does the s i g n a l i n c r e a s e but the a m p l i f i e r noise i s a minimum (Woll and Herscher, 1962). I n the present work, the magnitude of r was l i m i t e d by the r e s t r i c t i o n of working w i t h t i n and the d e t e r i o r a t i o n of t u nneling c h a r a c t e r i s t i c s a f t e r thermal c y c l i n g . From equation 3- 9 , i t may be seen that r <* exp(A(T)/kT) -179-which immediately suggests two ways i n which higher dynamic r e s i s t a n c e j u n c t i o n s could be r e a l i z e d i n f u t u r e : the use of m a t e r i a l s w i t h l a r g e r energy gaps (2A(T)) and the use of lower operating temperatures. Lead and niobium, f o r in s t a n c e , have r e l a t i v e l y l a r g e energy gaps (2.7 and 3.0 meV r e s p e c t i v e l y , Douglass and F a l i c o v , 1964) and have been used i n tunnel j u n c t i o n f a b r i c a t i o n (Pb f a i r l y r o u t i n e l y , see eg. Rowell (1963), but Nb l e s s so (Giaever, 1963)). Lower operating temperatures than 1 K 3 are r e a d i l y obtained w i t h l i q u i d He but, of course, at the expense of considerable c o m p l i c a t i o n i n the a n c i l l a r y cryogenics. The problems associated w i t h thermal c y c l i n g are e a s i l y obviated by design of a system i n which the specimens are kept at helium temperatures from the time t h e i r p r e p a r a t i o n i s complete u n t i l they are no longer of i n t e r e s t . In f a c t , should i t be necessary, there i s no fundamental reason why an evaporator could not be b u i l t i n t o a c r y o s t a t so that a set of j u n c t i o n s need never be exposed to anything but vacuum or i n e r t helium atmosphere. 2. Thermal Decoupling of Junctions from Substrate Two b e n e f i t s would accrue from thermally decoupling the tunnel j u n c t i o n from the su b s t r a t e : the recombination p r o b a b i l i t y W would be decreased and the ambiguity i n w caused by the d i f f u s i o n of heat from the sub s t r a t e to the j u n c t i o n would be e l i m i n a t e d . In connection w i t h experiments done on s i n g l e superconducting s t r i p p a r t i c l e detectors ( S p i e l et a l , 1965), C r i t t e n d e n (1968) has s u c c e s s f u l l y thermally i s o l a t e d a t h i n f i l m s t r i p from the substrate w i t h a phonon i n s u l a t o r o c o n s i s t i n g of about 300 A of e l e c t r o n - p o l i m e r i z e d v a r n i s h . This technique, based on the work of C h r i s t y (1960), c o n s i s t s of bombarding the substrate ( p r i o r to f i l m evaporation) w i t h 100-200 eV e l e c t r o n s i n the presence of s i l i c o n o i l vapour. 3. P r e a m p l i f i e r Improvements From the d e t a i l e d a n a l y s i s of chapter 7 i n which the pulse r e l a x a t i o n time T was ex t r a c t e d from photographic pulse data, i t i s evident much of the u n c e r t a i n t y could be reduced by the use of a wider bandwidth a m p l i f i e r p e r m i t t i n g d i r e c t measurement of the current pulse waveform to much sh o r t e r times. Furthermorej the p r e a m p l i f i e r used was the dominant noise source so that any innovation reducing t h i s f a c t o r would, of course, be b e n e f i c i a l as w e l l . -180-APPENDIX A JUNCTION PREPARATION A. Substrate P r e p a r a t i o n The substrates used were o r d i n a r y microscope s l i d e s cut on one s i d e to have o v e r a l l dimensions 2 cm x 7.6 cm and scored to d i v i d e i t i n t o f i v e areas of 1 cm x 2 cm as shown i n f i g u r e A - l ( a ) . In t h i s way, a l l f i v e j u n c t i o n s could be evaporated and dc t e s t e d at once and then the substrate could r e a d i l y be broken up i n t o the f i v e smaller substrates f o r pulse t e s t i n g of i n d i v i d u a l j u n c t i o n s ( c f . Chapter 4 ) . Though o r i g i n a l l y "process cleaned" by the manufacturer (Corning), the s l i d e s had to be cleaned again because of the handling i n v o l v e d i n c u t t i n g and s c o r i n g the packaged 1 i n . x 3 i n . s l i d e s . Cleaning was done w i t h "Teepol" detergent a f t e r which the substrates were r i n s e d s e q u e n t i a l l y i n hot, c o l d and d i s t i l l e d water followed by hot methanol. F a i r l y t h i c k , o probably 5,000-10,000 A, s i l v e r contacts were then evaporated through an appropriate mask i n the geometry shown i n f i g u r e A - l ( a ) f o l l o w i n g which the substrates were stored i n a d e s s i c a t o r u n t i l needed. P r i o r to being used i n the a c t u a l j u n c t i o n f a b r i c a t i o n , a completed substrate was r i n s e d again i n hot methanol and mounted i n the evaporator. B. Evaporation Procedure 1. Base F i l m ( f i g u r e A - l ( b ) ) Evaporation took place i n a CVC (Model CVE-15) o i l d i f f u s i o n -7 o pump evaporator w i t h a base pressure of l e s s than 5 x 10 Torr. The 2000 A —6 t h i c k bottom f i l m was evaporated at a pressure of 2-5 x 10 Torr and a r a t e P of approximately 2000 A/min. A shutter placed between the vapour source and the s u b s t r a t e masked the substrate from the evaporant u n t i l the source was uniformly hot, i m p u r i t i e s were d r i v e n o f f and evaporation was w e l l under way. -181-Microscope S l i d e (a) i o 6 6 6 t • • i i t-i l n i n ! n ! n 2 1 * -7.6 cm— ... * Contact Arrangement Vapour Deposited Ag contacts (b) -\ rh n rh h r 1 •1 P P P M L i i i i n 1 n • n ! n 1 n T 0.2 mm Base F i l m Evaporation . (c) • • I I 1 rH r~i i4"! rh r J- I,! y Lpl Lpl L i i i i r - i i r - i i r - i ! r - l ! n Oxidation 0.2 mm -ilr-(d) r - i T i rh »/' ™T "V I U 1 V 1 rh r J - I i i i r 1 n. i r T l i r ~t l Cross F i l m Evaporation Figure A - l J u n c t i o n Preparation -182-2. O x i d a t i o n Some 30 minutes a f t e r the base f i l m evaporation was complete, the d i f f u s i o n pump was valv e d o f f and dry oxygen was slowly leaked i n t o the b e l l j a r u n t i l the pressure was about 1/3 atmosphere. The substrate h o l d e r , (see f i g u r e A-2), a copper b l o c k w i t h i n t e r n a l e l e c t r i c a l heater, was then warmed s l o w l y ( i n about one hour) up to 110°C and kept at that temperature f o r 15-20 hours. A f t e r c o o l i n g to room temperature i n the oxygen atmosphere, the specimens were exposed to ambient room a i r f o r about one minute f o l l o w i n g which the b e l l j a r was evacuated. Presumably the b r i e f exposure to room a i r , t y p i c a l l y w i t h 40-50% r e l a t i v e humidity, provided the c a t a l y t i c a c t i o n required to make the o x i d a t i o n c o m p l e t e — t h e s i n g u l a r l a c k of success w i t h t i n j u n c t i o n s not submitted to t h i s phase of the procedure emphasizes i t s importance. ( I t should be pointed out that the b e n e f i c i a l e f f e c t of a b r i e f exposure to room a i r was not discovered u n t i l r a t h e r l a t e i n the experimental program so that i t s r o l e was not c a r e f u l l y i n v e s t i g a t e d . Those f a m i l i a r w i t h the vagaries of t h i n f i l m s w i l l a ppreciate that one h e s i t a t e s to tamper w i t h a r e c i p e that works.) 3. Top F i l m Evaporation A f t e r the pressure had been reduced to the usual base l e v e l -7 0 of l e s s than 5 x 10 Torr , the 2000 A t h i c k top f i l m was evaporated at the same pressure and r a t e as the lower f i l m i n the geometry shown i n f i g u r e A - l ( d ) . The sub s t r a t e was at room temperature throughout both evaporations. Samples f a b r i c a t e d f o l l o w i n g t h i s procedure were thus i n a reasonably c o n t r o l l e d environment from the time the sub s t r a t e was i n s t a l l e d u n t i l the completed j u n c t i o n s were removed from the evaporator to have the e l e c t r i c a l leads attached. A photomicrograph of a completed Sn-Sn j u n c t i o n i s shown i n f i g u r e A-3. (The white speck on the middle r i g h t i s a dust p a r t i c l e ) . 4. Attachment of E l e c t r i c a l Leads and Mounting E l e c t r i c a l connection to the t h i n f i l m s was made w i t h f i n e (0.003 i n . ) gold wires indium-soldered to the s i l v e r c o n t a c t s . (None of these j o i n t s , e a s i l y made w i t h a moderately warm, very c l e a n s o l d e r i n g i r o n ( f r e e of any f l u x ) , was ever observed to f a i l ) . The gold leads were then indium-soldered to the terminals of the dc or pulse t e s t holder (see f i g u r e s 4-2 and 4-3) a f t e r which the j u n c t i o n s were mounted i n the helium dewar -183-ready f o r p r e - c o o l i n g w i t h l i q u i d n i t r o g e n i n a dry n i t r o g e n or helium gas atmosphere. T y p i c a l l y , the time r e q u i r e d to e f f e c t the t r a n s f e r of specimens from evaporator to helium dewar w i t h a l l leads attached and e l e c t r i c a l c o n t i n u i t y checked was about 45 minutes; throughout t h i s p eriod the j u n c t i o n s were, of n e c e s s i t y , exposed to room a i r . C. Evaporation Apparatus 1. Masks The masks used to d e l i n e a t e the evaporated s t r i p s were made from 0.003 i n . Beryllium-Copper shim stock which combines ease of photoetching w i t h the d e s i r a b l e p r o p e r t i e s of low thermal expansion, low gas e m i s s i v i t y i n vacuum and r e s i s t a n c e to c r e a s i n g . To ensure that the masks would f i t c l o s e l y and uniformly against the s u b s t r a t e s , the masks re s t e d on a 1/4 i n . t h i c k s t a i n l e s s s t e e l t a b l e (ground f l a t on one s i d e to + 0.001 i n . ) and the substrate holder was so designed that the g l a s s s l i d e s l a y d i r e c t l y on the mask, held i n place by the weight of the hol d e r , (see f i g u r e A-2) The substrate holder moved only i n a v e r t i c a l plane; changing of masks w h i l e under vacuum was achieved by r o t a t i o n of the mask t a b l e i n a h o r i z o n t a l plane. The m o t i v a t i o n f o r t h i s f a i r l y s o p h i s t i c a t e d design was the d e s i r e to minimize the penumbra re g i o n on the f i l m s which, according to Rowell (1968), may give r i s e to v a r i o u s unfavourable j u n c t i o n c h a r a c t e r i s t i c s . I n s p e c t i o n of photomicrographs of specimens made w i t h t h i s apparatus ( f i g u r e A-3) i n d i c a t e s t h i s goal was l a r g e l y achieved. 2. Substrate Holder The substrate holder served a two-fold purpose: to a l i g n and f i r m l y press the glass s l i d e s on top of the masks and to heat the subs t r a t e uniformly to roughly 100°C i n vacuum. As constructed, see f i g u r e A-2, the holder was machined from copper w i t h an i n t e r n a l nichrome wire (.4°,/ft) heater. Mica sheets provided the required e l e c t r i c a l i n s u l a t i o n . Temperature measurement was e f f e c t e d w i t h a copper-constantan thermocouple with room temperature r e f e r e n c e , mounted d i r e c t l y on the holder body ( v i s i b l e on the photograph i n f i g u r e A-2). Energized by a f i l t e r e d dc power supply, the heater was capable of maintaining 110 ± 2°C f o r periods of 16-20 hours. -184-F i g . A - 2 : E v a p o r a t i o n A p p a r a t u s F i g . A-3: P h o t o m i c r o g r a p h o f Sn-Sn Tunnel J u n c t i o n ( M a g n i f i c a t i o n : X100) -185-3. Evaporation Sources Commercial evaporation sources or "boats", obtained from R. D. Mathis Co., 1345 Gaylord S t . , Long Beach, C a l i f o r n i a , were used l a t t e r l y f o r sample p r e p a r a t i o n . Made of W, Mo or Ta stock, the p r e f e r a b l e form of source was the s o - c a l l e d "open" type (No. S5) which co n s i s t e d of a s i n g l e piece of m a t e r i a l (approximately 0.005 x 1 x 4 i n . ) f o l d e d l o n g i t u d i n a l l y and bowed i n the middle as i n the plan view sketched below. A l l three r e f r a c t o r y metals were used w i t h l i t t l e to choose among them. Because of i t s b r i t t l e n e s s , W was d i f f i c u l t to bend i n t o a shape compatible w i t h the evaporator e l e c t r o d e s but, apart from t h a t , performed q u i t e acceptably. A second type of source found to be l e s s s a t i s f a c t o r y was the " p i n h o l e " type (No. S17A, see sketch) which tended to give non-uniform c o a t i n g s . The t r o u b l e a r i s e s from the f a c t that the "hot spot" of evaporation wanders p e r i o d i c a l l y along the length of the boat severely d i s t o r t i n g the c o n i c a l p a t t e r n over which the vapour i s i d e a l l y d i s t r i b u t e d . 4. F i l m Thickness Measurement A quartz c r y s t a l o s c i l l a t o r was used to measure f i l m thickness and r a t e of evaporation. The p r i n c i p l e of operation and p r a c t i c a l d e t a i l s are discussed i n a review by Behrndt (1966). There, i t i s shown that the s h i f t i n the fundamental frequency f of a c r y s t a l on which a f i l m of thi c k n e s s 6s has been deposited i s given by -186-where p i s the d e n s i t y of the m a t e r i a l being deposited, p ^ i s the d e n s i t y of quartz and N i s the frequency constant which, f o r AT-cut c r y s t a l s , i s 1670 kflz-mm. S u b s t i t u t i n g the appropriate values f o r the constants i n A - l y i e l d s the u s e f u l r e s u l t Sf= Hz (A-2) -3 ° f o r p i n g cm and 6s i n Angstroms. The commercial u n i t u s e d — a Sloan Instruments Corporation Deposit Thickness Monitor, Model DTM-3—included c r y s t a l , c r y s t a l o s c i l l a t o r h e r m e t i c a l l y sealed i n sensor head (not shown i n f i g u r e A-2), v a r i a b l e frequency o s c i l l a t o r and frequency meter. In p r a c t i c e , because geometrical c o r r e c t i o n s must be a p p l i e d to r e l a t e (6s) _ .. to (6s) . ',. i t i s more convenient to c a l i b r a t e c r y s t a l substrate the c r y s t a l d i r e c t l y i n terms of s u b s t r a t e t h i c k n e s s . Thus, one has 6f = ct(6s) , ^ t substrate (A-3) where a i s a constant which depends on p and the c r y s t a l - s u b s t r a t e -evaporation source geometry. To evaluate a(Pb), two independent methods of c a l i b r a t i o n were used. One method c o n s i s t e d of weighing the Pb f i l m s w i t h a micro-balance and determining t h e i r thickness from the f i l m area (as measured with a t r a v e l l i n g microscope) and the bulk d e n s i t y ; the other method i n v o l v e d the d i r e c t measurement of the f i l m thickness w i t h a Na l i g h t interferometer (a Sloan Instruments "Angstrometer", Model M100). The r e s u l t s are summarized below: Method ct(Pb)Hz/A Gravimetric I n t e r f e r o m e t r i e 3.1 3.2 o Applying approximate geometrical c o r r e c t i o n s to a(Pb) y i e l d s 7.3 Hz/A -187-which i s c o n s i s t e n t , w i t h i n experimental e r r o r , w i t h the value o p/1.78 = 6.4 Hz/A p r e d i c t e d from equation A-2. The value of a(Sn) was d e r i v e d from ot(Pb) by using equation A-2, g i v i n g a(Sn) = a(Pb) p. /p = 2.02 Hz/A -188-APPENDIX B dc CHARACTERISTICS OF LEAKY SPECIMENS A. I n t r o d u c t i o n This appendix i s a summary of work c a r r i e d out i n attempting to understand and systematize the experimental r e s u l t s obtained w i t h s o - c a l l e d " l e a k y " superconducting tunnel j u n c t i o n s . (In t h i s context, "leaky" i s used to des c r i b e any j u n c t i o n whose dc I-V c h a r a c t e r i s t i c i s incompatible w i t h the well-known q u a s i p a r t i c l e and Josephson t u n n e l i n g c h a r a c t e r i s t i c s d e scribed i n Chapter 2 ). I f such j u n c t i o n s are mentioned at a l l i n the l i t e r a t u r e , i t i s u s u a l l y only i n passing so that the new worker i n the f i e l d , who prepares a set of specimens and f i n d s t h e i r c h a r a c t e r i s t i c s to d i f f e r markedly from the " t y p i c a l " r e s u l t s given i n most a r t i c l e s ; i s l e f t to wonder what went wrong. The primary f u n c t i o n of t h i s s e c t i o n then i s to attempt to f i l l t h i s gap i n the l i t e r a t u r e f o r the newcomer by (1) d e s c r i b i n g some of the l i t t l e - p u b l i c i z e d "bad" tunnel j u n c t i o n r e s u l t s and (2) g i v i n g a guide to places i n the l i t e r a t u r e where i n d i v i d u a l p o i n t s are discussed. P a r t B i s devoted to a p r e s e n t a t i o n of the r e s u l t s and part C to t h e i r a n a l y s i s . Some of the unresolved problems i n an a l y z i n g the leaky j u n c t i o n c h a r a c t e r i s t i c s are o u t l i n e d i n part D. B. Results 1. I-V C h a r a c t e r i s t i c A r e p r e s e n t a t i v e I-V c h a r a c t e r i s t i c of a leaky Sn-Sn02~Sn tunne l j u n c t i o n i s given i n f i g u r e B - l . ( S i m i l a r r e s u l t s were obtained f o r the Pb-Pb 0 -Pb specimens discussed i n Appendix C.) The 4 ter m i n a l x y network by which these X-Y p l o t s were obtained i s s i m i l a r to that already described i n Chapter 5. I t should be noted that a l l these curves are taken i n zero a p p l i e d magnetic f i e l d but no e f f o r t was made to s h i e l d the specimens from the earth's f i e l d of approximately 0.2 G. -189-1.22 K — • 2. 01 K 2.97 K Figure B - l : T y p i c a l I-V C h a r a c t e r i s t i c s f o r "Leaky" Specimens -190-2. Magnetic F i e l d Dependence of Supercurrent The I-V c h a r a c t e r i s t i c s of most specimens showed some supercurrent, i e . the j u n c t i o n could pass a f i n i t e current without developing a voltage between the f i l m s . Because both the dc Josephson e f f e c t and superconducting m e t a l l i c f i l a m e n t s through the i n s u l a t i n g l a y e r could account f o r such c u r r e n t s , a v a l u a b l e d i a g n o s t i c t e s t i s to determine the magnetic f i e l d dependence of the maximum supercurrent (I ) c (see f i g u r e B-2). (The c i r c u i t f o r t h i s t e s t was l i k e that of f i g u r e 5-1, except that the input to the X a x i s was a voltage p r o p o r t i o n a l to the current passing through the Helmholtz c o i l s producing the magnetic f i e l d p a r a l l e l to the plane of the j u n c t i o n . ) To produce the ^ c r ^ t ~ ^ p l o t , the magnetic f i e l d was set at some l e v e l B^ w i t h the j u n c t i o n current I equal to zero; I was then sl o w l y increased u n t i l the voltmeter was observed to jump d i s c o n t i n u o u s l y from i t s zero v a l u e , the v e r t i c a l l i n e traced out by the recorder then being p r o p o r t i o n a l to I c r i t ( B ^ ) ' The f i e l d was then increased to B^ + and the operation repeated thereby e v e n t u a l l y generating the curve of f i g u r e B-2. C. A n a l y s i s of Results For purposes of d i s c u s s i o n i t i s convenient to consider the curves of f i g u r e B-1 i n two p a r t s : (a) 0.1 mV < V and (b) 0 $ V $ 0.1 mV. 1. I-V C h a r a c t e r i s t i c f o r V > 0.1 mV Theory p r e d i c t s and experiments show (see Chapters 2 and 6) that the s i n g l e q u a s i p a r t i c l e t u n n e l i n g current f o r b i a s voltages V<2A/e should decrease markedly w i t h decreasing temperature. I t i s immediately evident from a comparison of f i g u r e s B-1 and 6-1 that such behaviour i s not true f o r the leaky specimens. This suggests that the current i n t h i s r e g ion of the c h a r a c t e r i s t i c does not a r i s e from tunneling q u a s i p a r t i c l e s but i s i n s t e a d c a r r i e d p r i m a r i l y by m e t a l l i c f i l a m e n t s which have been d r i v e n normal but whose r e s i s t a n c e i s s t i l l considerably l e s s than that of the i n s u l a t i n g l a y e r when both f i l m s are normal (R ). One may t h e r e f o r e na'ively t h i n k of the j u n c t i o n as a p e r f e c t diode i n p a r a l l e l w i t h numerous fi l a m e n t s whose t o t a l r e s i s t a n c e i s R^. The I-V c h a r a c t e r i s t i c of such a model i s i l l u s t r a t e d i n f i g u r e B-3. I t would seem that such a model ex p l a i n s the general shape of the observed -191--192-Figure B-3: I-V C h a r a c t e r i s t i c f o r Model of I d e a l J u n c t i o n i n ; P a r a l l e l w i t h M e t a l l i c . Filaments -193-c h a r a c t e r i s t i c but there s t i l l e x i s t s an excess current the o r i g i n of which, as discussed i n s e c t i o n D, i s not f u l l y understood. That some of the current passing between the f i l m s was due to t u n n e l i n g i s i n d i c a t e d by the sharp increase i n current at b i a s voltages corresponding to twice the superconducting energy gap (see Chapter 2). The temperature dependence of the energy gap e x h i b i t e d by these specimens was checked by p l o t t i n g A(T)/A(0) against T/T , where A(T) i s one-half the energy gap at temperature T obtained experimentally be e x t r a p o l a t i n g to the voltage a x i s . Using the g e n e r a l l y accepted values of A(0) =1.1 meV and t r a n s i t i o n temperature T £ = 3.72 K, gave f a i r agreement w i t h the t h e o r e t i c a l curve ( f i g u r e 2-2). 2. Supercurrent (V = 0, Figure B^l) As mentioned e a r l i e r , I . -B p l o t s l i k e t hat of f i g u r e B-2 c r i t r ° were made i n an attempt to estimate the r e l a t i v e magnitudes of the dc Josephson and m e t a l l i c f i l a m e n t supercurrents. The reasoning behind t h i s t e s t i s that the dc Josephson current i s h i g h l y magnetic f i e l d dependent (see Chapter 2) and may be quenched f o r very small values of B. On the other hand, the c r i t i c a l f i e l d f o r the m e t a l l i c f ilament supercurrent should be very h i g h (Anderson, 1964) as the f l u x quanta are expected to pass through the i n s u l a t i n g spaces between the f i l a m e n t s l e a v i n g them super-conducting s h o r t s . Upon t h i s premise, the magnitude of the supercurrent due to superconducting m e t a l l i c f i l a m e n t s 1^ i n the j u n c t i o n s of f i g u r e B-2 was taken to be that supercurrent present at the lowest minimum detected i n the I . -B p l o t and the remainder was assumed to be dc Josephson current I T c r x t J (a) Josephson Current To confirm the assumption that dc Josephson current was indeed being observed, the magnitude of the maximum observed zero f i e l d current I T = I T n and the p e r i o d i c i t y of the magnetic f i e l d dependence of I were compared w i t h theory. The t h e o r e t i c a l maximum dc Josephson current i s given by ( c f . Chapter 2) I _ . = TrA(0)/2eR JM n -194-(For the j u n c t i o n s under c o n s i d e r a t i o n here, the low temperature normal t u n n e l i n g r e s i s t a n c e R was estimated by decomposing the a c t u a l I-V n -1 c h a r a c t e r i s t i c as shown i n f i g u r e B-3 and s e t t i n g (R ) equal to the slope n which the " i d e a l " j u n c t i o n c h a r a c t e r i s t i c approached a s y m p t o t i c a l l y f o r eV»2A). The r a t i o y = I ^ / I ^ was i n the range .07 <y< 0.5 which i s c o n s i s t e n t w i t h the r e s u l t s of other experiments. (Langenberg et a l , 1966). The p e r i o d i c i t y T' of the s t r u c t u r e i n I . -B p l o t s c r i t r l i k e f i g u r e B-2 may be estimated from the number of minima N 1 o c c u r r i n g i n an i n t e r v a l of magnetic f i e l d AB so that T' = AB/N*. From Chapter 2, the t h e o r e t i c a l p e r i o d between successive minima of the Josephson current i s -7 2 T=$ /2AW where $ = h/2e = 2 x 10 G cm i s the quantum u n i t of f l u x , X i i s o • o o the p e n e t r a t i o n depth of the magnetic f i e l d (=500 A i n Sn) and W i s the dimension of the j u n c t i o n normal to B (see i n s e t , Figure B-4). Both T' and T are p l o t t e d i n f i g u r e B-4 as a f u n c t i o n of W ^ and i t i s evident t h a t , f o r the three specimens considered, the agreement between T' and T i s q u i t e poor. I t must be remembered, however, that the sampling d e n s i t y i n the I . -B p l o t s — i e . the d i s t a n c e between the v e r t i c a l l i n e s on f i g u r e B-2— c r i t c 6 corresponded to r e l a t i v e l y l a r g e magnetic f i e l d increments of b - 0.23 G so that one does not expect to be able to r e s o l v e a l l the minima, e s p e c i a l l y when T i s close^b. This p o i n t i s i l l u s t r a t e d f o r a very simple case i n the sketch below. There, i t i s assumed that b = 2T/3 and i t i s )*- T-H observed form, N' = 3 •actual form, N = 6 q u i t e evident that N', the number of minima observed, would be about 1/2 the a c t u a l number N. When the data of f i g u r e B-4 are c o r r e c t e d approximately to take t h i s e f f e c t i n t o account, the agreement between experiment and theory i s q u i t e reasonable. In the l i g h t of these f i n d i n g s , i t may be concluded that dc Josephson supercurrent was present although i t s p r e c i s e magnitude remains somewhat obscured. (b) M e t a l l i c Filament Current The area A^ of m e t a l l i c f i l a m e n t s or "superleaks" -195-I 1 / / — J 0 0 v Fig.B-5: S t r u c t u r e i n I-V C h a r a c t e r i s t i c Near V =0 -196-r e q u i r e d to account f o r the observed supercurrent i s an extremely small f r a c t i o n of the geometrical j u n c t i o n area and i t i s e n t i r e l y p l a u s i b l e that imperfections of t h i s extent could occur i n the i n s u l a t i n g l a y e r . Consider, f o r example, the j u n c t i o n of f i g u r e B-2 and B-3. According to Anderson and Rowell (1963), the c r i t i c a l currents of 7 -2 f i n e l y d i v i d e d specimens never exceeds 10 A cm which, f o r I f = 1.7 mA -10 2 (see f i g u r e B-3) places a lower bound of about 1.7 x 10 cm on A f. -10 2 This i s compatible w i t h the value A f = L/aR f = 1.8 x 10 cm where ° 4 - 1 - 1 L = 10 A i s the assumed length of the f i l a m e n t s , a = 10 ft cm i s the assumed f i l a m e n t c o n d u c t i v i t y (Anderson and Rowell, 1963) and R^ i s the fi l a m e n t r e s i s t a n c e deduced from a n a l y s i s of the I-V c h a r a c t e r i s t i c i n f i g u r e B-3. The geometrical area A of that p a r t i c u l a r j u n c t i o n was -3 2 -5 2.4 x 10 cm so that A^/A^IO and there are no grounds f o r b e l i e v i n g that t h i s f r a c t i o n i s a t y p i c a l of the leaky j u n c t i o n s s t u d i e d . Bardeen (1962) f i n d s that the c r i t i c a l current of 2 3/2 sma l l specimens at temperature T i s p r o p o r t i o n a l to (1 - t ) where t = T/T^. I t might be expected that 1^ should behave s i m i l a r l y as a f u n c t i o n of temperature but u n f o r t u n a t e l y l c r ^ t ~ B p l o t s were made at only one temperature (1.2 K) making i t impossible to check that hypothesis. Further evidence of the i n f l u e n c e of m e t a l l i c f i l a m e n t s i s the constant current jump, shown i n f i g u r e B-1 at about 1.0 mA, which i s examined i n the next s e c t i o n . 3. I-V C h a r a c t e r i s t i c f o r 0<V$0.1 mV When I j Q i s exceeded i n an i d e a l t u n n e l i n g j u n c t i o n e x h i b i t i n g the dc Josephson e f f e c t , the t r a n s i t i o n from V = 0 to the ord i n a r y q u a s i -p a r t i c l e t u n n e l i n g c h a r a c t e r i s t i c should be a s i n g l e discontinuous step as shown i n f i g u r e 2- 7. Such behaviour was never observed i n leaky j u n c t i o n s as summarized i n f i g u r e B-5. The manner i n which the voltage increased from zero as the c r i t i c a l current was reached, f e l l i n t o the two cat e g o r i e s l a b e l l e d (a) and (b) i n f i g u r e B-5. The small jump, c a l l e d type ( a ) , w i l l be considered b r i e f l y here; type (b), i n which no d i s c r e t e jump i s detected, i s discussed i n s e c t i o n D. A q u a l i t a t i v e explanation of the small constant-current -197-v o l t a g e jump may be found by c o n s i d e r i n g the I-V c h a r a c t e r i s t i c of a superconducting wire which i s d r i v e n normal by a current i n excess of i t s c r i t i c a l current 1^. London (1950) shows that the v o l t a g e drop V per u n i t length of an i n f i n i t e l y long c y l i n d r i c a l wire c a r r y i n g current i £ i i s c v = (i/2)n o i [ i + (i-(ic/i)2)£] 2 where SlQ = p/na i s the low temperature normal s t a t e r e s i s t a n c e per u n i t l e n g t h of the w i r e . This r e l a t i o n p r e d i c t s that when i = i a v o l t a g e jump of ( l / 2 ) f t Q i c occurs and that the c h a r a c t e r i s t i c a s y m p t o t i c a l l y approaches the s t r a i g h t l i n e V = ift0« The p o i n t of t h i s d i s c u s s i o n i s that the low v o l t a g e r e g i o n of such a c h a r a c t e r i s t i c would resemble the i n i t i a l t r a c i n g of type ( a ) , f i g u r e B-5. The s i g n i f i c a n c e of t h i s s i m i l a r i t y i s d i f f i c u l t to assess i n view of the f a c t t h a t , g e o m e t r i c a l l y , the m e t a l l i c f i l a m e n t s being considered are the a n t i t h e s i s of an " i n f i n i t e " w i r e . Lacking b e t t e r evidence, however, one i s l e d to the t e n t a t i v e c o n c l u s i o n that the small constant-current v o l t a g e jump, which appears only at the lowest temperatures, i s the r e s u l t of s w i t c h i n g along the m e t a l l i c f i l a m e n t intermediate s t a t e l o a d l i n e . D. Unresolved Problems i n the A n a l y s i s This s e c t i o n o u t l i n e s two areas of d i f f i c u l t y i n i n t e r p r e t i n g the I-V c h a r a c t e r i s t i c s of leaky j u n c t i o n s which are not yet f u l l y understood. S p e c i f i c a l l y , these are the s o - c a l l e d "excess c u r r e n t " of f i g u r e B-3 and the s t r u c t u r e near V = 0 i n f i g u r e B-5. 1. Excess Currents In the leaky specimens a current was observed which exceeded that which could be accounted f o r on the b a s i s of a model c o n s i s t i n g of an i d e a l q u a s i p a r t i c l e tunnel j u n c t i o n i n p a r a l l e l w i t h m e t a l l i c f i l a m e n t s . C a r e f u l examination of the I-V c h a r a c t e r i s t i c of f i g u r e B-3 i n the region 0.15<V<2A/e shows that i n a d d i t i o n to the main current jump at eV = 2A , there are a d d i t i o n a l s i n g u l a r i t i e s l o c a t e d approximately at 2A/n, n = 2,3,4,5. Comparison of curves taken at 1.2 K and 2 K shows l i t t l e dependence of the amplitude of the s i n g u l a r i t i e s upon temperature. These s i n g u l a r i t i e s have been seen by other workers as w e l l and t h i s s e c t i o n i s a synopsis of the explanations they have proposed. -198-Taylor and B u r s t e i n (1963) f i r s t n o t i c e d the s i n g u l a r i t y at eV = A and, i n an accompanying paper, S c h r i e f f e r and W i l k i n s (1963) i n t e r p r e t e d i t s o r i g i n i n terms of second order tunneling e f f e c t s i n v o l v i n g double p a r t i c l e t u n n e l i n g v i a p a i r d i s s o c i a t i o n or recombination. The r a t i o of the magnitude of the d i s c o n t i n u i t y i n the current (J^) at eV = A f o r the double p a r t i c l e processes to the magnitude of the current d i s c o n t i n u i t y a t e V = ^A f o r the s i n g l e q u a s i p a r t i c l e processes i s ( S c h r i e f f e r and W i l k i n s , 1963) J A exp (-4<kx>d) J 2 A exp (-2<kj.>d) In B - l , <kJ>> i s an average component of wave number k perpendicular to the b a r r i e r and d i s the b a r r i e r t h i c k n e s s . For an i d e a l , uniform b a r r i e r J"A~ J^^/IO, but, as the authors p o i n t out, the e f f e c t i v e d f o r the two processes need not be equal f o r a patchy oxide l a y e r . The t h i n regions of the f i l m would be emphasized more by the double p a r t i c l e than the s i n g l e p a r t i c l e process because of the exponential f a c t o r s i n and J^A* ^ e double p a r t i c l e processes are e s s e n t i a l l y independent of the q u a s i p a r t i c l e d e n s i t i e s and are consequently, i n agreement w i t h experiment, e s s e n t i a l l y independent of temperature. Yanson et a l (1965) reported the higher order s i n g u l a r i t i e s 2A/n, f o r n up to 6, i n Sn-Sn j u n c t i o n s at 1.61 K i n a f i e l d of 33 Oe app l i e d to suppress the dc Josephson c u r r e n t . Strong evidence was a l s o presented c l a i m i n g t h e i r j u n c t i o n s to be f r e e of m e t a l l i c f i l a m e n t s across the i n s u l a t o r i mplying t h e r e f o r e that the s i n g u l a r i t i e s are associated w i t h m u l t i p l e p a r t i c l e t u n n e l i n g through the b a r r i e r and not w i t h processes i n v o l v i n g the presence of superleaks. The magnetic f i e l d dependence of the 2A/n s i n g u l a r i t i e s i n Pb-Pb tunnel j u n c t i o n s was studi e d by Marcus (1966, A,B) whose specimens e x h i b i t e d supercurrents due to both Josephson tunneling and m e t a l l i c f i l a m e n t s . He found that f o r a magnetic f i e l d normal to the plane of the j u n c t i o n ( B j . ) , the amplitude and sharpness of the s i n g u l a r i t i e s decreased -199-i n d i s c r e t e steps w i t h i n c r e a s i n g Bj. i n the range 10<B<1<100 G. The c o n c l u s i o n that Marcus draws from t h i s evidence, i n c o n t r a d i c t i o n to Yanson et a l , i s that the 2A/n s i n g u l a r i t i e s are due to a tunneling process tak i n g place i n the r e g i o n of the j u n c t i o n where the m e t a l l i c f i l a m e n t s e x i s t . Zawadowski (1966), on the other hand, i n t e r p r e t s the data of R o c h l i n and Douglass (1966), who observed s e v e r a l s e r i e s of 2A/n s i n g u l a r i t i e s i n Pb-Pb j u n c t i o n s , as being due to the break up of P p a i r s and the subsequent t u n n e l i n g or t r a n s i t i o n through m e t a l l i c f i l a m e n t s of n e l e c t r o n s i n the same higher order quantum mechanical processes w i t h a v o l t a g e t h r e s h o l d 2AP/n. Taking appropriate values of P and n, he was able to f i t a l l the s i n g u l a r i t i e s observed by R o c h l i n and Douglass. S t i l l another explanation of the subharmonic tunneling phenomenon i s suggested by Zawadowski i n the same paper. I t i s w e l l known that t u n n e l i n g j u n c t i o n s act as a microwave generator at f i n i t e a p p l i e d v o l t a g e because of the ac Josephson current; i t i s conceivable therefore that the s i n g u l a r i t i e s could be completely or p a r t i a l l y the r e s u l t of microwave photon-assisted q u a s i p a r t i c l e t u n n e l i n g . In a l a t e r paper, Zawadowski (1967) considers b r i e f l y the problem of higher order tunneling processes concluding that processes of t h i s type may be much more intense than expected i f the t r a n s i t i o n s of the q u a s i p a r t i c l e s through the b a r r i e r occur at some defect i n the b a r r i e r . Since Yanson et a l have observed m u l t i p l e p a r t i c l e t u n n e ling i n j u n c t i o n s apparently f r e e of m e t a l l i c f i l a m e n t s , i t would seem that the superleaks i n the samples of Marcus and those considered here are important only i n the sense of being defects i n the b a r r i e r at which higher-order tunneling i s enhanced. 2. S t r u c t u r e near V = 0 I n t e r e s t i n g behaviour i n the I T V c h a r a c t e r i s t i c near zero v o l t a g e was observed i n a l l leaky j u n c t i o n s ; i t appears to f a l l i n t o the two general c a t e g o r i e s of f i g u r e B-5. Retrace loops, whose area v a r i e d approximately i n v e r s e l y w i t h temperature, were always present but i n (a) supercurrent was r e s t o r e d at I = 1^ < I c r^ t» a n d i n (b) the supercurrent was r e s t o r e d at I = I . . S i m i l a r s t r u c t u r e to case (a) has been observed c r i t by Yanson et a l (1965) w i t h t h e i r m e t a l l i c f i l a m e n t - f r e e j u n c t i o n s at 2 K -200-i n zero magnetic f i e l d . The same workers a l s o found s t r u c t u r e resembling that of case (b) when they f i r s t a p p l i e d a magnetic f i e l d of tens of oersteds, turned i t o f f and ran an I-V curve at zero f i e l d . No mention i s made whether the change i s permanent. Yanson et a l a t t r i b u t e t h e i r r e s u l t s to magnetic f l u x being p a r t i a l l y trapped by inhomogeneities a t the edges of the superconducting f i l m s ; t h i s e xplanation can probably be extended to the present j u n c t i o n s f o r the m e t a l l i c f i l a m e n t s would serve mainly to inc r e a s e the f l u x t r a p p i n g . The i n f l u e n c e of f l u x t r a p p i n g on the d e t a i l e d behaviour of tunnel j u n c t i o n s has been f e l t by other workers as w e l l . J a k e l e v i c et a l (1965), f o r i n s t a n c e , d i s c l o s e that they found i t necessary to s h i e l d t h e i r j u n c t i o n s from even the earth's f i e l d w h i l e c o o l i n g them to prevent s e r i o u s a t t e n u a t i o n of the dc Josephson c u r r e n t . As mentioned i n paragraph 1, s i n g u l a r i t i e s were observed at eV = 2A/n w i t h n = 1,2,3 and as high as 5 i n some specimens. At b i a s voltages eV<2A/5-0.22 mV, however, these s i n g u l a r i t i e s were obscured by a much more complicated s t r u c t u r e (see f i g u r e s B-3 and B-5(b)) c o n s i s t i n g p a r t i a l l y of constant-voltage and constant-current steps; again, s i m i l a r behaviour has been reported by Yanson et a l (1965). I t seems qui t e reasonable that these steps are evidence of the ac Josephson current coupling w i t h the resonant modes of the j u n c t i o n which would act as an open ended p a r a l l e l p l a t e resonator. This e f f e c t was f i r s t s t u d i e d by F i s k e (1964, 1965) and Eck e t a l (1964, 1965); i t i s al s o discussed at some le n g t h by Langenberg, et a l (1966). (Note added i n proof: Since t h i s appendix was w r i t t e n , the author has become aware of a recent review ( W i l k i n s , 1969) of the subject of m u l t i p a r t i c l e t u n n e l i n g which deals w i t h some of the problems r a i s e d here.) -201-APPENDIX C OBSERVATIONS ON Pb-Pb TUNNEL JUNCTIONS I t was shown i n Chapter 3, that the most d e s i r a b l e tunnel j u n c t i o n s f o r use as a charged p a r t i c l e detector i s one w i t h lead f i l m s on each s i d e of the i n s u l a t i n g l a y e r (Pb-Pb). This appendix o u t l i n e s some of the ob s t a c l e s encountered i n the course of attempting to develop s u i t a b l e Pb-Pb j u n c t i o n s and e x p l a i n s why i t was not p o s s i b l e to use these j u n c t i o n s i n the detector experiment. Se c t i o n A i s a summary of the low temperature r e s u l t s obtained w i t h Pb-Pb specimens and the conclusions derived therefrom. Section B describes the phenomenon of nodule growth on the lead f i l m s and recounts the steps taken to understand and e l i m i n a t e t h i s undesirable s i d e e f f e c t . S e c t i o n C contains a very short survey of the l i t e r a t u r e concerning the o x i d a t i o n of metals i n general and of lead i n p a r t i c u l a r ; the plasma o x i d a t i o n technique i s suggested as an a t t r a c t i v e a l t e r n a t i v e to the conven-t i o n a l thermal o x i d a t i o n procedure. A. Tunnel J u n c t i o n Results 1. J u n c t i o n P r e p a r a t i o n The r e c i p e f o l l o w e d i n preparing these j u n c t i o n s , derived from that used by workers i n the B e l l Telephone Labo r a t o r i e s (Rowell, 1968), i s e s s e n t i a l l y the same as that given i n Appendix A f o r Sn-Sn specimens. In t h i s case, the i n s u l a t i n g l a y e r i s a thermally grown oxide of l e a d — probably PbO, see s e c t i o n C. (The technique of Taylor and B u r s t e i n (1963) of evaporating a very t h i n l a y e r of aluminum on top of the base f i l m and a l l o w i n g i t to o x i d i z e to form the tu n n e l i n g b a r r i e r was r e j e c t e d because of the p o s s i b i l i t y of incomplete conversion.) V a r i a t i o n s on the r e c i p e that were t r i e d i n an e f f o r t to make r e l i a b l e j u n c t i o n s w i l l be pointed out i n the d i s c u s s i o n of s e c t i o n B. -202-2. Low temperature dc C h a r a c t e r i s t i c s Without exception, a l l Pb-Pb tunnel j u n c t i o n s tested at l i q u i d helium temperatures—about 50 a l t o g e t h e r — w e r e leaky i n the sense defined i n Appendix B. In many cases the Pb-Pb j u n c t i o n s were worse than the leaky Sn-Sn j u n c t i o n s i n that no s i n g l e - q u a s i p a r t i c l e t u n n e ling was evide n t , as denoted by the absence of a sharp increase i n the current at V = 2A/e. Furthermore, the l a r g e supercurrent e x h i b i t e d by the Pb-Pb specimens was almost a l l due to superconducting m e t a l l i c f i l a m e n t s (or superleaks) as i t was i n excess of the expected maximum t h e o r e t i c a l dc Josephson c r i t i c a l current and, u n l i k e the dc Josephson c u r r e n t , showed no s i g n i f i c a n t dependence upon the a p p l i e d magnetic f i e l d . (The c r i t i c a l c u r r e n t f o r some specimens exceeded 150 mA—the maximum current output a v a i l a b l e from the dc supply used, see chapter 5—and i n s e v e r a l j u n c t i o n s , the t h i n f i l m s s e r v i n g as current leads were observed to go normal and l i t e r a l l y "burn out" before the j u n c t i o n would switch from the supercurrent c h a r a c t e r i s t i c . ) During an extensive experimental program aimed at improving the tunnel j u n c t i o n performance^ the e f f e c t s of v a r y i n g a l l the obvious parameters such as evaporation r a t e , f i l m t h i c k n e s s , o x i d a t i o n procedure and f i l m edge sharpness were stud i e d but without success. 3. Conclusions from Pb-Pb J u n c t i o n Studies The obvious c o n c l u s i o n , of course, i s that the lead oxide formed on the base f i l m was very patchy so t h a t , when the top f i l m was evaporated to complete the j u n c t i o n , a s i g n i f i c a n t f r a c t i o n of the overlapping area c o n s i s t e d of two lead f i l m s i n in t i m a t e contact forming an e s s e n t i a l l y continuous superconducting s t r i p . In s p i t e of the enormous e f f o r t put i n t o growing a s u i t a b l e oxide l a y e r and the a s s e r t i o n s of other workers concerning the ease of doing so, the dc measurements i n d i c a t e t h i s goal was never reached. Therefore, because no Pb-Pb j u n c t i o n s could be prepared i n which the supercurrent could be suppressed nor a s i n g l e q u a s i p a r t i c l e tunneling c h a r a c t e r i s t i c of s u i t a b l y high d i f f e r e n t i a l r e s i s t a n c e obtained, the idea of using Pb-Pb j u n c t i o n s i n the alpha p a r t i c l e d e t e c t i o n experiment had to be abandoned. One avenue of i n v e s t i g a t i o n which was f r u i t l e s s l y followed i n t h i s study was the e f f e c t on j u n c t i o n performance of nodules observed to -203-have developed on the surface of the t h i n f i l m s . The next s e c t i o n recounts the d e t a i l s of t h i s i n v e s t i g a t i o n . B. I n v e s t i g a t i o n of Nodule Growth on Lead Films M i c r o s c o p i c examination of Pb-Pb tunnel j u n c t i o n s , see f i g u r e s C - l and C-2, revealed that nodules, having one dimension as l a r g e as 4u, were randomly d i s t r i b u t e d over the Pb f i l m s . (Tin f i l m s , see eg. f i g u r e A-3, never formed these blemishes). Now i f these growths are merely surface features of the completed j u n c t i o n they are not very important but, i f they form on the base f i l m before the top f i l m i s evaporated and consequently p r o j e c t through i t , a mechanism f o r the formation of superleaks i s c l e a r l y suggested. The i n v e s t i g a t i o n s o u t l i n e d below showed that the nodules d i d indeed form on the base f i l m but when i t l a t e r became p o s s i b l e to f a b r i c a t e Pb-Pb j u n c t i o n s which were apparently nodule-free, they proved to be j u s t as leaky as t h e i r predecessors. I t seems, t h e r e f o r e , that the presence of nodules must c o n t r i b u t e to the formation of superleaks but i t i s not the fundamental reason why the oxide patchiness p e r s i s t s . (Oxide growth on t h i n f i l m s i s discussed b r i e f l y i n s e c t i o n C.) 1. D e s c r i p t i o n of Nodules That the white specks of f i g u r e C - l , photographed w i t h r e f l e c t e d l i g h t , were indeed growths on the surface of the f i l m s and not pinholes i n the f i l m s was e s t a b l i s h e d by the observation at higher m a g n i f i c a t i o n ( f i g u r e C-2) of the shadows they cast and by the f a c t that the specks were opaque to t r a n s m i t t e d l i g h t . An attempt was made w i t h an e l e c t r o n microscope to determine whether the nodules were metal that had been squeezed from surface f i s s u r e s or l a r g e i s l a n d s of metal oxide which had formed on the surface but the p r o j e c t proved f u t i l e because of the d i f f i c u l t y i n detaching the lead f i l m s from the glass substrate without d e s t r o y i n g them. Without t h i s i n f o r m a t i o n , probably no fundamental under-standing of these growths can be r e a l i z e d ; the purpose of the present study was simply to l e a r n how to prepare f i l m s f r e e of these d e f e c t s . 2. O r i g i n of Nodules-The Result of Thermal Treatment S u r p r i s i n g l y , the nodules were found to form whether a lead f i l m was heated to 395 K i n about 500 Torr of oxygen or cooled to 1.2 K i n a helium atmosphere. To observe t h i s , a set of j u n c t i o n s was prepared according to the schedule shown i n f i g u r e A - l and the f i l m s were examined m i c r o s c o p i c a l l y a f t e r the completion of stages b, c, d and a l i q u i d helium -204-(a) (b) F i g . C-1: P h o t o m i c r o g r a p h s o f Pb-Pb Tunnel J u n c t i o n (X100) (a) B e f o r e exposure t o L i q u i d H e l i u m (b) A f t e r exposure t o L i q u i d H e l i u m F i g . C-2: P h o t o m i c r o g r a p h showing Nodules ( M a g n i f i c a t i o n X1220) -205-exposure. No nodules were v i s i b l e on the r e c e n t l y formed base f i l m (stage b) but they were a f t e r the base f i l m was heated to expedite oxide formation (stage c ) . The nodules p e r s i s t , being v i s i b l e a l l along the base f i l m , and appear to penetrate the crossed f i l m when i t i s evaporated (stage d ) . (Photomicrograph C-l(a) was taken at t h i s p o i n t ; the v e r t i c a l f i l m i s the base f i l m . ) Subsequent c o o l i n g of the e n t i r e j u n c t i o n to l i q u i d helium temperatures caused f u r t h e r growth of the nodules on both f i l m s as i l l u s t r a t e d i n f i g u r e C - l ( b ) . As an a d d i t i o n a l check, f i v e other specimens were prepared simultaneously on one substrate and then separated such that three were heated and two were cooled (to 77 K); no s i g n i f i c a n t d i f f e r e n c e between the two groups, w i t h regard to the surface d e n s i t y of nodules formed, was d i s c e r n i b l e . This r e s u l t r a i s e s two problems. From a p r a c t i c a l viewpoint, a superconducting tunnel j u n c t i o n which d e t e r i o r a t e s upon c o o l i n g i s i n t o l e r a b l e ; from a t h e o r e t i c a l viewpoint, a chemical process that goes e q u a l l y w e l l at roughly 1 and 400 K i s hard to envisage. One explanation suggested by Young (1968) i s that the evaporated f i l m s are i n compression and that the nodules r e s u l t from the d i f f e r e n t i a l expansion and c o n t r a c t i o n of the glass and lead upon thermal c y c l i n g . Although the argument i s weakened somewhat by the absence of r e p o r t s of s i m i l a r e f f e c t s by other workers who have al s o used glass substrates f o r Pb-Pb j u n c t i o n s , i t i s c o n s i s t e n t w i t h the r e s u l t s of the next paragraph where i t i s seen that the nodules d i d not form i n f i l m s of thickness l e s s o than 1000 A. Such fil m s ^ presumably, would flow more f r e e l y w i t h the glass than t h i c k e r ones and be l e s s s u s c e p t i b l e to ruptures which could give r i s e to nodules. 3. Tests Made to I s o l a t e Nodule-Producing Parameters There are, of course, a host of parameters which a f f e c t a given evaporation. No c l a i m i s made that the l i s t of parameters tested here i s exhaustive f o r one i s concerned only w i t h those which could conceivably give r i s e to the nodule growth. A " t e s t " of the e f f e c t of a parameter change c o n s i s t e d of preparing a set of j u n c t i o n s ( u s u a l l y f i v e ) under the appropriate c o n d i t i o n s , examining the f i l m s m i c r o s c o p i c a l l y f o r the presence of nodules immediately f o l l o w i n g evaporation, c o o l i n g the j u n c t i o n s to 77 K i n a helium atmosphere and re-examining them f o r nodule growth. (Once the thermal--206-treatment o r i g i n of the nodules was e s t a b l i s h e d , c o o l i n g was chosen as the means of producing nodule growth f o r t e s t purposes, as the heating c y c l e during the o x i d a t i o n stage could conceivably be omitted i n f a b r i c a t i n g p r a c t i c a l j u n c t i o n s but never the c o o l i n g c y c l e i n using them. In other words, a t h i n f i l m that i s subject to nodule production at low temperatures i s useless whether or not nodules are produced at elevated temperatures.) F i r s t , the parameter v a r i a t i o n s of i n t e r e s t are l i s t e d and explained; the r e s u l t s are summarized i n the f i n a l paragraph. (Within experimental l i m i t a t i o n s only one parameter was changed at a time.) (a) Evaporant P u r i t y Two grades and forms of Pb were used as evaporant: 99.999% pure bar stock from Cominco and 99.99% pure p e l l e t s from F i s h e r Chemical. S i m i l a r evaporation sources were used i n each instance. (b) Evaporation Source Evaporations were performed w i t h "boats" of s i m i l a r geometry (type S5, Appendix A) but made of Ta, Mo or W. (c) Order of F i l m D e p o s i t i o n To t e s t the hypothesis that the nodules were due to contamination evolved from the i n i t i a l heating of a new charge of evaporant, the usual order of evaporating the f i l m s , see f i g u r e A - l , was reversed. (d) Substrate P r e p a r a t i o n To observe the e f f e c t of d i f f e r e n t c l e a n i n g techniques, a glass s u b s t r a t e of the type described i n Appendix A was prepared i n which 1/3 of the scored area had a f i n a l c l e a n i n g w i t h chromic a c i d and 1/3 w i t h very d i l u t e h y d r o f l u o r i c a c i d . The remaining 1/3 had been cleaned i n the usual manner which d i d not i n v o l v e the use of a c i d s . (e) Rate of Evaporation The e f f e c t of evaporation r a t e on nodule growth i s somewhat obscured by the unavoidable v a r i a t i o n s i n r a t e which occur during the course of a given evaporation. In general, an average d e p o s i t i o n r a t e o of about 1000 A/min was achieved. (f ) F i l m Thickness The importance of f i l m thickness to nodule formation -207-was studied f i r s t of a l l w i t h a " p i n - h o l e " source (see Appendix A) which gave f i l m s whose thickness v a r i e d continuously along the l e n g t h of the s u b s t r a t e but which otherwise were formed under i d e n t i c a l c o n d i t i o n s . L a t e r , j u n c t i o n s were made i n which the thickness of one f i l m was greater than o o 1000 A and the thickness of the other l e s s than 1000 A. (g) Summary The r e s u l t s of a l l the t e s t s except the l a s t were negative i n the sense that the nodule pop u l a t i o n produced was apparently i n v a r i a n t — a s recorded p h o t o g r a p h i c a l l y — t o those changes of evaporation parameter. I t was found, as stated e a r l i e r , that the nodules d i d not form, e i t h e r when the f i l m was heated or cooled, i n a f i l m of thickness l e s s than ° o 1000 A. Films t h i c k e r than 1000 A, prepared simultaneously to the thinner ones, formed nodules i n the usual manner. 4. Conclusions from Tests The consequences of t h i s discovery w i t h regard to the eventual use of Pb-Pb j u n c t i o n s f o r charged p a r t i c l e d e t e c t i o n are s e r i o u s . High d e t e c t i o n e f f i c i e n c y demands that the f i l m s of the tunnel j u n c t i o n be as t h i c k as p o s s i b l e (see Chapter 8) and a maximum p r a c t i c a l f i l m thickness o of 1000 A would be a severe handicap. I t should a l s o be mentioned that these tests do not r u l e out the p o s s i b i l i t y that the nodules are an hydroxide of lead formed when the j u n c t i o n s are transported i n room a i r . Personal experience has shown that water has a very c o r r o s i v e e f f e c t upon t h i n l e a d f i l m s and other workers (eg. Rowell (1968) and Schroen (1968)) have emphasized the need of avoiding exposure of lead f i l m s to water vapour. As there i s no apparent reason why o t h i s process would d i s c r i m i n a t e against f i l m s thinner than 1000 A, i t may be that h y d r o l y s i s i s supplementary to the d i f f e r e n t i a l expansion-contraction mechanism suggested e a r l i e r , the p o i n t s of rupture being more s u s c e p t i b l e to c o r r o s i o n than the surrounding " o x i d i z e d " surface. The c o r r o s i v e e f f e c t of mercury on metals l i k e lead i s a l s o w e l l known (see eg. S m i t h e l l s 1967) and i t i s p o s s i b l e , though u n l i k e l y , that mercury contamination i n the evaporator may not be n e g l i g i b l e i n t h i s r e s p e c t . (Although the evaporator used f o r these samples had an o i l d i f f u s i o n pump, hardware from a mercury d i f f u s i o n pumped system had been temporarily i n s t a l l e d i n i t at an e a r l i e r time, and the p o s s i b i l i t y of -208-r e s i d u a l mercury vapour contamination e x i s t s . ) C. Oxide Growth on Lead Films One question which i s not yet understood and upon which the preceding work sheds no l i g h t i s the reason why i t proved impossible to grow a continuous l a y e r of oxide on the lead f i l m s . Compounding the problem i s the f a c t that few data are a v a i l a b l e on the o x i d a t i o n of lead at temperatures below i t s m e l t i n g p o i n t . The purpose of t h i s s e c t i o n i s to c i t e some of the p u b l i c a t i o n s which are a v a i l a b l e and give a very b r i e f summary of t h e i r f i n d i n g s which are r e l e v a n t to tunnel j u n c t i o n f a b r i c a t i o n . o Anderson and Tare (1964) prepared t h e i r 2500 A t h i c k lead f i l m s by evaporation from a w e l l outgassed 99.999% pure lead charge at the o r e l a t i v e l y slow r a t e of about 250 A/min. Doses of oxygen were subsequently i n j e c t e d i n t o the evaporating v e s s e l and the i n i t i a l uptake of the oxygen by the f i l m s was found to be very r a p i d , (complete i n l e s s than one minute). An important r e s u l t was that the presence of water vapour or carbon d i o x i d e i n the o x i d i z i n g atmosphere had l i t t l e e f f e c t on the o x i d a t i o n r a t e and none on the oxide s t r u c t u r e which, as t e s t e d by e l e c t r o n - d i f f r a c t i o n techniques, was found to be orthorhombic lea d monoxide (PbO). Apart from the slow o evaporation r a t e used by these workers compared to the 1000 A/min r a t e of the present experiment, no s p e c i a l precautions appear to have been taken to ensure the growth of PbO that have been omitted i n work here. A l a t e r paper by the same authors (1965) extends the work on the o x i d a t i o n r a t e of pure lea d f i l m s and i n v e s t i g a t e s the r e d u c t i o n i n r a t e brought about by adding to the lead i m p u r i t i e s such as bismuth and t h a l l i u m . Several papers d e a l i n g g e n e r a l l y and t h e o r e t i c a l l y w i t h o x i d a t i o n problems i n metals are a v a i l a b l e as w e l l . (see eg. the four a r t i c l e s by T. N. Rhodin et a l , J . F. Chittum, D. W. Pashley and H. Fischmeister i n a conference report arranged by Benard, 1965). Of p a r t i c u l a r usefulness i s a review by U h l i g (1967) which i n c l u d e s a f a i r l y extensive b i b l i o g r a p h y . B r i e f l y , he s t a t e s that the i n t e r a c t i o n of oxygen w i t h a c l e a n metal surface f o l l o w s the sequence (1) p h y s i c a l adsorption of 0^ followed i n most instances by d i s s o c i a t i o n of 0^ and chemisorption of 0 atoms, (2) n u c l e a t i o n of metal oxides at d i s c r e t e s i t e s and (3) formation and growth of a continuous oxide l a y e r . The time required f o r t h i s to happen i s q u i t e short. For example, -209-a continuous Cu^O l a y e r i s formed i n about 6 sec i n 1 Torr of 0^ at 550°C; even s h o r t e r times are required at higher pressures or lower temperatures. C e r t a i n l y the 15-20 hours during which the lead f i l m s of t h i s experiment were heated to 100°C and exposed to 500 Torr of 0^ would seem to be ample time to form a continuous oxide l a y e r . The glow-discharge o x i d a t i o n technique described by Schroen (1968) i s perhaps the best way to avo i d the u n c e r t a i n t i e s inherent i n thermally grown oxides. While admi t t i n g that present t h e o r e t i c a l understanding of the processes i n v o l v e d i s based upon t e n t a t i v e models, Schroen claims that t h i s plasma method has been r e f i n e d to the p o i n t that r e p r o d u c i b l e and s t a b l e tunnel j u n c t i o n s are r o u t i n e l y made. -210-APPENDIX D EFFECT OF FINITE FILM RESISTANCE ON TUNNEL JUNCTION CHARACTERISTICS A. I n t r o d u c t i o n This appendix, of concern p r i m a r i l y to those who are i n t e r e s t e d i n problems connected w i t h t h i n f i l m tunnel j u n c t i o n f a b r i c a t i o n , describes two l i t t l e - p u b l i c i z e d e f f e c t s which were observed e a r l y i n the experiment as w e l l as the t h e o r e t i c a l and experimental steps taken i n attempting to understand these e f f e c t s . Though an i n t e r e s t i n g s i d e l i g h t , the problems discussed here are not germane to an understanding of the tunnel j u n c t i o n charged p a r t i c l e d e t e c t o r . The f i r s t e f f e c t observed (Section B) was that many c r o s s e d - f i l m samples, when te s t e d w i t h the conventional 4-terminal technique sketched i n f i g u r e D-l(a) d i s p l a y e d an I-V curve w i t h negative slope. Pedersen and Vernon (1967) observed s i m i l a r behaviour i n the p a r a l l e l - f i l m type of j u n c t i o n (Sn-Sn) sketched i n f i g u r e D - l ( b ) . The second e f f e c t discovered was that the slope of the I-V curve depended on <|>, the angle of i n t e r s e c t i o n of the f i l m s . T h e o r e t i c a l ( S e c t i o n C) and experimental model s t u d i e s (Sections D and E) of these phenomena show that they may be a t t r i b u t e d to the f i n i t e r e s i s t a n c e of the t h i n f i l m s c o n s t i t u t i n g the j u n c t i o n . B. Experimental Observations of E f f e c t s w i t h Tunnel Junctions 1. "Negative Resistance" Figure D-l(c) shows the I-V c h a r a c t e r i s t i c s obtained w i t h two t y p i c a l Sn-Sn c r o s s e d - f i l m tunnel j u n c t i o n s ; s i m i l a r r e s u l t s were obtained w i t h Al-Pb and Pb-Pb j u n c t i o n s . I t i s evident that at c e r t a i n temperatures the slope of the I-V curve i s negative, (with I and V measured as shown i n f i g u r e D - l ( a ) ) , so that the j u n c t i o n e x h i b i t s "negative r e s i s t a n c e . " In a d d i t i o n , the magnitude of the slope i s temperature dependent so that the j u n c t i o n r e s i s t a n c e goes from a l a r g e negative value at 296 K, through -211-#1 V (mV) #11 V n T > V ) Figure D - l ( c ) : T y p i c a l I -V C h a r a c t e r i s t i c s of 2 Sn Tunneling Junctions | -212-zero around 4 K to a small p o s i t i v e average value at 3 K and below. These r e s u l t s are incompatible w i t h the simple one-dimensional e q u i v a l e n t c i r c u i t o f t e n used to describe four t e r m i n a l measurement of tunnel j u n c t i o n s as shown i n f i g u r e D-2. In the f i g u r e , and R 2 a r e the I A - v V ^ / --AVvV- D ~1 ?R 2 - W W -iR2 -o< B Figure D-2 Four-Terminal Equivalent C i r c u i t of Tunnel J u n c t i o n r e s i s t a n c e of t h i n f i l m s 1 and 2 r e s p e c t i v e l y and R^ i s the t o t a l tunneling r e s i s t a n c e . For the p o l a r i t i e s shown, which are i d e n t i c a l w i t h those of f i g u r e D - l ( a ) , i t i s c l e a r that the measured j u n c t i o n r e s i s t a n c e Rj = Vp-B^c-A = R T > 0 f o r a 1 1 t e m P e r a t u r e s T > T c - (Below the t r a n s i t i o n temperature T^, the I-V c h a r a c t e r i s t i c s of tunnel j u n c t i o n s composed of superconducting f i l m s have small regions of negative slope which a r i s e from the r e l a t i v e magnitudes of the energy gaps of the superconductors i n v o l v e d (see eg. Figure 2-6); such e f f e c t s are not of i n t e r e s t i n the present d i s c u s s i o n ) . As there i s no way i n which the c i r c u i t of f i g u r e D-2 can e x p l a i n the r e s u l t s shown i n f i g u r e D-l(c) a more s o p h i s t i c a t e d model i s req u i r e d : the t h e o r e t i c a l approach i s discussed i n s e c t i o n C and the experimental approach, c o n s i s t i n g of j u n c t i o n s i m u l a t i o n s t u d i e s , i s discussed i n s e c t i o n D. Several other features i n the I-V curves of f i g u r e D-l(c) should be noted. Breaks appearing i n the I-V curve at 3 K between 0.6 and 1.1 mV are caused by the f i l m s being temporarily d r i v e n normal by the measuring cu r r e n t . The r e v e r s a l of slope shown to occur at about 1.2 mV f o r sample I I i s caused by the measuring current having d r i v e n the f i l m s completely normal. The f a c t that the r e s i s t a n c e i s approximately the same at 1.25 K and 3.0 K f o r both specimens i n d i c a t e s m e t a l l i c f i l a m e n t s were s h o r t i n g out the i n s u l a t i n g l a y e r so that only a very small component of the current f l o w i n g between the f i l m s was due to tun n e l i n g . I t must be emphasized, however, that w h i l e the s o - c a l l e d "negative" r e s i s t a n c e i s -213-a s s o c i a t e d w i t h low j u n c t i o n r e s i s t a n c e , as shown i n s e c t i o n C, i t i s not confined s o l e l y to j u n c t i o n s i n which m e t a l l i c shorts are present. Evidence of t h i s comes from the r e s u l t s of Pedersen and Vernon (1967) whose j u n c t i o n s d i s p l a y e d conventional t u n n e l i n g behaviour below the t r a n s i t i o n temperature. 2. S t r i p I n t e r s e c t i o n Angle-Dependence of Slope Let <K^ 90°) be the angle of i n t e r s e c t i o n of the two t h i n f i l m s t r i p s s e r v i n g as current terminals (see f i g u r e D - l ( a ) ) . Thus, depending on the choice of terminals used i n the measurements, a given j u n c t i o n may be c h a r a c t e r i z e d by two r e s i s t a n c e s , R C<(>) and Py. (180-<J>). I d e a l l y , w i t h reference to f i g u r e s D-l(a) and D-2, ^ ( 0 ) = R..(180-<|>) as the e f f e c t i v e r e s i s t a n c e should be the t u n n e l i n g r e s i s t a n c e R^ , which depends on the overlap area and i s independent of the choice of current t e r m i n a l s . E x p e r i m e n t a l l y , i t was found that at room temperature, R. (<)>) was not equal to R.. (180-<j>). Table D-l summarizes the experimental r e s u l t s where p = Rj (<J) l a r g e ) / R j (<}> small) i s introduced as a convenient measure of the j u n c t i o n r e s i s t a n c e asymmetry. This angle dependence of the r e s i s t a n c e was i n v e s t i g a t e d w i t h an experimental model study, the r e s u l t s of which are discussed i n s e c t i o n E. Sample (Pb-Al) No. R. (ft) J <j> = 125° Rjft) $ = 55° R.Ofr = 125°) P R. (<j> = 55°) J 66-3 -.2 -.008 25 66-11 -.2 -.004 50 66-15 -.1 +.01 -10 66-17 + .24 +.3 .8 66-18 -.2 -.003 66 66-19 -.1 +.01 -10 66-21 + .11 +.17 .65 66-29 +.016 + .03 .53 16 others .3+130 .3+130 1.0 Table D - l : Dependence of J u n c t i o n Resistance on <j> C. T h e o r e t i c a l I n v e s t i g a t i o n s The simplest j u n c t i o n to consider i n d e t a i l i s the p a r a l l e l - f i l m type f o r , i f uniform thickness and width are assumed f o r the f i l m s and -214-i n s u l a t o r , the a n a l y s i s i s one-dimensional. I t turns out that the behaviour of t h i s j u n c t i o n can be described a n a l y t i c a l l y whereas the behaviour of the c r o s s e d - f i l m j u n c t i o n must be i n v e s t i g a t e d n u m e r i c a l l y . As shown i n f i g u r e D - l ( b ) , the current I i s fed i n t o t e r m i n a l A on f i l m 1 and removed from t e r m i n a l C on f i l m 2 w i t h the j u n c t i o n voltage V being measured from t e r m i n a l D to B. This case, which corresponds to <}> = 180° ( c f . f i g u r e s D-l(a) and ( b ) ) , has been analyzed by Pedersen and Vernon (1967) who f i n d that R_. = V j / I i s given by R..(180o) =a£Rr[csch(a£) + 2R ; [R 2(R 1 + R 2 ) ~ 2 t a n h (aA/aJ-R^/ ( R ^ R ^ ; (D-l) where A i s the le n g t h of the j u n c t i o n and 2 a = g ( r x + r 2 ) R-. = r, A 1 1 R 0 = r„A 2 2 . . -1 Here, g i s the tunnel conductance per u n i t length and r ^ , r 2 are the r e s i s t a n c e s per u n i t length of f i l m s 1 and 2 r e s p e c t i v e l y ; R i s the t o t a l t u n n e l i n g r e s i s t a n c e . F i r s t of a l l , i t i s c l e a r that when both f i l m s are superconducting, R^ = R 2 = 0 = a, the measured r e s i s t a n c e R. becomes the a c t u a l tunneling r e s i s t a n c e R^,. Obviously, f o r f i n i t e values of R^ and R 2 > R ^ Rp and furthermore, because of the negative term, there w i l l e x i s t values of R^ and R 2 f o r which R i s negative. The e f f e c t of f i n i t e f i l m r e s i s t a n c e on R i s more r e a d i l y observed i f equation D-l i s s i m p l i f i e d to the case of a symmetric j u n c t i o n , i e . R = R 2 = R. Then otA = (R/2R T) and Rj(sym) = (RRT/2)£ coth (R/2Rj,)^ - R/2, <j>=180° (D-2) -215-The p l o t of equation D-2, i n f i g u r e D-3 shows that f o r R / R ^ = 2.88, R^(sym) i s zero and, f o r l a r g e r r a t i o s of R / R ^ , R^ (sym) i s negative. The v a l i d i t y of t h i s model was checked by Pedersen and Vernon w i t h e x c e l l e n t agreement being obtained between the experimentally measured t h i n f i l m j u n c t i o n r e s i s t a n c e and that c a l c u l a t e d from equation D-l using experimental values f o r R^, R^ and R^' Another j u n c t i o n c o n f i g u r a t i o n that may be t r e a t e d as a one dimensional problem i s the cj> = 0° case ( c f . f i g u r e D-l(b)) when the current I i s fed i n t o t e r m i n a l A on f i l m 1 and removed from terminal B on f i l m 2 w i t h the j u n c t i o n v o l t a g e V_. being measured from t e r m i n a l D to C. Using the same equivalent c i r c u i t as Pedersen and Vernon (1967) and appl y i n g the appropriate boundary c o n d i t i o n s y i e l d s , f o r a symmetric j u n c t i o n R. = R „ = R, J 1 2 * V° 0 ) = V 1 = (2RV* c s c h (2R/V* ( D _ 3 ) Equation D-3, p l o t t e d i n f i g u r e D-3, shows that the measured r e s i s t a n c e R^ becomes the tu n n e l i n g r e s i s t a n c e as R-H) and that R . -K) i n the l i m i t of la r g e f i l m r e s i s t a n c e . U n f o r t u n a t e l y , i t i s not p o s s i b l e to deal w i t h the c r o s s e d - f i l m j u n c t i o n i n such a s t r a i g h t f o r w a r d manner as the lumped equivalent c i r c u i t i s a three-dimensional network. Consequently, the vo l t a g e V(x,y) between the f i l m s , cannot be expressed i n a simple, closed form and i t would be necessary to r e s o r t to some form of numerical a n a l y s i s such as the F i n i t e D i f f e r e n c e or the Monte Carlo method. (Binni s and Lawrenson, 1963). While d i f f e r i n g i n d e t a i l from those of the c r o s s e d - f i l m j u n c t i o n , the r e s u l t s of the <J> = 180° p a r a l l e l - f i l m case lea d to some general conclusions f o r l i m i t i n g cases concerning the e f f e c t of f i n i t e f i l m r e s i s t a n c e on the measured crossed f i l m j u n c t i o n r e s i s t a n c e . For R < < K - r r , » which implies a r e l a t i v e l y t h i c k i n s u l a t i n g l a y e r , f i l m s 1 and 2 are e f f e c t i v e l y i n s u l a t e d from one another so that terminals D and B have e s s e n t i a l l y the same p o t e n t i a l as A and C r e s p e c t i v e l y making R . = (V - V D ) / I > 0; f o r R >> R „ , f i l m s 1 j u a l and 2 are v i r t u a l l y a s i n g l e sheet w i t h , as i s confirmed experimentally i n -216--217-the f o l l o w i n g s e c t i o n , t e r m i n a l D being negative w i t h respect to B such that R = (V D - V B ) / I < 0. D. Experimental S i m u l a t i o n of Crdssed-Film Junctions 1. Graphite Coated Paper To simulate the j u n c t i o n i n which R >> R^ ,, a sheet of c o l l o i d a l g r a p h i t e coated paper, which i s conducting on one s i d e , was cut i n t o a shape resembling two f i l m s " c r o s s i n g " at an acute angle. (see f i g u r e D-4; t h i s model corresponds i n f a c t to R^ = 0 ) . E q u i p o t e n t i a l l i n e s were p l o t t e d by supplying a steady current as shown and measuring the voltage at every i n t e r s e c t i o n of a reasonably f i n e g r i d l a i d out on the graphite s u r f a c e . The distance from the i n t e r s e c t i o n to the end of the " f i l m s " was about 10 times the " f i l m " width so that by t a k i n g care to place the current e l e c t r o d e s c e n t r a l l y on terminals A and C a uniform current flow was obtained. Voltage was measured w i t h a Fluke D i f f e r e n t i a l voltmeter, Model 881A, having e s s e n t i a l l y i n f i n i t e input impedance at the n u l l p o i n t . I t i s c l e a r from f i g u r e D-4 t h a t R. = (V_ - V_,)/I. < 0. A r e s i s t a n c e of the same s i g n but j u a A—L greater magnitude was obtained by passing the current from A to B and grounding t e r m i n a l C. For t h i s " j u n c t i o n " = R(102°) = -135ft =  P R(78°) -99ft ' * This simple t e s t provides f u r t h e r c o n f i r m a t i o n that the reason that a c t u a l tunnel j u n c t i o n s d i s p l a y e d an angle dependent negative r e s i s t a n c e (Table D-l) was because t h e i r tunnel r e s i s t a n c e was s u f f i c i e n t l y s m a l l as to make them look e l e c t r i c a l l y l i k e a s i n g l e conducting sheet. 2. Soldered Manganin S t r i p s As a c l o s e r approximation to a r e a l specimen i n which R >> Rj,, two t h i n s t r i p s of manganin (6 x 1 x .012 i n . ) were soldered together at t h e i r middle region w i t h care being taken to make the solder l a y e r as uniform as p o s s i b l e . (A rough estimate shows R/RT = 10 f o r t h i s " j u n c t i o n " ) ' '• E q u i p o t e n t i a l l i n e s mapped out using the technique o u t l i n e d above were s i m i l a r to those of the conducting sheet. Thus, the p r e d i c t i o n s of the p a r a l l e l - f i l m s t r u c t u r e i n the region of very small t u n n e l i n g r e s i s t a n c e are borne out i n models of c r o s s e d - f i l m j u n c t i o n s . -218--219-3. Compressed Nichrome S t r i p s In order to study the behaviour of c r o s s e d - f i l m j u n c t i o n s f o r other values of R/R^ ,, a l u c i t e j i g was prepared i n which two nichrome s t r i p s (1/8 x 4 x .021 i n . ) could be constrained to l i e one on top of the other and i n t e r s e c t at an angle of 53°. The " t u n n e l i n g r e s i s t a n c e " between the two s t r i p s could then be v a r i e d by e x e r t i n g pressure normally on the i n t e r s e c t i o n w i t h a s m a l l b a k e l i t e plunger. (see f i g u r e D-5) (A Budd S t r a i n I n d i c a t o r , Model P-350, was used to measure the load developed by the departmental workshop h y d r a u l i c press.) Before being placed i n the j i g , the surfaces of the s t r i p s which were to be superimposed were cleaned and p o l i s h e d w i t h f i n e carborundum paper (No. 4/0). The nichrome s t r i p " j u n c t i o n " r e s i s t a n c e R. was measured J w i t h the 4-terminal network shown i n f i g u r e D-5 f o r the two c o n f i g u r a t i o n s cb = 53° and cb = 127°. I t was found that R. = V_ „/l. „ was p o s i t i v e f o r j D-B A-C s m a l l loads on the plunger and negative f o r l a r g e r loads i n both c o n f i g u r a -t i o n s . To understand t h i s r e s u l t q u a l i t a t i v e l y , i t i s assumed that the e f f e c t i v e " t u n n e l i n g r e s i s t a n c e " R^, between the two s t r i p s i s i n v e r s e l y r e l a t e d to the load p r e s s i n g the surfaces together and c e r t a i n l y much more s e n s i t i v e to changes i n pressure than the bulk r e s i s t a n c e R of the nichrome s t r i p s . For l a r g e loads (small R T ) , the s t r i p s are e s s e n t i a l l y a continuous piece l i k e the g r a p h i t e coated paper and R.. i s negative; f o r small loads ( l a r g e R^), the s t r i p s are i n s u l a t e d from each other making R^  p o s i t i v e . The adequacy of the p a r a l l e l f i l m model to e x p l a i n these crossed nichrome s t r i p r e s u l t s q u a n t i t a t i v e l y was explored i n the f o l l o w i n g manner. Figure D-3 i n d i c a t e s that i n the l i m i t of cj> = 0, R j / R r p > 0 f o r a l l R/RT and i n the l i m i t <j> = 180°, R j / R T < 0 f°r R j / R T > 2.88. I t seems reasonable to assume ther e f o r e that f o r s t r i p s c r o s s i n g at a r b i t r a r y <j>, (0 $ <j) $ 180°), the measured " j u n c t i o n " r e s i s t a n c e R.. may be expressed as a l i n e a r sum of the two l i m i t i n g cases (equations D-2 and D-3). Thus, i n s i m p l i f i e d form, B = (l-g(cb))[(A/2)* coth(A/2)* -A/2] + B($)(2A)* csch(2A)* , (D-4) where A = R(<b)/RT> B = R j / R T a n c* the mixing parameter g(cb) l i e s i n the range 0 $ 6(cb) £ 1 w i t h l i m i t i n g values g(0) = 1 and 0(180) = 0. The e f f e c t i v e bulk s t r i p r e s i s t a n c e R - R(cb) i s taken to be a f r e e parameter -220-Press B a k e l i t e Plunger Nichrome S t r i p L u c i t e J i g Load c e l l Figure U-V. Pressed Nirhrnme S r r i n " J n n r r i rms" -221-because the current d i s t r i b u t i o n i n the s t r i p s i n the overlap region at a r b i t r a r y a) i s not known; i n the extreme cases, R(0) = R(180) = 1.9 mfi/sq. f o r the s t r i p s used. To r e l a t e equation D-4 to the simulated j u n c t i o n measurements i t i s necessary to make some assumption concerning the r e l a t i o n between Rj, and the compressing l o a d L. The form chosen, which i s simplest to t r e a t mathematically and yet has the c o r r e c t asymptotic behaviour, i s Rj, = k(o))/L where k=k(<j>) i s a parameter to be determined experimentally. (The <f> dependence of k, l i k e that of R(<J>), a r i s e s from the unknown angular dependence of the current d e n s i t y i n the " j u n c t i o n " region.) For a given lo a d L., t h e r e f o r e 1 A± = R(q ))L i/k(q)) and B± = R ( c f O l ^ / k ^ ) . With equal weight assumed f o r the experimental values B^(expt) - R ^ ( e x p t , <f>)L^/k(a>), equation D-4 was l e a s t squares f i t t e d to the data obtained f o r the two nichrome s t r i p " j u n c t i o n " c o n f i g u r a t i o n s . The values of the parameters 8, R and k which minimized ^ [ B ± ( e x p t ) - B ± ( t h e o r y ) ] 2 are given i n t a b l e D-2. • 3 R(mft) k(mft l b . ) 53° .96+.01 2.8+.1 71+2 127° .05+.02 3.3+.1 100+10 Table D-2 Least Squares F i t Parameters Figure D-6 summarizes,the r e s u l t s . The s o l i d curves represent equation D-4 evaluated f o r the values of 3 given i n t a b l e D-2. (For reference purposes, the l i m i t i n g behaviour of equation D-4 i s shown by the dashed curves.) The experimental r e s u l t s have been reduced, as shown on the graph, by the appropriate best f i t values of R ( t f > ) and k(a>). -222-^ = ( 1 - 8 ) [ ( R / 2 R T ) J c o t h(R/2R T) 2-R/2R T] + 3 W ^ ) * c s c h ( 2 R / R T ) J Figure D-6: Comparison of P a r a l l e l f i l m theory with Crossed Nichrome S t r i p " J u n c t i o n " data. -223-I t i s evident that t h i s simple model succeeds q u i t e w e l l i n i n t e r p r e t i n g the nichrome s t r i p " j u n c t i o n " measurements. P a r t i c u l a r l y s a t i s f y i n g are the r e s u l t s (1) that the e f f e c t i v e s t r i p r e s i s t a n c e R i s f a i r l y c o n s i s t e n t i n the two c o n f i g u r a t i o n s , being reasonably c l o s e to the value 1.9 mft/sq. c a l c u l a t e d f o r the p a r a l l e l f i l m geometry, and (2) that only a s m a l l change i n B from the cb = 0 and cb = 180° c o n f i g u r a t i o n values i s r e q u i r e d to f i t the 53° and 127° data. The important c o n c l u s i o n to be drawn from these s t u d i e s i s that the nichrome s t r i p " j u n c t i o n " can simulate, the behaviour of t h i n f i l m j u n c t i o n s and c l e a r l y demonstrate the i n t e r a c t i o n of the e f f e c t i v e f i l m and t u n n e l i n g r e s i s t a n c e s i n determining the s i g n and magnitude of the measured j u n c t i o n r e s i s t a n c e f o r a given t e r m i n a l c o n f i g u r a t i o n . The angular dependence of the j u n c t i o n r e s i s t a n c e i s merely another m a n i f e s t a t i o n of the same e f f e c t . E. Angular Dependence of the Thin F i l m J u n c t i o n Resistance This s e c t i o n demonstrates that the observed cb dependence of the magnitude of the t h i n f i l m j u n c t i o n r e s i s t a n c e (see Table D-l) may be under-stood from the nichrome s t r i p s i m u l a t i o n s . A convenient measure of the asymmetry i n the measured j u n c t i o n r e s i s t a n c e s due to angle dependence i s the q u a n t i t y p which i s defined i n s e c t i o n B and appears i n the r i g h t hand column of Table D - l . (In the l i m i t of R << R^, the f i l m s are e s s e n t i a l l y e q u i p o t e n t i a l s so that R.. i s expected to be independent of the choice of current and v o l t a g e terminals and p should tend to 1 as i t does f o r 16 of the specimens of Table D - l . In the l i m i t of R >> R j , the continuous sheet approximation, i t i s expected that p w i l l be d i f f e r e n t from 1, i t s p r e c i s e value depending on the j u n c t i o n geometry and t u n n e l i n g conductance.) By drawing smooth curves through the data obtained from two d i f f e r e n t nichrome s t r i p " j u n c t i o n s " (samples a and b ) , i t i s p o s s i b l e to f i n d p as a f u n c t i o n of the l o a d — s e e f i g u r e D-7. Three regions are i n evidence, depending on the s i g n of Rj(cb). The s i g n i f i c a n t part of t h i s r e s u l t i s t h a t , as set out i n Table D-3, a l l the v a r i o u s asymmetries observed i n a c t u a l t h i n f i l m j u n c t i o n s can be simulated by the nichrome s t r i p model. 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