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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 . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1963 M.A.Sc, U n i v e r s i t y o f 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 t h e Department of Physics  We a c c e p t t h i s t h e s i s as conforming t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1969  In  presenting  an  advanced  the  Library  I further for  this  thesis  degree shall  agree  scholarly  a t the U n i v e r s i t y make  that  purposes  h i s representatives.  of  this  written  i tfreely  permission  by  thesis  in partial  may  of  PHYS ICS Columbia  copying  b y t h e Head  It i s understood gain  Columbia,  f o r reference  f o rextensive  be g r a n t e d  for financial  The University of British V a n c o u v e r 8, Canada  of British  available  permission.  Department  f u l f i l m e n t of the requirements f o r  shall  that  n o t be a l l o w e d  and  that  Study.  of this  o f my  copying  I agree  thesis  Department o r or  publication  without  my  ii  ABSTRACT  A superconducting  t h i n f i l m t u n n e l j u n c t i o n (Sn-SnO -Sn) of -4 2 t o t a l t h i c k n e s s 4000 A, a r e a 7 x 10 cm and normal (4.2 K) r e s i s t a n c e o  77 mft was was  p r e p a r e d on a g l a s s s u b s t r a t e .  When c o o l e d t o 1.2  K the j u n c t i o n  b i a s e d a t 0.3 mV where, the Josephson s u p e r c u r r e n t h a v i n g been  suppressed  w i t h a magnetic f i e l d , t h e j u n c t i o n dynamic r e s i s t a n c e had i t s maximum v a l u e o f 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 t h e r e s u l t i n g p u l s e s induced i n the t u n n e l i n g c u r r e n t were observed have a m p l i t u d e s up to 19 times the p r e a m p l i f i e r - d o m i n a t e d level.  rms output  to  noise  | For purposes of a n a l y s i s , i t was  p u l s e had the form i ( t ) = i  assumed t h a t the induced c u r r e n t  e x p ( - t / t ) , t >, 0.  With t h i s form of the  !  c u r r e n t p u l s e and the known t r a n s f e r f u n c t i o n o f the 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, i t was  c a l c u l a t e d t h a t f o r a l l p u l s e s T = (1.38±.33)xl0 ^ sec  and t h a t f o r the l a r g e s t a m p l i t u d e p u l s e s , c o r r e s p o n d i n g to an energy l o s s AE  £ 2.75 MeV, i l a y i n the range 20 $ i " £ 26 a ' o ° o v a l u e o f 22 uA.  8.2  x 10  -3  uA w i t h a most p r o b a b l e r  With t h i s v a l u e of i  and AE = 2.75 MeV, an upper l i m i t of ° ot eV has been a s s i g n e d to t h e ' v a l u e .of'w(Sn), the average energy  expended by the a l p h a 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 superc o n d u c t i n g t i n a t 1.2  K.  A t e n t a t i v e t h e o r y of the s u p e r c o n d u c t i n g  t u n n e l 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 i s g i v e n and the c r y o g e n i c and e l e c t r o n i c apparatus r e q u i r e d f o r the measurements are d e s c r i b e d . 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 t e c h n o l o g y i n t e r p r e t a t i o n of dc e x p e r i m e n t a l r e s u l t s are d i s c u s s e d i n f o u r  and  appendices.  iii  TABLE OF CONTENTS  Page Chapter 1 - INTRODUCTION  ......  A.  Motivation  B.  P r i n c i p l e of Operation . . ,  1 , . . .  1.  Phenomenological  2.  M i c r o s c o p i c Viewpoint  3.  D i f f e r e n c e from I n t e r m e d i a t e S t a t e Bolometers  C.  Expected  D.  Noise  E.  Preview of R e s u l t s  F.  Thesis Outline  1  Treatment  4 4 6  . .  Signal  7 7  ,  .  9 9 11  Chapter 2 - THEORETICAL ASPECTS OF.THE SUPERCONDUCTING TUNNEL JUNCTION  13  A.  Introduction  13  B.  T u n n e l i n g Between Normal M e t a l s  13  C.  D.  1.  Tunneling Current  . . . . .  2.  Dependence o f Tunnel C u r r e n t upon B a r r i e r Parameters  Review o f S u p e r c o n d u c t i v i t y Theory  15 18 21  1.  Two F l u i d Model  . .  2.  Energy Gap . . .  22  3.  D i s t r i b u t i o n Function for Quasiparticles  27  4.  Density of States  27  S i n g l e Q u a s i p a r t i c l e Tunneling  21  . .  28  iv 1.  Page  T u n n e l i n g Between a Normal M e t a l and a Superconductor  E.  F.  28  2.  T u n n e l i n g Between Two Superconductors  3.  The S ~ S 1  2  . .  29  Tunneling Current  30  Josephson T u n n e l i n g  32  1.  U n f a v o u r a b l e A s p e c t s o f t h e Josephson E f f e c t  2.  Theory o f t h e dc Josephson E f f e c t  3.  E f f e c t o f Magnetic  4.  S u p p r e s s i o n o f Josephson S u p e r c u r r e n t  ...  33 35  F i e l d on dc Josephson C u r r e n t  M u l t i p a r t i c l e T u n n e l i n g Phenomena  .  36  . . . . . . .  39  . .  39  Chapter 3 - OPERATING PRINCIPLES OF THE SUPERCONDUCTING CHARGED PARTICLE DETECTOR A.  Introduction  40 40  1.  P r e s e n t Methods o f Charged P a r t i c l e Spectrometry  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 o f Superconducting  B.  ...  Devices  .  . .  41  42  D e t e c t i o n o f Microwave R a d i a t i o n w i t h Tunnel J u n c t i o n s  .  45  1.  Microwave and I n f r a r e d Photon D e t e c t i o n . . . . . .  45  2.  Microwave Phonon D e t e c t i o n  48  i  C.  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  D.  Optimum J u n c t i o n Type and C o n s t i t u e n t M a t e r i a l  E. F.  . . . . . . . .  49  . . . . .  50  1.  Type o f J u n c t i o n t o be Used as a D e t e c t o r  50  2.  Type o f Superconductor i  58  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 Small S i g n a l Equivalent C i r c u i t  62 .  63  1.  G e n e r a l Treatment o f 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 o f S m a l l S i g n a l Parameters  67  G.  v  Page  Estimate of the Signal Size  68  1.  M a c r o s c o p i c o r P h e n o m e n o l o g i c a l Approach  69  2.  "Microscopic"  71  Approach  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  K.  ;• •i  77  1.  Shot N o i s e on B i a s i n g ; C u r r e n t  78  2.  Johnson N o i s e  78  3.  Generation-Recombination Noise  4.  V a r i a t i o n s i n Insulator Thickness .  78  5.  Fluctuations i n Signal  79  ...  Summary  78  79  Chapter 4 - CRYOGENIC APPARATUS . .  81  A.  Introduction  .|  ,  81  B.  Dewars and Dewar Cap  81  C.  Pumps  84  D.  Pressure-Temperature  E.  Sample Mounts  Measurements  84 r  . . .  Chapter 5 - ELECTRICAL MEASUREMENT TECHNIQUES A.  84  89  dc Measurements  89  1.  Introduction  . . .  89  2.  Power Supply  3.  Shielding  4.  D i s p l a y o f I-V C h a r a c t e r i s t i c s  89 ;.  B.  H e l m h o l t z C o i l s f o r Magnetic B i a s i n g  C.  Pulse Detection  Electronics .  .  89 91  . . . .  91 93  vi  Page  1.  B i a s i n g and P u l s e 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.  Ancillary Electronics  96  D.  I n t e r f e r e n c e V e t o i n g 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 Line  99  F.  1.  Introduction  . .  99  2.  E f f e c t i v e L i n e Impedance  101  3.  L i n e 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.  C o n s i s t e n c y Check o f Measured Values w i t h Theory. .  107  6.  Propagation V e l o c i t y  107  :  . . . . .  Summary  110  Chapter 6 - RESULTS A.  i  I l l  dc C h a r a c t e r i s t i c s  I l l  1.  Temperature Dependence o f T u n n e l i n g C u r r e n t . . . .  I l l  2.  Magnetic F i e l d Dependence o f T u n n e l i n g C u r r e n t  I l l  3.  D e t e r m i n a t i o n o f Dynamic R e s i s t a n c e (8V/8I) as a  . .  Function of Voltage  B.  C.  4.  D e t e r i o r a t i o n o f Specimens a f t e r Thermal C y c l i n g  5.  Magnetic F i e l d Dependence of Energy Gap  O b s e r v a t i o n o f P u l s e s from A l p h a P a r t i c l e s  .  ........  117 119 121  1.  Counting of Pulses  121  2.  Pulse Characteristics . .  125  3.  P u l s e H e i g h t Spectrum . .  127  Noise 1.  129 Observed N o i s e  .  ;  '  2. O r i g i n o f N o i s e . Determination of J u n c t i o n Capacitance t  D.  115  129 129 136  vii  E.  Page  1.  P a r a l l e l P l a t e Model.  136  2.  N o i s e Measurements  138  Summary  143  Chapter 7 - ANALYSIS OF RESULTS  145  A.  Introduction  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  2.  Laplace Transform Representation  3.  Inverse Transform  4.  J u s t i f i c a t i o n o f Approximate T r a n s m i s s i o n  146 . .  150 Line  Treatment C.  D.  E.  Determination  152  o f C u r r e n t P u l s e Parameter x  152  1.  E x t r a c t i o n of t h e x ^ and o\ from Photographs  2.  M* C a l c u l a t i o n . .  154  3.  V a l i d i t y o f t h e Assumed Input C u r r e n t P u l s e . . . .  157  . . .  Estimate of Current Pulse Amplitude ( i ^ ) -  153  157  1.  T r a n s f e r F u n c t i o n f o r Input C u r r e n t Step P u l s e  . .  159  2.  E v a l u a t i o n o f GR' . . . . .  , .  160  3.  Evaluation of i o  . .. .  160  :  Energy Loss p e r Q u a s i p a r t i c l e . . .  161  1.  Number o f Q u a s i p a r t i c l e s Produced  2.  Energy L o s s Corresponding  3.  161  t o Maximum A m p l i t u d e  Pulse  F.  149  . . .  Estimate of w  164 165  Energy Loss and D i f f u s i o n . . . 1.  Energy Loss P r o c e s s e s  . . . . .  2. 3.  Energy T r a n s f e r . . . . . . . Evidence f o r Heat C o n t r i b u t i o n from S u b s t r a t e . . .  166 166 168 170  viii G.  Chapter  Page  Summary  172  8 - CONCLUSIONS  174  A.  Major R e s u l t s  174  B.  E v a l u a t i o n o f t h e D e v i c e as a N u c l e a r Spectrometer  C.  . . .  175  *  175  1.  Energy R e s o l u t i o n  2.  Linearity  176  3.  Stopping E f f i c i e n c y  177  4.  Advantages over C o n v e n t i o n a l Spectrometers  5.  Disadvantages  f o r O p e r a t i o n as a Spectrometer  ....  177.  . . .  178  .  178  1.  Use o f J u n c t i o n s w i t h L a r g e r Dynamic R e s i s t a n c e . .  178  2.  Thermal D e c o u p l i n g of J u n c t i o n from S u b s t r a t e . . .  179  3.  P r e a m p l i f i e r Improvements  179  F u t u r e Work  Appendix A - JUNCTION PREPARATION .-  180  A.  Substrate Preparation .  180  B.  E v a p o r a t i o n Procedure  180  C.  . . . .  1.  Base F i l m  180  2.  Oxidation  182  3.  Top F i l m E v a p o r a t i o n  182  4.  Attachment o f E l e c t r i c a l Leads and Mounting . . . .  182  E v a p o r a t i o n Apparatus  183  1.  Masks  183  2.  S u b s t r a t e Holder  3.  E v a p o r a t i o n Sources .  4.  F i l m T h i c k n e s s Measurement  . . . . . ;  .  183 185 185  ix  Page  Appendix B - dc CHARACTERISTICS OF LEAKY SPECIMENS  188  A.  Introduction  188  B.  Results  188  1.  I-V C h a r a c t e r i s t i c  188  2.  M a g n e t i c F i e l d Dependence o f S u p e r c u r r e n t  190  C.  D.  Analysis 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  Unresolved  .  Problems i n t h e A n a l y s i s . .  1.  Excess C u r r e n t s  .  2.  S t r u c t u r e near V = 0  197 -.  B.  C.  197 199  Appendix C - OBSERVATIONS ON Pb-Pb TUNNEL JUNCTIONS A.  196  Tunnel J u n c t i o n R e s u l t s ,  201 201  1.  Junction Preparation  201  2.  Low Temperature dc C h a r a c t e r i s t i c s  202  3.  Conclusions  202  From Pb-Pb J u n c t i o n S t u d i e s  I n v e s t i g a t i o n of Nodule Growth on Lead F i l m s  203  1.  D e s c r i p t i o n o f Nodules. . . i  203  2.  O r i g i n o f N o d u l e s — T h e R e s u l t o f Thermal Treatment.  203  3.  T e s t s Made t o I s o l a t e N o d u l e - P r o d u c i n g Parameters .  205  4.  Conclusions  207  from T e s t s ;  Oxide Growth on Lead F i l m s  ,  208  Appendix D - EFFECT OF FINITE FILM RESISTANCE ON TUNNEL JUNCTION CHARACTERISTICS A.  Introduction  B.  Experimental  210  . . Observations  210 of E f f e c t s w i t h Tunnel J u n c t i o n s  210  x  Page  1.  "Negative R e s i s t a n c e "  210  2.  S t r i p I n t e r s e c t i o n Angle-Dependence of S l o p e . . . .  213  C.  Theoretical Investigations  213  D.  Experimental Simulation of Crossed-Film Junctions . . . .  217  E.  1.  G r a p h i t e Coated Paper  217  2.  S o l d e r e d Manganin S t r i p s  217  3.  Compressed Nichrome S t r i p s  219  A n g u l a r Dependence o f 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 . . .  Bibliography  ;  223  226  xi  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  for Several Materials . . .  2- 1  Comparison o f T h e o r e t i c a l and E x p e r i m e n t a l  3  Values  o f 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 . . . .  3-2  Comparison of F i g u r e s of M e r i t f o r Sn and Pb  3- 3  T u n n e l i n g P r o b a b i l i t y per sec f o r S e v e r a l T u n n e l i n g  57 61  Thicknesses  76  4- 1  Comparison o f T r a n s m i s s i o n L i n e 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 E x p e r i m e n t a l and T h e o r e t i c a l T r a n s m i s s i o n L i n e Parameters Comparison o f E x p e r i m e n t a l and T h e o r e t i c a l Maximum  108  Supercurrent  113  6- 2  Specimen C h a r a c t e r i s t i c s B e f o r e and A f t e r Thermal C y c l i n g  118  7- 1  Estimate of i  161  6- 1  f o r Maximum Amplitude P u l s e s o  D-l  Dependence o f J u n c t i o n R e s i s t a n c e on I  . .  213  D-2  L e a s t Squares F i t Parameters.  221  D-3  J u n c t i o n R e s i s t a n c e Asymmetry Observed  225  xii  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  Structure.  ...  16  2-3  I n s u l a t o r T h i c k n e s s v s Tunnel J u n c t i o n R e s i s t a n c e  ....  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  ....  2-6 2-7  31 31  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) 2-8(b) 3-1  37 Dependence of I  ....  37  Schematic Energy Diagrams D e p i c t i n g E f f e c t s of Microwave Photon o r Phonon A b s o r p t i o n and Phonon G e n e r a t i o n ....  47  m  a  x  Upon A p p l i e d Magnetic F i e l d B  3-2 3-3  52 Energy l o s s o f 5.1 MeV a - P a r t i c l e T r a v e r s i n g 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 V a r i o u s 65  3-5  S m a l l S i g n a l E q u i v a l e n t 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 5-4  Schematic of J u n c t i o n B i a s i n g and P u l s e D e t e c t i o n  Circuit  94 95  xiii Figure < 5-5  Page P r e a m p l i f i e r Input Impedance  97  5-6  B l o c k Diagram o f E l e c t r o n i c U n i t s  5-7  Schematic of Veto D i s c r i m i n a t o r  100  5-8  R e s i s t a n c e and Reactance of T r a n s m i s s i o n L i n e  103  5- 9  Impedance o f T r a n s m i s s i o n L i n e 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 "Sn Tunnel J u n c t i o n , B=100 G. 2  112  6-2  I-V C h a r a c t e r i s t i c s o f Sn-SnO^-Sn Tunnel J u n c t i o n f o r B i n P l a n e of J u n c t i o n  114  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 Field  98  Magnetic  ;  116  6-4  Dynamic R e s i s t a n c e vs V o l t a g e f o r Specimen J-5.  6-5  Magnetic  6-6  Detector B i a s i n g Conditions  122  6-7  J u n c t i o n - S o u r c e Geometry  124  6-8  P u l s e s Observed a t Optimum Bandwidth  124  6-9  Current S e n s i t i v i t y  126  6-10  P u l s e Amplitude vs Dynamic R e s i s t a n c e  6-11  P u l s e H e i g h t Spectrum  128  6-12  Output N o i s e v s J u n c t i o n Dynamic R e s i s t a n c e  130  6-13  Output N o i s e vs E x t e r n a l : C a p a c i t a n c e  130  6-14  E q u i v a l e n t C i r c u i t s of D e t e c t o r and P r e a m p l i f i e r w i t h Noise Generators Included  131  6-15  . . . . .  F i e l d Dependence of Energy Gap  116 120  Calibration .....  126  O v e r a l l E q u i v a l e n t C i r c u i t f o r D e t e c t o r and P r e a m p l i f i e r With N o i s e  134  6-16  E q u i v a l e n t C i r c u i t f o r S i g n a l to N o i s e R a t i o E s t i m a t e . .  134  6-17  E q u i v a l e n t 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 w i t h N o i s e Sources Only (no a l p h a p u l s e s ) 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 N o i s e Output vs Dynamic R e s i s t a n c e of J u n c t i o n . .  6- 18 7- 1  E q u i v a l e n t C i r c u i t of P u l s e D e t e c t i o n System  139 142 147  xiv Figure  Page  7-2  M* as F u n c t i o n o f x, t  7-3  90%  7-4  M* (minimum) v s R e l a x a t i o n Time x  158  7-5  P a r t i c l e T r a c k Geometry  167  7-6  T h e o r e t i c a l P u l s e H e i g h t ,Spectrum  167  A-l  Junction Preparation  181  A-r2  Evaporation  184  A-3  P h o t o m i c r o g r a p h of Sn-Sn Tunnel J u n c t i o n  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  B^2  Typical I  B-4  and C  C o n f i d e n c e Volume i n C  Jtr  B-3  o  Jf  T  155  t , T Space  156  J  Apparatus .  184 ....  . P l o t f o r "Leaky" Specimens crit  189 .  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 with M e t a l l i c Filaments. . Josephson C u r r e n t  P e r i o d v s ( J u n c t i o n Width)  A p p l i e d Magnetic F i e l d  191  192  in  ...  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  P h o t o m i c r o g r a p h s of Pb-Pb Tunnel J u n c t i o n  204  C-2  Photomicrograph  204  D-l  C r o s s e d - 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  Showing Nodules  C h a r a c t e r i s t i c s of 2 Sn Tunnel J u n c t i o n s  211  D-2  F o u r - T e r m i n a l E q u i v a l e n t 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 R e s i s t a n c e Resistance  (Rj)  (R) on Measured  Junction  ,  216  D-4  G r a p h i t e Coated Paper " J u n c t i o n "  D-5  Pressed  D-6  Comparison o f P a r a l l e l F i l m Theory w i t h Crossed  Nichrome  ...  Strip "Junctions"  " S t r i p " Data D-7  218  p v s Load f o r Nichrome  220 Nichrome 222  Strip "Junctions" '  224  XV  ACKNOWLEDGEMENTS  S p e c i a l thanks a r e 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 t h e d u r a t i o n o f t h i s p r o j e c t , e x p e c i a l l y d u r i n g times o f d e s p a i r 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 d u r i n g the e a r l y hours o f numerous mornings when many o f t h e measurements were made.  V a l u a b l e d i s c u s s i o n s w i t h t h e o t h e r members o f my s u p e r v i s o r y committee, D r s . R. E. B u r g e s s , G. Jones and P. W. Matthews, a r e acknowledged with gratitude.  The a s s i s t a n c e o f members o f t h e departmental,Van de G r a a f f 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 t h e  d e s i g n and c o n s t r u c t i o n o f a p p a r a t u s i s much a p p r e c i a t e d .  My h e a r t f e l t thanks go t o my w i f e L i n d a f o r h e r s u s t a i n i n g encouragement and h e r i n v a l u a b l e a s s i s t a n c e b o t h i n t h e t y p i n g o f t h e complete m a n u s c r i p t and t h e f i n a l p r e p a r a t i o n o f many o f t h e f i g u r e s . Mr. K. T a y l o r i s a l s o d e s e r v i n g o f thanks f o r h i s h e l p i n t h e p r e p a r a t i o n o f 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 r e c e i v e d from t h e N a t i o n a l Research C o u n c i l i n t h e form o f one b u r s a r y and t h r e e s t u d e n t s h i p s as w e l l as t h e c o n t i n u i n g s u p p o r t from t h e Van de G r a a f f group i n t h e form o f a r e s e a r c h assistantship i sgratefully  acknowledged.  The l o a n o f i n s t r u m e n t a t i o n from t h e B. C. V o c a t i o n a l School h e l p e d g r e a t l y w i t h some o f t h e impedance measurements.  i  -1-  CHAPTER 1  INTRODUCTION A.  Motivation Much of the p r e s e n t  n u c l e u s has been o b t a i n e d  knowledge of the s t r u c t u r e of the atomic  t h r o u g h 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 p r o d u c t s .  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 d a t a , c o n s i d e r a b l e 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 n u c l e a r p a r t i c l e spectrometry.  (See eg. Yuan, 1961;  D e a r n a l e y & N o r t h r o p , 1966). the i d e a l d e t e c t o r  of  Ajzenberg-Selove,  1960;  From the n u c l e a r p h y s i c i s t ' s p o i n t of v i e w ,  (or s p e c t r o m e t e r ) i s one w i t h h i g h 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 h a v i n g v e r y 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 t o 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 a t the d e t e c t o r at v e r y n e a r l y the same time, and  with  h i g h 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 " d e t e c t o r " i s o f t e n used synonymously w i t h " s p e c t r o m e t e r " . )  T h i s t h e s i s d e s c r i b e s the theory of  p r e l i m i n a r y r e s u l t s from a new,  fast, highly sensitive particle  i n the form of a s u p e r c o n d u c t i n g o f two  t h i n superconducting  (see f i g u r e 2-1).  The  and  detector  t u n n e l j u n c t i o n — a "sandwich" c o n s i s t i n g  f i l m s separated  by a v e r y t h i n i n s u l a t i n g l a y e r  d e t e c t o r 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 , w h i c h are c u r r e n t l y the  best  nuclear detectors a v a i l a b l e .  penetrates  When an e n e r g e t i c p a r t i c l e such as a p r o t o n or a l p h a  particle  a medium, i t l o s e s energy v i a a s e r i e s of i o n i z i n g  collisions  w i t h the atoms c l o s e to i t s p a t h .  F o r each g i v e n s u b s t a n c e , (gas or ;  i t i s c o n v e n i e n t t o d e f i n e a q u a n t i t y w,  solid),  the average energy w h i c h the charged  p a r t i c l e must l o s e i n t h a t 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 .  C o n s e q u e n t l y , i f an e n e r g e t i c  particle  -2l o s e s energy AE i n a s u b s t a n c e , 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 a m p l i t u d e ) i s N = AE/w. p a r t i c l e always had  I f the  energetic  t o 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  s u b s t a n c e , then a p e r f e c t d e t e c t o r made from t h a t substance c o u l d d i s t i n g u i s h between e n e r g i e s  AE^  = Nw  the d e t e c t o r would be w.  and AE^  = (N ± l ) w , and  the energy r e s o l u t i o n of  However, s i n c e w i s an average over a v e r y  number of e x c i t a t i o n s , most of w h i c h w i l l i n v o l v e energy l o s s e s from w,  large  differing  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  c e r t a i n energy i n p u t i s s u b j e c t t o f l u c t u a t i o n s . where the e x c i t a t i o n s are c o n s i d e r e d  I n the s i m p l e s t  a  case  to be i n d e p e n d e n t l y 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 t a k e n 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.  Hence, u n c e r t a i n t y of o r d e r  2  energy of a p a r t i c l e , f o r now case, which very o f t e n occurs,  v^/N  i s introduced  AE = wN(l  ±  v^/N).  i n determining  the  I n a more r e a l i s t i c  the e x c i t a t i o n p r o c e s s e s are not  independent,  but are c o r r e l a t e d by the f a c t t h a t the p a r t i c l e must l o s 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 g i v e n by F N N, 2  where F, the Fano' f a c t o r (Dearnaley and N o r t h r o p , 1966)  0 and  1 and,  by 1-10  MeV  f o r example, has  e l e c t r o n s i n germanium a t 78 K.  The viewpoint,  the v a l u e F $ 0.16  2  r a t h e r than  l i e s between  f o r e x c i t a t i o n s produced  (Mann et a l , 1966).  u n c e r t a i n t y d e c r e a s e s 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  the b e s t energy r e s o l u t i o n w i l l be o b t a i n e d  from the  detector  i n w h i c h f o r g i v e n energy l o s s , N i s ' l a r g e s t o r , i n o t h e r words, i n which w i s smallest. The  motivation  f o r c o n s i d e r i n g the s u p e r c o n d u c t i n g t u n n e l j u n c t i o n  as a p a r t i c l e d e t e c t o r i s apparent from the comparison of 7N/N media i n t a b l e Two  1-1. o t h e r 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 i n c l u d e d i n the t a b l e . (-dE/dx) and  for several  The  s p e c i f i c r a t e of energy l o s s  the range, c a l c u l a t e d f o r a 5 MeV  a l p h a p a r t i c l e , are  optimal  i n Pb which means t h a t to s t o p p a r t i c l e s of a g i v e n energy a s m a l l e r of Pb would be needed than of any  conventional  detecting material.  thickness Smaller  s i z e , of c o u r s e , g e n e r a l l y i m p l i e s f a s t e r r e s p o n s e . The not new;  i d e a of u s i n g s u p e r c o n d u c t o r s to d e t e c t n u c l e a r p a r t i c l e s i s  what i s n o v e l i s the e x p l o i t a t i o n of the f a v o u r a b l e  character-  i s t i c s of the s u p e r c o n d u c t i n g t u n n e l i n g j u n c t i o n f o r t h i s purpose.  -3Andrews e t a l (1949) bombarded a t h i n s t r i p o f niobium n i t r i d e , h e l d v e r y c l o s e t o i t s s u p e r c o n d u c t i n g t r a n s i t i o n temperature, w i t h a l p h a p a r t i c l e s  Material  w(eV)  N for E=l MeV  -dE/dx (MeV/cm)  Range (5 MeVa)  Hydrogen-gas  37  2.7  x  10  4  6.1xl0~  3  Krypton-gas  24  4.2  x  10  4  4.9xl0~  3  1.25  Silicon-semi-con.  3.6  2.8  x  10  5  1.9xl0~  3  1500  2.5 x  10"  3  Germanium-semi-con  2.9  3.5 x  10  5  1.7xl0  2000  1.5 x  10"  3  Tin*-super-con.  .003*  3.3 x  10  8  5.5xl0~  5  2300  1.6  x  10~  3  .005*  2.x  7.1xl0"  5  2600  1.3 x  10"  3  Lead*-super-con.  10  8  T a b l e 1-1: P a r t i c l e D e t e c t i o n Parameters *  15.5  cm  2.81  for Several Materials  The magnitude o f w used f o r the s u p e r c o n d u c t o r s i s based on the assumption t h a t , i n analogy w i t h s e m i c o n d u c t o r s , (Sherman, 1964) w may be 3-5 times the energy gap w i d t h — an assumption l a t e r shown t o 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 c h a p t e r and c h a p t e r 7.)  and observed strip.  - 3  .18  the r e s u l t i n g v o l t a g e p u l s e s developed a l o n g the l e n g t h of the 1  A s i m i l a r experiment  u s i n g evaporated narrow t h i n f i l m s r a t h e r than  b u l k s t r i p s were l a t e r proposed S p i e l e t a l (1965).  by Sherman (1962,B) and performed  by  When an a l p h a p a r t i c l e impinged on the superconductor  s m a l l , c y l i n d r i c a l r e g i o n s u r r o u n d i n g the: t r a c k was d r i v e n normal.  a  The  d i a m e t e r of t h i s r e g i o n i n c r e a s e d as a f u n c t i o n of time a t a r a t e determined by the ambient c u r r e n t c a r r i e d by the f i l m and the energy l o s t by the a l p h a p a r t i c l e , u n t i l i t became e q u a l to f i l m w i d t h ; a t 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 t r a n s v e r s e normal r e g i o n , a p u l s e observed  i n t h e v o l t a g e between the ends of the f i l m .  was  The b a s i c d i f f e r e n c e s  between the s u p e r c o n d u c t i n g s i n g l e s t r i p and the t u n n e l j u n c t i o n d e t e c t o r s w i l l be o u t l i n e d a t the end of the n e x t s e c t i o n ; more d e t a i l on the superc o n d u c t i n g s t r i p d e t e c t o r s i s g i v e n i n s e c t i o n A, c h a p t e r I t was  B u r s t e i n e t a l (1961) who  f i r s t proposed  3. and a n a l y z e d the  use o f t h e s u p e r c o n d u c t i n g t u n n e l j u n c t i o n s , which had r e c e n t l y been s t u d i e d i n the p i o n e e r i n g work o f G i a e v e r (1960,61), as d e t e c t o r s of e l e c t r o magnetic r a d i a t i o n .  Such a d e t e c t o r i s of g r e a t p r a c t i c a l i n t e r e s t because  the f r e q u e n c y range to which i t i s s e n s i t i v e , a p p r o x i m a t e l y 85 to 650  GHz,  -4i s v i r t u a l l y i n a c c e s s i b l e by c o n v e n t i o n a l d e t e c t i o n t e c h n i q u e s . processes  The two  o f 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 tunneling.  energy gap and " p h o t o n - a s s i s t e d "  I t t u r n s out t h a t t u n n e l j u n c t i o n s may a l s o s e r v e as g e n e r a t o r s  and d e t e c t o r s of microwave sound (Lax and Vernon, 1965; G o l d s t e i n e t a l , 1965; and Eisenmenger and Dayem, 1967) v i a an analogous p r o c e s s known as "phononassisted" tunneling. Taylor  A b r i e f r e v i e w o f these a p p l i c a t i o n s i s g i v e n by  (1968). The  scene was now s e t f o r work t o b e g i n on t h e s u p e r c o n d u c t i n g  t u n n e l 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 s i d e was t h e n u c l e a r  s p e c t r o s c o p i s t w i t h h i s c o n t i n u i n g demand f o r more s e n s i t i v e d e t e c t o r s ; on t h e o t h e r was t h e low temperature p h y s i c i s t w i t h a new d e v i c e  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 s t e p s i n an attempt t o adapt t h i s p o t e n t i a l and supply t h e demand a r e described i n this B.  thesis.  P r i n c i p l e of O p e r a t i o n 1.  1  P h e n o m e n o l o g i c a l Treatment For a g i v e n s u p e r c o n d u c t i n g  t h i n f i l m tunnel j u n c t i o n at  t e m p e r a t u r e 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 , t h e r e 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 t h e I-V c h a r a c t e r i s t i c — b e t w e e n t h e dc c u r r e n t I p a s s i n g from one f i l m t o t h e o t h e r via  e l e c t r o n t u n n e l i n g and the v o l t a g e V developed a c r o s s t h e i n s u l a t i n g  layer.  A t b i a s v o l t a g e s 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 t h e temperature dependent s u p e r c o n d u c t i n g (discussed i n chapter  energy gap w i d t h  2 ) , t h e c u r r e n t i s p r o p o r t i o n a l t o t h e d e n s i t y of  t h e r m 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 which depends a p p r o x i m a t e l y temperature a c c o r d i n g t o e x p ( - A ( T ) / k T ) . fi  Boltzmann's c o n s t a n t . )  (See a l s o f i g u r e 6-1; k i s g  T h i s statement n e g l e c t s t u n n e l c u r r e n t c o n t r i b u t i o n s  a r i s i n g from the t u n n e l i n g 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 a r e d i s c u s s e d i n Appendix B. present  upon t h e  N e g l e c t i n g f o r the  t h e temperature dependence of A, i t i s e v i d e n t t h a t any s t i m u l u s ,  (eg. t h e energy l o s t by a n u c l e a r charged p a r t i c l e i n t r a v e r s i n g o r coming to  r e s t i n a t u n n e l j u n c t i o n ) , w h i c h i n c r e a s e s t h e f i l m temperature w i l l  increase the tunneling current. question.  That such an e f f e c t e x i s t e d was not i n  What was i n q u e s t i o n , however, was whether or not t h e change i n  t u n n e l i n g c u r r e n t due t o a 5 MeV a l p h a p a r t i c l e , f o r example, was  observable  -5and,  i f s o , 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? B e f o r e g o i n g f u r t h e r , the r o l e p l a y e d by the energy s h o u l d be s t r e s s e d .  gap  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 t h a t A s h o u l d be as s m a l l as p o s s i b l e (A . ). mxn  On  the  o t h e r hand, the " l e a k a g e " or t h e r m a l q u a s i p a r t i c l e t u n n e l i n g c u r r e n t through the j u n c t i o n s h o u l d be m i n i m a l to reduce shot n o i s e and t h i s i m p l i e s t h a t A s h o u l d be l a r g e (A  ).  C l e a r l y , f o r a g i v e n l o w e s t o p e r a t i n g temperature,  a s u p e r c o n d u c t o r w i t h a compromise v a l u e r e q u i r e d , but A n  c  should favour A  max  degraded by a f a c t o r (1 + A /'A^  (A ) such t h a t A . < A < A is c min c max f o r the energy r e s o l u t i o n i s o n l y &  1  whereas the thermal c u r r e n t i s i n c r e a s e d  f r o m the o p t i m a l v a l u e by a f a c t o r e x p f H A ^ - A  )/kgT].  When a t u n n e l 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 , t h e p a r t i c l e s t r a v e r s e one or b o t h of the f i l m s g i v i n g up some of energy t o the f i l m s p r e d o m i n a n t l y The  v i a a s e r i e s of i o n i z i n g :  their  collisions.  amount of energy d e p o s i t e d d i r e c t l y i n the j u n c t i o n AE i s g i v e n by  AE = jf " (dE/dx)dx where l a y e r s and initial  i s the path l e n g t h through the  superconducting  (dE/dx) i s the s p e c i f i c r a t e o f energy l o s s w h i c h depends on  energy and type o f p a r t i c l e and  are prepared.  the  the m a t e r i a l from which the f i l m s  For purposes of a n a l y s i s , i t i s assumed t h a t the r e s u l t i n g  d i s t u r b a n c e 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 v a l u e i s a maximum immediately  f o l l o w i n g the passage of the a l p h a p a r t i c l e  and  t h a t 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 i n c r e a s e i n the t u n n e l i n g c u r r e n t  occurs  w h i c h , a f t e r a p p r o p r i a t e a m p l i f i c a t i o n and s h a p i n g , appears as a s i g n a l p u l s e superimposed on the ambient thermal may  be a n a l y z e d  and  the time at which i t was  electronics.  tunneling current.  This  pulse  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 produced, u s i n g s t a n d a r d n u c l e a r d a t a a c q u i s i t i o n  I t i s found e x p e r i m e n t a l l y  ( c h a p t e r 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 ( c h a p t e r 3) t h a t the s i g n a l t o n o i s e r a t i o f o r the o v e r a l l system ( d e t e c t o r p l u s a m p l i f i e r s ) i s a maximum when the dynamic resistance r = (3V/9I)  T  i s as l a r g e as p o s s i b l e .  The dc o p e r a t i n g c u r r e n t  o f the j u n c t i o n was  t h e r e f o r e chosen so t h a t ( 3 V / 8 I )  magnitude of I  not c r i t i c a l , as tjhe j u n c t i o n n o i s e sources were s m a l l  was  compared w i t h p r e a m p l i f i e r n o i s e .  T  was  Questions concerning  a maximum; the  the amplitude  and  d u r a t i o n of such a p u l s e 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  -6and d i s c u s s e d a t l e n g t h i n c h a p t e r s 3 and 2.  7.  Microscopic Viewpoint The p u r e l y phenomenological d i s c u s s i o n g i v e n i n the p r e c e d i n g  s e c t i o n i s n e c e s s i t a t e d by the extreme c o m p l e x i t y of the m i c r o s c o p i c p i c t u r e s k e t c h e d 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 t u n n e l j u n c t i o n l e a d s to the g e n e r a t i o n 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 a l o n g t h e p a r t i c l e t r a c k which exceed t h o s e p r e s e n t when the f i l m s a r e i n t h e r m a l e q u i l i b r i u m w i t h a c o n s t a n t temperature b a t h .  The energy i s t r a n s p o r t e d 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 t h a t the t r a n s p o r t p r o c e s s e s might be a d e q u a t e l y d e s c r i b e d by c l a s s i c a l heat d i f f u s i o n theory. the  As a consequence  of the temperature d i s t r i b u t i o n w h i c h o b t a i n s ' i n  superconducting f i l m s , excited q u a s i p a r t i c l e d e n s i t i e s N  w h i c h a r e i n excess o f the t h e r m a l e q u i l i b r i u m v a l u e .  are produced  q  The excess i s maximum  i m m e d i a t e l y f o l l o w i n g the p e n e t r a t i o n o f the a l p h a p a r t i c l e and i t decays back t o z e r o 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 a t l e a s t o  two c o m p l i c a t i n g f a c t o r s .  J  F i r s t o f a l l , the s u p e r c o n d u c t i n g energy  gap  (2A(T)) a t a p o i n t i n the s u p e r c o n d u c t o r i s u n l i k e the band gap i n semic o n d u c t o r s i n t h a t i t i s s t r o n g l y temperature dependent and v a n i s h e s , of c o u r s e , a t the s u p e r c o n d u c t i n g - n o r m a l t r a n s i t i o n temperature T .  Now  large  temperature e x c u r s i o n s a r e 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 t h a t for  p e r i o d s of tens o f 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 m e t a l i n t h i s v e r y s m a l l r e g i o n i s , i n f a c t , i n the normal s t a t e — n o f i x e d v a l u e of A may be assumed.  therefore  S e c o n d l y , the phonons e m i t t e d when two e x c i t e d q u a s i p a r t i c l e s  combine t o form a Cooper p a i r may b r e a k up o t h e r p a i r s .  not be i g n o r e d f o r t h e y , i n t u r n ,  may  Rothwarf and T a y l o r (1967) have shown t h a t the r a t e  at w h i c h t h e s e r e c o m b i n a t i o n phonons a r e l o s t from the energy  range  nu) £ 2A(T) o f t e n determines the r a t e a t 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.  I n a s m a l l volume l i k e the t u n n e l j u n c t i o n , i t i s  expected t h a t t r a n s m i s s i o n of the phonons t o the g l a s s s u b s t r a t e and s u p e r f l u i d h e l i u m w i l l p l a y an i m p o r t a n t p a r t . chapter 7).  (For f u r t h e r d i s c u s s i o n , see  -7C e r t a i n l y , the s i t u a t i o n i n superconducting  tunnel junctions  i s even more c o m p l i c a t e d  than i n semiconductor d e t e c t o r s o f p r a c t i c a b l e  geometry where, a l t h o u g h  t h e energy gap may be c o n s i d e r e d c o n s t a n t and t h e  e f f e c t o f b o u n d a r i e s i g n o r e d , t h e a n a l y s i s i s q u i t e i n v o l v e d (see eg. Shockley, 3.  1961). D i f f e r e n c e from I n t e r m e d i a t e  S t a t e 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 t h e t u n n e l j u n c t i o n d e t e c t o r and t h e s i n g l e s u p e r c o n d u c t i n g workers.  s t r i p used by p r e v i o u s  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.  energy below some t h r e s h o l d v a l u e  (E ^ ' ) — d e t e r m i n e d  Particles with  by t h e b i a s c u r r e n t and  s t r i p d i m e n s i o n s , m a t e r i a l and t e m p e r a t u r e — a r e unable t o c o n v e r t a s u f f i c i e n t l y l a r g e r e g i o n o f t h e s t r i p i n t o t h e normal s t a t e so t h a t no v o l t a g e p u l s e o c c u r s ; p a r t i c l e s w i t h energy E > E ^ do generate a p u l s e but i t s amplitude  i s i n s e n s i t i v e t o t h e a c t u a l magnitude o f IE and i s n o t s u i t a b l e  f o r multichannel spectrometry.  I n t h e j u n c t i o n d e t e c t o r , on t h e o t h e r 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 l o n g as E i s g r e a t e r than a few times 2A(T), s a y , so t h a t t h e energy t h r e s h o l d i s s e t by t h e j u n c t i o n a m p l i f i e r n o i s e and f l u c t u a t i o n s i n t h e s i g n a l . or e x p e r i m e n t a l  (Insufficient  e v i d e n c e i s a v a i l a b l e y e t t o make any statements  the l i n e a r i t y between p u l s e a m p l i t u d e  concerning  and energy d e p o s i t e d i n t h e j u n c t i o n . )  By a n a l o g y , i t might be s a i d t h a t the s i n g l e " s u p e r c o n d u c t i n g t u n n e l j u n c t i o n d e t e c t o r as t h e g e i g e r c o u n t e r C.  theoretical  s t r i p i s t o the  i s t o t h e p r o p o r t i o n a l counter.  Expected S i g n a l As a s t a r t i n g p o i n t , t h e time dependence o f t h e c u r r e n t  pulse  r e s u l t i n g from t h e passage o f an a l p h a p a r t i c l e through t h e j u n c t i o n i s assumed t o be  • • i ( t ) = i exp(-t/-r) o  where t h e peak c u r r e n t i  • , t * o  (1-1)  i s p r o p o r t i o n a l t o t h e 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 t h e c h a r a c t e r i s t i c time mentioned e a r l i e r . To e v a l u a t e i i n e q u a t i o n 1-1, i t i s convenient o i  N  o o  = 61  to define  (9I/8T)„6T v  = 5n = O n ( T ) / 3 T ) 6 T = 2AE/w ( V o l )  -8where n(T) i s the t h e r m a l 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 c h a p t e r 3 ) .  Eliminating  6T between t h e s e e x p r e s s i o n s g i v e s  1  o  91 9T  9T 9n  2AE w- ( V o l )  U  Rough e s t i m a t e s of the v a r i o u s parameters that i  Q  - 15 uA f o r AE = 0.5 MeV  i n t i n a t 1.2  K.  i n e q u a t i o n 1-2  L )  indicate  ( I t s h o u l d be p o i n t e d out  t h a t the v a l u e s o f (91/91)^ and  (9T/9n) used i n t h i s e s t i m a t e are c o n s t a n t  and do not r e p r e s e n t i n any way  the t r u e , c o m p l i c a t e d b e h a v i o u r of these  two q u a n t i t i e s which are s t r o n g l y temperature g r o s s s i m p l i f i c a t i o n s , t h i s e s t i m a t e of i  Q  dependent.)  I n s p i t e of these  i s r e a s o n a b l y c o n s i s t e n t w i t h the  most p r o b a b l e v a l u e o f 22 yA deduced e x p e r i m e n t a l l y (see s e c t i o n E f o l l o w i n g and c h a p t e r 7 ) . E s t i m a t e s of the time c o n s t a n t 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 o c c u r s e i t h e r by t u n n e l i n g (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.  quasiparticle pair.  (Recombination, see c h a p t e r 7, i s t a k e n to i n c l u d e  t h o s e p r o c e s s e s 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 e n s i t y i n the s u p e r c o n d u c t i n g f i l m s , i t does not i n c l u d e merely the r e c o m b i n a t i o n 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 e m i s s i o n of a phonon, s i n c e such a phonon may  i n t u r n c r e a t e 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.) i s W = W_  Hence the r a t e a t which the excess q u a s i p a r t i c l e d e n s i t y decays + W  I  =  T "  1  .  K  W^  >  may  be e s t i m a t e d v i a a method due to G i n s b e r g (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 o f the r e c o m b i n a t i o n phonons mentioned e a r l i e r , i t i s expected t h a t W  s h o u l d not be e s t i m a t e d 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 r e c o m b i n a t i o n p r o b a b i l i t y W  which i n c l u d e s 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. L e v i n e and H s i e h , 1968) who  f i n d that  -9for  s u p e r c o n d u c t o r s and temperatures of i n t e r e s t 2 x 10^ sec ^ > W ' > 5 - 1 5 x 10 sec . A l t h o u g h the s t e a d y s t a t e c o n d i t i o n s under which W ' was R  R  measured d i f f e r from those i n t u n n e l 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 r e c o u r s e but to use W + W f o r an o r d e r o f _1 R T magnitude e s t i m a t e o f T . . 1  to 1.4  W i t h i n the r e l i a b i l i t y —6 x 10 sec. In  i(t) = i  of these e s t i m a t e s , ! ranges from 4 x  summary, the expected  c u r r e n t i s assumed to be o f the form  expC-t/x) w i t h i = 15uA  t i n j u n c t i o n a t 1.2  K) and 4 x 1 0 ~  10  ( f o r energy AE = 0.5 MeV < T < 1.4 x 1 0 ~  8  6  d e p o s i t e d i n the  sec.  i D.  Noise I t was  found  t h e p r e s e n t experiment  ( c h a p t e r 6) t h a t the dominant source of n o i s e i n was  magnitude of the n o i s e was  the 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 and 3.5  that the !  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.'  ( T h i s n o i s e l e v e l t u r n e d out to be e q u i v a l e n t to the p u l s e a m p l i t u d e  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 t u n n e l j u n c t i o n s h a v i n g h i g h e r output impedances than  t h e 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 , n o i s e 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 c h a p t e r 3, may  assume an i m p o r t a n t r o l e i n  d e t e r m i n i n g j u n c t i o n performance. The requirement  t h a t the a m p l i f i e r s e n s i t i v i t y had t o meet was  t h a t p u l s e s h a v i n g the parameters p r e d i c t e d above s h o u l d produce an a m p l i f i e r output e q u a l to o r g r e a t e r than t h r e e t i m e s the a m p l i f i e r n o i s e l e v e l . a m p l i f i e r was E.  c o n s t r u c t e d which met  this  An  criterion.  Preview of R e s u l t s The  experiment  r e p o r t e d i n t h i s t h e s i s demonstrates f o r the f i r s t  time t h a t the s u p e r c o n d u c t i n g as'a d e t e c t o r of charged  t h i n f i l m t u n n e l j u n c t i o n may  particles.  indeed be used  S p e c i f i c a l l y , i t shows t h a t the p u l s e  o f t u n n e l i n g c u r r e n t 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 a l p h a p a r t i c l e s i s r e a d i l y o b s e r v a b l e . An upper l i m i t o f 8.2 -3 10 eV has been p l a c e d on w(Sn), the average energy l o s s by a charged  x  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 s u p e r c o n d u c t i n g 1.2  t i n at  K w h i c h , w i t h r e f e r e n c e to t a b l e 1-1, may be seen to agree r e a s o n a b l y  w e l l w i t h the v a l u e d e r i v e d from a v e r y naiilve analogy w i t h  semiconductors.  -10(Because of a m b i g u i t y caused by heat d i f f u s i n g back from the s u b s t r a t e to c o n t r i b u t e t o the p u l s e s as d i s c u s s e d i n c h a p t e r 7 — t h i s agreement perhaps be f o r t u i t o u s , and i n f a c t w(Sn) analogy would  may  may  be s m a l l e r than the n a i v e  suggest.)  The p u l s e s observed t h e rms n o i s e l e v e l .  ( f i g u r e 6-8)  had a m p l i t u d e s up t o 19  Because t h e i r amplitude decreased w i t h d e c r e a s i n g  j u n c t i o n dynamic r e s i s t a n c e as p r e d i c t e d , b e c a u s e w i t h a mechanical  times  they c o u l d be t u r n e d o f f  s h u t t e r p l a c e d between the j u n c t i o n and s o u r c e , and because  the p u l s e count r a t e agreed w i t h t h a t c a l c u l a t e d from the source s t r e n g t h and the e x p e r i m e n t a l geometry, i t was  concluded  t h a t the p u l s e s c e r t a i n l y  were the r e s u l t of a l p h a 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 f r o m bombardment of the l e a d - i n f i l m s , or the s u b s t r a t e a l o n e . l e a s t squares  A  f i t of the t h e o r e t i c a l p u l s e shape ( m o d i f i e d by the t r a n s f e r  f u n c t i o n o f the a m p l i f i e r system) to t h a t r e c o r d e d 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 t h e e r r o r s correspond  to a 90% c o n f i d e n c e  limit.  W i t h 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 t h e l a r g e s t a m p l i t u d e p u l s e s observed  20 $ i  i s deduced t o l i e i n the range  $ 26 uA w i t h a most p r o b a b l e v a l u e of 22 uA.  c o r r e s p o n d i n g to t h i s v a l u e o f i  Q  The v a l u e of  i s d i s c u s s e d below.  The  AE  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 a l p h a p a r t i c l e i s c a l c u l a t e d t o be  N  where W and W^  o  =  W — — W„,e  i ( t ) d t = i /We o T  « 3.A  x  10  8  have been d e f i n e d 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.  C o n s i d e r a b l e 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 A E  a  t h a t the a l p h a p a r t i c l e d e p o s i t s ' i n 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 c h a p t e r 7) g i v e s _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 , b o t h i n s o f a r as the understanding  of the b a s i c p h y s i c s 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 w o r t h w h i l e n u c l e a r r a d i a t i o n d e t e c t o r s i s concerned.  F u r t h e r t h e o r e t i c a l and e x p e r i m e n t a l work, which i s b e i n g  c o n t i n u e d 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 r e q u i r e d 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 p r o c e s s e s  i n v o l v e d and probe such  s u b j e c t s as energy r e s o l u t i o n , l i n e a r i t y of response performance (see c h a p t e r 8 ) .  and r e p r o d u c i b i l i t y o f  I n 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 b e i n g made  -11concerning  t h e 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 t h e manner i n w h i c h the heat s p i k e 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 t h e r e f o r e t r e a t e d v e r y b r i e f l y i n this F.  thesis. Thesis  Outline Chapter 2 i n t r o d u c e s t h e s u b j e c t o f t u n n e l i n g 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 m e t a l s .  T h i s i s f o l l o w e d by  a:short  r e v i e w o f t o p i c s f r o m s u p e r c o n d u c t i v i t y t h e o r y which a r e of r e l e v a n c e to the t u n n e l 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 .  An o u t l i n e o f 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 t u n n e l i n g t h e o r y concludes  the  chapter. The o p e r a t i n g p r i n c i p l e s of t h e s u p e r c o n d u c t i n g detector are given i n chapter 3 .  charged p a r t i c l e  'To p l a c e the t u n n e l j u n c t i o n d e t e c t o r i n  p e r s p e c t i v e t h r e e commonly used d e v i c e s f o r measuring charged p a r t i c l e energy s p e c t r a , t h e 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 , a r e 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 o f t u n n e l 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 a r e examined because o f t h e i r analogy t o 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 f a v o u r a b l e 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 s u p e r c o n d u c t o r s energy gap.  which have h i g h s t o p p i n g power and a l a r g e i  i  i  S  The n a t u r e 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 n t h e s u p e r c o n d u c t o r by a t r a v e r s i n g charged p a r t i c l e i s d i s c u s s e d  after  w h i c h t h e parameters f o r a s m a l l s i g n a l e q u i v a l e n t c i r c u i t f o r t h e d e t e c t o r a r e d e r i v e d and an e s t i m a t e o f the s i g n a l s i z e o b t a i n e d .  This i s followed  by n o t e s on t h e importance o f t h e 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 the p r a c t i c a l o p e r a t i o n o f t h e d e t e c t o r . of the n o i s e sources  Concluding  the chapter  i s a survey  i n t h e t u n n e l j u n c t i o n d e t e c t o r 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, w h i c h d e s c r i b e t h e c r y o g e n i c apparatus and e l e c t r i c a l measurement t e c h n i q u e s i n t e r e s t and may be o v e r l o o k e d  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  by the reader u n i n t e r e s t e d i n these  A d i s c u s s i o n o f the e x p e r i m e n t a l 6.  details.  r e s u l t s may be found i n chapter  Of c h i e f c o n c e r n a r e t h e dc c h a r a c t e r i s t i c s of t h e 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 o b s e r v a t i o n and p h y s i c a l c h a r a c t e r i s t i c s o f p u l s e s i n t h e t u n n e l i n g c u r r e n t which were induced by a l p h a  particle  -12bombardment and  the e s t i m a t i o n of the j u n c t i o n c a p a c i t a n c e from n o i s e  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 d e t e r m i n a t i o n and w.  of the parameters of i n t e r e s t T , i  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 r i g o r o u s c a l c u l a t i o n of  the amount of energy d e p o s i t e d c l o s e s the  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  section. F i n a l l y , i n c h a p t e r 8, the i m p o r t a n t r e s u l t s are summarized  placed  i n perspective  by e v a l u a t i n g  charged p a r t i c l e s p e c t r o m e t e r . c h a p t e r and  interest.  the t u n n e l j u n c t i o n as a p o s s i b l e  Guidelines  f o r f u t u r e work b r i n g  the main body of the t h e s i s to a  The  and  the  conclusion.  f o u r appendices d e a l w i t h m a t t e r s of s p e c i f i c t e c h n i c a l  Appendix A o u t l i n e s t h i n f i l m t u n n e l j u n c t i o n  t e c h n i q u e s ; Appendix B a n a l y z e s the I-V  preparation  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 s y n o p s i s of phenomena observed i n the preparation  of Pb-Pb j u n c t i o n s ; Appendix D i s an account of s t u d i e s 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 r e s i s t a n c e o f the j u n c t i o n measured w i t h a s t a n d a r d 4 - t e r m i n a l  the  technique.  CHAPTER 2  THEORETICAL ASPECTS OF THE SUPERCONDUCTING A.  TUNNEL JUNCTION  Introduction I n i t s most common form, t h e s u p e r c o n d u c t i n g  tunnel j u n c t i o n  c o n s i s t s of two superimposed e v a p o r a t e d t h i n m e t a l f i l m s separated  by  an e x t r e m e l y 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 t h e r m a l l y grown o x i d e 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 d e f i n e d as t h e  region of overlap of the f i l m s .  T y p i c a l l y , t h e f i l m s a r e a few t e n t h s  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 ; t h e i n s u l a t i n g l a y e r t h i c k n e s s i s u s u a l l y 10 t o 20 angstroms. f l o w from one s u p e r c o n d u c t i n g  The mechanism f o r c u r r e n t  f i l m t o the o t h e r i s , as t h e name i m p l i e s ,  electron tunneling. To i n t r o d u c e the e s s e n t i a l s of t h e s u b j e c t o f t u n n e l i n g , t h e problem o f t u n n e l i n g between normal m e t a l s i s c o n s i d e r e d Before  first  g o i n g on t o d i s c u s s t u n n e l i n g between s u p e r c o n d u c t o r s ,  (section B). a pause i s  made ( s e c t i o n C) f o r t h e purpose of r e v i e w i n g b r i e f l y those t o p i c s from the t h e o r y of s u p e r c o n d u c t i v i t y which a r e r e l e v a n t t o t h e  superconducting  t u n n e l 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  t h r e e s e c t i o n s a r e devoted t o a d i s c u s s i o n o f t u n n e l i n g between superconductors:  section D considers s i n g l e q u a s i p a r t i c l e tunneling, section E  i n v e s t i g a t e s Josephson o r p a i r t u n n e l i n g  (because of i t s n u i s a n c e r a t h e r  than i n t r i n s i c v a l u e t o t h e t u n n e l j u n c t i o n d e t e c t o r ) 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 s t u d y i n g t u n n e l i n g phenomena i n g e n e r a l  i s r e f e r r e d t o a v e r y r e c e n t and comprehensive r e v i e w e d i t e d by B u r s t e i n and L u n d q v i s t B.  Tunneling  (1969). 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 t h e e l e c t r o n t u n n e l c u r r e n t through a m e t a l - i n s u l a t o r - m e t a l  (M-I-M) t u n n e l i n g  -15diode.  D u r i n g the p a s t 36 y e a r s , c o n s i d e r a b l e  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 s u b j e c t can be found i n S t r a t t o n (1962), Simmons (1963A,B), Kuhn (1966) and Duke ( 1 9 6 9 ) — b u t the a c t u a l p h y s i c a l and  s t u d y of t h i n f i l m t u n n e l j u n c t i o n s was  h i g h vacuum and  t h i n f i l m t e c h n o l o g y had  d e l a y e d u n t i l 1960  preparation when  s u f f i c i e n t l y advanced f o r  G i a e v e r (1960) to make Att-AA^O^-Pb t u n n e l i n g j u n c t i o n s .  In t h i s s e c t i o n ,  the g e n e r a l r e s u l t f o r the t u n n e l i n g c u r r e n t between normal m e t a l s i s quoted f o l l o w e d by an a p p r o x i m a t i o n due  t o 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 t u n n e l c u r r e n t 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 r e f e r e n c e s 1.  the c a l c u l a t i o n s may  (1961) and  c i t e d above.)  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  j u n c t i o n i s shown i n f i g u r e 2-2. the g r o s s f e a t u r e s of the I-V of meV),  be found i n H a r r i s o n  As l o n g as one  i s i n t e r e s t e d only i n  c h a r a c t e r i s t i c (of the o r d e r of hundreds  i t i s p e r m i s s i b l e t o 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 p r o c e s s (see eg. Duke, 1969).  ( I n 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 b o t h w i t h themselves and i n the m e t a l and  tunnel  the b a r r i e r are n e g l e c t e d ;  the phonons  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.) one  dimensional  b a r r i e r and  specular  ( i e . the t r a n s v e r s e wave number k^ e l e c t r o n s t a t e s ) and  If a  t r a n s m i s s i o n through i t are assumed  i s the same f o r the i n i t i a l and  final  i f i t i s f u r t h e r assumed t h a t the m a t r i x element f o r 2  the t r a n s i t i o n from m e t a l 1 to m e t a l 2 i s such t h a t  |Mi | 2  2  2  = | 211 M  =  M  the t u n n e l c u r r e n t 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 ,  I  4ire  | M | p ( E ) p ( E + eV)[f 2  l x  2 x  (E) - f^E  + eV)]dE  (2-1)  k.  In equation  2-1,  e i s the e l e c t r o n i c c h a r g e , h = "h2ir i s P l a n c k ' s  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 ) X  constant, is a  1  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 h a v i n g x d i r e c t e d energy between E  x  and E + dE x x  w i t h f i x e d t r a n s v e r s e momentum k . t  approximations stated, equation  2-1  Within  i s the fundamental e q u a t i o n  t u n n e l i n g through a o n e - d i m e n s i o n a l b a r r i e r and  the  for  a p p l i e s e q u a l l y w e l l to  , 1961)  -16-  vacuum  F i g u r e 2-2:  P o t e n t i a l Energy Diagram f o r I d e a l M-I-M ( a f t e r Simmons, 1963A)  Structure  -17M-I-M s t r u c t u r e s , whether t h e m e t a l i s normal or s u p e r c o n d u c t i n g .  A  d i s t i n c t i o n between t h e two cases must be made, as w i l l be seen s h o r t l y , when t h e m a t r i x element i s e v a l u a t e d . ,2  ,  Evaluating  |M|  by assuming t h a t t h e 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 t o one s i d e o f t h e b a r r i e r o r the o t h e r and u s i n g a WKB a p p r o x i m a t i o n t h r o u g h t h e b a r r i e r r e g i o n , H a r r i s o n |M|  2  = T(E ) t ^ P 2  x  S u b s t i t u t i o n i n equation  ( )P E  l x  2 x  (  E  +  e  V  )]  _  (1961) found  1  2-1 g i v e s CO T(E  ) [ f ( E ) - f ( E + eV)]dE  (2-2)  o where T(E ) i s t h e WKB r e s u l t f o r t h e b a r r i e r t r a n s m i s s i o n o f a f r e e x e l e c t r o n i n c i d e n t on t h e b a r r i e r . Examination of equation  2-2 r e v e a l s t h a t t h e d e n s i t y o f  s t a t e s f a c t o r s a p p e a r i n g i n 2-1 have c a n c e l l e d w i t h those c o n t a i n e d i n the m a t r i x element so t h a t the t u n n e l i n g c u r r e n t no l o n g e r depends on them.  T h i s r e s u l t i s c o n s i s t e n t w i t h t h e observed b e h a v i o u r o f normal  M-I-M t u n n e l j u n c t i o n s b u t c o n t r a r y t o the b e h a v i o u r o f j u n c t i o n s i n which one  o r b o t h o f t h e m e t a l s a r e s u p e r c o n d u c t i n g (see eg. f i g u r e 2-6).  Briefly,  t h i s absence o f t h e d e n s i t y o f s t a t e s from the r e s u l t f o r s u p e r c o n d u c t o r s a r i s e s from t h e f a i l u r e o f t h e independent, p a r t i c l e model t o a d e q u a t e l y represent  superconductors.  The d i s c r e p a n c y  may be removed by assuming  ( H a r r i s o n , 1961) t h a t t h e d e n s i t y o f s t a t e s e n t e r i n g e q u a t i o n i  i  corresponds t o that of q u a s i p a r t i c l e s but that g i v e n by t h e normal d e n s i t y o f s t a t e s . has been e s t a b l i s h e d from m i c r o s c o p i c 1964.)  2-1  2  |M|  «  isstill  •:  satisfactorily  (The v a l i d i t y o f t h i s assumption t h e o r y , see eg. Douglas & F a l i c o v ,  As a r e s u l t , c a n c e l l a t i o n o f t h e s e f a c t o r s i s removed and i  agreement w i t h experiment o b t a i n e d . between s u p e r c o n d u c t o r s i s r e s e r v e d With e q u a t i o n  i  Further d i s c u s s i o n of tunneling f o r section D of this  chapter.  2-2, the problem o f f i n d i n g t h e t u n n e l  c u r r e n t between two normal m e t a l s i s s o l v e d i n p r i n c i p l e b u t f u r t h e r a p p r o x i m a t i o n s a r e r e q u i r e d t o o b t a i n an e x p l i c i t c u r r e n t v o l t a g e  relation  -18for  comparison to e x p e r i m e n t .  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 t u n n e l c u r r e n t t o the t h i c k n e s s o f  the  insulator. 2.  Dependence Of Tunnel C u r r e n t UpOn B a r r i e r Parameters I t w i l l be shown i n c h a p t e r  3 t h a t one c o n d i t i o n f o r  optimum performance of the t u n n e l 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 i s t h a t the b a r r i e r be as t h i n as p o s s i b l e .  C o n s e q u e n t l y , from an  e x p e r i m e n t a l i s t ' s p o i n t of v i e w , an e x p r e s s i o n r e l a t i n g  effective  b a r r i e r t h i c k n e s s t o the measurable I-V c h a r a c t e r i s t i c w i l l s e r v e as a u s e f u l guide i n f a b r i c a t i n g t u n n e l j u n c t i o n s . been g i v e n by Simmons (1963A,B) who  Such c a l c u l a t i o n s have  has r e l a t e d the c u r r e n t d e n s i t y to  the t h i c k n e s s , average v a l u e of the i n s u l a t o r energy gap and c o n s t a n t o f the b a r r i e r . his  dielectric  I t i s the purpose of t h i s paragraph to g i v e  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  o v e r k^ i s changed to an i n t e g r a l over the  a p p r o p r i a t e s u r f a c e i n k space and i t i s assumed t h a t 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 c o n d u c t i o n band of the m e t a l or t u n n e l i n g through the i n s u l a t o r , e q u a t i o n 2-2 the form (Kuhn, 1966;  Simmons, 1 9 6 3 A — e q u a t i o n E  J  4irme  =  m  T(E  x  )dE  [f(E)  x  may  be r e - w r i t t e n i n  7)  (2-3)  - f ( E + eV)]dE  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 and E = im(v + v ). E v a l u a t i o n of e q u a t i o n 2-3 t o determine the c u r r e n t r y z v o l t a g e 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 c o m p l i c a t e d m  behaviour  o f T(E ) w h i c h , w i t h r e f e r e n c e to f i g u r e 2-2, may x  T(E  x  ) =  exp  -4TT  be w r i t t e n  rs, {2m(<f>(x) - E ) } x  h  2  dx  s. Simmons' c o n t r i b u t i o n was  to express T(E^)  v a r i o u s b a r r i e r shapes <{> (x) and s u b s e q u e n t l y  i n an approximate form f o r c a l c u l a t e the t u n n e l i n g  c u r r e n t d e n s i t y (J) a t T = 0 K f o r low, i n t e r m e d i a t e and h i g h v o l t a g e s between e l e c t r o d e s of s i m i l a r and d i s s i m i l a r m e t a l s .  For s i m i l a r  metals—  -19the t y p e o f 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  r e c 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 o f low voltages  (V) i s  J V  (2m) s  =  <|> exp(-A<j> ) 2  L  (2-4)  L  where 1.15As S  £n  2" 1 S  and s. = i s [ l - ( l - 1 . 1 5 X / K ( f . ) ] * 3(eV A)/K<j> , 2  O  J.  O  K<t> » o r  A a (4irs/h)  (2m)  m  e l e c t r o n mass  e  e l e c t r o n charge  s  thickness of i n s u l a t i n g region =  h = Planck's T  1.15X  s  2  + s  i  constant  mean b a r r i e r h e i g h t o X = e £n(2)/167rKe s 2  z = 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 o f empty space "o K = d i e l e c t r i c c o n s t a n t = e/e Equation  2-4 has been e v a l u a t e d  p l o t t e d i n f i g u r e 2-3.  f o r s e v e r a l v a l u e s of K and .<)> and  Such c u r v e s p r o v i d e a c o n v e n i e n t e s t i m a t e o f the  i n s u l a t o r t h i c k n e s s once t h e room (or l o w e r ) temperature j u n c t i o n conductance i s known. I t i s important  experimental  (X =4 the q u a n t i t y used by Simmons,  1963A)  t o note t h a t t h e t h i c k n e s s S^, = s d e r i v e d  from t u n n e l i n g r e s i s t a n c e measurements (R ) and e q u a t i o n n  2-4 ( V / J = R 'A) n  i s the t h i c k n e s s o f an i d e a l , u n i f o r m b a r r i e r whose a r e a A i s the g e o m e t r i c a l a r e a o f the j u n c t i o n . constant  The t h i c k n e s s o f a r e a l b a r r i e r , however, i s not  b u t 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 (Hurych, 1966).  -20io r 1 7  J  Dielectric Constant (K)  H h — • —  1  — 10  F i g u r e 2-3:  — • — « — i — i i i 20' I n s u l a t o r T h 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 vs Tunnel J u n c t i o n 1  —  j  •  J/V  i 30  •  -21Now  the t h i n n e r r e g i o n s of the i n s u l a t o r w i l l be emphasized because of  the e x p o n e n t i a l dependence o f r e s i s t a n c e upon t h i c k n e s s so t h a t the e f f e c t i v e a r e a o f 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 e q u a l to A ( g e o m e t r i c a l ) nor i s t h e average t h i c k n e s s <S>A = n e c e s s a r i l y e q u a l t o S^.  S(x,y)dx  dy  T h i s p o i n t i s of r e l e v a n c e £ 0 c h a p t e r 3 where the  dependence o f 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 t h i c k n e s s i s d i s c u s s e d 1  and to c h a p t e r 6 where the j u n c t i o n c a p a c i t a n c e i s e s t i m a t e d . C.  Review o f S u p e r c o n d u c t i v i t y Theory As mentioned e a r l i e r , t h i s s e c t i o n i s devoted  t o those  aspects  o f s u p e r c o n d u c t i v i t y t h e o r y which a r e germane to an u n d e r s t a n d i n g superconducting  of t h e  t u n n e l 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 d e t e c t o r of  charged p a r t i c l e s .  T o p i c s such as z e r o e l e c t r i c a l r e s i s t a n c e and  the  M e i s s n e r e f f e c t , w h i l e of i n t e r e s t i n g e n e r a l , w i l l t h e r e f o r e be i g n o r e d ( a p a r t from the f a c t t h a t the s t a n d a r d 4 - t e r m i n a l network f o r measuring I-V c h a r a c t e r i s t i c s o f t u n n e l j u n c t i o n s i s e x a c t o n l y i n the l i m i t of z e r o f i l m r e s i s t a n c e ; see appendix D). Many r e v i e w s a r e now  a v a i l a b l e d e a l i n g w i t h the s u b j e c t of  s u p e r c o n d u c t i v i t y i n g r e a t d e t a i l so t h a t no attempt w i l l be made t o j u s t i f y o r d e r i v e the r e s u l t s g i v e n i n the s u c c e e d i n g paragraphs. i n t e r e s t e d reader i s r e f e r r e d t o : Blatt  Kuper (1968), R i c k a y z e n  (1964), S c h r i e f f e r (1964) o r B o g o l i u b o v  are g i v e n by Tinkham (1965) and Lynton  The  (1965),  (1962); c o n c i s e p r e s e n t a t i o n s  (1964).  The prominent p a r t p l a y e d by the energy gap, as f a r as the t u n n e l j u n c t i o n d e t e c t o r i s concerned, was s e c t i o n t h e r e f o r e w i l l be concerned dependence upon temperature,  s t r e s s e d i n c h a p t e r 1.  This  p r i m a r i l y w i t h the energy gap and i t s  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 e x p e r i m e n t a l r e v i e w of the energy gap i n superconductors  has been g i v e n by Douglass and F a l i c o v (1964).)  i m p o r t a n t concepts  Other  such as the two f l u i d model, t h e q u a s i p a r t i c l e d e n s i t y  o f 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 i n t r o d u c e d at  t h i s time f o r c l a r i t y and subsequent r e f e r e n c e . 1.  Two  F l u i d Model The v e r y s u c c e s s f u l Bardeen, Cooper, S c h r i e f f e r (BCS,  t h e o r y v i e w s the s u p e r c o n d u c t o r fluids:  1957)  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  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  -22Cooper p a i r s w h i c h are p a i r s of e l e c t r o n s , phonon-mediated i n t e r a c t i o n and (k>, -k+)  within a thin shell  Fermi s u r f a c e .  (0  l o o s e l y "bound" v i a a  o c c u p y i n g a p a i r of e l e c t r o n (+(1^0 / E ) ^  states  = + .004k ) s t r a d d l i n g  the  F  = 200 K i s a t y p i c a l Debye t e m p e r a t u r e , k D  is B  Boltzmann's c o n s t a n t )  I t i s the Cooper p a i r s w h i c h g i v e the  i t s c h a r a c t e r i s t i c l o n g range o r d e r and without d i s s i p a t i o n .  The  i t s a b i l i t y to c a r r y  superconductor current  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  excitations  from the s u p e r c o n d u c t i n g ground s t a t e , h a v i n g 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 h o l e e x c i t a t i o n s i n a normal m e t a l .  Much as i n a normal  m e t a l , 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  t h e i r energy by a r b i t r a r i l y s m a l l amounts.  main d i f f e r e n c e  e x c i t a t i o n s i n s u p e r c o n d u c t o r s and  The  are a b l e to change  i n m e t a l s l i e s i n the  between  relationship  i between energy and  momentum, as d e s c r i b e d below.  As the temperature of a m e t a l c a p a b l e of becoming a s u p e r c o n d u c t o r 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 u n s t a b l e a g a i n s t the f o r m a t i o n of Cooper p a i r s . ' F u r t h e r d e c r e a s i n g of the temperature p e r m i t s 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 a b s o l u t e zero no  unpaired  q u a s i p a r t i c l e s , whose e n e r g i e s are w i t h i n a p p r o x i m a t e l y k 9 D  energy are p r e s e n t .  T h i s h i g h l y c o r r e l a t e d BCS  s u p e r c o n d u c t o r l e a d s n a t u r a l l y to an energy gap  of the  ground s t a t e of or band of  Fermi  U  the  forbidden  e n e r g i e s 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,  may  be  the l o w e s t e x c i t e d s t a t e of the  found by c o n s i d e r i n g  i n the s t a t e let  superconductor  the e f f e c t of adding to the system an  where i t s mate -k+  i s assumed to be empty.  of t h i s p r o c e s s i s to remove the s t a t e  (k+,  effect  -k-t-) from the m a n i f o l d 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  responsible  t o g e t h e r the Cooper p a i r s ; r e m o v i n g t h i s s t a t e r e q u i r e s I f the B l o c h energy of the added e l e c t r o n i s the F e r m i energy, the energy r e q u i r e d  The  electron  for  binding  a f i n i t e energy  A^.  measured w i t h r e s p e c t to  to c r e a t e 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 s u p e r c o n d u c t o r i s then g i v e n by.  • E  k=  9  9  1  +  \  i >'  (2-5)  -23I n c o n t r a s t t o the s i t u a t i o n i n normal m e t a l s a t 0 K, t h e e f f e c t o f t h e p a i r i n g i n t e r a c t i o n i n s u p e r c o n d u c t o r s i s to e s t a b l i s h a f i n i t e  probability  t h a t an e l e c t r o n may be added t o a s t a t e below as w e l l as above t h e Fermi surface.  I n e i t h e r c a s e , t h e e x c i t a t i o n energy E, i s p o s i t i v e . ->-  Similarly,  k  the energy r e q u i r e d t o 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 c r e a t e a h o l e i n let ) i s found t o be g i v e n by e q u a t i o n 2-5 and t o be p o s i t i v e whether  |ic| i s g r e a t e r o r l e s s than t h e Fermi momentum k„.  The  r  minimum energy r e q u i r e d t o i n j e c t o r 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). T h i s gap i n t h e 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 t h a t which would be observed i n e x p e r i m e n t s w h i c h d e p o s i t o r withdraw e l e c t r o n s from t h e s u p e r c o n d u c t o r (as i n normal m e t a l - i n s u l a t o r - s u p e r c o n d u c t o r t u n n e l j u n c t i o n s ) . I n experiments w h i c h do n o t 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 e l e c t r o m a g n e t i c a b s o r p t i o n work) t h e energy gap i n t h e 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 c r e a t e d :  (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 q u a s i h o l e i s c r e a t e d i n k ^ t ) r e q u i r i n g energy E^l.; ^ energy  a n  electron i s created i n state k^t requiring  The t o t a l e x c i t a t i o n energy r e q u i r e d i s then  E ^ + E^2 ^ 2A  = E^  Paragraphs (a) t o ( c ) f o l l o w i n g c o n s i d e r b r i e f l y  the t e m p e r a t u r e , specimen s i z e and magnetic f i e l d dependence of the energy gapB e f o r e g o i n g on t o t h e s e t o p i c s , one i m p o r t a n t p o i n t s h o u l d be s t r e s s e d .  S i n c e q u a s i p a r t i c l e s must be c r e a t e d i n p a i r s t o c o n s e r v e ,  on t h e a v e r a g e , t h e t o t a l number o f e l e c t r o n s , i t i s t e m p t i n g t o c o n s i d e r t h e s e 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 t h e s i t u a t i o n i n normal m e t a l s .  Such an a n a l o g y must be used w i t h c a u t i o n  however as t h e e x c i t a t i o n s from t h e ground s t a t e a r e r e a l l y  linear  c o m b i n a t i o n s of " h o l e s " and " e l e c t r o n s " becoming h o l e - l i k e o r 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„)  -  r  r  k„  .  (The  r  e x c i t a t i o n s from t h e ground s t a t e a r e c o n v e n i e n t l y d e s c r i b e d by the Bogoliubov operators 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 e q u a t i o n s 2-6, c ,  ->•  i n s t a t e s k+ and -k+  respectively.  In  and c^ a r e the c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s  +  f o r Fermions, v = i(l-e^/E, ) i s the p r o b a b i l i t y t h a t ( k t , -k+ ) i s 9 2 ••-»•-»• o c c u p i e d and ur = 1-v, i s t h e p r o b a b i l i t y t h a t (k+ , -k+ ) i s unoccupied. C l e a r l y , f o r k » k , v£ •> 0 and y •* c ) . The importance o f t h i s fc  p  k +  f c +  d i s t i n c t i o n w i l l become e v i d e n t i n s e c t i o n D where the use of t h e so c a l l e d s e m i c o n d u c t o r model t o r e p r e s e n t s u p e r c o n d u c t o r s i s b r i e f l y discussed. (a) Temperature Dependence o f the Energy  Gap  At T = 0 K, BCS f i n d the magnitude of the energy gap t o be 2A(0) = 3 . 5  k T g  c  where T^ i s the s u p e r c o n d u c t i n g - n o r m a l t r a n s i t i o n temperature mentioned earlier.  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 t h a t t h e gap d e c r e a s e s v e r y s l o w l y f o r T <  T /3 c  so t h a t  2A(T) = 3.5 L I B c For  0 $ T $  0.3T  c  temperatures n e a r T^, t h e gap w i d t h i s approximated by  2A(T) =3.2  k T (1-T/T ) * Be c  T a b u l a t i o n s of A(T)/A(0) a r e g i v e n by Parmenter and B e r t o n (1964) and M u h l s c h l e g e l (1959). T a b l e 2-1 g i v e s 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 e x p e r i m e n t a l v a l u e s f o r the energy gap f o r f o u r commonly used s u p e r c o n d u c t o r s ( c f . a l s o Douglass & F a l i c o v , For  1964).  Sn a t 1.2 K i t i s c l e a r , from f i g u r e 2-4, t h a t the gap w i d t h may  be t a k e n t o be 2A(0) w i t h n e g l i g i b l e  error.  (b) Specimen-Size Dependence of the Energy  Gap  E x p r e s s i o n s d e r i v e d from the BCS m i c r o s c o p i c t h e o r y 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 p r o p e r boundary c o n d i t i o n s s h o u l d be c o n s i d e r e d . Anderson's t h e o r y of the " d i r t y "  -25-  0  F i g u r e 2-4:  .2  .4  .6  .8  1.0  T/T  Temperature Dependence of Energy Gap parameter A(T)  -26s u p e r c o n d u c t o r (Anderson, 1959) and the good agreement  of energy gap  2A(0) Superconductor  T  (°K)  BCS  c  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  T a b l e 2-1:  meV  Comparison of T h e o r e t i c a l and E x p e r i m e n t a l Values of 2 A ( 0 ) .  v a l u e s whether determined e x p e r i m e n t a l l y from b u l k o r t h i n f i l m samples ( L y n t o n , 1964), i n d i c a t e t h a t i n z e r o 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 t h a t i t i s assumed h e n c e f o r t h that  A(films) - A(bulk),  When H ^ 0, however, c o m p l i c a t e d as seen i n the f o l l o w i n g  H = 0  the s i t u a t i o n becomes more  section.  (c) M a g n e t i c F i e l d Dependence of Energy  Gap  I t i s w e l l known (see eg. L y n t o n (1964) o r Kuper (1968)) t h a t 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 b u l k specimen may quenched by a p p l y i n g a magnetic f i e l d of s u f f i c i e n t s t r e n g t h if H  c  = H  o  be  so t h a t ,  f o r T = 0, H  (T) c H  = 1 o  As i t has been shown, the p r e s e n c e of an energy gap i s fundamental t o the s u p e r c o n d u c t i n g s t a t e which i m p l i e s , t h a t i f s u 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  the energy gap w i d t h must depend  magnetic f i e l d p r e s e n t . No complete m i c r o s c o p i c t h e o r y of the e f f e c t of a  magnetic f i e l d on the energy gap i s y e t a v a i l a b l e .  Experimental  on  -27e v i d e n c e w i t h t h i n f i l m s (Meservey and D o u g l a s s , 1964) i n d i c a t e s t h a t the gap according  taken a t T =  0.12T  goes c o n t i n u o u s l y to zero w i t h magnetic f i e l d  H  to  ^ A(0)  = 1 - (H/H  )  (2-7)  2  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 v a l u e  of 4 w h i c h i s a p p r o p r i a t e t o the f i l m s used i n the p r e s e n t experiment b u t , as d e s c r i b e d i n s e c t i o n A, c h a p t e r p r e d i c t e d by 2-7  decrease f o r H = 100  Oe  agrees f a v o u r a b l y w i t h t h a t which we measured f o r t i n .  (Here, d i s the f i l m t h i c k n e s s and o f the o r d e r o f 500 A i n t i n at 1.2 3.  6, the 7% gap  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 K.)  D i s t r i b u t i o n Function f o r Quasiparticles 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 a r e independent  and obey-Fermi D i r a c s t a t i s t i c s w i t h the r e s u l t t h a t the 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  g(E) = [1 + e x p ( B E ) ] "  distribution  Fermi d i s t r i b u t i o n  (2-8)  1  where 8 = 1/k^T  and a l l the o t h e r symbols have been d e f i n e d p r e v i o u s l y  (Note:  0)  4.  E >, M  Density of The  States 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 g i v e n by BCS  N (E) b Q  as  = N (0) |E| m  (E  2  - A ( T ) ) " * , E ^ 4(1)  (2-9)  2  where N (0) i s the d e n s i t y o f 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 s u r f a c e of the normal m e t a l . From e q u a t i o n 2-9, i t i s e v i d e n t t h a t N_(E) b  reduces t o N (0) f o r E >> m  A(T)  as i t s h o u l d .  (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 o n l y s t a t e s w i t h i n a few kT of the Fermi energy and  those  the q u a s i p a r t i c l e and  B l o c h s t a t e s s h o u l d c o i n c i d e at e n e r g i e s f a r from the Fermi s u r f a c e ) .  normal 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 t h a t s t a t e s "squeezed o u t "  -28of the gap must be accommodated a t the edges, to s a t i s f y the t h a t the t o t a l number of s t a t e s be D.  requirement  conserved.  Single Q u a s i p a r t i c l e Tunneling I n the d i s c u s s i o n o f t u n n e l i n g between normal m e t a l s g i v e n i n  s e c t i o n B i t was p o i n t e d out t h a t , i n o r d e r t o o b t a i n agreement w i t h the r e s u l t s o b t a i n e d when one o r b o t h of the m e t a l s a r e s u p e r c o n d u c t i n g , i t was n e c e s s a r y o n l y t o 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 t h a t o f the e l e c t r o n s i n the normal m e t a l . t o b r i e f l y o u t l i n e why energy diagram may  The purpose of t h i s s e c t i o n i s  t h i s a s s e r t i o n i s v a l i d and why  the  semiconductor  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 .  C o n c l u d i n g the s e c t i o n i s an e x p r e s s i o n f o r 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 c u r r e n t between s u p e r c o n d u c t o r s d e r i v e d 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 t u n n e l i n g c u r r e n t s between a normal m e t a l and a s u p e r c o n d u c t o r were r e p o r t e d by G i a e v e r (1960).  Soon  a f t e r w a r d , N i c o l e t a l (1960) and G i a e v e r and Megerle  the  (1961) extended  study t o the case of t u n n e l i n g between two s u p e r c o n d u c t o r s .  S i n c e then a  v a s t e x p e r i m e n t a l and t h e o r e t i c a l e f f o r t has gone i n t o u n d e r s t a n d i n g  and  e x p l o i t i n g t h i s phenomenon; the p r e s e n t s t a t e of the s u b j e c t i s summarized by B u r s t e i n and L u n d q v i s t 1.  (1969).  T u n n e l i n g Between a Normal M e t a l and a  Superconductor  Cohen e t a l (1962) computed the c u r r e n t I g ^ i n an t u n n e l s t r u c t u r e a t f i n i t e temperature t w o f o l d degeneracy  f a c t o r s u^ and v ^ from e q u a t i o n 2 6  |M|  2  X  SM  =  C  (The degeneracy | k1 2  <  k  p  d i s a p p e a r e d y i e l d i n g the s i m p l e r e s u l t  M  s  M  such t h a t E^  = E^  2  and u  + u^.  2  2  f c  £ u^ h ( k ) +  = 1 = v^  there i s a  p 2  + v  k 2  .  2  i s the m a t r i x  on each s i d e o f the i n s u l a t o r , f  the Fermi f a c t o r f o r the normal m e t a l ,  N M  I n summing  h'(k) reduce to  I n e q u a t i o n '2-10, C i s a c o n s t a n t , |M|  element r e l a t i n g normal e l e c t r o n s  (2-10)  g  a r i s e s from the f a c t t h a t f o r each|k^|> k  h(k) + h'(k).)  coherence  [ f ( E - eV). - f ( E ) ] N (E-eV) N ( E ) d E  over a l l k, terms o f the form I  and found t h a t , because of the  of energy l e v e l s i n s u p e r c o n d u c t o r s , the _  M-S  is  ( E ) i s the d e n s i t y o f e l e c t r o n  -29s t a t e s i n the normal m e t a l , N ( E )  i s the d e n s i t y  g  i n the s u p e r c o n d u c t o r ( e q u a t i o n 2-9)  and  levels.  c  The  occupation p r o b a b i l i t y f ( E )  f _ ( E ) = g(E)  of q u a s i p a r t i c l e  V i s the d i f f e r e n c e i s defined  states  i n the Fermi  such t h a t  for E > 0 (2-1D  = l-g(|E|) for E < 0 where g(E)  i s the q u a s i p a r t i c l e o c c u p a t i o n p r o b a b i l i t y g i v e n i n e q u a t i o n  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 o b t a i n e d by Cohen et a l i s  t h a t i t c o u l d have been d e r i v e d  from.the g e n e r a l t u n n e l i n g  s i m p l y 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 remembering t h a t the one the t o t a l d e n s i t y  dimensional density 1  of s t a t e s , and  equation  of s t a t e s f o r m e t a l 1  i n the t u n n e l i n g  the m e t a l s i s s u p e r c o n d u c t i n g i s i n the d e n s i t y be i g n o r e d and  In o t h e r  current  when one  of s t a t e s f a c t o r — t h e  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  c o n d u c t o r r e g a r d e d i n one-to-one correspondence w i t h e x c i t a t i o n s i n the normal s t a t e .  to  w i t h the assumption of  s p h e r i c a l symmetry of the dependence of E upon wave number.  and v ^ may  say,  of s t a t e s i s p r o p o r t i o n a l  summing over  words, the o n l y e s s e n t i a l d i f f e r e n c e  2-1  of u  super-  the e l e c t r o n and  hole  T h i s view; of the superconductor i s , of  c o u r s e , the s o - c a l l e d semiconductor model f i r s t i n t r o d u c e d by G i a e v e r 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 e x p e r i m e n t a l r e s u l t s .  and  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) c o n f i r m e d their r e s u l t , providing  f u r t h e r j u s t i f i c a t i o n f o r u s i n g the  model i n c a l c u l a t i n g s i m p l e 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 the s i m u l t a n e o u s t u n n e l i n g (see eg. W i l k i n s , 1969)  of two  semiconductor currents.  (When  or more q u a s i p a r t i c l e s i s c o n s i d e r e d  the coherence f a c t o r s u^ and v ^ do not c a n c e l  the s i m p l e independent q u a s i p a r t i c l e model w i t h a m o d i f i e d d e n s i t y  and  of  states i s inadequate.) 2.  T u n n e l i n g Between Two Following  computed the t u n n e l i n g found i t to be of the  J  SS  Superconductors  the approach of Cohen et a l (1962), Josephso'n (1962)  current  between two  s u p e r c o n d u c t o r s (S^-S2) and  form  * o J  +  * 1 1 J  S  + S  2  +  * ! 2 J  S  + S  1  ( 2  "  1 2 )  1  -30where S  i s an o p e r a t o r d e f i n e d i n such a way  ground s t a t e of s u p e r c o n d u c t o r  t h a t when a c t i n g on  i w i t h N e l e c t r o n s i t c r e a t e s a new  c o n d u c t i n g 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  the super-  i s similar o  to the e x p r e s s i o n o b t a i n e d by Cohen et a l and reduces because o f c a n c e l l a t i o n of t h e coherence f a c t o r s to a form l i k e e q u a t i o n 2-10 N ( E ) ->- N (E) and f ( E ) + f M  M  r e p r e s e n t a new  e f f e c t , now  S L  (E).  The o t h e r terms i n e q u a t i o n  chapter.  i m p o r t a n t r e s u l t from the f o r e g o i n g 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 due  o r S^-S^  s t r u c t u r e s i s t h a t the t u n n e l c u r r e n t  t o t h e r m 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 may  the simple semiconductor-type conductor. S^-S^  be s a t i s f a c t o r i l y d e r i v e d from  s i n g l e p a r t i c l e a p p r o x i m a t i o n of a super-  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  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 r e l e v a n c e t o the  t u n n e l j u n c t i o n t o be used f o r charged 3.  The  2-12  known as the Josephson e f f e c t , w h i c h i s  d i s c u s s e d f u r t h e r i n s e c t i o n E of t h i s The  with  S -S- T u n n e l i n g 1 2  f o r an  superconducting  particle detection.  Current  a  When a s u p e r c o n d u c t i n g t u n n e l j u n c t i o n l i k e t h a t of f i g u r e 2-1  i s b i a s e d a t a v o l t a g e V, i t may  t h e semiconductor  a r e r e p r e s e n t e d by the o c c u p i e d s t a t e s above and  s t a t e s below the gap.  i s g i v e n by N  (cf. equation  S ±  In  model s k e t c h e d t h e r e , the t h e r m 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  i n each s u p e r c o n d u c t o r unoccupied  be r e p r e s e n t e d by f i g u r e 2-5.  ( E ) = N.(0)  the  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  2-9) ti.(E) = N ( 0 ) | E | / ( E - A . ( T ) ) ' ,|E| >A.(T) 2  2  2  i  (2-13) =0  , |E|<  A.(T)  and they are o c c u p i e d w i t h p r o b a b i l i t y f g ( E ) g i v e n by e q u a t i o n 2-11. s i n g l e q u a s i p a r t i c l e c u r r e n t f l o w i n g from s i d e 1 to s i d e 2 may  then  The be  w r i t t e n (see eg. G i a e v e r and M e g e r l e (1961) or S h a p i r o et a l , (1962)) as  h-2  |M|  2  -  A  '  N (E) f S 1  where A' i s a c o n s t a n t , JMI  g l  (E) N  g 2  (E + e V ) ( l - f ( E + eV))dE s 2  i s the m a t r i x element f o r the t r a n s i t i o n  and  -31-  r  eler.tron current  kl  ' 2 A  Fermi s u r f a c e  ATITT  _t.  ( T )  E=0  (£„, ) Fl'  A (T)  F2»  1  |A (T) 2  T >  N (E). S 2  F i g u r e 2-5:  1  -^N +j.-  S 1  0  K  (E)  i  Q u a s i p a r t i c l e Energy Diagram f o r S^-I-S  Junction  3r-  ..Calculated c u r r e n t .  1  2 voltage  F i g u r e 2-6:  Experimental points ( f i t t e d at p o i n t marked by arrow)  (mV)  I-V C h a r a c t e r i s t i c f o r Asymmetric J u n c t i o n . ( a f t e r S h a n i m kt a l . 1Q67^  -32e_„  i s the Fermi energy of m e t a l 2.  A s i m i l a r r e s u l t o b t a i n s f o r I„ , 2  so t h a t by assuming the d e n s i t y of s t a t e s 1^(0)  and | M|  a r e c o n s t a n t near the Fermi s u r f a c e , t h e net c u r r e n t may 1  "  I 1  _  ~ V l  2  =  n (E)  A | M |  where A i s a c o n s t a n t .  x  n (E+eV)[f (E) 2  (For m a t h e m a t i c a l  i n t e g r a t i o n has been extended  t o -<*>.  s  (see Bardeen, be w r i t t e n  - f (E+eV)]dE  (2-14)  g  convenience,  1961)  the lower l i m i t of  Because o f the a c t i o n of the Fermi  f u n c t i o n s the i m p o r t a n t r e g i o n o f i n t e g r a t i o n i s c o n f i n e d t o an i n t e r v a l of a few k T about the Fermi l e v e l . ) e q u a t i o n 2-14  The s i m i l a r i t y between the r e s u l t s of  d e r i v e d from the phenomenological  those of e q u a t i o n 2-10  semiconductor  picture  and  d e r i v e d from an e x a c t m i c r o s c o p i c approach i s  evident. The agreement of e q u a t i o n 2-14 demonstrated  by S h a p i r o et a l (1962) who  w i t h experiment  has been  e v a l u a t e d the e q u a t i o n n u m e r i c a l l y  and o b t a i n e d a c l o s e f i t w i t h t h e i r e x p e r i m e n t a l d a t a (see f i g u r e E.  2-6).  Josephson T u n n e l i n g The  t u n n e l i n g c u r r e n t between two s u p e r c o n d u c t o r s as c a l c u l a t e d  by Josephson (1962) was  g i v e n i n e q u a t i o n 2-12.  Two  new  effects  i n t h i s r e s u l t were i n d i c a t e d by Josephson a t t h a t time and l a t e r experimentally: current J  =0,  o  confirmed  (1) a t 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 but a dc s u p e r c u r r e n t up to a maximum of | j . I 1  (2) a t f i n i t e v o l t a g e s the u s u a l dc c u r r e n t J a h i g h f r e q u e n c y ac c u r r e n t o f a m p l i t u d e corresponds  implicit  t o 483.6 MHz).  |j  q  can o c c u r ;  appears but t h e r e i s a l s o  | and f r e q u e n c y 2eV/h (luV  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 s u p e r c o n d u c t i n g t u n n e l j u n c t i o n charged  particle  d e t e c t o r and w i l l not be d i s c u s s e d f u r t h e r ; the i n t e r e s t e d r e a d e r i s r e f e r r e d t o the book e d i t e d by B u r s t e i n and L u n d q v i s t  (1969).  The o t h e r 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 a c r o s s the i n s u l a t o r i n which 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  the  i s not d i s t u r b e d .  P h y s i c a l l y , t h e e f f e c t a r i s e s when two s u p e r c o n d u c t o r s a r e arranged o  s u f f i c i e n t l y c l o s e t o g e t h e r (=10 A) as to become weakly c o u p l e d i n the sense t h a t t h e r e l a t i v e phase between the Cooper p a i r s i n each of the super-  -33conductors  i s no l o n g e r a r b i t r a r y .  I n o t h e r words, the Cooper p a i r  c o r r e l a t i o n s extend t h r o u g h the i n t e r v e n i n g d i s t a n c e so t h a t the superconductors  behave i n some r e s p e c t s as a s i n g l e b l o c k .  The  two Josephson  e f f e c t i s thus not n e c e s s a r i l y a t u n n e l i n g phenomenon but t u n n e l j u n c t i o n s have been used as a means o f s t u d y i n g 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 w h i c h 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 c h a p t e r , the dc Josephson s u p e r c u r r e n t of i t s n u i s a n c e  i s of i n t e r e s t to the p r e s e n t experiment o n l y because  value.  o u t l i n e d i n p a r a g r a p h 1.  The way  Paragraphs 2 and  the dc Josephson s u p e r c u r r e n t may  i n which t h i s e f f e c t i s undesirable i s 3 b r i e f l y r e v i e w the t h e o r y  and paragraph 4 c o n s i d e r s how  be s u p p r e s s e d w i t h a magnetic  the  of  supercurrent  field.  I n a d d i t i o n t o the B u r s t e i n , L u n d q v i s t book (1969) a r e c e n t c o n c i s e r e v i e w of the s u b j e c t has been g i v e n by Anderson (1967); the most r e c e n t comprehensive b i b l i o g r a p h y i s g i v e n by Schroen 1.  probably (1968).  U n f a v o u r a b l e A s p e c t s o f the Josephson E f f e c t W i t h r e f e r e n c e t o f i g u r e 2-7  i t i s seen t h a t 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 t u n n e l i n g f e a t u r e s a l a r g e s u p e r c u r r e n t w h i c h may  exceed the magnitude o f the t h e r m a l l y -  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 c u r r e n t d i s c u s s e d 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) 5 to 10.  When the s u p e r c u r r e n t  i n f i g u r e 2-7), by a f a c t o r of  reaches a c r i t i c a l v a l u e  (I . ) i t i s crit r e p l a c e d 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 v o l t a g e V = 2A/e appears a c r o s s i  the j u n c t i o n . Now i t i s e v i d e n t t h a t a Josephson j u n c t i o n b i a s e d a t I < I . might be made t o s w i t c h from V = 0 t o V = 2A/e when bombarded crxt ° 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 t h 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 n a t u r e makes t h i s response unsuitable for multichannel  spectrometry.  I t i s shown i n c h a p t e r of the tunnel j u n c t i o n detector  3 t h a t the d e s i r e d o p e r a t i n g p o i n t  (denoted by Q i n f i g u r e 2-8(b)) i s i n the  r e g i o n o f h i g h 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 tunneling curve.  As t h i s b i a s i n g p o i n t cannot be reached r e p r o d u c i b l y i n  a j u n c t i o n e x h i b i t i n g a Josephson s u p e r c u r r e n t , a way prevent  p a i r t u n n e l i n g from o c c u r i n g .  paragraph 4 f o l l o w i n g .  must be found to  Such a t e c h n i q u e  i s outlined in  -34-  I  load l i n e switching L  crit  z dc Josephson ,  supercurrent  _  E_xp_e rimen t a l Q  t  Idea r " " j  _L 1.1 (2A/e)  F i g u r e 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  -352.  Theory o f t h e dc Josephson E f f e c t The  (Scalapino  d e t a i l e d microscopic  t h e o r y o f t h e Josephson e f f e c t  (1969), Josephson (1965), Anderson (1964)) shows t h a t t h e super-  c u r r e n t d e n s i t y J p a s s i n g between two s u p e r c o n d u c t o r s c o m p r i s i n g j u n c t i o n i s g i v e n by ( c f . e q u a t i o n  a tunnel  2-12)  J = J  s i n <j>  (2-15)  where <J> i s t h e quantum m e c h a n i c a l phase d i f f e r e n c e between t h e p a i r wavef u n c t i o n s i n t h e two s u p e r c o n d u c t o r s and J ^ i s a measure o f t h e t u n n e l i n g p r o b a b i l i t y through t h e 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 a l s o depends on t h e temperature.  Ambegoakar and B a 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 s u p e r c o n d u c t o r s on e i t h e r s i d e o f t h e b a r r i e r — a n d  f i n d i t t o be A.(T) 2k T  :n  B  where e i s t h e e l e c t r o n i c c h a r g e ,  i s t h e l o w temperature, l o w v o l t a g e  r e s i s t a n c e o f t h e j u n c t i o n when i n i t s normal s t a t e and A i s t h e j u n c t i o n overlap area.  When T = 0 K, J ^ i s a maximum g i v e n by  ,  n  -  and f o r t y p i c a l t i n specimens used i n t h i s e x p e r i m e n t , -4 2 (R - 0.1 A = 7 x 10 cm ) I = J 'A - 10 mA. n 1 1  n,  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 s u p p l y — p e r h a p s a b a t t e r y i n s e r i e s w i t h a variable resistor  (see eg. f i g u r e 5 - 1 ) — a n d t h e phase d i f f e r e n c e <j> a d j u s t s  i t s e l f as i n e q u a t i o n  (2-15) t o c a r r y t h i s v a l u e o f c u r r e n t .  Consequently  as t h e c u r r e n t determined by t h e e x t e r n a l c i r c u i t i s i n c r e a s e d ,  <j> i n c r e a s e s  u n t i l i t reaches i n a t which p o i n t the j u n c t i o n i s c a r r y i n g t h e maximum o r c r i t i c a l dc Josephson c u r r e n t I I , the j u n c t i o n switches recorder  c  = J^(max) A.  For currents greater  than  t o t h e q u a s i p a r t i c l e I-V c u r v e as shown i n the  t r a c i n g o f f i g u r e 2-7.  The d o t t e d l i n e on t h e c h a r a c t e r i s t i c o f  the f i g u r e i n d i c a t e s r e t r a c e b e h a v i o u r common t o almost a l l specimens.  -36When t h e c u r r e n t  ( I ) through the j u n c t i o n i s decreasing, the j u n c t i o n  becomes u n s t a b l e f o r J. < I supercurrent disturbance.  and s w i t c h e s o v e r t o the z e r o - v o l t a g e o r  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 The o r i g i n and d e t a i l e d shape o f t h i s h y s t e r e s i s loop a r e  s t i l l an open q u e s t i o n  (Schroen,  1968).  B a s i c a l l y , i t i s governed by t h e  q u a s i p a r t i c l e t u n n e l i n g c u r r e n t and some workers (March and B l a c k f o r d , 1967)  suggest t h a t t h e power d i s s i p a t e d i n t h e j u n c t i o n may p l a y an important  role. 3.  E f f e c t o f M a g n e t i c F i e l d on dc Josephson C u r r e n t A f u r t h e r r e s u l t of the microscopic  t h e o r y i s t h a t t h e dc  Josephson c u r r e n t s h o u l d depend on t h e v e c t o r p o t e n t i a l s i n c e t h e quantum m e c h a n i c a l phase d i f f e r e n c e depends on t h e v e c t o r p o t e n t i a l . mks u n i t s , e q u a t i o n  Hence, i n  2-15 s h o u l d be ( J a k l e v i c e t a l , 1965) r  2e J = J^ sin * - t r x  f  2  i A* 'dt  1  (2-17)  -  1  where t h e l i n e i n t e g r a l term from s i d e 1 t o s i d e 2 i s i n c l u d e d t o ensure t h a t t h e c u r r e n t d e n s i t y i s i n v a r i a n t under a gauge t r a n s f o r m a t i o n . Equation  2-17 has been e v a l u a t e d by J a k l e v i c e t 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 p l a c e d i n a u n i f o r m magnetic f i e l d B which i s p a r a l l e l to  t h e o x i d e l a y e r ; they  find  J(x) = J  s i n '[<f>(6) +C2e/h)Bx(t + 2X)]  (2-18)  where t i s t h e o x i d e t h i c k n e s s , X i s t h e p e n e t r a t i o n depth o f t h e magnetic f i e l d and x i s t h e d i s t a n c e a l o n g t h e j u n c t i o n p e r p e n d i c u l a r t o B (x = 0 a t the c e n t e r o f t h e j u n c t i o n , see f i g u r e 2-8(a).)  Equation  2-18, which g i v e s  the magnetic f i e l d dependence o f t h e Josephson c u r r e n t , i n d i c a t e s t h a t the c u r r e n t a t one p o i n t can f l o w o p p o s i t e t o t h e way i t f l o w s a t another. f o l l o w s from e q u a t i o n 2-18 t h e r e f o r e t h a t f o r a j u n c t i o n o f g i v e n  It  size  t h e r e a r e c e r t a i n v a l u e s o f B* f o r which the maximum dc Josephson c u r r e n t I(max) c a r r i e d by t h e j u n c t i o n w i l l be z e r o .  T h i s may be seen by i n t e g r a t i n g  2-18 from -£W to, +£W (W i s t h e l e n g t h of the j u n c t i o n p e r p e n d i c u l a r t o S)  -37-  F i g u r e 2-8 ( a ) :  Schematic C r o s s - S e c t i o n of T u n n e l i n g . J u n c t i o n I  max  (Arbitrary units) ]  -38and s e t t i n g I  = J • A so t h a t  I - \  where  ^ X E l I .  s  i  n  ,,,(0)  (2-19)  p = $ . /$ 3  °  $. = B(2X  and  3 $  =  h/2e  + t ) W = f l u x e n c l o s e d 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 = 2.07 x 10 G cm = quantum u n i t o f magnetic flux  As mentioned above, t h e phase d i f f e r e n c e <f> a d j u s t s t o t h e e x p e r i m e n t a l c o n d i t i o n s so t h a t maximum p o s i t i v e c u r r e n t f l o w i s o b t a i n e d <J>(0)  =  +?TT  with  making  I(max) = 1^  s i n (pip pir  (2-20)  I t i s c l e a r f r o m (2-20) t h a t I(max) = 0 whenever $. e q u a l s some i n t e g r a l m u l t i p l e o f h/2e and a p l o t o f I(max) a g a i n s t B w i l l resemble t h e f a m i l i a r Fraunhoffer and  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  X =500 A so t h a t t h e p e r i o d of I(max) i n B i s about 0.7 G.  of t h i s e f f e c t was f i r s t confirmed  by R o w e l l  (1963) u s i n g  The e x i s t e n c e  symmetrical  Pb-Pb j u n c t i o n s . I n j u n c t i o n s o f s m a l l d i m e n s i o n o r low c r i t i c a l  current,  the magnetic f i e l d due t o t h e t u n n e l i n g c u r r e n t can be n e g l e c t e d .  Such i s  not t h e case i f t h e j u n c t i o n w i d t h becomes comparable t o o r l a r g e r than a c h a r a c t e r i s t i c l e n g t h 2X^, t h e Josephson p e n e t r a t i o n d e p t h , g i v e n by ( i n mks u n i t s )  X = (h/4ey A J . ) * J o 1 T  where u  o  -  (2-21)  i s t h e p e r m e a b i l i t y o f f r e e space and i t i s assumed t h a t  2X + t - 2X .  The s i g n i f i c a n c e o f Xj i s t h a t when t h e j u n c t i o n w i d t h W i s  l a r g e compared t o X^, t h e a p p l i e d magnetic f i e l d i s screened from t h e i n t e r i o r o f t h e j u n c t i o n due t o t h e f l o w o f t h e Josephson c u r r e n t s themselves w i t h t h e r e s u l t t h a t most o f t h e d i r e c t c u r r e n t f l o w s w i t h i n a d i s t a n c e 2X  T  o f t h e 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  -39t y p i c a l j u n c t i o n s i n t h e l i t e r a t u r e range from 0.1 t o 1.0 mm w h i c h i s s u f f i c i e n t l y l a r g e compared t o t h e j u n c t i o n dimensions used i n t h i s e x p e r i ment, t h a t u n i f o r m p e n e t r a t i o n o f t h e magnetic  f i e l d may be s a f e l y assumed.  I n s t a n c e s where Xj i s i m p o r t a n t have been d i s c u s s e d by R o w e l l Owen and S c a l a p i n o (1967) and P r i t c h a r d and Schroen 4.  S u p p r e s s i o n o f Josephson  (1963),  (1968).  Supercurrent  E q u a t i o n 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 magnetically 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 t h a t p i s an i n t e g e r ( i e . B = n$ (W(2X + t ) ) Q  s u f f i c i e n t l y large f i e l d  , n = 1,2  ) making I(max) = 0 o r a p p l y a  t h a t t h e (pir) ^ dependence reduces I (max) t o  negligible proportions. I m m e d i a t e l y , t h e f i r s t o p t i o n . i s r u l e d o u t because o f t h e f a c t t h a t , i n t h e j u n c t i o n s used i n t h i s e x p e r i m e n t , no sharp minima i n which t h e s u p e r c u r r e n t v a n i s h e d were ever o b s e r v e d .  (Even i f such minima had been  o b t a i n e d , an e x t r e m e l y w e l l - r e g u l a t e d magnetic  f i e l d would, o f c o u r s e , be  required to maintain that condition.) E x p e r i m e n t a l l y , i t t u r n s o u t (see s e c t i o n 1, c h a p t e r 6) t h a t t h e second approach  i s t h e one t o t a k e i n t h a t magnetic  f i e l d s o f 30-100 G a r e  s u f f i c i e n t t o e f f e c t i v e l y suppress t h e s u p e r c u r r e n t w i t h a maximum ' r e d u c t i o n i n t h e energy gap w i d t h o f l e s s than F.  about.6%.  M u l t i p a r t i c l e T u n n e l i n g Phenomena F o r t h e sake o f completeness,  i t s h o u l d be mentioned  here t h a t  h i g h e r o r d e r o r m u l t i p a r t i c l e t u n n e l i n g p r o c e s s e s may a l s o occur i n superconducting tunneling j u n c t i o n s .  As t h e 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 t h e name g i v e n t o t h e s i m u l t a n e o u s t r a n s f e r o f more than one q u a s i p a r t i c l e through t h e t u n n e l j u n c t i o n , b a r r i e r . 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 i 14  ( P r o b a b l y no h i g h e r o r d e r ( W i l k i n s , 1969).)  This  second o r d e r p r o c e s s ( p r o p o r t i o n a l t o |M| ) goes o n l y i f t h e r e i s an i n t e r a c t i o n between t h e p a r t i c l e s  (as i n a s u p e r c o n d u c t o r ) and i s m a n i f e s t e d as  a sharp i n c r e a s e i n t h e c u r r e n t a t eV = A (see T a y l o r and B u r s t e i n ,  1963).  An e x c e l l e n t r e v i e w o f t h i s t o p i c has r e c e n t l y been g i v e n 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 a l s o be found i n appendix B o f t h i s t h e s i s where d e f e c t i v e o r " l e a k y " j u n c t i o n s a r e c o n s i d e r e d .  -40-  CHAPTER 3  OPERATING PRINCIPLES OF THE SUPERCONDUCTING CHARGED PARTICLE DETECTOR  A.  Introduction T h i s c h a p t e r o u t l i n e s t h e p r i n c i p l e s and p r a c t i c a l  requirements of the superconducting detector.  operation  t u n n e l j u n c t i o n as a charged p a r t i c l e  To p l a c e t h i s a p p l i c a t i o n o f t u n n e l j u n c t i o n s i n p e r s p e c t i v e ,  i t i s convenient  to review b r i e f l y :  (1) t h r e e o f t h e p r e s e n t  techniques  f o r measuring charged p a r t i c l e energy s p e c t r a 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 o f i t s  r e l a t i o n t o t h e d e t e c t i o n o f charged p a r t i c l e s , p r e v i o u s work on the d e t e c t i o n o f microwave r a d i a t i o n i s a l s o reviewed The  (section B).  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 t h e purpose o f s t i p u l a t i n g t h e a t t r i b u t e s w h i c h t h e t u n n e l j u n c t i o n ' s response t o such r a d i a t i o n s h o u l d have i f t h e j u n c t i o n i s t o be u s e f u l as a s p e c t r o m e t e r .  I n s e c t i o n D i t i s e s t a b l i s h e d t h a t t h e most  f a v o u r a b l e t u n n e l j u n c t i o n c o n f i g u r a t i o n i s t h e symmetric one w i t h l e a d f i l m s as t h e s u p e r c o n d u c t o r .  The n a t u r e o f t h e 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 g e n e r a t e d by an a l p h a p a r t i c l e i n t h e s u p e r c o n d u c t i n g j u n c t i o n i s c o n s i d e r e d i n s e c t i o n E.  tunnel  A small signal equivalent c i r c u i t f o r  the d e t e c t o r i s g i v e n 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) t h e assumed form of t h e r e s u l t i n g s i g n a l c u r r e n t p u l s e i s s t a t e d a l o n g w i t h  estimates  of i t s parameters. Because o f t h e r e s t r i c t i o n i t p l a c e s on t h e maximum a c c e p t a b l e i n s u l a t i n g l a y e r thickness, the tunneling p r o b a b i l i t y s e c t i o n H.  i sdealt with i n  A n o t e on t h e p r a c t i c a l o p e r a t i o n o f t h e d e t e c t o r i s g i v e n i n  s e c t i o n I and, c o n c l u d i n g t h e c h a p t e r , i s a s h o r t survey o f t u n n e l j u n c t i o n n o i s e sources  (section J ) .  -411.  P r e s e n t Methods o f Charged P a r t i c l e  Spectrometry  B r o a d l y s p e a k i n g , t h e methods c a n be d i v i d e d i n t o t h r e e major c a t e g o r i e s :  t h e measurement o f momentum, energy l o s s o r v e l o c i t y .  G e n e r a l r e v i e w s o f t h e s e methods can be found i n books by Yuan (1961) and A j z e n b e r g - S e l o v e  (1960); a s h o r t survey i s g i v e n by Wood (1965).  Since  the s u p e r c o n d u c t i n g " d e t e c t o r " i s e s s e n t i a l l y an e n e r g y - l o s s d e v i c e , o n l y t h o s e d e t e c t o r s o f t h e "energy l o s s measurement" type w i l l be d i s c u s s e d here. When a charged p a r t i c l e passes through m a t t e r , i t l o s e s energy through t h e e x c i t a t i o n and i o n i z a t i o n o f atoms c l o s e t o t h e p a t h o f t h e particle.  T h i s p r o p e r t y has been e x p l o i t e d i n t h e development o f s e v e r a l  types of p a r t i c l e spectrometers.  Three i n common use a r e t h e 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 o r glass chamber, equipped w i t h an anode and cathode, and f i l l e d w i t h an a c c u r a t e l y d e f i n e d volume o f gas.  The passage o f an e n e r g e t i c charged p a r t i c l e  through  a s u i t a b l e "window" i n t o t h e gas volume causes e l e c t r o n s t o be removed from some o f t h e gas m o l e c u l e s thereby p r o d u c i n g e l e c t r o n - p o s i t i v e - i o n pairs.  Under t h e i n f l u e n c e o f t h e e l e c t r i c f i e l d , t h e e l e c t r o n s m i g r a t e  t o t h e anode and t h e p o s i t i v e i o n s t o t h e cathode.  Because t h e average  amount o f energy l o s t by t h e charged p a r t i c l e t o produce an i o n p a i r i s a p p r o x i m a t e l y independent  o f t h e energy o f t h e p a r t i c l e , t h e a m p l i t u d e of  the r e s u l t i n g charge p u l s e i s p r o p o r t i o n a l t o t h e energy l o s t i n t h e gas by t h e 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 consists 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 o r g a s — o p t i c a l l y coupled t o 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 t h e phosphor, t h e  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 t h e m a t e r i a l t o be e x c i t e d i n t o t h e c o n d u c t i o n band.  Then, a t i m p e r f e c t i o n s i t e s i n t h e  c r y s t a l , a l a r g e f r a c t i o n o f t h e e x c i t e d e l e c t r o n s f a l l back t o t h e i r s t a t e and emit a photon  i n a process c a l l e d fluorescence.  e m i t t e d d u r i n g f l u o r e s c e n c e s t r i k e t h e photocathode  ground  The photons  of the p h o t o m u l t i p l i e r  tube whose o u t p u t s i g n a l i s p r o p o r t i o n a l t o t h e number o f photons generated or t h e energy l o s t by t h e charged p a r t i c l e i n t r a v e r s i n g t h e phosphor. I n t h i s c a s e , 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 o f t h e  -42average energy l o s s per photon g e n e r a t e d . D u r i n g the p a s t t e n y e a r s , semiconductor d e t e c t o r s 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 D e a r n a l e y and N o r t h r o p (1966) and T a y l o r (1963) d e a l t h o r o u g h l y w i t h t h e s e d e t e c t o r s g i v i n g d e t a i l s on t h e o r y and a p p l i c a t i o n ; a r e c e n t r e v i e w by Tavendale (1967) s u r v e y s the p r e s e n t s t a t e of d e t e c t o r t e c h n o l o g y . E s s e n t i a l l y , the semiconductor d e t e c t o r 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 f a c e o f a s l a b of germanium or s i l i c o n .  Typically  the j u n c t i o n i s w i t h i n 0.5 p of the one f a c e and i s v e r y a b r u p t .  When the  j u n c t i o n i s r e v e r s e b i a s e d , a d e p l e t i o n r e g i o n i s c r e a t e d 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 h o l e s generated by the  passage of a charged p a r t i c l e through t h i s r e g i o n d r i f t t o the e l e c t r o d e s 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 t o a charge p u l s e whose a m p l i t u d e i s l i n e a r l y r e l a t e d t o the p a r t i c l e energy, the l i n e a r i t y b e i n g due t o the approximate energy independence of the average energy l o s s per electron-hole pair. I t i s of i n t e r e s t to note t h a t the average e n e r g i e s w ;  r e q u i r e d t o 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 a r e  3.6  and 2.9 eV r e s p e c t i v e l y , compared to an average v a l u e of about 30 eV per i o n p a i r i n gases and about 300 eV per p h o t o e l e c t r o n a t the photocathode of a s c i n t i l l a t i o n c o u n t e r .  (Dearnaley and N o r t h r o p , 1966).  This implies,  t h e r e f o r e , t h a t f o r a g i v e n 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 s e m i c o n d u c t o r 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 o t h e r c o u n t e r s and t h a t the s t a t i s t i c a l f l u c t u a t i o n s , e x p r e s s e d as a p e r c e n t a g e of the s i g n a l (see c h a p t e r 1) a r e c o r r e s p o n d i n g l y s m a l l e r . 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 S u p e r c o n d u c t i n g D e v i c e s A l t h o u g h 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 supercon-  d u c t i n g t u n n e l i n g j u n c t i o n s has never been r e p o r t e d p r i o r t o the p r e s e n t e x p e r i m e n t , 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 u s i n g o t h e r s u p e r c o n d u c t i n g ;  devices. Andrews e t a l (1949) bombarded a s t r i p of n i o b i u m n i t r i d e (3.5 x 0.4  x 0.006 mm,  T^ = 15.6 K ) , w i t h a l p h a p a r t i c l e s from a polonium  s o u r c e and found t h a t c o u n t a b l e p u l s e s were produced, one f o r each impact. The NbN  s t r i p was m a i n t a i n e d a t 15.5 K, j u s t below i t s t r a n s i t i o n  tem-  -43p e r a t u r e , a d i r e c t c u r r e n t was a p p l i e d and t h e p o t e n t i a l between t h e ends o f the s t r i p m o n i t o r e d w i t h a p u l s e 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 t h a t t h e c o u n t i n g r a t e was a maximum when t h e ambient c u r r e n t  was  40 mA and t h e specimen was m a i n t a i n e d  between t h e normal and s u p e r c o n d u c t i n g  a t t h e mid p o i n t o f i t s t r a n s i t i o n  s t a t e s — a change o f ± 0.1 K reduced  the s i g n a l t o t h e same l e v e l as t h e n o i s e . -4 p u l s e s l a s t e d f o r about 10  I t was e s t i m a t e d  s e c w i t h an a m p l i t u d e  maximum s i g n a l t o n o i s e r a t i o was 3 t o 1.  that the  o f about 0.1 uV.  The  No t h e o r e t i c a l e x p l a n a t i o n o f  t h e i r f i n d i n g s was g i v e n by t h e a u t h o r s b u t , s i n c e t h e s t r i p was presumably i n t h e i n t e r m e d i a t e s t a t e , t h e temperature r i s e accompanying the passage of an a l p h a p a r t i c l e would be m a n i f e s t e d  as an i n c r e a s e i n t h e s t r i p  r e s i s t a n c e and t h e r e f o r e (assuming c o n s t a n t  c u r r e n t b i a s i n g ) as an i n c r e a s e  i n t h e v o l t a g e a c r o s s t h e specimen. A p r o p o s a l t o use t h e c r y o t r o n , o r s u p e r c o n d u c t i n g to  d e t e c t i o n i z i n g p a r t i c l e s was p u t forward by Sherman (1962,A).  switch, The  c r y o t r o n , i t was s u g g e s t e d , s h o u l d c o n s i s t o f two gate l o o p s , made o f d i s s i m i l a r superconductors  such as l e a d and t i n w i t h a c o n t r o l l o o p o f  n i o b i u m on t h e l e a d g a t e .  I f the d e v i c e were m a i n t a i n e d  a t a temperature  j u s t below t h e l o w e s t o f t h e t h r e e c r i t i c a l temperatures (T =3.72 K f o r S n ) , c i t was argued t h a t t h e c r y o t r o n c o u l d be kept i n a m e t a s t a b l e 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 c u r r e n t  l e s s than t h e c r i t i c a l c u r r e n t a t which s u p e r c o n d u c t i v i t y  slightly  would be quenched.  The passage o f a charged p a r t i c l e i n t o t h a t l o o p might then e s t a b l i s h a s m a l l r e g i o n o f normal m e t a l a c r o s s t h e t i n f i l m and cause t h e c u r r e n t t o be suddenly s w i t c h e d  i n t o t h e l e a d l o o p w h i c h , p r i o r t o t h i s i n s t a n t , would  be i n t h e s u p e r c o n d u c t i n g  s t a t e b u t n o t c a r r y i n g any c u r r e n t .  A p u l s e would  then r e s u l t on t h e n i o b i u m c o n t r o l w i n d i n g w h i c h c o u l d be used t o 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 t h e c u r r e n t back i n t o t h e t i n l o o p ready f o r t h e next p a r t i c l e . Sherman (1962,B) made a second p r o p o s a l f o r a  superconducting  n u c l e a r p a r t i c l e d e t e c t o r which was t o c o n s i s t o f a narrow ( l O u ) , t h i n o  (1000 A) f i l m o f c u r r e n t - c a r r y i n g s u p e r c o n d u c t o r c o o l e d w e l l below i t s t r a n s i t i o n temperature and p l a c e d i n s e r i e s w i t h a s m a l l r e s i s t o r . was  It  argued t h a t an a l p h a p a r t i c l e i m p i n g i n g on a t i n f i l m o f such dimensions  would cause t h e r e s i s t a n c e t o change from zero t o about 0.2  by i n s e r t i n g  a s m a l l t r a n s v e r s e s t r i p o f normal m e t a l i n t h e path o f t h e c u r r e n t .  The  -44r e d u c t i o n of c u r r e n t would be d e t e c t e d a c r o s s the l o a d r e s i s t o r as a v o l t a g e p u l s e e s t i m a t e d t o be about 1 mV f o r a 5 mA load.  The r i s e time o f the p u l s e was  t r a n s p o r t c u r r e n t and 10 ft  e s t i m a t e d to be r o u g h l y l O " ^ sec  and  -  the f a l l time to be about 10 ^ s e c . R e c e n t l y , S p i e l e t a l (1965), s u c c e s s f u l l y d e t e c t e d 5.3 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  MeV  similar  i n s p i r i t t o t h a t o u t l i n e d by Sherman (1962,B) but d i f f e r e n t i n d e t a i l i n t h a t t h e 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 c u r r e n t a l o n g t h e l e n g t h of t h e f i l m r a t h e r than i n a l o a d r e s i s t a n c e .  The  e f f e c t was  observed  i n b o t h 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. experiment  S i m i l a r l y to the  r e p o r t e d by Andrews (1949), v o l t a g e p u l s e s o c c u r r e d o n l y f o r  temperatures  near the c r i t i c a l  c r i t i c a l current.  temperature  and t r a n s p o r t c u r r e n t s near the  For an I n f i l m c a r r y i n g a c u r r e n t of 5.5 mA a t 3.3  K,  the v o l t a g e p u l s e s had a peak a m p l i t u d e o f a p p r o x i m a t e l y 200 pV r i s i n g i n about 15 nsec and d e c a y i n g i n r o u g h l y 70 nsec.  The r a d i u s 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 a l p h a p a r t i c l e was be 14 p under these c o n d i t i o n s .  e s t i m a t e d to  The s w i t c h i n g a c t i o n i n these d e v i c e s  c o u l d be used as a p a r t i c l e d e t e c t o r and energy t h r e s h o l d 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 t o m u l t i c h a n n e l  spectrometry.  A s u b j e c t t h a t might be r e - - i n v e s t i g a t e d , 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  i n t e r m e d i a t e 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 : h a v i n g h i g h s e n s i t i v i t y f o r heat p u l s e d e t e c t i o n a r e now  b e i n g developed. o  For example, Ackerman e t a l (1969)  ,  r e p o r t the development o f 400 A t h i c k Zn-Cd f i l m s f o r temperature detection with a s e n s i t i v i t y superconducting (AR/RAT) - 5  t r a n s i t i o n (T  pulse  (AR/RAT) = 4 0 0 n e a r the c e n t r e of the - 0.5  f o r carbon f i l m s . )  K).  (For  normal-  comparison,  I t i s c o n c e i v a b l e t h a t such a f i l m ,  mounted on a s u i t a b l e s u b s t r a t e and biased- e l e c t r i c a l l y , m a g n e t i c a l l y , and t h e r m a l l y a t i t s m i d - t r a n s i t i o n p o i n t , c o u l d be made to respond to heat p u l s e s r e s u l t i n g from e n e r g e t i c charged the s u b s t r a t e .  linearly  p a r t i c l e s i m p i n g i n g upon  As the c r o s s - s e c t i o n a l a r e a of these f i l m s must be  kept  s m a l l 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 c o n c e n t r a t e d heat p u l s e i s l i m i t e d . f i l m c o m p l e t e l y normal,  Consequently,  i t would be n e c e s s a r y  t o a v o i d s w i t c h i n g the  t o d i s s i p a t e the o r i g i n a l heat  p u l s e s i n a r e l a t i v e l y l a r g e volume w h i c h , of c o u r s e , i n c r e a s e s the  response  -45time.  The s t o p p i n g 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 a n o t h e r  a l t e r n a t i v e but now  the h e a t i n g would no l o n g e r be m a c r o s c o p i c and the  complex b e h a v i o u r of mixed s t a t e domains i n the presence o f e l e c t r i c a l , magnetic and t h e r m a l g r a d i e n t s must be c o n s i d e r e d . B.  D e t e c t i o n o f Microwave R a d i a t i o n With Tunnel J u n c t i o n s S i n c e the p i o n e e r i n g work of G i a e v e r (1960,1961) on t u n n e l i n g  between s u p e r c o n d u c t o r s , much a t t e n t i o n has been g i v e n t o u s i n g t h i s phenomenon t o s t u d y the more fundamental a s p e c t s 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 s u p e r c o n d u c t i v e  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 t h a t many attempts have been made t o d e v e l o p u s e f u l d e v i c e s based on the t u n n e l i n g p r i n c i p l e .  A recent  r e v i e w by T a y l o r (1968) summarizes work t h a t has been done i n t h i s a r e a . Of p a r t i c u l a r i n t e r e s t , because the p r o c e s s e s i n v o l v e d a r e somewhat analogous t o those t a k i n g p l a c e i n the d e t e c t i o n o f charged p a r t i c l e s w i t h t u n n e l 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  The aim of t h i s s e c t i o n i s t o  acknowledge  h i s t o r i c a l p r i o r i t y of t h e s e t u n n e l 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 t h e r e b y those i d e a s and r e s u l t s from p r e v i o u s workers which have b e n e f i t e d the p r e s e n t e x p e r i m e n t .  For s i m p l i c i t y , a symmetric supercon-  d u c t i n g j u n c t i o n w i l l be c o n s i d e r e d . 1.  Microwave and I n f r a r e d Photon D e t e c t i o n The f e a s i b i l i t y o f u s i n g t u n n e l i n g j u n c t i o n s as  microwave  and f a r i n f r a - r e d quantum d e t e c t o r s was f i r s t proposed and a n a l y z e d by B u r s t e i n e t a l (1961A,B).  The p r o c e s s i n 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 a c r o s s the energy gap o f the  s u p e r c o n d u c t o r 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 d i o d e s .  I n the former  c a s e , the lower f r e q u e n c y (a)) l i m i t of d e t e c t a b l e r a d i a t i o n would be governed by the r e q u i r e m e n t t h a t 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 films i s  hw >, 2A(T) ;  which i s the energy r e q u i r e d t o 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  -46f i g u r e 3-1(a). due  F o l l o w i n g a s h o r t b u r s t of r a d i a t i o n , the e x t r a c u r r e n t  t o 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 p u l s e superimposed on the background t u n n e l i n g c u r r e n t w h i c h , f o r b i a s v o l t a g e s eV < 2A(T)  (as seen i n c h a p t e r  2 ) , i s propor-  t i o n a l t o the d e n s i t y o f t h e r m 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 . The  magnitude  and d u r a t i o n of t h i s p u l s e 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 . i s t h a t the frequency 650 GHz  An i m p o r t a n t  f e a t u r e o f such d e t e c t o r s  range they c o v e r , a p p r o x i m a t e l y  85 GHz  (A£)  to  ( P b ) , i s one w h i c h 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 .  the same t o k e n , because of the p a u c i t y of l a b o r a t o r y g e n e r a t o r s  By  i n this  r a n g e , the p r o p e r t i e s of the B u r s t e i n d e v i c e have not been w i d e l y s t u d i e d ; the o n l y c l a i m to the e f f e c t b e i n g found e x p e r i m e n t a l l y seems to be unpublished  work of Dayem and M i l l e r  the  (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 o n l y  one  p o s s i b l e , no change i n the t u n n e l i n g c u r r e n t 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 e x p e r i m e n t a l l y , however, t h a t  c o n s i d e r a b l e i n t e r a c t i o n between the microwave f i e l d and take p l a c e f o r fiu < 2A(T) photon-assisted  the j u n c t i o n d i d  v i a a p r o c e s s which has come to be known as  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 a c r o s s the gap d e s c r i b e d above, the r a d i a t i o n b r e a k s 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 o f which remains i n the normal f l u i d i n S  a  w h i l e the o t h e r t u n n e l s through the b a r r i e r to become p a r t o f the normal  f l u i d o f S^.  I t i s c l e a r t h a t 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 e n e r g i e s l e s s than 2A(T)  and,  i n fact, for a  j u n c t i o n b i a s e d a t v o l t a g e V, the t h r e s h o l d f o r a one-photon process i s  fun = 2A(T)-eV  In a d d i t i o n , multiphoton  ..  p r o c e s s e s , having, a t h r e s h o l d g i v e n  nfuo = 2A(T)-eV. (n an  by  integer)  were o b s e r v e d by Dayem and M a r t i n and more, r e c e n t l y by Cook and (1967).  Everett  .,. Perhaps the most u s e f u l r e f e r e n c e at the p r e s e n t t o the  -47-  F i g u r e 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 A b s o r p t i o n (a,b). and Phonon G e n e r a t i o n (c)  -48s u b j e c t of p h o t o n - a s s i s t e d t u n n e l i n g i s the work o f Cook and E v e r e t t c i t e d above who  g i v e e x t e n s i v e e x p e r i m e n t a l 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 o f power a t 36 G H z — a n d 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 o f T i e n and Gordon (1963) and Cohen, e t a l (1963). 2.  Microwave Phonon D e t e c t i o n I n the t w o - f l u i d p i c t u r e o f 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 pairs.  P a i r s a r e c o n t i n u a l l y b e i n g b r o k e n up by a b s o r b i n g t h e r m a l phonons  of energy g r e a t e r than or e q u a l t o 2A(T) recombining  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  i n t o p a i r s v i a the e m i s s i o n of phonons of energy e q u a l to  2A(T)—according  t o B u r s t e i n , e t a l (1961) and Rothwarf and Cohen (1963)  phonon e m i s s i o n r a t h e r than photon e m i s s i o n i s the dominant decay mode. Because of t h i s p r o p e r t y , t u n n e l j u n c t i o n s can be used as e i t h e r d e t e c t o r s o r g e n e r a t o r s of v e r y h i g h frequency hundreds of  phonons—several  GHz. D e t e c t i o n t a k e s p l a c e v i a a p r o c e s s 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 ) . g r e a t e r than 2A(T)  The a b s o r p t i o n of phonons of energy  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 o f Cooper p a i r s  i n t o q u a s i p a r t i c l e s w h i c h t u n n e l and g i v e 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 c o n v e n t i o n a l means. Eisenmenger and Dayem (1967) overcame t h i s d i f f i c u l t y and were a b l e t o observe phonons.  the e f f e c t by u s i n g another t u n n e l j u n c t i o n as the source of A c o u s t i c c o u p l i n g was  p r o v i d e d by e v a p o r a t i n g the two Sn-SnO^-Sn  j u n c t i o n s on o p p o s i t e ends o f a 1 cm l o n g s a p p h i r e c r y s t a l ; the g e n e r a t i n g u n i t was  b i a s e d a t eV £ 2A(T)  and the r e c e i v i n g one a t eV <2A(T).  The d e t a i l s o f phonon g e n e r a t i o n a r e i l l u s t r a t e d s c h e m a t i c a l l y 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  into S 3  g e n e r a l two r e l a x a t i o n p r o c e s s e s . of the energy gap  undergo i n D  F i r s t , t h e r e i s a r e l a x a t i o n to the top  i n a c h a r a c t e r i s t i c time T ^ , w i t h the e m i s s i o n of a phonon  of energy fuo = eV-2A f o l l o w e d by a r e c o m b i n a t i o n 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 e m i s s i o n of a phonon of energy "hco = 2A.  Recent e x p e r i m e n t a l r e s u l t s  ( M i l l e r & Dayem, (1967)  (1968)) i n d i c a t e 5 x 10 < T -1 depending on the t e m p e r a t u r e , and x < 10 x . T R and L e v i n e and H s i e h  r  < 2 x 10  sec,  Microwave phonons can a l s o i n t e r a c t w i t h t u n n e l j u n c t i o n s to produce p h o n o n - a s s i s t e d  t u n n e l i n g which i s analogous t o the  t u n n e l i n g d e s c r i b e d above and shown i n f i g u r e 3-1 ( b ) . Dayem (1967  , see a l s o Eisenmenger, 1969)  photon-assisted  Eisenmenger and  found the e f f e c t t o be  negligible  a t the f r e q u e n c i e s w i t h which they were w o r k i n g but o t h e r w o r k e r s , eg. and Vernon (1965), A b e l e s and G o l d s t e i n (1965) and G o l d s t e i n et a l u s i n g f r e q u e n c i e s i n the X band (9 GHz) the e f f e c t .  (1966),  have demonstrated the e x i s t e n c e of  Because the f r e q u e n c i e s were too low t o produce c u r r e n t  s t e p s a t v o l t a g e s eV = 2A(T)-nnu) as i n the quantum p h o t o n - a s s i s t e d experiments,  the p r o c e s s was  manifested  tunneling  s i m p l y as an excess c u r r e n t  dependent on b o t h 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 junction.  Lax  the  Agreement w i t h theory i s o n l y q u a l i t a t i v e and s e v e r a l s e r i o u s  d i s c r e p a n c i e s between experiment and t h e o r y remain. C.  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 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 d u r i n g w h i c h 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 d e v i c e s which respond to n u c l e a r r a d i a t i o n i n the sense j u s t d e f i n e d and a d e v i c e l i k e the g e i g e r c o u n t e r w h i c h 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 o t h e r than i t must exceed some t h r e s h o l d v a l u e .  Photographic  emulsions,  on the o t h e r  hand, y i e l d i n f o r m a t i o n c o n c e r n i n g  the energy o f i n d i v i d u a l p a r t i c l e s  but  are not a b l e to d i s t i n g u i s h i n time between the a r r i v a l of one p a r t i c l e 1  any o t h e r .  To d e t e c t charged p a r t i c l e s , then, the s u p e r c o n d u c t i n g  and  tunnel  j u n c t i o n must respond i n a manner t h a t i s not o n l y 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 i m p i n g i n g 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 m i n i m i z e the time r e s o l u t i o n — a measure of the s m a l l e s t time i n t e r v a l which can o c c u r between the a r r i v a l of two p a r t i c l e s t h a t are d i s t i n g u i s h e d and r e c o r d e d The  as s e p a r a t e  entities.  p r i n c i p l e upon w h i c h the s u p e r c o n d u c t i n g  d e t e c t o r i s based has been o u t l i n e d i n c h a p t e r 1. confirmed  experimentally  (see c h a p t e r  d u r i n g w h i c h the s i g n a l c u r r e n t pulse of 10  1  - 10  ^  sec.  ionizing  particle  I t i s shown t h e r e  and  7) t h a t x, the c h a r a c t e r i s t i c time 1  r e l a x e s back to z e r o , i s of the  As the c u r r e n t p u l s e i s assumed to r i s e  order  instantaneously,  -50-  ( i n s t a n t a n e o u s l y meaning i n times much s h o r t e r than x ) the o v e r a l l  pulse  w i d t h i s expected to be s u f f i c i e n t l y s h o r t to make p o s s i b l e the r e s o l u t i o n i n time o f i n d i v i d u a l e v e n t s i n f l u x e s as h i g h as 1 0 p a r t i c l e s / s e c . 6  I n the t h r e e types of p a r t i c l e d e t e c t o r d e s c r i b e d e a r l i e r i n s e c t i o n A, t h e r e was p a i r s , photoelectrons signal.  some t y p i c a l energy w r e q u i r e d to generate the i o n or e l e c t r o n - h o l e p a i r s w h i c h c o n s t i t u t e d the d e t e c t i o n  T h i s concept was  i n chapter  c a r r i e d over t o the s u p e r c o n d u c t i n g  tunnel j u n c t i o n  1 where, i f energy AE i s d e p o s i t e d i n the j u n c t i o n by a n u c l e a r  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 t o be  N  o  =  AE/w  Since i n c o n v e n t i o n a l p a r t i c l e spectrometers  w does not v a r y 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 ( a l t h o u g h t h e r e i s no e v i d e n c e y e t to subs t a n t i a t e t h i s hope) t h a t the amplitude  of the excess c u r r e n t p u l s e 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 d e t e c t o r . 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 t h a t i n p r i n c i p l e the s u p e r c o n d u c t i n g  t u n n e l j u n c t i o n may  possibly satisfy  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 c o n s i d e r e d a p a r t i c l e d e t e c t o r .  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 t u n n e l j u n c t i o n w h i c h d e t e r m i n e i t s performance as a d e t e c t o r and d e s c r i b e how j u n c t i o n may D.  such a  be used i n p r a c t i c e .  Optimum J u n c t i o n Type and C o n s t i t u e n t 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 f o u r types or c o n f i g u r a t i o n s of t u n n e l j u n c t i o n w h i c h c o u l d 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 m e t a l -  normal m e t a l j u n c t i o n (M-M), the normal m e t a l - s u p e r c o n d u c t o r 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 s u p e r c o n d u c t o r on e i t h e r s i d e 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 superconductors. mathematically of the S^-S  2  discussion.)  different  (Because o f 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 and because the M-S  and S-S  j u n c t i o n s r e p r e s e n t l i m i t i n g cases  j u n c t i o n , i t i s not c o n s i d e r e d e x p l i c i t l y i n the f o l l o w i n g I t t u r n s out t h a t the S-S  j u n c t i o n i s the most f a v o u r a b l e .  -51The  purpose o f t h i s s e c t i o n i s t o 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 t h e S-S c o n f i g u r a t i o n . for  F i r s t of a l l , general  criteria  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 t h e c o u p l i n g o f t u n n e l  j u n c t i o n s t o a common base p r e a m p l i f i e r a r e i n t r o d u c e d  (paragraph a) a f t e r  w h i c h t h e 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 a r e e s t i m a t e d theory  (paragraph b) and i n t e r - c o m p a r e d (paragraph c ) .  from  (A common base  p r e a m p l i f i e r was chosen o v e r t h e common e m i t t e r c o n f i g u r a t i o n because t h e lower i n p u t impedance o f t h e former was more c o m p a t i b l e w i t h t h e 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 v a r i o u s and  specimens  c o u l d be expected t h e o r e t i c a l l y . ) I t must be emphasized t h a t t h i s a n a l y s i s i n no way p u r p o r t s  t o be a g e n e r a l  t h e o r y o f t h e optimum c o u p l i n g o f t u n n e l 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 n o r i s i t i m p l i e d t h a t t h e common base t r a n s i s t o r p r e a m p l i f i e r used was t h e optimum p r e a m p l i f i e r . " W h i l e such a g e n e r a l t h e o r y o f s o u r c e - a m p l i f i e r c o u p l i n g i s o f 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 o f 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 s t a g e o f i n v e s t i g a t i o n because o f t h e l a c k o f 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 p u l s e r i s e and f a l l  times and because o f t h e p r e s e n t i n a b i l i t y t o  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 3-2 where t h e t u n n e l j u n c t i o n d e t e c t o r i s r e p r e s e n t e d i n p a r a l l e l w i t h t h e j u n c t i o n dynamic r e s i s t a n c e r . for  the detector  t h i s chapter;  i s considered  c i r c u i t of figure  as a c u r r e n t  generator  (The e q u i v a l e n t  circuit  i n more d e t a i l i n a subsequent p o r t i o n o f  t h e 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 c h a p t e r s 6 and 7.)  I n a n t i c i p a t i o n o f t h e r e s u l t s from t h e  n o i s e c a l c u l a t i o n and measurements o f c h a p t e r 6, n o i s e s o u r c e s a s s o c i a t e d w i t h t h e d e t e c t o r , such as Johnson n o i s e i n t h e b i a s r e s i s t a n c e and shot n o i s e on t h e b i a s i n g c u r r e n t I , have been o m i t t e d in  comparison t o t h e t r a n s i s t o r n o i s e The  as they a r e n e g l i g i b l e  sources.  s i g n a l current i  g  i s t r e a t e d as b e i n g due t o t h e  momentary h e a t i n g o f t h e m a t e r i a l on b o t h s i d e s o f t h e 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 t o T + ST wbere 6T <<  T, then  -52-  F i g u r e 3-2:  E q u i v a l e n t C i r c u i t of D e t e c t o r  and  Preamplifier  -53where I = I ( V , T ) i s the t h e r m a l 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 , o n l y  cases a r e c o n s i d e r e d temperature.  The  t u n n e l c u r r e n t f o r the j u n c t i o n b i a s e d a t those  i n which b o t h s i d e s of the j u n c t i o n a r e a t the same  r e s t r i c t i o n t h a t 6T <<  p a r t i c l e t r a c k v i c i n i t y immediately  T c l e a r l y does not h o l d i n the  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 s m a l l s i g n a l approach i s a p p l i c a b l e o n l y f o r times  sufficiently  l o n g f o l l o w i n g a p a r t i c l e bombardment t h a t the energy l o s t by the p a r t i c l e may  be c o n s i d e r e d  to have d i f f u s e d throughout a l a r g e enough volume t h a t  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 d e t e c t o r t o be used w i t h 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 o t h e r would be  I =  T  I  2  Here, £ i s a measure o f the fundamental s i g n a l - t o - n o i s e r a t i o of  the  d e t e c t o r and, o f c o u r s e , i t i s d e s i r a b l e t h a t £ s h o u l d be as l a r g e as possible. I n p r a c t i c e , however, one d e a l s w i t h n o i s y 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 t h a t £ should be l a r g e i s unchanged, the r o l e p l a y e d by the j u n c t i o n r e s i s t a n c e r becomes i m p o r t a n t .  With  r e f e r e n c e t o the n o i s e 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 e q u a t i o n 6-6) to r l ,  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  F decreases monotonically  i n the l i m i t of r-*».  with r , decreasing  to i t s l o w e s t  Thus, from the p o i n t of v i e w of m i n i m i z i n g  value F,  the d e s i r e d type o f j u n c t i o n and b i a s i n g p o i n t a r e 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 c o n s i d e r a t i o n s  too d e t a i l e d t o be g i v e n i n t h i s p r e l i m i n a r y work) h a v i n g t o do w i t h  the  r e q u i r e d v a l u e of the system response time and  the  the s p e c i f i c v a l u e of  energy l o s s AE, e n t e r i n t o the d e t e r m i n a t i o n of an optimum (presumably f i n i t e ) r. Nonetheless, what one  '  since r  ^ i s probably opt r  J  f a r i n excess o f  i s a b l e to o b t a i n p r e s e n t l y w i t h t u n n e l j u n c t i o n s , the performance  c r i t e r i a may be maximum.  be a d e q u a t e l y  summarized by the statement t h a t r and  £ should  -54-  (b) D e r i v a t i o n o f the Parameters D e t e r m i n i n g ( i ) M-M  Junction The  t u n n e l c u r r e n t i n the M-M  found from the g e n e r a l r e s u l t o f e q u a t i o n 2-14 2 |M|  1  j u n c t i o n may  = n (E 2  be  s i n c e the m a t r i x element  i s t a k e n t o be the same whether the m e t a l s on e i t h e r s i d e of  i n s u l a t o r a r e normal or s u p e r c o n d u c t i n g . n (E)  Performance  the  When b o t h m e t a l s a r e normal,  + eV) = 1 so t h a t I =  A|M|  [ f ( E ) - f ( E + eV)]dE  2  2 I t i s convenient  t o r e d e f i n e the c o n s t a n t A|M|  t o be G/e  so t h a t , upon  i n t e g r a t i o n , the c u r r e n t i s I  where $ = 1/kT  B  i  |  *n(e& ) = e V  G V  and e i s the e l e c t r o n i c charge.  The  c o n s t a n t G may  be i n t e r p r e t e d as the low temperature normal s t a t e conductance of  thus the  tunnel j u n c t i o n g i v i n g  r  MM  =  ( 3 V / 9 I  (3I/3T)  V  >T  =  G _ 1  ( 3  "  1 }  =0  I m m e d i a t e l y , 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 c o n t e n t i o n  a s , t o f i r s t o r d e r at l e a s t , the t u n n e l c u r r e n t i s i n s e n s i t i v e to temperature change making i  = 0. ( i i ) M-S  Junction A g a i n , the t u n n e l c u r r e n t i s g i v e n by  2-14  w i t h say n ( E 2  + eV) = 1 but w i t h n^(E)  equation  g i v e n by e q u a t i o n 2-13.  Hence  -55e  J 9 i (E - A )  G e  n ( E ) [ f (E-eV) - f (E + eV)]dE  [f(E) - f ( E + eV)]dE  1  MS  Z  2  ?  1  To go further requires some approximations.  At T = 1.2 K (3 = 10 (meV) ) -1  the minimum value of E i s 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.  I  M S  = (2G/e) n  T  Thus  sinh (3eV)  where, f o r future reference, the thermal equilibrium density of quasiparticles i n the superconductor i s  N (0)n m i  n (T) o  2 N (0) m  n(E)  exp(-BE)dE  A  -BA(T) = 2N (0)e m  P + A(T) x  -e  3  p  dp (3-2)  (p^ + 2 A(T))' P  = 2 N (0) A(T) K (BA(T)) m  X  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 i s convenient to set K^(x) =  a (x)exp(-x) where a(x) varies slowly with x (for x =  MS  1  =  ( 2 G / e )  A  (  T  ) a  BA >> 1)  so that  < ( > / ) exp(-A(T)/kT) sinh(eV/kT), A  T  kT  0 <eV < A/2 T = 1.2 K. As mentioned  i n 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 s l i g h t l y with i t s argument, i t may also be treated approximately as a constant giving  -56I  = (2GAa/e) exp(-BA)  M S  sinh(BeV) (3-3)  = 2GAag exp(-gA) cosh(3eV) MS and  8T  MS  f-2GAcx] exp(-BA)[-Asinh(BeV) ekl  V  + Vcosh(3eV)]  2  ( i i i ) S-S J u n c t i o n Once more, the t u n n e l c u r r e n t i s g i v e n by e q u a t i o n 2-14 w i t h n ( E ) and n ( E + eV) g i v e n by 2-13 w i t h t h e r e s u l t 1  2  |EJ |E 4- eV| [ f ( E ) - f ( E + eV)]dE SS  e  (E  2  - A ( T ) ) ^ [(E + e V ) - A ( T ) ] ^ 2  2  2  U s i n g s i m i l a r r e a s o n i n g and n o t a t i o n t o t h e M-S c a s e , one f i n d s I section  (see  E ( e q u a t i o n 3-8) of t h i s c h a p t e r ) t o be g i v e n a p p r o x i m a t e l y by  I  s  s  = ( 2 G A a / e ) [ l - exp(-BeV)] exp(-3A) 0 < T < 2 K i  A <eV < 2 A  The o t h e r q u a n t i t i e s  of i n t e r e s t are  r  = 2GAct3 exp(-8(A  + eV))  (3-4)  ss  and 81  SS  w  V  f -2GActl exp(-3A)[exp(-8eV)(eV 2 ekT  + A) -A ]  (c) Comparison o f Parameters Because of t h e r e s t r i c t i o n s on V i n e q u a t i o n s 3-3 and 3-4, i t i s convenient  t o e v a l u a t e t h e MS parameters a t eV = A/2 and t h e SS  -57parameters a t 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 h a l f w a y between 0 and the  level  a t w h i c h the l a r g e c u r r e n t jump o c c u r s . ) . I t i s then easy to show t h a t I (MS) I(SS)  |exp(i3A)  r (MS) r(SS)  2exp(-3(3A/2)  S(MS) S(SS)  *exp(iBA) -  and  These r a t i o s , w h i c h have been e v a l u a t e d (A = 0.55  meV),  are g i v e n i n t a b l e  Parameter I (MS) KSS) S(MS) S(SS)  T a b l e 3-1:  .  i ^ 1  KSS)  f o r Pb  Pb  meV)  and  Sn  Sn  260  7.8  130  3.9  1.4xl0~  5 (MS) 5(SS)  8.1  8 ;  5.2xl0"  4  1.4  Comparison of Parameters f o r M^S important  (A = 1.25  3-1.  r(MS) r(SS)  The  1  and  S-S  Junctions  c o n c l u s i o n to be drawn from these r e s u l t s  i s t h a t 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 g r e a t e r i n the M-S  j u n c t i o n t h i s advantage i s o v e r w h e l m i n g l y o f f s e t by the  much l o w e r dynamic r e s i s t a n c e of the M-S  junction.  relatively  Balancing a s l i g h t loss  of s i g n a l a g a i n s t more e f f i c i e n t c o u p l i n g of t h a t s i g n a l to the a m p l i f i e r a l o n g 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 choose the symmetric j u n c t i o n as the most p r o m i s i n g  device.  i s compelled to As  confirmed  -58-  i n the n e x t s e c t i o n , i t i s a l s o e v i d e n t  t h a t the j u n c t i o n s h o u l d be made of  Pb f i l m s ( o r a s u p e r c o n d u c t o r h a v i n g an even l a r g e r energy gap)  for  optimal  performance. 2.  Type of S u p e r c o n d u c t o r To p r o v i d e  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 s u p e r c o n d u c t o r s t o n u c l e a r p a r t i c l e bombardment, i t i s c o n v e n i e n t t o d e f i n e a f i g u r e of  merit  F' = N / (N + N ) o T o' where N  q  2  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 V o l = J u n c t i o n Volume, i s the T o  thermal  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 p r e s e n t i n the j u n c t i o n a t 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' s h o u l d 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 n o i s e i s p r o p o r t i o n a l i o t o (N^ + N ) . To d e a l w i t h F' i n g e n e r a l , however, i s i n c o n v e n i e n t so 2  q  t h a t two  l i m i t i n g forms of F' w i l l be c o n s i d e r e d (a) N  q  «  N  instead,  T  T h i s c a s e , w h i c h c o r r e s p o n d s t o the s i t u a t i o n when a v e r y 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 w i t h the use  be  studied  of F = N /N ^ Q  (3-5)  t  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 required  to d i s s o c i a t e a Cooper p a i r i n t o two  for 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  o  = AE/w  q u a s i p a r t i c l e s , then  - AE/vA(T)  under the assumption ( f o l l o w i n g Sherman, 1962,A) t h a t w i s some m u l t i p l e of the gap  energy.  ( I n a semiconductor l i k e s i l i c o n , f o r example, w ( S i )  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 N o r t h r o p , 1966.)  N  i s g i v e n by e q u a t i o n  3-2,  =  -59n (T) o  = N / V o l . = 2 N ( 0 ) A(T) T  (3A(T))  m  At t h e t y p i c a l o p e r a t i n g temperature o f 1.2 K, 3 - 10 (meV) ^ so t h a t BA= 5.5 and 12.5 r e s p e c t i v e l y f o r Sn and Pb, t h e two s u p e r c o n d u c t o r s o f interest.  (see t a b l e 3-2.)  As b e f o r e , f o r arguments o f t h i s magnitude, i t  i s u s e f u l t o w r i t e K^(x) = a ( x ) exp (-x) which g i v e s  AE exp(£BA(T))  F  A ^ ( T ) (2N (0) • V o l • a ( 3 A ( T ) ) ) m 3/  Y  ?  The energy AE g a i n e d by a s u p e r c o n d u c t o r i n s l o w i n g down a t r a v e r s i n g n u c l e a r 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 in  that m a t e r i a l .  loss  Considerable data i n t h i s area are a v a i l a b l e i n nuclear  d a t a t a b l e s (eg. M a r i o n , 1960).  From these and o t h e r t a b l e s ,  Blackmore  (1967) c o n s t r u c t e d a s e m i - e m p i r i c a l e x p r e s s i o n from which c o u l d be c a l c u l a t e d , via of  a program w r i t t e n f o r the Van de G r a a f f PDP-8 computer, t h e energy p a r t i c l e s i n p a s s i n g through a t h i n f o i l .  loss  T h i s program was used t o  e s t i m a t e t h e energy l o s s o f 5.1 MeV a l p h a 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 , a l o n g w i t h those f o r A£, a r e p l o t t e d i n f i g u r e 3-3. ( F o r p r a c t i c a l reasons h a v i n g t o do w i t h t h e i r ease o f p r e p a r a t i o n as t h i n f i l m s and t h e w o r k i n g temperatures c o n v e n i e n t l y a v a i l a b l e , Pb and Sn were t h e o n l y superconductors given serious c o n s i d e r a t i o n . ) No e v i d e n c e i s a v a i l a b l e y e t c o n c e r n i n g t h e v a r i a t i o n o f y = w/A(T) from one s u p e r c o n d u c t o r t o another b u t perhaps some guidance may be sought from the s i t u a t i o n i n s e m i c o n d u c t o r s . (1966) l i s t  t h e gap energy E  Dearnaley and Northrop  and e x p e r i m e n t a l v a l u e s 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 o f w/E 2.9 and 4.6.  v a r i e s o n l y between  There appears t h e r e f o r e t o be some j u s t i f i c a t i o n i n assuming  the change i n y from s u p e r c o n d u c t o r t o s u p e r c o n d u c t o r may be s u f f i c i e n t l y s m a l l t h a t y can be t a k e n as a c o n s t a n t . N ( 0 ) , t h e energy d e n s i t y o f s t a t e s a t t h e Fermi l e v e l when m t h e m e t a l i s i n t h e normal s t a t e , i s s i m i l a r l y not v e r y w e l l known b u t , as Ziman (1965) p o i n t s o u t ,  N  ( 0 ) f ° most m e t a l s except t h e t r a n s i t i o n ones r  m  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 v a l u e n ( E ) = const x (E )  2  r  where E  i s the Fermi energy. I t w i l l be assumed t h e r e f o r e t h a t N (0).«n(E ) F m r where E_, = 4.4 eV f o r b o t h Pb and Sn.  -60-  -61The r e l a t i v e v a l u e s o f F a r e s u f f i c i e n t f o r a comparison of t h e two s u p e r c o n d u c t o r s ; t h e r e l e v a n t f a c t o r s a r e summarized i n t a b l e 3-2 f o r f i l m s o f i d e n t i c a l t h i c k n e s s and a r e a . AE(relative units) F i g . 3-3  Superconductor  A(T)meV  Sn  3  .55  Pb  4  1.25  T a b l e 3-2:  3A(T)  a(0A)  5.5  F ( r e l a t i v e t o Sn)  .57  12.5  .36  1 16.5  Comparison o f F i g u r e s o f M e r i t f o r Sn and Pb.  C l e a r l y , from t h i s p o i n t o f v i e w , Pb i s s u p e r i o r t o Sn as a m a t e r i a l from which t o make a s u p e r c o n d u c t i n g i o n i z i n g p a r t i c l e d e t e c t o r . The temperature 1.2 K was chosen as a b a s i s on which t o compare t h e s u p e r c o n d u c t o r s 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 s e r v e o n l y t o i n c r e a s e t h i s f i g u r e o f  m e r i t i n f a v o u r o f Pb whose t r a n s i t i o n temperature ( T ) i s h i g h e r than S n — c  because F <* exp ( I S A ) = exp(.875 T /T) I c  T h i s l a s t r e s u l t c o u l d g i v e t h e f a l s e i m p r e s s i o n t h a t f o r optimum performance, the  energy gap s h o u l d be as l a r g e as p o s s i b l e and one s h o u l d perhaps use  semiconductors.  That such i s n o t t r u e may be seen from c o n s i d e r i n g t h e case  when AE i s v e r y l a r g e so t h a t N (b)  N  Q  »  N  Q  >> N^,.  T  I n t h i s i n s t a n c e , t h e " g e n e r a l f i g u r e o f m e r i t approaches  F"  = N *  =  (AE/YA(T))^  :  so t h a t F"(Sn) F"(Pb)  1.25 .55  = 1.3  -62and Sn i s seen t o be s l i g h t l y f a v o u r e d o v e r Pb because o f t h e former's s m a l l e r energy gap. (c)  Conclusions I t i s e v i d e n t t h a t , i n g e n e r a l , t h e c h o i c e o f t h e optimum  s u p e r c o n d u c t o r w i l l depend upon t h e expected energy l o s s i n t h e superconduct o r s and t h e a c c e s s i b l e o p e r a t i n g t e m p e r a t u r e s .  On these grounds, a symmetric  Pb-Pb j u n c t i o n c o o l e d t o 1.2 K would appear t o be t h e b e t t e r c h o i c e f o r the p a r t i c l e e n e r g i e s o f t h i s experiment.  U n f o r t u n a t e l y , as d i s c u s s e d  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 e l i n g  characteristics  proved p r o h i b i t i v e l y d i f f i c u l t t o make so t h a t an unhappy compromise was n e c e s s a r y ; 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 t h e  d e t e c t i o n experiments. E.  E x c i t a t i o n s i n t h e 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 Now t h a t i t has been e s t a b l i s h e d t h a t a symmetric  superconducting  j u n c t i o n i s t h e 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 n e c e s s a r y t o c o n s i d e r t h e e x c i t a t i o n s produced p a r t i c l e as i t passes  i n t h e s u p e r c o n d u c t i n g f i l m s by t h e charged  through.  Throughout i t s t r a v e r s a l o f one o r b o t h o f t h e f i l m s c o m p r i s i n g t h e j u n c t i o n , t h e charged p a r t i c l e l o s e s energy p r i m a r i l y by t h e i o n i z a t i o n and e x c i t a t i o n o f e l e c t r o n s a s s o c i a t e d w i t h t h e atoms a l o n g i t s p a t h . I t seems r e a s o n a b l e (as d i s c u s s e d i n more d e t a i l i n c h a p t e r 7, s e c t i o n F) -9 t h a t w i t h i n a time t h e o r d e r o f l e s s than 10  s e c a f t e r t h e passage o f t h e  p a r t i c l e , t h e 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 s l e n d e r c y l i n d e r c o - a x i a l w i t h t h e p a r t i c l e t r a c k have r e l a x e d t o a common l o c a l  temperature  w h i c h i s i n excess o f t h e j u n c t i o n e q u i l i b r i u m (or h e l i u m b a t h )  temperature.  The aim o f t h i s s e c t i o n i s t o c o n s i d e r t h e 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 o c c u r i n t h e s u p e r c o n d u c t o r s a f t e r t h i s t h e r m a l i z a t i o r t has t a k e n p l a c e . At t h i s p o i n t i t i s u s e f u l t o r e c a l l t h e phonon d e t e c t i o n e x p e r i ments d e s c r i b e d i n s e c t i o n B o f t h i s c h a p t e r where t h e d e t e c t i o n p r o c e s s was d e s c r i b e d i n terms o f t h e 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 c o u l d s u b s e q u e n t l y t u n n e l . analogous.  The p r e s e n t s i t u a t i o n i s  Here, t h e phonons i n excess o f t h e t h e r m a l e q u i l i b r i u m number  c o r r e s p o n d t o t h e i n j e c t e d phonons and, as. l o n g as these excess phonons remain i n t h e energy range tuo £ 2 A ( T ) , t h e y may generate q u a s i p a r t i c l e s v i a the b r e a k i n g up o f p a i r s .  -63From the m i c r o s c o p i c p o i n t o f v i e w , when one c o n s i d e r s an i n d i v i d u a l event i n w h i c h a p a i r i s broken up by a phonon, the r e s u l t i n g may  excitation  be d e s c r i b e d as t h e c r e a t i o n o f 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 o f c o u p l e d p a i r s b e f o r e and a f t e r the (phonon-pair) i n t e r a c t i o n . t h e p a i r b r o k e n up had k, k' > k^,  Such a p r o c e s s i s p o s s i b l e because even though  (say ( k t , -k+), w i t h k t g o i n g to k ' t ) might have  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 m e t a l or s u p e r c o n d u c t o r , to u n i q u e l y determine i t s e l e c t r o n or hole nature. ( e q u a t i o n 2-6)  ( I n the  B o g o l i u b o v r e p r e s e n t a t i o n mentioned  i n chapter 2  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 a m p l i t u d e s f o r an e l e c t r o n and h o l e . )  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 p r o c e s s 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 t h a t i t d e f i n i t e l y c r e a t e s an e l e c t r o n and d e f i n i t e l y creates a hole.  On the average, however, p a r t i c l e number must  be conserved so t h a t 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.  U s i n g the semiconductor model of a s u p e r c o n d u c t o r ,  which  i s a t h e r m a l 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 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  view  particle  as e q u a l numbers of e l e c t r o n s (above the gap) and h o l e s (below the gap). These may  s u b s e q u e n t l y t u n n e l , obeying Fermi's "Golden R u l e " and t h e  c o n s e r v a t i o n o f energy, as do the q u i e s c e n t t h e r m 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 c o n s i d e r e d i n the d e r i v a t i o n of the t u n n e l c u r r e n t between two c o n d u c t o r s g i v e n i n c h a p t e r 2 ( e q u a t i o n 2-14).  super-  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 e s t i m a t e s to f o l l o w i n s e c t i o n F. B e f o r e 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 s h o u l d be c o n s i d e r e d .  As mentioned  b r i e f l y i n c h a p t e r 1 and more f u l l y i n  c h a p t e r 7, x depends p r i m a r i l y on the r a t e a t w h i c h t h e excess q u a s i p a r t i c l e s t u n n e l and the r a t e at which phonons a r e l o s t from the energy range fico >, 2A(T); x was e s t i m a t e d ( c h a p t e r 1) to l i e i n the range -8 -6 4 x 10  < x < 1.4x10  sec which i n c l u d e s the v a l u e o f a p p r o x i m a t e l y  1.4  x 10 ^ sec deduced from t h i s experiment.  F.  Small Signal Equivalent C i r c u i t  (see c h a p t e r  7.)  L o g i c a l l y , t h e next t o p i c to be d i s c u s s e d s h o u l d be the expected s i g n a l s i z e b u t , to m o t i v a t e and c l a r i f y such a d i s c u s s i o n , i t i s c o n v e n i e n t t o i n t e r j e c t a t t h i s p o i n t (1) a g e n e r a l t e c h n i q u e f o r o b t a i n i n g the s i g n a l  -64c u r r e n t and  (2) an a n a l y s i s of the s m a l l 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  tunnel j u n c t i o n detector. 1.  G e n e r a l Treatment of S i g n a l F o l l o w i n g a p a t h o f t e n taken by e l e c t r i c a l engineers  in  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 d e v i c e l i k e a t r i o d e or t r a n s i s t o r , one may dc I-V  c o n s i d e r f i g u r e 3-4 where a t y p i c a l f a m i l y of t u n n e l j u n c t i o n  c u r v e s t a k e n a t v a r i o u s temperatures T i s p l o t t e d .  The  load l i n e ,  w h i c h passes t h r o u g h the chosen b i a s i n g p o i n t Q — t h e c r i t e r i o n f o r Q being that r = (9V/9I)  T  s h o u l d be maximum as mentioned e a r l i e r — h a s a  n e g a t i v e s l o p e e q u a l to the r e c i p r o c a l o f the l o a d r e s i s t a n c e tunnel j u n c t i o n . T , q  U s i n g f i g u r e 3-4,  the c o r r e s p o n d i n g  choosing  one  seen by  sees t h a t as T i s i n c r e a s e d above  tunnel current i s uniquely determined.  A g e n e r a l macroscopic method o f d e t e r m i n i n g  the c h a r a c t e r -  i s t i c s o f the c u r r e n t p u l s e a r i s i n g from charged p a r t i c l e bombardment i s evident.  the  now  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 t h a t the manner i n w h i c h 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 chapter  7).  d e s c r i b e d by the c l a s s i c a l d i f f u s i o n e q u a t i o n  Thus, i n p r i n c i p l e , one may  (see  c a l c u l a t e T ( x , y, t ) where x and y  l i e i n the p l a n e of the j u n c t i o n and t i s the time e l a p s e d s i n c e the "instantaneous"  thermalization.  For an i s o t h e r m a l j u n c t i o n a t temperature  T w i t h ( f o r the sake o f s i m p l i c i t y ) a p e r f e c t l y u n i f o r m b a r r i e r of a r e a  A,  the t u n n e l c u r r e n t 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 t h a t the s i g n a l t u n n e l i n g c u r r e n t observed at  time t i s g i v e n  by  I(t)  J ( T ( x , y, t ) ) dx  =  dy  A  W h i l e 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 equation  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  (see c h a p t e r  7) and by the f a c t t h a t a r e a l b a r r i e r i s not p e r f e c t l y  u n i f o r m making the t u n n e l c u r r e n t d e n s i t y a f u n c t i o n of x and y as well>as o f T.  C a l c u l a t i o n s a l o n g these l i n e s have been i n i t i a t e d a t 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  thesis.  L a c k i n g d e t a i l e d i n f o r m a t i o n t h e r e f o r e about the time dependence of T, i t w i l l be n e c e s s a r y  to assume some time dependence of  the  -65-  T > T. > T. c 1 i-1  F i g u r e 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 V a r i o u s Bath Temperatures T.  m  i—i pq  c  R.  O  6  •rH /-s  u »  c>  CO  <  6i  R_  6e  W PQ  JO  Actual Circuit R  B  =  R  BIAS'  r  <  =  R  F i g u r e 3-5:  TheVenin E q u i v a l e n t  B  >  9 V / 9 I )  >  r  T'  '  C  6  6 =  C  (  3  V  /  3  T  - > T  Junction  +  6  T  C  '  6  1  Norton Equivalent °  (3I/3T) 6T V  stray  S m a l l S i g n a l E q u i v a l e n t C i r c u i t f o r Tunnel J u n c t i o n  -66s i g n a l c u r r e n t i ( t ) — a s was  done i n c h a p t e r 1 — a n d e s t i m a t e 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 a g a i n where the j u n c t i o n i s b i a s e d a t  Q w h i c h c o r r e s p o n d s to some c u r r e n t I temperature e x c u r s i o n  = I (  q  V Q  , T ).  I f the average peak  (6T) o f 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 s m a l l compared w i t h T , and w i t h the d i f f e r e n c e T o  - T ,  t h e change i n c u r r e n t i s g i v e n a p p r o x i m a t e l y  the  T a y l o r ' s s e r i e s expansion  by the f i r s t two  T o  (3I/3T)  V o  6T +. ..,  eV  where i t i s assumed, i n c o n s i s t e n c y w i t h e x p e r i m e n t a l  6T.  T q  and  (3I/3T)  V q  are reasonably  An e q u i v a l e n t form may  I q  6T  evidence,  2A(T)  that and  (3V/3TV . 6T +. .. Io, be r e p r e s e n t e d  as a v o l t a g e  i n s e r i e s w i t h an impedance r = (3V/3I)^, o r ,  by N o r t o n ' s theorem, as a c u r r e n t g e n e r a t o r an impedance r = (3V/3I)^,.  <  c o n s t a n t o v e r a s m a l l range o f 6V  Thus, by Thevenin's theorem, the j u n c t i o n may 6e = ( 3 V / 3 T )  Q  be w r i t t e n  6V = ( 3 V / 3 I ) _ 61 + To  generator  terms of  O  of I ( V , T )  61 * ( 3 # V ) 6 V +  (3I/9V)  C  6 i = (3I/3T) 6T i n p a r a l l e l with V  Both r e p r e s e n t a t i o n s are shown i n f i g u r e  (Note t h a t the c u r r e n t source form was  used i n s e c t i o n D o f t h i s  3-5.  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 p o s s i b l e was (3I/3T)  V q  i m p o r t a n c e o f h a v i n g b o t h r and  s t r e s s e d i n s e c t i o n D.  (3I/3T)  V q  as l a r g e as  I t might be n o t e d i n p a s s i n g  that  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  quasiparticle tunneling.  Those u n i t s  1  h a v i n g m e t a l l i c f i l a m e n t s or " b r i d g e s "  j o i n i n g the f i l m s through i m p e r f e c t i o n s 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 c u r r e n t a t f i n i t e v o l t a g e s w h i c h i s temperature i n s e n s i t i v e (see appendix B) making them u s e l e s s as p o t e n t i a l d e t e c t o r s o f charged particles. I n p r a c t i c e , the parameters ( 3 I / 3 T ) ^ and r may  be  estimated  e x p e r i m e n t a l l y from a s e r i e s o f I-V c h a r a c t e r i s t i c s taken over a range of t e m p e r a t u r e s near T ; they may  a l s o be e s t i m a t e d r o u g h l y from q u a s i p a r t i c l e  -67t u n n e l i n g t h e o r y as o u t l i n e d below. 3.  D e r i v a t i o n Of S m a l l S i g n a l Parameters The  c u r r e n t i n a s y m m e t r i c a l j u n c t i o n ( c f . e q u a t i o n 2-14)  i s , i n t h e n o t a t i o n o f c h a p t e r 2,  i =  A|M|  n ( E ) n ( E + e V ) ( f ( E ) - f ( E + eV)) dE  1  (3-7)  2  As was done i n s e c t i o n D o f t h i s c h a p t e r , i t i s c o n v e n i e n t t o s e t 2 A|M|  = G/e and, as w i l l be e v i d e n t s h o r t l y , t o use the a p p r o x i m a t i o n  f ( E ) = (1 + e x p ( B E ) ) "  = exp(-BE).  1  The range o f v a l i d i t y o f n ( E ) i s  r e s t r i c t e d by e q u a t i o n 2-13 so t h a t , f o r eV < 2 A ( T ) , 3-7 may be r e - w r i t t e n as  I  -  E(E-eV)[(f(-E)-f(-E+eV)]dE  £  e  (E -A (T))*((E-eV) -A (T))* 2  A+eV  +  2  2  E(E+eV)[(f(E)-f(E+ev)]dE (E -A (T)) ((E+eV) -A (T))*  2  2  2  2  2  2  Thus, t o a good a p p r o x i m a t i o n f o r Sn w i t h 0/ < T < 2K, t h e c u r r e n t i s  I =  G e  E(E-V)e~ dE ,2 .2 2 .2, (E -A ) ( ( E -V ) -A*") 3E  (e -l) 6 V  d-e"  3 V  )  2  A+V  e  (Int  1  n  E(E+V)e" dE o o 1 o o l (E -A ) ( ( E + V ) -A ) eE  +  2  + Int ) 2  where, f o r c o n v e n i e n c e ,  i t i s understood  t h a t V-s-eV and A = A ( T ) .  t = E-V-A and p e r f o r m i n g some elementary m a n i p u l a t i o n s y i e l d s In^  -BA . -BV= e (1-e ) 1  (t+A)e"  g t  (t +2At)^ 2  which, since q =  t+V+A  < 1, can be w r i t t e n  1-  A t+V+A  dt  Setting  -68l n  h  -  e-  g A  (l-e-  p V  (t+A)e  )  -Bt c dt  i  (t"+2At) A s i m i l a r s u b s t i t u t i o n , t = E-A,  -r 2G I = —  1  P  2 , 3  4 ,  .  2  i n I n t ^ reveals that I n t  2  = I n t ^ or  _ 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 n e c e s s a r y t o t e r m i n a t e a f t e r the f i r s t term which may  o n l y 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. i n t e g r a l i s t h a t of e q u a t i o n  3-2  With t h i s approximation,  the  making ( w i t h eV e x p l i c i t l y r e - i n s e r t e d )  0 < T < 2 I =  the s e r i e s  (l-exp(-3eV))AK (BA) 1  K  : ,  (3-8) A < eV < 2 A  and r"  1  = (3I/9V)  To c a l c u l a t e ( 3 l / 3 T ) , the  = 2GB  T  exp(~3eV)AK  (8A)  (3-9)  identity  v  | (K (Z)) Z  =  X  -K (Z) Q  Z  _ 1  K (Z) 1  i s used w i t h the r e s u l t  V  -2GAg eT  K (BA) X  (BeV+l)e-  g e V  -l  -AK  o  (BA)(l-e"  e e V  )  (3-10) where K G.  o  i s a zero o r d e r m o d i f i e d  Estimate  B e s s e l f u n c t i o n of the second k i n d ,  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 t h a t the c u r r e n t  pulse  superimposed on the ambient t u n n e l i n g c u r r e n t i s of the form i(t) = i  Q  exp(-t/x)  t * 0  (3-11)  -69where i  i s p r o p o r t i o n a l to the magnitude of the excess  quasiparticle  d e n s i t y and T (see s e c t i o n E of t h i s c h a p t e r ) 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. A t the o u t s e t i t must be emphasized t h a t the a c t u a l time dependence i s p r o b a b l y much more c o m p l i c a t e d than the s i m p l e e x p o n e n t i a l form of e q u a t i o n 3-11.  As p o i n t e d 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 y e t s u f f i c i e n t l y advanced t o j u s t i f y the use of a more complex form and, as i t t u r n s out i n c h a p t e r 7, the q u a l i t y o f the d a t a p r e s e n t l y a v a i l a b l e does not seem to w a r r a n t the use of such a form.  I t must be  remembered, however, t h a t the form of i ( t ) chosen w i l l a f f e c t the v a l u e of i  deduced from the e x p e r i m e n t a l measurements ( c h a p t e r 7) and  will  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 v a l u e found f o r w. F o r d e f i n i t e n e s s , a l p h a 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 a c c o r d i n g to the geometry o f f i g u r e 1.  M a c r o s c o p i c o r Phenomenological  6-7.  Approach  As d i s c u s s e d p r e v i o u s l y i n t h i s c h a p t e r , the s m a l l s i g n a l a p p r o x i m a t i o n i s t a k e n t o a p p l y a t times s u f f i c i e n t l y l o n g a f t e r the a l p h a p a r t i c l e t r a v e r s a l t h a t a s m a l l volume of the j u n c t i o n (2XA') may e r e d t o be a t an average temperature  T  be c o n s i d -  + 6T where <ST i s s u i t a b l y s m a l l . O  (A' i s the a r e a o f the " h o t " 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 o f the p r e c e d i n g s e c t i o n were d e r i v e d on the assumption t h a t the e n t i r e j u n c t i o n i s a t 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 a r e a A ( t y p i c a l l y 4 x 10  cm ) i s p a s s i n g c u r r e n t ;  i t seems r e a s o n a b l e t h e n t h a t the peak s i g n a l c u r r e n t may i  = (3I/ T) 6T  o  9  V  be w r i t t e n  - Al  I f n (T) i s the t h e r m a l 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 ( e q u a t i o n o t h e n the change i n the q u a s i p a r t i c l e d e n s i t y '(in the volume 2XA') t h e momentary temperature  due  3-2)  to  r i s e 5T caused by the a l p h a p a r t i c l e i s 9  n (T) o  6n =  6 T  3T  ,  2  N  _JL 2XA':  AV  . _AE_ =  wXA'  ( 3  _  1 2 )  -70S o l v i n g 3-12 f o r 6T y i e l d s  1  o  =  f9T  ('  43 T f ^  V  1 9T 3n (T)  ATT AE wXA  S T  o  (3-13)  The maximum energy l o s t by t h e a l p h a 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 w h i c h c o r r e s p o n d s t o an a l p h a p a r t i c l e i n c i d e n t on t h e j u n c t i o n a t an a n g l e o f 80° w i t h a p a t h l e n g t h i n t h e t i n o f a p p r o x i m a t e l y 23,000 A ( c f . f i g u r e 3-3).  C a l c u l a t i n g ( 3 n ( T ) / 9 T ) from o  e q u a t i o n 3-2 and ( 3 I / 9 T ) ^ from e q u a t i o n 3-10 a t a temperature o f 1.2 K = R = 0.1 U and u s i n g t h e v a l u e w(Sn) - .003 eV n as g i v e n i n t a b l e 1-1 y i e l d s an e x p e c t e d maximum c u r r e n t a m p l i t u d e for a j u n c t i o n having G  i  o  = 15 uA  •  w h i c h i s n e a r t h e range o f v a l u e s found e x p e r i m e n t a l l y i n c h a p t e r 7. The c o r r e s p o n d i n g temperature i n c r e a s e i s ST - 0.12 K. I t s h o u l d be noted t h a t t h e s e e s t i m a t e s a r e based on t h e assumption t h a t t h e energy d e p o s i t e d by the a l p h a p a r t i c l e i n t h e s u b s t r a t e does n o t c o n t r i b u t e t o the e x p e c t e d s i g n a l .  I t t u r n s out (see c h a p t e r 7)  t h a t 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 ) > A E = 0.5 MeV i m p l y i n g t h e r e f o r e that i i s underestimated. Furthermore, the values of (9I/9T) o V and (9n (T)/9T) used a r e c o n s t a n t and a r e v a l i d ( c f . f i g u r e 3-4) o n l y i n the o l i m i t o f s m a l l temperature changes. TT  The manner i n which t h e a m p l i f i e r s e n s i t i v i t y was checked t o ensure d e t e c t i o n o f p u l s e s 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 c h a p t e r 5. [Note: an a l t e r n a t i v e method o f e s t i m a t i n g 6T i s t o s e t ( c f . e q u a t i o n 3-12) 6T = AE/pc XA' P -3 where p i s t h e d e n s i t y and c t h e heat c a p a c i t y . T a k i n g p = 7.29 g cm _c P . and c (1.2 K) = 9 x 10 J g ~ l ( d e g r e e K) (Johnson, 1960) and the same P _  i  v a l u e s f o r t h e o t h e r q u a n t i t i e s as b e f o r e g i v e s - i r e a s o n a b l y w e l l w i t h t h a t found above.  - 24 uA which agrees  (This l a t t e r value of i i s  -71p r o b a b l y an o v e r e s t i m a t e  s i n c e c^ i n c r e a s e s r a p i d l y w i t h i n c r e a s i n g temp-  erature).] 2.  "Microscopic" The  Approach  c o n s i d e r a t i o n s of t h i s p a r a g r a p h p r o v i d e  t i o n f o r the s i m p l e  some  justifica-  form o f i ( t ) w h i c h has been assumed.  As s t a t e d e a r l i e r , i m m e d i a t e l y f o l l o w i n g the passage of alpha p a r t i c l e there e x i s t s a density N w h i c h i s i n excess of n (T).. two p a t h s :  q  an  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  Decay of the excess d e n s i t y proceeds  along  (1) the q u a s i p a r t i c l e s t u n n e l through the i n s u l a t i n g l a y e r or  (2) they recombine t o form Cooper p a i r s and emit a phonon of energy "hed * 2 A ( T ) . that N  q  <<  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 a t t h i s p o i n t i f i t i s assumed n 0  (T)  and t h a t the r e c o m b i n a t i o n phonons may  Rothwarf and T a y l o r v a l i d and  be i g n o r e d .  As  (1967) s t a t e , however, t h i s a p p r o x i m a t i o n i s r a r e l y  they have g i v e n 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 u n i f o r m j u n c t i o n geometry.  S u b s e q u e n t l y , they a r r i v e at a p a i r of c o u p l e d e q u a t i o n s d e s c r i b i n g 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 number of phonons w i t h energy Tito > 2A). modified  the  (the  total  Ideally, a similar analysis,  t o 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 t u n n e l j u n c t i o n p a r t i c l e d e t e c t o r but t h i s would r e q u i r e c a r e f u l t r e a t m e n t o f 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 b o u n d a r i e s and  t h e r m a l impedances a t those b o u n d a r i e s .  Such  a program w i l l not be u n d e r t a k e n h e r e ; i t i s mentioned a t t h i s p o i n t  only  t o 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 g i v e n 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 p u l s e g e n e r a t o r to be i n s e r t e d i n f i g u r e Thus, one  ignores  the time dependence of the r a t e W  at w h i c h the  p a r t i c l e s w i l l recombine and assumes t h a t the t u n n e l i n g r a t e and s u f f i c i e n t l y s m a l l w i t h r e s p e c t pendent of one  another.  The  to W  3-5. quasi-  is  t h a t they are e f f e c t i v e l y  constant inde-  decay of the excess q u a s i p a r t i c l e d e n s i t y  t h e n be viewed as s k e t c h e d below.  may  -72I f R ( N, n; t ) i s t h e j o i n t p r o b a b i l i t y  distribution  f u n c t i o n t h a t a t time t , t h e r e a r e 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 t u n n e l e d , then  R(N,n,.t + d t ) = R(N,n;t)(1-NWdt) + R ( N + l , n ; t ) ( N + l ) W dt D  + R ( N + l , n - l ; t ) ( N + l ) W dt T  Solving  f - Vl,n< > R \ l , n - l < N+1  W  +  N+1) W  +  T "  \n  W  yields <N(t)> = N  q  e"  (3-14)  W t  <n(t)> = -2-1 (x-e-Wt) W  ( 3  .  1 5 )  The c u r r e n t p u l s e due t o t h e f r a c t i o n o f t h e excess q u a s i p a r t i c l e p a i r s which have t u n n e l e d i n time t i s thus  i ( t )  =  ed^t)>  eN W  =  o  T  exp(-Wt)  ( 3  _  l f i )  where e i s t h e e l e c t r o n i c charge. [Note: t h e c u r r e n t i s n o t 2e d<n(t)>/dt because, a t t h e u s u a l b i a s i n g p o i n t eV - A, o n l y one member o f each q u a s i p a r t i c l e p a i r can t u n n e l t o t h e o t h e r s u p e r c o n d u c t o r , due t o t h e a c t i o n of t h e energy gaps.]  Setting N W e  Q  T  = i  Q  yields  i ( t ) = i exp(-t/-r) o w h i c h , o f c o u r s e , i s t h e form assumed e a r l i e r  , t * 0  ( e q u a t i o n 3-11) w i t h o u t  -73justification. As i t s h o u l d be, t h e q u a n t i t y i  j u s t i n t r o d u c e d i s the  same as t h a t found by t h e p h e n o m e n o l o g i c a l treatment may be seen by s u b s t i t u t i n g f o r W  X  •  o  -  G  eN (0) m  ( e q u a t i o n 3-13).  This  v i a e q u a t i o n 3-17 t o g i v e  . M_  wXA  w h i c h i s i d e n t i c a l t o e q u a t i o n 3-13 f o r s i m p l e , approximate forms o f n (T) Q  H.  ( e q u a t i o n 3-2) and I ( V = A/e,T) ( e q u a t i o n 3-8).  Tunneling  Probability  From e q u a t i o n 3-16, i t can be seen t h a t , i g n o r i n g t h e 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 , t h e maximum s i g n a l i t u n n e l i n g p r o b a b i l i t y p e r sec o f superconductor  i s maximum.  (N  i s o b t a i n e d when t h e  i s f i x e d by t h e c h o i c e  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 t h e o t h e r hand, from the s m a l l s i g n a l e q u i v a l e n t  c i r c u i t and e q u a t i o n 3-10, i t i s e v i d e n t t h a t t h e 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. T h i s s e c t i o n w i l l demonstrate t h e e q u i v a l e n c e o f these two r e quirements and g i v e 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 p l a c e on t h e acceptable i n s u l a t i n g layer thickness. A rough e s t i m a t e o f W Ginsberg  (1962).  may be made u s i n g a method suggested by  F o r b i a s v o l t a g e s eV >>  2A(T), the c u r r e n t should be  I = e N (0) AXW V m 1 2  where N (0) i s t h e normal s t a t e d e n s i t y o f f r e e e l e c t r o n s t a t e s p e r u n i t m volume near t h e Fermi s u r f a c e and t h e o t h e r symbols have t h e i r u s u a l meaning. I n t h i s range o f b i a s v o l t a g e , t h e d i f f e r e n t i a l r e s i s t a n c e i s v e r y n e a r l y c o n s t a n t and e q u a l t o t h e low temperature,  normal s t a t e r e s i s t a n c e (R ) o f  the j u n c t i o n so t h a t W  = ( e N (O)XAR ) m n 2  T  m  -  1  = G(e N (O)XA)" m 2  1  (3-17)  -74and,  like  (8I/8T)  V >  W  i s p r o p o r t i o n a l t o G.  T  The number o f f r e e e l e c t r o n  s t a t e s p e r u n i t volume per u n i t energy range f o r b o t h s p i n s i s i/Z  N(e)de = 4ir(2m)  e  1  h  2  _3  i de = k e d e 2  where m i s t h e e l e c t r o n mass and h i s P l a n c k ' s c o n s t a n t , average v a l u e o f N(e) near t h e Fermi s u r f a c e is  (3-18) so t h a t t h e  (e - e_) i n a range 6e « r  e F  e +6e/2 F  i  < F>  N  fe  =  ke'de = ke  6T  (3-19)  2  r  e -6c/2 F  Now, t h e number o f f r e e e l e c t r o n s N^ i s s i m p l y  N  f  =  N(e.)de = 2keJ '  /3  (3-20)  o  w h i c h g i v e s , upon combining e q u a t i o n 3-19 and 3-20,  N (0) m  = N(e ) = y N / p  f  E q u a t i o n 3-21 may then be e v a l u a t e d  N  f  F  by t a k i n g  = nN p/A A  (3-21)  £  (3-22)  t  a n d ( W i l s o n , 1965) e'  F  where  = (h^/8m)(3N /Tr)  3  f  n = number o f f r e e e l e c t r o n s p e r atom N^= Avogadro's number p = density A = a t o m i c weight t  For Sn, n = 1.1 ( W i l s o n , 1965) so t h a t  (3-23)  -75N  = 4.05 x 1 0  e l s cm"  2 2  3  Ep = 4.4 eV  (3-24)  N (0) = 1.4 x 1 0 m .  2 2  eV^cm"  3  o  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 R  n  = G  _ 1  = 0.1ft  yielding W  = 5.6 x 1 0  T  5  -4  2 cm and  sec" . 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 t h a t t h e maximum s i g n a l i  Q  i s greatest  for  "large".  To p l a c e t h i s statement on a q u a n t i t a t i v e  b a s i s , c o n s i d e r e q u a t i o n s 3-15 and 3-16 w h i c h may be r e w r i t t e n as  <  n  (  t  )  >  =  i(t) respectively.  N  o  w+w" R  = eN  W  Q  T  [i-exp(-t(w +w ))] T  R  T exp(-t(W +W )) T  R  Assuming t h a t t h e 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 t h e s i g n a l  i  go a c c o r d i n g t o < n ( t ) > , i t i s i m p e r a t i v e t h a t <n(t)> be as l a r g e as p o s s i b l e w h i c h means W s h o u l d a t l e a s t be comparable t o W . (W i s t h e r a t e a t 2  T  ...  which the q u a s i p a r t i c l e population tunneling,  see c h a p t e r 7.)  K  K  i s reduced by p r o c e s s e s o t h e r than  Two p r a c t i c a l c o n s i d e r a t i o n s  p l a c e an upper  bound on W ; (1) minimum p r a c t i c a b l e t h i c k n e s s  o f 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 o f r e a l a m p l i f i e r s  would decay b e f o r e an a m p l i f i e r c o u l d Unfortunately,  (if  > 10  sec  respond).  no f i r m t h e o r e t i c a l e s t i m a t e of W  yet for tunnel junctions  , the s i g n a l i s available  s u b j e c t e d t o charged p a r t i c l e bombardment. F o r  purposes o f e s t a b l i s h i n g t h e o r d e r s o f magnitude i n v o l v e d , however, t h e v a l u e o f 2 x 10^ s e c  > W  > 2 x 10"* s e c \ as determined  experimentally  R by t h e s t e a d y s t a t e i n j e c t i o n o f q u a s i p a r t i c l e s i n s u p e r c o n d u c t o r s ( M i l l e r and Dayem, 1967 and L e v i n e and H s i e h , 1 9 6 8 — s e e a l s o chapter 1) may be used. By i n s p e c t i o n o f e q u a t i o n 3-17, i t i s c l e a r t h a t f o r a g i v e n type of s u p e r c o n d u c t o r , W  T  may be i n c r e a s e d  by d e c r e a s i n g e i t h e r t h e j u n c t i o n  -76volume (XA) o r t h e t u n n e l i n g 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 t h e 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 , t h e volume should be as l a r g e as p o s s i b l e .  J u n c t i o n s w i t h lower t u n n e l i n g r e s i s t a n c e may be  produced by d e c r e a s i n g t h e t h i c k n e s s o f t h e i n s u l a t i n g l a y e r (see t a b l e 3-3) but o n l y a t t h e c o s t o f g i v i n g r i s e t o an unwanted dc Josephson c u r r e n t and o  running  t h e r i s k o f t h e o x i d e b e i n g patchy ( a t 10 A i t i s o n l y 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 w h i c h become superconducting  shorts.  The f i r s t o b j e c t i o n i s n o t s e r i o u s as a  low magnetic f i e l d w i l l quench t h e s u p e r c u r r e n t  reasonably  (see c h a p t e r s  2 or 6);  the second problem i s v e r y s e r i o u s and has proved t o be a f o r m i d a b l e to making a u s a b l e  obstacle  detector.  T a b l e 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 o f dimensions g i v e n above, i l l u s t r a t e s t h a t t u n n e l i n g t h i c k n e s s e s  (as d e r i v e d from f i g u r e 2-3)  o  no g r e a t e r than 10 A a r e p e r m i s s i b l e i f W (Here, S  i s t o be comparable t o W„. T R i s t h e 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 )  S (A) T  G =R  ft  -1  in  T a b l e 3-3: T u n n e l i n g I.  6  8xl0~  8  1.6xl0~  4  2  •  -1 W sec T  7.0xl0  7  3.5xl0  6  10  0.27  2.1xl0  5  12  5.4  l.OxlO  4  P r o b a b i l i t y per sec f o r S e v e r a l T u n n e l i n g  Thicknesses.  P r a c t i c a l Operation of Detector I t i s convenient  a t 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 t h e manner i n w h i c h t h e s u p e r c o n d u c t i n g  t u n n e l j u n c t i o n would  operate  in practice. The  s i m p l e s t method of bombarding a t u n n e l j u n c t i o n w i t h 249  p a r t i c l e s i s t o mount an a p p r o p r i a t e r a d i o a c t i v e source  (Pu  alpha  , 5.13 MeV  a l p h a s was t h e one chosen) i n s i d e t h e h e l i u m dewar i n c l o s e p r o x i m i t y ' t o 1  the j u n c t i o n (see f i g u r e s 4-1 and 4-3). Now t h e energy l o s s p e r u n i t path  -77l 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 -15 140 MeV/cm c a l c u l a t e d on the b a s i s of the s t o p p i n g power e = 6.4 x 10 eV _3 a  for helium a t 1.2  (Whaling, 1958)  K ( W i l k s , 1967).  and d e n s i t y p = 0.145  g cm  for liquid  helium  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 d i s t a n c e would be about 2 o r 3 mm;  hence, 5 MeV  a l p h a p a r t i c l e s would be stopped  Consequently,  i n the  liquid.  the specimen and source must be r a i s e d above the  h e l i u m b a t h l e v e l , b e i n g m a i n t a i n e d a t 1.2  K by a t h i n f i l m of s u p e r f l u i d  h e l i u m and by c o n t a c t 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 . 0.5  At 1.2  K, the vapour p r e s s u r e above the b a t h i s about  T o r r w h i c h l e a d s to a s p e c i f i c energy l o s s o f a p p r o x i m a t e l y 25 keV/cm;  t h e 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  h e l i u m atmosphere would t h e r e f o r e be a n e g l i g i b l e 0.5%.  Of c o u r s e ,  the  a l p h a p a r t i c l e s must a l s o pass through the s u p e r f l u i d f i l m w h i c h would cover —6 the j u n c t i o n but t h i s f i l m i s v e r y t h i n , the o r d e r of 4 x 10 h e i g h t of 0.5  cm above a 1.2  K helium bath  ponding energy l o s s i s o n l y about 0.6 keV.  ( W i l k s , 1967), and the c o r r e s 'For a l l p r a c t i c a l purposes  t h e r e f o r e , the a l p h a 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 monoenergetic, h a v i n g energy 5.1  cm a t a  be  considered  MeV.  P r e v i o u s d i s c u s s i o n s have emphasized t h a t 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 t h e r e f o r e l a r g e (91/91)^. o p e r a t i n g temperature  as to have a h i g h t u n n e l i n g conductance and Furthermore,  i t has been observed  s h o u l d be as low as p o s s i b l e and the o p e r a t i n g b i a s  v o l t a g e must be chosen such t h a t r i s as l a r g e as p o s s i b l e . suppress  t h a t the F i n a l l y , to  the unwanted dc Josephson s u p e r c u r r e n t w h i c h 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 n e c e s s a r y  s u p p l y a moderate magnetic f i e l d  G) i n the plane o f the j u n c t i o n .  (A d e t a i l e d account  (<100  of problems a s s o c i a t e d w i t h a t t a i n i n g these o p e r a t i n g  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 s e c t i o n ' A , chapter J.  to  6.)  Noise T h i s s e c t i o n o u t l i n e s v e r y b r i e f l y the n o i s e sources t h a t s h o u l d  be c o n s i d e r e d i n d e s c r i b i n g the performance:of  the t u n n e l j u n c t i o n as a  detector. I n the p r e s e n t experiment  the magnitude o f the j u n c t i o n n o i s e i s  not i m p o r t a n t f o r i t i s shown ( s e c t i o n C, c h a p t e r 6) t h a t t h e dominant  cm'  -78n o i s e source was t h e common base p r e a m p l i f i e r . As t e c h n o l o g y  improves  and lower n o i s e p r e a m p l i f i e r s a r e d e s i g n e d — p e r h a p s o p e r a t i n g i n s i d e the h e l i u m dewar a t 4 K — t h e n o i s e a s s o c i a t e d w i t h t h e t u n n e l j u n c t i o n w i l l become more i m p o r t a n t and f o r t h a t r e a s o n i s d i s c u s s e d a t t h i s time.  1.  Shot N o i s e on B i a s i n g C u r r e n t ( I ) The  shot n o i s e , which a r i s e s from t h e c o r p u s c u l a r  nature  of t h e c u r r e n t , i s g i v e n by  i  2  s  - 2el6f  where 6f i s t h e bandwidth i n t e r v a l . 2.  Johnson (Thermal) N o i s e For a t u n n e l j u n c t i o n i n thermal e q u i l i b r i u m , t h e Johnson  o r t h e r m a l n o i s e i s g i v e n by t h e e x p r e s s i o n  i  where V  Q  2  J  = 4kT«5f/ o : r  i s t h e z e r o - v o l t a g e j u n c t i o n dynamic r e s i s t a n c e , k i s  Boltzmann's c o n s t a n t and T i s t h e a b s o l u t e temperature o f t h e j u n c t i o n . Note:  t h i s e x p r e s s i o n i s v a l i d o n l y f o r an i s o t h e r m a l j u n c t i o n a t zero ~2 b i a s v o l t a g e V; when V ^ 0 t h e n o i s e i s g i v e n by i above. s 3. G e n e r a t i o n - R e c o m b i n a t i o n (GR) Noise GR n o i s e i s due t o f l u c t u a t i o n s i n t h e d e n s i t y o f t h e r m 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 ' brought about by v a r i a t i o n s i n t h e g e n e r a t i o n and r e c o m b i n a t i o n 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 t u n n e l j u n c t i o n , t h e i n s u l a t i n g l a y e r t h i c k n e s s  -79i s not u n i f o r m but may  be  o v e r the j u n c t i o n a r e a .  thought o f as v a r y i n g  As d i s c u s s e d i n s e c t i o n H the t u n n e l i n g  depends s t r o n g l y on b a r r i e r t h i c k n e s s v a r i a t i o n i n i n s u l a t o r thickness ambient t u n n e l i n g  current.  w h i c h i m p l i e s t h a t the  Calculations  Fluctuations  in  n o i s e i n the ambient c u r r e n t : thickness,  later.  the  to the same mechanisms w h i c h produce  g e n e r a t i o n and  to r a d i a t i o n - e x c i t e d and  the  of the magnitude of t h i s e f f e c t  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  f o l l o w i n g s o u r c e s , w h i c h are due due  stochastic  Signal  Fluctuations  current  1966)  probability  w i l l be m a n i f e s t e d as f l u c t u a t i o n s i n  are i n p r o g r e s s a t U.B.C. and w i l l be r e p o r t e d 5.  s t o c h a s t i c a l l y (Hurych,  r e c o m b i n a t i o n n o i s e i n the  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 layer  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  originally  e x c i t e d by the bombarding charged p a r t i c l e . K.  Summary B e g i n n i n g the c h a p t e r was  ventional detectors.  g a s - f i l l e d , s c i n t i l l a t i o n and The  d e t e c t o r s was junctions  a v e r y b r i e f d e s c r i p t i o n of the  use  semiconductor charged p a r t i c l e  of s i n g l e s t r i p s u p e r c o n d u c t i n g b o l o m e t e r s as  then o u t l i n e d f o l l o w e d  by an account of the use of  as d e t e c t o r s of microwave photons and In d i s c u s s i n g  charged p a r t i c l e d e t e c t o r i s based, i t was  radiation tunneling  phonons.  the p r i n c i p l e s upon which the t u n n e l  the b e s t c h a r a c t e r i s t i c s was  con-  junction  shown t h a t the d e v i c e h a v i n g  a Pb-Pb symmetric s u p e r c o n d u c t i n g j u n c t i o n .  (These u n i t s c o u l d not be used i n p r a c t i c e however, because of the  difficulty  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  p a r t i c l e i n the s u p e r c o n d u c t o r s were c o n s i d e r e d next and  i t was  concluded  t h a t , on the average, they c o u l d be regarded as e q u a l numbers of and h o l e s a p p e a r i n g above and  below the energy gap  alpha  electrons  r e s p e c t i v e l y i n the  semi-  c o n d u c t o r model. Following  a brief discussion  the s i g n a l from the d e t e c t o r , j u n c t i o n d e t e c t o r was and  r were d e r i v e d  of the g e n e r a l approach to  a s m a l l s i g n a l e q u i v a l e n t c i r c u i t of  found and  e x p r e s s i o n s f o r the parameters  from the t u n n e l i n g  current  analyzing the  (31/31)^  I = I _(V,T) between s i m i l a r  -80superconductors. The  c u r r e n t p u l s e i ( t ) , w h i c h f o l l o w s the passage o f a charged  p a r t i c l e t h r o u g h the j u n c t i o n and c u r r e n t , was  i s superimposed on the thermal  tunneling  assumed to be o f the form  i(t) = i T h i s form of i ( t ) was  exp(-t/-r)  , t * 0  (3-11)  found to be t h a t g i v e n by a v e r y s i m p l e model i n  w h i c h the excess q u a s i p a r t i c l e d e n s i t y decayed v i a two decoupled and independent c h a n n e l s — t u n n e l i n g The  and  time  recombination.  s i z e o f the s i g n a l c u r r e n t i  was  e s t i m a t e d on the b a s i s  o f 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 o p e r a t i n g c o n d i t i o n s to be approxi m a t e l y 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  p u l s e s o f t h a t form and magnitude was The was  s u f f i c i e n t to d e t e c t  checked e x p e r i m e n t a l l y .  importance and means of o b t a i n i n g a h i g h t u n n e l p r o b a b i l i t y  then d i s c u s s e d on the b a s i s of a s i m p l e e s t i m a t e of W  due  to  T Ginsberg  (1967) W  - G(e N (O)XA)" m 2  i  (3-17)  1  A s h o r t d e s c r i p t i o n of the p r a c t i c a l o p e r a t i o n of the j u n c t i o n was then g i v e n e m p h a s i z i n g the need to have the a c t u a l o v e r l a p area  slightly  above the h e l i u m b a t h l e v e l and the need of a magnetic f i e l d i n the  plane  o f 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 n o i s e s o u r c e s , i n the t u n n e l j u n c t i o n d e t e c t o r i s given.  They a r e not c o n s i d e r e d  t h a t the p r e a m p l i f i e r was  i n depth as i t t u r n s out  the p r i n c i p a l source of n o i s e .  experimentally  CHAPTER 4  . CRYOGENIC APPARATUS A.  Introduction T h i s c h a p t e r i s of t e c h n i c a l i n t e r e s t o n l y and may  be o m i t t e d 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 . I n essence, the c r y o g e n i c equipment c o n s i s t s of a d o u b l e - w a l l e d 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 h e l i u m 4 — r e f e r r e d t o h e r e a f t e r as the h e l i u m d e w a r — s i t t i n g inside a larger, similar vessel f i l l e d with l i q u i d nitrogen, r e f e r r e d t o 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  b o t h as a p r e - c o o l a n t and as a heat s h i e l d f o r the l i q u i d h e l i u m b a t h . specimens a r e immersed i n the h e l i u m b a t h whose temperature about 1.2 B.  K t o 4.2  The  can be v a r i e d  from  K through r e g u l a t i o n of the vapour p r e s s u r e above the l i q u i d .  Dewars and Dewar Cap Both the n i t r o g e n and h e l i u m dewars, c o n s t r u c t e d by the  departmental  g l a s s b l o w e r , a r e of c o n v e n t i o n a l d e s i g n except t h a t a pyrex t e e i s a f f i x e d t o t h e top o f the d o u b l e - w a l l e d p o r t i o n of the h e l i u m dewar.  The t e e  was  n e c e s s i t a t e d by the f a i r l y u n c o n v e n t i o n a l manner i n which the dewars were installed.  They were mounted i n a f i r m l y anchored  t a b l e , the top of the tee  b e i n g about f o u r f e e 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 o r remove t h e specimens w i t h o u t d i s t u r b i n g the dewars.  One p o r t of  the t e e thus p r o v i d e d an opening f o r pumping on the h e l i u m dewar; the o t h e r p o r t p r o v i d e d a r e a d i l y demountable vacuum c o u p l i n g f o r the dewar cap. The b r a s s dewar cap p r o v i d e s a vacuum t i g h t cover f o r t h e h e l i u m dewar w i t h p o r t s f o r the t r a n s f e r s i p h o n , p r e s s u r e s e n s i n g , e l e c t r i c a l f e e d t h r o u g h and, of c o u r s e , the specimens.  Because i t was n e c e s s a r y 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 designed to accept e i t h e r  one.  was  -83-  KEY TO FIGURE 4-1  G l Marsh D i a l Gauge, Type 100  0-760 T o r r  G2 Edwards D i a l Gauge, Type C.G.3  0-20 T o r r  G3 Edwards V a c u s t a t , Model 2E, 0-1 T o r r , Lowest Reading S e n s i t i v i t y = 0.001 T o r r G4 Marsh D i a l Gauge, (no type No.)  0-760 T o r r  G5 A s h c r o f t D i a l Gauge, Type 1850  0-760 T o r r ( v a c ) , 0-380 T o r r ( p r e s s , gauge)  1. Sample mount. 2. E l e c t r i c a l  lead  feedthroughs.  3. Sample mount s u p p o r t and c o n t r o l t u b e s . 4. Dewar c a p . 5. P y r e x Tee (3 x 3 x 2 i n . ) . 6. H e l i u m Dewar I n t e r s p a c e 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 . l o n g .  8. N i t r o g e n Dewar, g l a s s , 116 mm i . d . , 150 mm o.d., 9. L i q u i d N i t r o g e n . 10. L i q u i d  Helium.  11. Sample. 12. H e l m h o l t z  coils.  13. He r e t u r n  line.  14. Welch R o t a r y Pump, Model 1402B, 5 cfm. 15. Stokes Pump, Model 49-10, 80 cfm. 16. Welch R o t a r y Pump, r o u g h i n g purposes, Model 1405. 17. Mercury manostat (0-100 T o r r ) . 18. B o t t l e d d r y N  gas; f l u s h i n g  19. T r a n s f e r s i p h o n p o r t .  purposes.  approx. 38 i n . l o n g .  -84C.  Pumps The vacuum pumps used t o reduce t h e vapour p r e s s u r e o f t h e l i q u i d  h e l i u m a r e 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 o f about 2 K c o u l d  be o b t a i n e d w i t h t h e Welch 1402B r o t a r y pump and about 1.2 K w i t h t h e S t o k e s , Model 49-10, Low Temperature l a b o r a t o r y "community" pump. D.  Pressure-Temperature  Measurement  Temperatures a r e measured by o b s e r v i n g t h e h e l i u m vapour p r e s s u r e and r e l a t i n g i t t o t h e a b s o l u t e temperature w i t h a s t a n d a r d t a b l e . pressure-temperature  The vapour  r e l a t i o n s h i p o f l i q u i d h e l i u m 4, t h e s u b j e c t o f e x t e n s i v e  i n v e s t i g a t i o n , can be found i n many s t a n d a r d r e f e r e n c e s , eg. Mendelssohn (1960). A mercury v a c u s t a t and two d i a l gauges, see f i g u r e 4-1, p r o v i d e adequate coverage E.  o f t h e p r e s s u r e range from 0.12 T o r r (1.0 K) t o 760 T o r r (4.2 K ) .  Sample Mounts I n s p i t e o f t h e g r e a t c a r e t a k e n i n t h e i r f a b r i c a t i o n , as d e s c r i b e d  i n Appendix A, i t was conceded t h a t p r o b a b l y no more than 25-50% o f t h e t u n n e l i n g j u n c t i o n s p r e p a r e d would have t h e s u i t a b l e c h a r a c t e r i s t i c s , as d i s c u s s e d i n an e a r l i e r c h a p t e r , f o r a s e n s i t i v e t e s t o f t h e i r response t o charged p a r t i c l e bombardment. a r e needed:  F o r t h i s r e a s o n , two types o f specimen mount  one t o check t h e I-V c u r v e s and magnetic f i e l d b e h a v i o u r o f a l l  f i v e j u n c t i o n s a t once, t h e r e b y d e t e c t i n g t h e good specimens; t h e o t h e r t o . i n v e s t i g a t e t h e charged p a r t i c l e response o f i n d i v i d u a l specimens. f e a t u r e o f t h e two mounts i s t h a t , i n o r d e r t o change samples,  A unique  only these pieces  o f a p p a r a t u s 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 h o l d e r i s shown i n f i g u r e 4-2. P e r m u t a t i o n o f t h e t w e l v e e l e c t r i c a l l e a d s connected  to the substrate permits  f o u r - t e r m i n a l t e s t i n g o f any o f t h e f i v e j u n c t i o n s . F i g u r e 4-3 shows t h e i m p o r t a n t f e a t u r e s o f t h e s i n g l e j u n c t i o n sample h o l d e r and a l p h a p a r t i c l e source s u p p o r t .  P r o v i s i o n i s made f o r e i g h t e l e c t r i c a l  l e a d s , t h e g l a s s - t o - m e t a l s e a l s b e i n g o b t a i n e d by u s i n g t h e base o f a d i s c a r d e d 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 p e r m i t s r o t a t i o n o f t h e  c o n t r o l tube which a c t i v a t e s t h e s h u t t e r s e p a r a t i n g t h e r a d i o a c t i v e from t h e t u n n e l j u n c t i o n .  source  A s m a l l 2 Kft S p e c t r o l p o t e n t i o m e t e r , Model 60-1-1,  s e r v e s p r i m a r i l y as a b e a r i n g f o r t h e c o n t r o l . t u b e b u t can be i n t r o d u c e d as a c i r c u i t element i f r e q u i r e d .  The s h u t t e r i s a t h i n p i e c e o f s t a i n l e s s  steel  -85-  15 P i n Cannon P l u g s  12 P i n Kovar S e a l (x2)  .Brass F l a n g e 12 Leads (Advance, #38 AWG)  4" D i a . S/S Tube  Teflon Plate (1/8" T h i c k )  Glass Substrate w i t h Tunnel J u n c t i o n s -  Teflon-Insulated, Feedthroughs ( x l 2 )  .003" D i a . Au Wire  © !  F i g u r e 4-2:  dc Sample H o l d e r  -86-  S l i d i n g Seal  I  Thin Wall S/S Tubing  I  =-#33 AWG Forme1 covered Cu w i r e  ii  "Q Dope Bead  Octal Plug  Transmission Line S/S C o n t r o l Tube (.125 x .006 i n . ) Transmission Line (see i n s e t )  /S Support Tube (.250 x .006 i n . ) Radiation Shield Shutter  T h i n W a l l S/S Tubing  Pivot He l e v e l  a - P a r t i c l e Source & Support Sample  T h i n W a l l Cu Tubing  Cold F i n g e r ( E l e c t r o l y t i c Tough P i t c h Copper)  F i g u r e 4-3:  P u l s e Test, Holder  -87t o w h i c h a t h i n l a y e r of Pb was  s o l d e r e d to ensure the stoppage of 5 Mev  a l p h a p a r t i c l e s and t o make the s t a b l e s h u t t e r 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 l o w e r i n g the s h u t t e r ' s c e n t r e of g r a v i t y below t h e p i v o t p o i n t . A s o u r c e o f 5.13 Mev  alpha p a r t i c l e s  p i e c e of s t a i n l e s s s t e e l  (1/4" x 1/4")  s u p p o r t as shown i n t h e diagram. be 1.1 x 10^ dpm  (Pu 239) was p r e p a r e d on a t h i n , smooth which was  then s o l d e r e d t o the copper  The s t r e n g t h of the s o u r c e was measured to  i n t o 4ir s t e r a d i a n s o r about 5  yCi.  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 t o v a r y t h e a n g l e a t which the a l p h a 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 r o u g h l y 1.0 cm—much g r e a t e r t h a n the range o f a l p h a p a r t i c l e s i n l i q u i d h e l i u m (see c h a p t e r 3 ) — i t i s n e c e s s a r y t h a t the h e l i u m b a t h 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 o t h e r hand, the b a t h must be k e p t as c l o s e as p o s s i b l e  t o the j u n c t i o n f o r maximum c o o l i n g .  A compromise between t h e s e c o n d i t i o n s  was  reached i n t h e f o l l o w i n g manner. The sample s u b s t r a t e was clamped to a copper bar of h i g h t h e r m a l 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 g r a d e — m o s t of which was l i q u i d h e l i u m to s e r v e as a c o l d f i n g e r .  immersed i n  I n a d d i t i o n , the f o u r e l e c t r i c a l  l e a d s a r e passed around the bottom of the sample mount b e f o r e c o n n e c t i o n to t h e sample, so t h a t heat t r a n s f e r r e d down the e l e c t r i c a l l e a d s and support tubes i s dumped d i r e c t l y i n t o the b a t h .  C o n s e q u e n t l y , the j u n c t i o n may  somewhat above t h e b a t h l e v e l but i t s temperature s h o u l d be but  be  slightly  g r e a t e r t h a n t h a t of the b a t h because i t i s t h e r m a l l y anchored t o the b a t h v i a t h e copper b a r . cooling  S u p e r f l u i d f l o w o v e r the j u n c t i o n s e r v e s as an a d d i t i o n a l  agent. The l e v e l o f the h e l i u m b a t h f a l l s , of c o u r s e , d u r i n g an  experiment  so t h a t 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 — w h i c h , when i n t e r p o s e d between the sample mount -flange and dewar cap, p e r m i t t e d t h e j u n c t i o n to be lowered i n s t e p 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 v o l t a g e l e a d s 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  w i r e s spaced an average d i s t a n c e o f .06 i n . a p a r t and kept c e n t r a l i n the s h i e l d by beads o f Q-dope p l a c e d every few i n c h e s a l o n g i t s e n t i r e l e n g t h of 1.62m.  I t was not p r a c t i c a b l e to use t h e more obvious c o - a x i a l l i n e con-  f i g u r a t i o n o f a copper c e n t r e w i r e and s t a i n l e s s . s t e e l s h i e l d because the  -88r e s u l t i n g c o n t a c t p o t e n t i a l would be l a r g e r than the bucking v o l t a g e a v a i l a b l e w i t h t h e dc v o l t m e t e r m o n i t o r i n g the j u n c t i o n b i a s . s m a l l diameter of  commercial  i t s relatively  transmission l i n e  Using a  c a b l e , such as RG No. 174/U, was r u l e d out because  l a r g e heat l e a k .  A comparison  o f these t h r e e types o f  i s g i v e n i n t a b l e 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 pair (.125 i n . diam S. S t e e l tube, 2 x .0071 i n . Cu w i r e )  7.8xl0~  3  .011  170  Co-axial (.125 i n . diam S. S t e e l tube, 1 x .0071 i n . Cu w i r e )  5.8xl0"  3  .008  150  C o - a x i a l , RG 174/U  80xl0  - 3  • 115  T a b l e 4-1: Comparison of T r a n s m i s s i o n L i n e C h a r a c t e r i s t i c s  *White (p.198) and Hoare, e t a l (p. 142) g i v e the f i g u r e 1.43 £/hr/watt as the b o i l o f f r a t e o f l i q u i d helium.  50  -89-  CHAPTER 5  ELECTRICAL MEASUREMENT TECHNIQUES A.  dc Measurements 1.  Introduction B e f o r e a t t e m p t i n g t o use any t h i n - f i l m j u n c t i o n t o d e t e c t  charged p a r t i c l e s , i t i s n e c e s s a r y as d i s c u s s e d i n Chapter 3 t h a t 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 t h e dynamic  r e s i s t a n c e , r = (BV/DI)^ 0<V<2A/e, and t h e magnetic f i e l d dependence o f any s u p e r c u r r e n t which t h e j u n c t i o n may pass.  E x a m i n a t i o n o f t h e dc I-V curves  o f v a r i o u s specimens thus s e r v e s a double purpose:  to eliminate quickly  those j u n c t i o n s h a v i n g u n d e s i r a b l e c h a r a c t e r i s t i c s and t o p r o v i d e a b a s i s f o r d e c i d i n g o p t i m a l use o f specimens h a v i n g s u i t a b l e  characteristics.  The major components o f the dc c i r c u i t r y a r e , as shown i n f i g u r e 5-1, a b a t t e r y power s u p p l y , c u r r e n t - and v o l t - m e t e r s w i t h r e c o r d e r o u t p u t s and an X-Y r e c o r d e r . 2.  '  Power Supply The power s u p p l y i s e n e r g i z e d by a b a t t e r y because t u n n e l i n g  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 t h e Josephson e f f e c t , a r e h i g h l y n o n - l i n e a r d e v i c e s making i t i m p e r a t i v e t h a t the r i p p l e i n t h e b i a s i n g c u r r e n t be a b s o l u t e l y m i n i m a l .  P o t e n t i o m e t e r R l s e r v e s as a c o a r s e c u r r e n t  c o n t r o l ; p o t e n t i o m e t e r R2 s e r v e s as a v o l t a g e d i v i d e r such t h a t o n l y a f r a c t i o n o f t h e n o i s e generated i n the t e n - t u r n p o t e n t i o m e t e r appears a c r o s s the specimens. For some " l e a k y " specimens, the maximum output c u r r e n t (16 mA) from the r e g u l a r s u p p l y was i n s u f f i c i e n t t o o b t a i n m e a n i n g f u l I-V curves.  I t was t o t e s t such j u n c t i o n s t h a t t h e s i m p l e , h i g h c u r r e n t (150 mA)  a u x i l i a r y s u p p l y was employed. 3.  Shielding I n p r a c t i c e , the v o l t a g e a p p l i e d t o the samples i s v e r y  ulti-strand  3x12V cells (#292)  Auxiliary  Supply F i g u r e 5-1:  dc C i r c u i t r y  (Schematic)  Crvostat  -91-  s m a l l — o n e o r two m i l l i v o l t s — a n d comparable to t h a t 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 .  C o n s e q u e n t l y , c a r e 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 o n l y e x c e p t i o n b e i n g a o n e - f o o t s e c t i o n of p a t c h - c o r d s used to connect the power s u p p l y and v o l t m e t e r t o the m u l t i - s t r a n d s h i e l d e d c a b l e l e a d i n g to the cryostat. 4.  D i s p l a y of I-V  Characteristics  A Moseley X-Y current-voltage curves.  r e c o r d e r , Model 2DR-2, was used t o p l o t out the  Because i t s maximum s e n s i t i v i t y was  o n l y 0.5  mV/in.—  the maximum j u n c t i o n v o l t a g e 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 n e c e s s a r y t o p r o v i d e some a m p l i f i c a t i o n , i e . a Hewlett 425A v o l t m e t e r w i t h r e c o r d e r o u t p u t , f o r a r e a s o n a b l e d i s p l a y . 5-1).  Packard  (see f i g u r e  The c u r r e n t was measured w i t h a K e i t h l e y , Model 600A, b a t t e r y - o p e r a t e d  e l e c t r o m e t e r w h i c h p r o v i d e d a v o l t a g e 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 p a s s i v e meters  h a v i n g s u i t a b l y wide c u r r e n t 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 r e a s o n a b l y p r i c e d l i n e - p o w e r e d meters were n o t - a d e q u a t e l y i s o l a t e d ground.  Low  from  f r e q u e n c y r i p p l e , m o s t l y 60 Hz, p r e s e n t on the output of both  t h e v o l t a g e and c u r r e n t meters i n t r o d u c e d h y s t e r e s i s i n the curves t r a c e d out by the r e c o r d e r .  The amount o f h y s t e r e s i s was  p r o p o r t i o n s by p l a c i n g a low-pass  filter,  reduced t o n e g l i g i b l e  7.5 Kft and 64uF, a c r o s s the outputs  of b o t h meters but the i n t r o d u c t i o n of t h i s 0.5  sec time c o n s t a n t r e q u i r e d  t h a t the c u r v e s be t r a c e d out q u i t e s l o w l y and u n i f o r m l y . B.  H e l m h o l t z C o i l s f o r Magnetic  Biasing  As p o i n t e d out i n Chapter 3, i t i s n e c e s s a r y t o p r o v i d e a magnetic f i e l d of up t o 100 G i n the p l a n e of the j u n c t i o n to suppress the tunneling current.  Josephson  H e l m h o l t z c o i l s were chosen as b e i n g t h e s i m p l e s t and  most e c o n o m i c a l means of p r o d u c i n g a u n i f o r m , low l e v e l magnetic  field.  Space l i m i t a t i o n s r e q u i r e d t h a t the c o i l s b e p l a c e d o u t s i d e the n i t r o g e n dewar 1  s e t t i n g t h e r e f o r e a l o w e r l i m i t of about 6.5  i n c h e s on the c o i l  radius.  C o n s e q u e n t l y , a p a i r o f c o i l s were designed and c o n s t r u c t e d 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 : Power Supply  .r  H e w l e t t - P a c k a r d , Model 6268A, 40V-30A, operated i n c o n s t a n t v o l t a g e mode.  -92-  Magnetic I n d u c t i o n F i g u r e 5-2:  (G)  C a l i b r a t i o n of Helmholtz C o i l s  ( S e r i e s Mode)  -93Maximum F i e l d  100 G, S e r i e s 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 w a t t s , s e r i e s mode 1150 w a t t s , p a r a l l e l mode  Number o f Turns  298 p e r c o i l  C o i l Diameter  14 i n . i . d . , 17 i n , , o . d .  C o i l Width  1.31 i n .  C o i l Form  Aluminum  Wire  13 gauge, heavy Formel covered  The power s u p p l y was o p e r a t e d than the constant was  copper.  i n the constant v o l t a g e rather  c u r r e n t mode because t h e quoted r i p p l e and n o i s e ( l u V rms)  much l e s s i n t h e f o r m e r . The  c o i l s were wound i n such a way t h a t c u r r e n t c o u l d be passed  t h r o u g h a l l 298 t u r n s connected i n s e r i e s o r two 149 t u r n superimposed c o i l s connected i n p a r a l l e l  (see i n s e t o f f i g u r e 5-2).  I n 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 t o almost 200 G, w i t h o u t h a v i n g a higher voltage  t o use  supply.  C a l i b r a t i o n o f 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, I n c r e m e n t a l cations.  Gaussmeter, showed t h e c o i l s t o meet d e s i g n  specifi-  F i g u r e 5-2 c o n t a i n s t h e c a l i b r a t i o n curve f o r t h e c o i l s connected  for series operation. C.  Pulse Detection E l e c t r o n i c s 1.  B i a s i n g and P u l s e  Circuitry  J u n c t i o n s t o be used f o r a l p h a p a r t i c l e d e t e c t i o n were mounted on t h e s o - c a l l e d " p u l s e t e s t " sample h o l d e r d e s c r i b e d i n f i g u r e 4-3; t h e c i r c u i t r y r e q u i r e d t o p r o v i d e dc b i a s and c o n n e c t i o n a m p l i f i e r i s shown i n f i g u r e 5-3.  to the pulse  The f i l t e r box s e r v e s t o i s o l a t e t h e  dc a m p l i f i e r s from unwanted ac s i g n a l s and p r o v i d e a l o w impedance path t o the p u l s e 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  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 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 F i g u r e 5-4 i s a schematic o f t h e low n o i s e , c u r r e n t 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  Filter  For d e t a i l s of t h i s Dotted R e g i o n , See F i g . 5-1  Box C r y o s t a t - P u l s e Holder (Shielded)  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  Jr=d. F i g u r e 5-3:  I Schematic of J u n c t i o n B i a s i n g and P u l s e D e t e c t i o n  Circuit  1  -O " 4  6V  100 3.3K0 470 Input d)  1h 1.0  .01  Hl-o—* t70  CB  16G2  dro.i  <j>  16G2  27Kft  1.0 Output 220  ;56 ) - 6V  CE  Q)  ii  Note: C a p a c i t a n c e s i n uF Resistances i n Q unless otherwise noted.  F i g u r e 5-4:  Preamplifier  Schematic  -96as impedance t r a n s f o r m e r .  (The common e m i t t e r (CE) p r e a m p l i f i e r (Z. = 400 Q) in shown i n t h e diagram was c o n s t r u c t e d on t h e same c h a s s i s t o p r o v i d e 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 s h o u l d t h e need have a r i s e n . ) The i n p u t impedance (Z. m = R.i n + j Xi. n ) o f t h e CB rp r e a mrp l i f i e r> , determined d i r e c t lJ y r  J  w i t h a Hewlett-Packard  RF V e c t o r Impedance Meter (Model 4815A), i s p l o t t e d  as a f u n c t i o n o f f r e q u e n c y  i n f i g u r e 5-5. An independent check o f t h e i n p u t  impedance o f t h e p r e a m p l i f i e r and t h e 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 g e n e r a t o r by  measuring t h e phase and a m p l i t u d e voltage.  r e l a t i o n s between i n p u t c u r r e n t and  Some o f t h e i n d u c t i v e r e a c t a n c e may be a t t r i b u t e d t o t h e l y F  c o u p l i n g c a p a c i t o r s i n t h e f i l t e r box and p r e a m p l i f i e r w h i c h , when checked s e p a r a t e l y , were found t o l o o k l i k e l u F i n s e r i e s w i t h 0.5uH, h a v i n g a s e l f r e s o n a t i n g frequency The  o f about 0.9 MHz.  10-90% r i s e t i m e ( T - ) was determined by o b s e r v i n g t h e  response o f t h e a m p l i f i e r t o an i n p u t p u l s e r i s i n g i n l e s s than 1 n s e c . I t was found t h a t T upper 3db frequency 3.  d  = 15 nsec which i s c o n s i s t e n t w i t h t h e measured  o f 28 MHz.  Ancillary Electronics A p a r t from t h e p r e a m p l i f i e r s and v e t o d i s c r i m i n a t o r , t h e  remaining  a m p l i f y i n g and p u l s e h a n d l i n g d e v i c e s used were o f commercial  m a n u f a c t u r e ; f i g u r e 5-6 i s a b l o c k diagram i l l u s t r a t i n g t h e arrangement of the various instruments.  T a b l e 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 o f t h e a m p l i f i e r system s e n s i t i v i t y was performed by u s i n g i t t o a m p l i f y known amplitude  p u l s e s developed by a p u l s e  generator  i n a lumped c o n s t a n t model o f t h e s m a l l s i g n a l e q u i v a l e n t c i r c u i t o f t h e specimen.  The system was c o n s i d e r e d  t o 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 " i n p u t p u l s e — t h a t p u l s e whose  corresponding  o u t p u t s i g n a l was e q u a l t o 3 times t h e rms n o i s e 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 t h e  t u n n e l j u n c t i o n due t o a l p h a p a r t i c l e bombardment. D.  Interference Vetoing  System  Such a h i g h g a i n 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  interference.  I n s p i t e of c a r e f u l s h i e l d i n g ,  s p u r i o u s s i g n a l s were d e t e c t e d which were found t o 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 t h e Helmholtz c o i l power s u p p l y .  -98-  Bias Current  <  — W W } R  Interference Pick-up Antennae  CE Preamp ( F i g . 5-4) B u w  Specimen  0  1  Amp (A- 2) Pulse Height Analyzer (E)  c Amp. (B-2)  Amp. (D)  Wideband VTVM ' (F)  O  C OJ  0) <4-l  u 0J  Amp. (A-D  Amp. (B-1)  00  >  CB Preamp.  or d e t a i l s above t h i n e , see f i g u r e s 5and 4.  CO  •H  Filter Box  dc VTVM  Veto Discrimina t o r (K)  0J  Amp & Bandpass Filter (C)  U CO  >% CO 60  C  • H •4-1  (X 6  Del ay (500 n sec)  Pulse  < B 60 • H CO  Gate  Discriminator Scaler (G) D i s c . Output Scaler (H)  Oscilloscope (I) ^Camera  F i g u r e 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 a r e l i s t e d i n Table 5-1  -99Th e a m p l i t u d e  and number of these unwanted s i g n a l s were such t h a t  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 p u l s e s coming from the  j u n c t i o n d e t e c t o r 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 o r 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 R e s e a r c h Systems, Dual L i n e a r A m p l i f i e r , Model B - Hewlett-Packard,  L i n e a r A m p l i f i e r , Model 462A,  C - O r t e c , L i n e a r A m p l i f i e r , Model D - Hewlett-Packard, E - Nuclear-Data,  107F  (2)  410  L i n e a r A m p l i f i e r , Model 460A  P u l s e Height A n a l y z e r , Model 101 (Note: T h i s u n i t was out o f o r d e r when the p u l s e d a t a were a c t u a l l y a c q u i r e d from specimen.)  F - K e i t h l e y , Wideband VTVM, Model  120  G - O r t e c , D i s c r i m i n a t o r - S e a l e r , Model H - O r t e c , S c a l e r , Model  429  430  1 ~ T e k t r o n i x , O s c i l l o s c o p e , Model J - O r t e c , P u l s e S t r e t c h e r , Model  454 411  K - U.B.C., I n 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, was  Pulse Generator,  Model 106A,  (not shown on f i g .  5-6)  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 o f the Veto system i n  Chapter 6 ) .  C o n s e q u e n t l y an " I n t e r f e r e n c e V e t o i n g " system was  (see f i g u r e 5-6)  which o p e r a t e d  devised  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 a p p r o p r i a t e antennae to p i c k up the same i n t e r f e r e n c e as the main s y s t e m — a n d put out g a t i n g p u l s e s to p a r a l y z e the s c a l e r and p u l s e  height  a n a l y z e r whenever the i n t e r f e r e n c e s i g n a l exceeded a t h r e s h o l d v a l u e . (The t h r e s h o l d l e v e l i s c o n v e n i e n t l y s e t by s t o p p i n g the a l p h a  particles  b e f o r e they r e a c h the j u n c t i o n by l o w e r i n g i t below the h e l i u m 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 " z e r o " counts a r e observed on  the  gated  scaler.) F i g u r e 5-7  by "K" E.  i n figure  i s a schematic of the v e t o d i s c r i m i n a t o r u n i t  5-6.  E l e c t r i c a l P r o p e r t i e s of T r a n s m i s s i o n 1.  designated  Line  Introduction 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 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 used i n t h i s experiment was c h a p t e r 4.  given i n  T h i s 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 o f  the  -°+12V  754A 914  -0+5.2V  of  •220ft iMC408  iMC408  iMC408  2N1304  1N100  O r t e c Veto +4.8  13  100 pF-»-  2  •  1N100  1  220ft  2N2925 3.3Kft  ND 101 Veto  lKft  lOKftC  ilKft  -• •2N1305  F i g u r e 5-7:  Schematic o f Veto D i s c r i m i n a t o r  »S>»U -0-6V  2N1304  -101l i n e needed f o r the n o i s e and p u l s e shape a n a l y s i s of c h a p t e r 7 and d e s c r i b e s how  they were d e t e r m i n e d . To a r e a s o n a b l e a p p r o x i m a t i o n , the l i n e  (assumed to be  l i n e a r ) may be r e p r e s e n t e d by lumped c o n s t a n t s f o r i t s l e n g t h (£ =1.6m) i s much l e s s than t h e minimum wavelength o f 85m c o r r e s p o n d i n g to t h e upper 3db-down f r e q u e n c y 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 t e r m i n a t e d i n Z = R + i X I t t t b  simply Z  i  is  J  = R^ + R  + j ( X ^ + X ) where Z^ = R^ + j X ^ i s t h e e f f e c t i v e s e r i e s fc  impedance of the l i n e . The f o l l o w i n g two paragraphs d e s c r i b e the measurement of R. and X. and the e s t a b l i s h m e n t o f 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 configurations.  P a r a g r a p h 5 i s an i n v e s t i g a t i o n made t o check the c o n s i s t e n c y  between t h o s e parameters measured and t h o s e which may  be e s t i m a t e d  a p p r o x i m a t e l y from t h e o r y . 2.  E f f e c t i v e L i n e Impedance Although Z  may have been e s t i m a t e d from t h e o r y , l i t t l e JO  c o n f i d e n c e c o u l d be p l a c e d i n the r e s u l t s because of the inhomogenity o f the  line  (see paragraph 5) and the temperature range i t spanned.  was  immersed i n 1.2 K l i q u i d h e l i u m w h i l e the o t h e r end was  anchored a t room temperature 295 K.)  (One  end  thermally  C o n s e q u e n t l y , R^ and X^ were determined  e x p e r i m e n t a l l y over the a m p l i f i e r passband by s h o r t i n g the " d e t e c t o r end" of the t r a n s m i s s i o n l i n e  (making R  = X t  =0)  and measuring Z. = Z =R 1  U  w i t h a H e w l e t t - P a c k a r d Model 4815A RF V e c t o r Impedance Meter t h r e e d i f f e r e n t temperature c o n d i t i o n s , namely: temperature  Jo  +jX J6  Xr  (Z meter) f o r  complete l i n e a t room  (295 K ) , one end a t 295 K w i t h the r e m a i n i n g 90% c o o l e d to  77 K w i t h h e l i u m gas, one end a t 295 K w i t h the o t h e r a t 1.2 K as i n the a c t u a l experiment.  The r e s u l t s a r e p l o t t e d i n f i g u r e 5-8 where the e r r o r  b a r s shown a r e t y p i c a l and a r i s e from t h e a c c u r a c y l i m i t s o f the Z meter. (For  a r e a d i n g of Z = |z| exp  ( j <j>), the a c c u r a c y l i m i t s quoted by the  m a n u f a c t u r e r a r e AZ = ± 4% F.S.D. and 6 = Aq> = + 3° so t h a t 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 R e s i s t a n c e 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 b o t h temperature and f r e q u e n c y dependent.  A l e a s t squares f i t of R  = R  + R * f t o the d a t a n  -102i 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 r e f e r e n c e , when  the l i n e i s s a i d to be a t temperature T, i t i s to be u n d e r s t o o d t h a t end  i s a t 295 K and  the o t h e r a t T; eg. R  Temperature Range  ori  ( 7 7 K) = 1.73  ho™  (K)  one  ±.3452 ) .  R ( f t MHz  1  £ 1  )  295-295  2.47  +  .49  .875 +  0.17  295-77  1.73  +  .34  .534  +  .10  295-1.2  1.60  +  .32  .557  +  .11  (b) E f f e c t i v e I n d u c t a n c e Within experimental of  accuracy,  the r e a c t i v e component  i s t e m p e r a t u r e independent and p u r e l y i n d u c t i v e so t h a t i t may  written  =  2 T T f L ^  where  i s the e f f e c t i v e i n d u c t a n c e  the bandwidth of i n t e r e s t .  Setting  2TTL^  s t r a i g h t l i n e t h r o u g h the e x p e r i m e n t a l  be  o f t h e l i n e over  e q u a l to the s l o p e of the b e s t  p o i n t s f o r a l l t h r e e temperatures  yields  = 1.63  T h i s v a l u e of  ± .13'uH  a g r e e s c l o s e l y w i t h the v a l u e o f 1.6uH  obtained  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) h i g h frequency 2 -1 f i n d i n g the r e s o n a n t f r e q u e n c y a> = (LC) 3.  L i n e Input  by  c a p a c i t o r and  Impedance w i t h P r a c t i c a l Loads  On the b a s i s o f the p r e c e d i n g r e s u l t s , i t may  be s a i d t h a t ,  f o r the f r e q u e n c i e s used i n t h i s e x p e r i m e n t , the t r a n s m i s s i o n l i n e i n Z- may  be regarded  terminated  as a t w o - t e r m i n a l network whose i n p u t impedance i s  Z^ = R + j X where, f o r temperature T  R = R  T  + R  £ Q  (T) +  R (T)-0)/2TT u  (5-1) X - X I f e q u a t i o n 5-1  t  +  j L . U  4  i s to be used t o d e s c r i b e the i n p u t impedance l o o k i n g i n t o  -103-  R„A = R„^ ,*f £0 + R £1 n  Transmission Line Temperature (T)  X  i  a  r—\ a  T  co u c  CO 4J W •H W OJ  »1  ft  r  X  4r  i  Frequency  35  (T) X 295 K O 77 K £ 1.2 K  V^ £  short  X  £ "  2  ^  f L  -  (MHz)  £  2!  30  25  X 295 K  O 77 K Al-2 K  ft  20 X I CJ  OJ  a  c  cd o co <u  21  15 10 L  L  *  6  0  _L  1  "2  ' H """  Frequency F i g u r e 5-8:  Resistance  (MHz)  & Reactance of T r a n s m i s s i o n  Line  -104e l t h e r end o f t h e l i n e , i t must be assumed t h a t t h e e f f e c t i v e l i n e impedance  = R^(T) + J  o r " c o l d " end.  w L  £ is  same whether determined from the warm  t n e  F o r T = 295 K, t h i s a s s u m p t i o n was v e r i f i e d  but i t c o u l d n o t be c o n f i r m e d ,  experimentally  o f c o u r s e , a t T = 77 o r 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 a r e o f i n t e r e s t t o the i n t e r p r e t a t i o n of the experimental  results.  For p u l s e  shape a n a l y s i s , t h e d e s i r e d impedance i s t h a t seen by t h e 1.2 K j u n c t i o n as i t l o o k s i n t o t h e c o l d end o f t h e t r a n s m i s s i o n l i n e when i t i s t e r m i n a t e d w i t h t h e p r e a m p l i f i e r a t 295 K ( s i m i l a r l y t o 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. , = Z. . i,cold xc n  r  For noise c a l c u l a t i o n s , the '  r e q u i r e d impedance i s t h a t seen by t h e p r e a m p l i f i e r as i t l o o k s i n t o t h e warm end o f t h e t r a n s m i s s i o n l i n e when i t i s t e r m i n a t e d w i t h a c o l d t u n n e l 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. . x,warm IW To t e s t t h e a c c u r a c y o f e q u a t i o n 5-1 i n p r e d i c t i n g Z^ and J  c  Z. f o r t h e a c t u a l j u n c t i o n , s t u d i e s were made ( a t 295 K) o f t h e l i n e xw t e r m i n a t e d w i t h t h e p r e a m p l i f i e r and w i t h t h e l i n e t e r m i n a t e d c o n s t a n t model o f a t u n n e l j u n c t i o n .  i n a lumped  To i l l u s t r a t e these t e s t s , t h e  r e s u l t s w i t h t h e p r e a m p l i f i e r as l o a d a r e g i v e n below. (a) Z. f o r L i n e w i t h P r e a m p l i f i e r as Load x The p r e a m p l i f i e r i n p u t impedance Z. = R. + j X . was xn xn xn c  determined i n d e p e n d e n t l y  (see f i g u r e 5-5) so t h a t by e q u a t i o n 5-1, t h e  p r e d i c t e d impedance l o o k i n g i n t o t h e l i n e i s Z. = R. + R„(295 K,o>) + j ( X . + ooL.) x xn I xn V  (5-2)  J  T h i s may be compared t o Z^ = R' + j X ' o b t a i n e d w i t h t h e Z meter by c o n n e c t i n g the p r e a m p l i f i e r t o t h e l i n e as shown i n t h e i n s e t o f f i g u r e 5-9. model o f e q u a t i o n 5-1 i s v a l i d i t i s expected t h a t Z^ = The r e s u l t s a r e g i v e n i n f i g u r e 5-9. measured r e a c t a n c e X' agrees w e l l w i t h t h a t o b t a i n e d The agreement between the c o r r e s p o n d i n g  I f the  Z\ C l e a r l y , the  from e q u a t i o n 5-2.  r e s i s t i v e components o f t h e impedance  i s n o t as good b u t may be made c o n s i s t e n t by u s i n g t h e s l i g h t l y  smaller  average v a l u e o f R. = 7.5ft i n s t e a d o f 7.9ft as determined from f i g u r e 5-5. in (Such a change i s w i t h i n t h e a c c u r a c y l i m i t s s t i p u l a t e d by t h e Z meter 5  -105-  Z'  Trans Line  r—j  l i i  ! J 1  ^  X+R'  P. A.  (Measured)  F i l ter Box  Z' = R' + j X ' E x p t ' l : R. + A t t e n u a t i o n (R. = 7.9 ft) in l n  14 13  E x p t ' l : R. + in Attenuation (R. = 7.5 n) D i r e c t Measurement (R')  01 CJ  C to 4-1  CO •H CO OJ  Pd  12 11 R.  10  (See eq'n 5-3)  IC  -A"  -L 4  Freq.  (MHz)  A.-*X' (Meas.) O -»X=X. +X„ i n SL  35 30  X. from F i g . 5-5 in X  £  from F i g . 5-8  20 0)  o c cfl 4-1 (J  fS OJ  Pi  10  4  F i g u r e 5-9:  Impedance of T r a n s m i s s i o n  Freq.  (MHz)  L i n e w i t h P r e a m p l i f i e r as Load  -106manufacturer.) (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 o f the Z ^ symmetry assumption  stated  e a r l i e r and t h e agreement between measurement and the model o f t h e l i n e r e p r e s e n t e d by e q u a t i o n 5-1, Z^  may be w r i t t e n  Z. = R. + R„.(1.2 K,u>) + jtoL xc in £ c  = R.  J  where R. = 7.5ft and L = L. + L = 1.69yH. m c £ preamp. p l o t t e d i n f i g u r e 5-9.  + jX i c c  (5-3)  J  F o r r e f e r e n c e , R. i s ' ic  (c) Z ^ f o r T r a n s m i s s i o n l i n e + Tunnel J u n c t i o n By t h e same r e a s o n i n g as g i v e n above, Z^  Z  iw  =  R  j  +  V* 1  ' '  2  K  w ) +  j  (  X  j "  H j j L  £  may be w r i t t e n  )  ( 5  "  4 )  where Z.. = R.. + j X ^ i s t h e t u n n e l j u n c t i o n output impedance. 4.  Characteristic  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 i n d u c t a n c e (L ) w h i c h have been deduced a r e a l l t h a t i s r e q u i r e d t o r e p r e s e n t t h e l i n e i n t h e lumped c o n s t a n t s e r i e s impedance a p p r o x i m a t i o n t h a t has been chosen. To a i d i n u n d e r s t a n d i n g t h e l i n e and t o ensure t h a t t h e measured  parameters  f o r t h e l i n e a r e s e l f - c o n s i s t e n t , t h e c h a r a c t e r i s t i c impedance Z o f t h e o l i n e was a l s o determined. Z  q  was found by t e r m i n a t i n g t h e t r a n s m i s s i o n l i n e i n a p u r e l y  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 i n p u t impedance o f t h e l i n e ( Z ^ ) as measured by t h e Z meter e x h i b i t e d no r e a c t i v e component.  (Note: " p u r e l y  r e s i s t i v e " and "no r e a c t i v e component" mean t h a t w i t h i n t h e + i£° e m p i r i c a l l y determined phase a n g l e s e n s i t i v i t y o f t h e i n s t r u m e n t near z e r o , t h e observed phase a n g l e i n t h e two impedances was z e r o . )  The v a l u e o f Z  q  thus deduced  was Z  o  = 168 + 17ft -  where t h e e r r o r s a r i s e from t h e a c c u r a c y l i m i t s on the phase a n g l e (as j u s t  -107s t a t e d ) and on the magnitude (as g i v e n i n paragraph  2) f o r b o t h  and  Z.. 1  5.  C o n s i s t e n c y Check of Measured Values w i t h Theory The p r e c e d i n g paragraphs have shown t h a t a l l the parameters  needed t o 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 s u c c e e d i n g c h a p t e r s be determined  experimentally.  may  No f a i t h c o u l d be p l a c e d i n these q u a n t i t i e s  c a l c u l a t e d from t h e o r y because the l i n e operated over a s i g n i f i c a n t range o f temperatures  and b o r e but f a i n t resemblance to the i d e a l , s i m p l e c o n f i g u r a -  t i o n s f o r w h i c h 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 b e s the v e r y n o n - i d e a l 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 e s t i m a t e d f o r the s h i e l d e d parallel pair approximates  (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 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 e x p e r i m e n t a l l i n e was  not  o p e r a t e d i n b a l a n c e d mode f o r b o t h the " s h i e l d " and one o f the w i r e s were grounded.  Being homemade, i t had a non-uniform  because o f space l i m i t a t i o n s , the " s h i e l d " was  structure.  For example,  o n l y 2/3 as l o n g as the c e n t e r  c o n d u c t o r s ; the two w i r e s were somewhat unevenly  spaced b o t h  w i t h r e s p e c t to  each o t h e r and to the " s h i e l d " ; t h r e e d i e l e c t r i c s were p r e s e n t : l a y e r (= 0.001 continuous  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 -  lumps of Q-dope, which o c c u p i e d perhaps 20-40% of the volume  i n s i d e the " s h i e l d " , air  (1) a t h i n  (the Q-dope was  used to space the w i r e s ) and  (3) room  o r h e l i u m which f i l l e d the r e m a i n i n g volume. Table 5-2  summarizes the comparison between the e x p e r i m e n t a l l y  and t h e o r e t i c a l l y e s t i m a t e d parameters.  Considering  the inhomogeneous n a t u r e  of the l i n e , the agreement i s q u i t e r e a s o n a b l e . 6.  Propagation  Velocity  From the f o r e g o i n g d i s c u s s i o n , i t i s e v i d e n t t h a t a r e a s o n a b l e r e p r e s e n t a t i o n o f the l i n e has been o b t a i n e d i n t h a t impedances measured w i t h t h i s model a r e 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 e s t i m a t e d from t h e o r y . a s p e c t o f the t r a n s m i s s i o n l i n e remains w h i c h i s not y e t understood  However, one completely  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 o f l a t e r  c h a p t e r s — t h e p r o p a g a t i o n v e l o c i t y u = (LC)  2  = Z /L - (.56  ±.l)c  seems to  -108be i n o r d i n a t e l y l o w . T h i s v a l u e o f u = c(K'K') i m p l i e s t h a t K'K* = 3.2 „ m m - .9 where K' = e'/e i s t h e e f f e c t i v e d i e l e c t r i c c o n s t a n t 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 t o p e r c e i v e how such 2  r  Parameter Resistance  (R, 3 MHz)  Inductance  (L)  Characteristic Impedance Z o T a b l e 5-2:  Experiment 3.1 ± .6ft/m * 1.01 ± .08pH/m * 168 ± 17ft  Comparison of E x p e r i m e n t a l Parameters.  Theory (Model) 2.1ft/m  (spp)  1.3uH/m (pp) t 190ft (spp) §  and T h e o r e t i c a l T r a n s m i s s i o n  Line  *  The p h y s i c a l l e n g t h o f t h e 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 e s t i m a t e o f L i s a v a i l a b l e f o r t h e spp.  §  T h i s v a l u e o f Z depends on t h e v a l u e o f e f f e c t i v e d i e l e c t r i c c o n s t a n t 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 t h e l i n e — s e e paragraph 6.  a l a r g e p r o d u c t o b t a i n s i n v i e w o f t h e f a c t t h a t t h e 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 t h e r e a r e no o b v i o u s l y 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 t o a i r , t h e o t h e r d i e l e c t r i c s p r e s e n t i n t h e l i n e were Formel (a p o l y v i n y l a c e t a t e r e s i n ) w i t h K = 2.92 (von H i p p e l , 1954) and Q-dope (mostly p o l y s t y r e n e ) w i t h K = 2.52-2.65 as s t a t e d by the manufacturer (GC E l e c t r o n i c s , R o c k f o r d , I l l i n o i s ) and checked e x p e r i m e n t a l l y .  The frequency  s t a b i l i t y o f 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 t h e f a c t t h a t K ( p o l y s t y r e n e ) i s e s s e n t i a l l y independent o f f r e q u e n c y (Formel) decreases  over t h e range 1 - 3000 MHz and K  by o n l y 5% over t h e same range (von H i p p e l , 1954).  Although  the Formel and Q-dope occupy o n l y a r e l a t i v e l y  s m a l l f r a c t i o n o f t h e 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 c o n s t a n t 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 t o t h e volume. T h i s a r i s e s from t h e f a c t t h a t these substances  are located close  -109t o the w i r e s u r f a c e s 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 s t o r e d 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 , c o n s i d e r a l i n e w i t h  s i m p l e c o - a x i a l geometry w i t h r a d i i a ( w i r e ) and c ( s h i e l d ) and a d i e l e c t r i c surrounding  the w i r e out t o a r a d i u s b such t h a t f o r a<r$b, K>1  b<r£c, K = 1.  The  K' = l n ( c / a ) ( K  _ 1  K  and f o r  e f f e c t i v e d i e l e c t r i c c o n s t a n t o f such an arrangement i s l n ( b / a ) + l n ( c / b ) ) ; s e t t i n g a = .0035 i n . , b = .03 i n . , _ 1  c = .06 i n . and K = 2.6,  as a v e r y rough a p p r o x i m a t i o 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 from the v a l u e K'(volume) = 1.37  to the s i t u a t i o n i n  which d i f f e r s  significantly  c a l c u l a t e d on the assumption t h a t 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  e x p e c t e d t h a t t h i s e f f e c t would be even more pronounced i n the t w i n w i r e 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 c o n c e n t r a t e d Q-dope f i l l e d r e g i o n k e e p i n g c o n s t a n t as l a r g e as 2.3  the w i r e s a p a r t .  Thus, an e f f e c t i v e  (the l o w e r . l i m i t on K'K^)  There seems t o be no way  i n the dielectric  i s perhaps c o n c e i v a b l e .  i n which the t r a n s m i s s i o n l i n e c o u l d  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 g r e a t e r than I t i s p o s s i b l e t h a t 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 e e l used i n the s h i e l d was and  one.  stainless  g r e a t e r than o n e — d u e to c o l d w o r k i n g e f f e c t s  the i r r e v e r s i b l e p a r t i a l t r a n s f o r m a t i o n 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 w h i c h takes p l a c e 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 r e p e a t e d l y c o o l e d to h e l i u m temperatures (Reed and M i k e s e l l , 1 9 6 0 ) — b u t because of i t s r e l a t i v e l y s m a l l 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 o f 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 o f the l i n e i s expected to be n e g l i g i b l e . p o s s i b l e to t e s t K  (Because i t was  f o r the t u b i n g used as the " s h i e l d " w i t h o u t  not  destroying  the l i n e , a p i e c e o f s i m i l a r " o f f the s h e l f " t u b i n g (not known to have been c o o l e d t o h e l i u m t e m p e r a t u r e s ) was K  >  checked but i t gave no e v i d e n c e o f  having  1.)  m From these c o n s i d e r a t i o n s , i t appears t h a t the v e l o c i t y deduced from Z  q  propagation  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  v a l u e s of the e f f e c t i v e d i e l e c t r i c c o n s t a n t and r e l a t i v e p e r m e a b i l i t y but t h a t the maximum v a l u e of u a l l o w e d by t h e - e r r o r s does approach a v a l u e which i s c o n s i s t e n t with theory.  T h i s problem was  not pursued f u r t h e r as i t was  not c o n s i d e r e d c e n t r a l to the purpose o f the experiment.  To  satisfactorily  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 w h i c h , because o f i t s u n w i e l d y  geometry and horrendous m i x t u r e o f d i e l e c t r i c  i n t e r f a c e s , presents a formidable  challenge.  -110F.  Summary The  a p p a r a t u s f o r t r a c i n g out the dc I-V c u r v e s ,  magnetically  b i a s i n g t h e specimens and o b s e r v i n g the p u l s e s o u t p u t from t h e d e t e c t o r has been d e s c r i b e d .  Q u a n t i t i e s o f p a r t i c u l a r i n t e r e s t t o 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! + j x ! where, i n t h e a m p l i f i e r bandpass, in in in ' r » r  R! =' 7.9ft = R. in  and X! m m  =  coL - (toC)"  (5-5)  1  w i t h L = 0.23uH and C = 0.5yF. 2.  Transmission  L i n e + Load E q u i v a l e n t  Circuit  For t h e t r a n s m i s s i o n l i n e at. temperature T t e r m i n a t e d i n Z  = R  fc  + jX » t h e e f f e c t i v e i n p u t impedance i s a p p r o x i m a t e l y t  Z^ = R + j X  where  R = R  t  + R^co.T)  X - X  t  + j o ^  w i t h co b e i n g t h e a n g u l a r inductance  and R^  =  R^  frequency, n  + R^  L^ = 1.63uH t h e e f f e c t i v e  "10/211  t h e frequency  line  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 d e s c r i b e d 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 c h a p t e r s , i t was n e c e s s a r y  to obtain  the dc I-V c h a r a c t e r i s t i c s o f each j u n c t i o n a t h e l i u m temperatures t o d e t e r m i n e i t s s u i t a b i l i t y as a d e t e c t o r .  The q u a n t i t i e s o f 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 t h e dynamic r e s i s t a n c e r = (3V/8I)^, and  t h e degree t o w h i c h t h e s u p e r c u r r e n t  t i o n o f a magnetic f i e l d .  c o u l d be suppressed by t h e a p p l i c a -  T h i s 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  a g a i n s t " 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 s h o r t s through t h e 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 to  i s v e r y much l e s s s e n s i t i v e  a magnetic f i e l d than i s t h e dc Josephson s u p e r c u r r e n t .  "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 a r e d e s c r i b e d b r i e f l y i n Appendix B. 1.  Temperature Dependence o f T u n n e l i n g  Current  F i g u r e 6-1 i l l u s t r a t e s t h e v a r i a t i o n i n t h e dc I-V c h a r a c t e r 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 d e c r e a s e i n c u r r e n t w i t h d e c r e a s i n g temperature and t h e f a c t t h a t a f i e l d o f 100 G has c l e a r l y suppressed t h e z e r o - v o l t a g e c u r r e n t i n d i c a t e t h a t t h e observed c u r r e n t f l o w i s t h a t o f 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 o b t a i n e d from a  t y p i c a l l e a k y 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. I n a d d i t i o n , i t may be seen t h a t t h e s l o p e ( 9 I / 3 V ) the I-V curve d e c r e a s e s as t h e temperature i s lowered. need t o o p e r a t e  t h e j u n c t i o n a t t h e l o w e s t convenient  T  of  T h i s p o i n t s out the temperature so t h a t  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 t o maximize the s i g n a l to n o i s e r a t i o i n accord w i t h the t h e o r y o u t l i n e d i n Chapter 3. 2.  M a g n e t i c F i e l d Dependence o f T u n n e l i n g The  Current  observed v a r i a t i o n of t h e j u n c t i o n c u r r e n t w i t h a magnetic  -112-  0  0.2  0.4  0.6  0.8  1.0  V(mV) F i g u r e 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  1.2  -113f l e l d p a r a l l e l t o t h e p l a n e o f t h e j u n c t i o n i s shown i n t h e composite X-Y r e c o r d e r t r a c i n g s o f f i g u r e 6-2.  ( I n p r a c t i c e , p l o t s o f 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 t h e c o n v e n t i o n a l I ( c r i t ) - B o r F r a u n h o f f e r 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 o f t h e S u p e r c u r r e n t  (B=0)  For a g i v e n j u n c t i o n a t a f i x e d temperature, t h e observed to  maximum z e r o - v o l t a g e c u r r e n t ( I ) was found t o v a r y from t r a c i n g £  t r a c i n g and t o be o n l y a s m a l l f r a c t i o n o f t h e t h e o r e t i c a l maximum  supercurrent  (see e q u a t i o n 2-16  I The  )  g i v e n by  = J (T)-A=TrA(0)/2eR o n  cm  o b s e r v a t i o n s a r e summarized i n t a b l e 6-1.  THEORETICAL SPECIMEN  ''"cm  H-4  3mA  H-5  4.5mA  J-5  11.2mA  Icm c  m  I  That t h e observed  0 B S  cm  ,.04±.02mA  T a b l e 6-1: Comparison o f E x p e r i m e n t a l  smaller  AVERAGE OBSERVED I  T H Y  '  .013  .045±.005mA  .01  .35±.15mA  .031  & T h e o r e t i c a l Maximum  -  Supercurrent.  1^ i s almost two o r d e r s o f magnitude  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 t o the f a c t t h a t no attempt was made t o m a g n e t i c a l l y s h i e l d t h e specimens.  Workers s t u d y i n g t h e dc Josephson e f f e c t and d e t e c t i n g 1^ as  h i g h as 50-90% o f I  ( t h e o r e t i c a l ) , r e p o r t ( J a k l e v i c e t a l , 1965) f i n d i n g  i t necessary  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 t h e e a r t h ' s  to cool  magnetic f i e l d .  Otherwise,  f l u x was a p p a r e n t l y trapped i n the f i l m s o r  j u n c t i o n s which s e v e r e l y a t t e n u a t e d t h e maximum Josephson c u r r e n t .  In  a d d i t i o n , t h e c o n d i t i o n "B- = 0" i n t h e p r e s e n t experiment i n d i c a t e s o n l y t h a t the f i e l d due t o t h e Helmholtz  c o i l s c a n c e l l e d out t h e component o f t h e  e a r t h ' s f i e l d l y i n g i n the j u n c t i o n plane t h e r e f o r e , a t "B" = 0"  (see f i g u r e 5-2).  Most  likely,  s t r a y magnetic f i e l d s were p r e s e n t w h i c h , as  -115d i s c u s s e d i n Chapter  2, need o n l y be of s m a l l magnitude t o 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 c h a p t e r 2, I i n c r e a s i n g magnetic f i e l d a c c o r d i n g to  s h o u l d decrease  sinCrcXB)  w n  with  e r e B i s the  field  TTAJJ  magnitude and X i s a term c h a r a c t e r i s t i c of a g i v e n j u n c t i o n and o f the o r d e r 1.5  G .  (For B=70 G, I  _1  $ I  /IOOTT)  c  .  typically  Q u a l i t a t i v e agreement  cm  w i t h t h e o r y i s c l e a r l y e v i d e n t i n f i g u r e 6-2 w h i c h i n d i c a t e s t h a t 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 . p o i n t , as f a r as the p r e s e n t experiment  i s concerned,  v o l t a g e s u p e r c u r r e n t can be e f f e c t i v e l y suppressed  The  important  i s t h a t the z e r o -  by t h e a p p l i c a t i o n o f a  magnetic f i e l d w h i c h i s much s m a l l e r than the c r i t i c a l f i e l d o f the conducting f i l m s .  (See paragraph  super-  5 of t h i s ' s e c t i o n . )  (c) V a r i a t i o n o f 3V/3I w i t h Magnetic  Field  For a g i v e n magnetic f i e l d B and temperature  T,  there  i s a p o i n t (V , I ) a t w h i c h r = 3V/31 (eV$2A) i s a maximum. I t i s m m e v i d e n t i n f i g u r e 6-2 t h a t r = (3V/3I), ,, v a r i e s somewhat w i t h magnetic m vnij Im f i e l d , d e c r e a s i n g s h a r p l y a t 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 e d u c t i o n m r  7  T  i n the s u p e r c o n d u c t i n g energy gap w i d t h 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 t o the decrease  i n r ^ brought about by i n c r e a s i n g the  t h e r e b y r e d u c i n g the g a p — a s i s e v i d e n t i n f i g u r e  temperature—  6-1.  (d) Optimum B i a s i n g P o i n t The  criteria  o p e r a t i n g p o i n t a r e now f i g u r e 6-2,  clear.  f o r and means of d e t e r m i n i n g t h e optimum From a s e r i e s o f curves l i k e those i n  taken a t the lowest a c c e s s i b l e temperature,  I , V and B a r e found a t w h i c h For example, had specimen H-4 d i s c u s s e d i n paragraph  t h e v a l u e s of  i s maximum and t h e s u p e r c u r r e n t i s minimum. not degenerated  w i t h thermal c y c l i n g — t o  be  4 o f t h i s s e c t i o n — t h e b i a s i n g p o i n t chosen would  have been I = 9.5uA, V = 0.22mV and B = 70 G where r  = 100ft a t 1.2  K.  m 3.  D e t e r m i n a t i o n of Dynamic R e s i s t a n c e (3V/3I) as a F u n c t i o n o f V o l t a g e 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  X-Y  plot.  For purposes o f s e l e c t i n g the o p e r a t i n g p o i n t i t was  a c c u r a t e t o draw tangents to the I-V curves and determine  an  sufficiently  t h e i r slope.  It  -116-  Specimen  H-4  . T = 1.2 K  e *6  100  -o-  CN  «  50  0 0  X  X  20  X  40  60  X  X  80  100  B(G)  F i p n r p 6-3: V a r i a t i o n o f maximum dynamic r e s i s t a n c e w i t h magnetic f i e l d  id  = (av/a^n T = 1.2 K B = 30 G  X  X  .2 .3 Junction Voltage  (mV)  .5  F i g u r e 6-4: Dynamic r e s i s t a n c e v s v o l t a g e f o r specimen J-5  .6  -117t u r n e d o u t , however, t h a t i n o r d e r t o understand  t h e manner i n which t h e  n o i s e 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 o f 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 o f r = 3V/9I was n e c e s s a r y f o r sample J - 5 , t h e sample w i t h w h i c h t h e a l p h a p a r t i c l e s were d e t e c t e d . 4 A f o u r t h order polynomial I = E f i t t e d t o each o f two e x p e r i m e n t a l c u r v e s T = 1.2 K and B = 30 G f o r t h i s specimen. was  a V n  was l e a s t  squares  ° (0$V$.8 mV) o b t a i n e d a t  n  F o r each c u r v e , r = ( 9 I / 9 V ) "  1  c a l c u l a t e d w i t h t h e mean v a l u e b e i n g 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 o f Specimens a f t e r Thermal C y c l i n g Because o f t h e h i g h percentage  t e s t i n g apparatus was d e s i g n e d  (see Chapter  of f a u l t y j u n c t i o n s , the  4) so t h a t f i v e specimens c o u l d  be " d c " t e s t e d a t one time i n t h e so c a l l e d "dc h o l d e r " t o determine t h e specimen w i t h t h e 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 . (24-36 h o u r s ) up t o room temperature  A f t e r warming s l o w l y  i n a h e l i u m atmosphere, t h e specimens  were removed from t h e "dc h o l d e r " , t h e b e s t specimen was s e p a r a t e d from t h e o t h e r s and mounted i n t h e " a c " o r " p u l s e t e s t h o l d e r " (see f i g u r e 4-2 and 4-3).  T h i s e x t r a h a n d l i n g of t h e specimens was n e c e s s i t a t e d by t h e f a c t  t h a t t h e " p u l s e t e s t h o l d e r " , which was equipped and s h u t t e r , was o r i g i n a l l y d e s i g n e d  w i t h the alpha  source  t o take o n l y one 1 cm x 2 cm s u b s t r a t e  with i t s four e l e c t r i c a l leads. I t was d i s c o v e r e d — w h e n Sn specimens were f i n a l l y made h a v i n g 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 b e i n g warmed up t o room temperature,  i n s t a l l e d i n t h e " p u l s e t e s t h o l d e r " , and c o o l e d a g a i n t o  helium temperatures,  a specimen s u f f e r e d c o n s i d e r a b l e , a p p a r e n t l y  irrever-  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 r e n d e r i n g i t u s e l e s s as a detector.  Of t h e 6 samples f o r w h i c h d a t a a r e a v a i l a b l e , t h e g e n e r a l  symptoms were a d e c r e a s e i n t h e maximum dynamic r e s i s t a n c e , l i t t l e o r no temperature  dependence o f t h e c u r r e n t ( 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 o f t h e s u p e r c u r r e n t t o a magnetic f i e l d  ( u n l i k e f i g u r e 6-2).  T a b l e 6-2 l i s t s t h e p r o p e r t i e s o f one such j u n c t i o n (H-4) w h i c h was c y c l e d 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 a r e v e r y  similar  to those o f t h e " l e a k y " specimens d i s c u s s e d i n Appendix B w h i c h i m p l i e s t h a t d u r i n g the c y l i n g p r o c e d u r e , minute m e t a l l i c f i l a m e n t s o r s u p e r c o n d u c t i n g s h o r t s had developed  through t h e i n s u l a t i n g o x i d e .  The f a c t t h a t these  -118f 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 e x p l a i n e d  as f o l l o w s .  If R  i s the normal s t a t e  resistance  n  H-4 Before Cycling  SPECIMEN Property R  n  I  (4.2 K)  After  .286ft  (B)  .289ft  m  .lmA(lOOG)  64ft  4ft  F i t s Theory  Insensitive to Temp.  Not  I  I  =3V/3I  Temp. Dependence  B Dependence  I  quenchable  not c quenchable  After  .077ft  .0003mA(100G)  c r  J-5 Before C y c l i n g  .076ft  .001mA(40G)  .01mA(40G)  70ft  9.3ft  Measured  Reasonable  quenched  I  quenched  T a b l e 6-2: Specimen C h a r a c t e r i s t i c s B e f o r e and A f t e r Thermal C y c l i n g .  b e f o r e 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 R ' n  1  1  +  R  '  Rr.-i  n  f ii  where R,., i s the r e s i s t a n c e o f the f i l a m e n t s when i n t h e i r normal s t a t e , f ii Rough e s t i m a t e s of R J . Q o b t a i n e d the f i l a m e n t s I f . Q i s g i v e n  by assuming t h a t the c u r r e n t  c a r r i e d by  by  I . . . = I(V , T , a f t e r cycling) - I(V , T , before cycling) = fil o'o' <o'o' 6  J  &  R  fil  i n d i c a t e R,..., = 3-6ft which means t h a t , w i t h i n e x p e r i m e n t a l e r r o r , R '=R . fil n n I n an attempt to p r e v e n t 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 w h i c h was s u c c e s s f u l i n s o f a r as t h e 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 s e v e r e as t o p r e v e n t  -119i t s b e i n g s u c c e s s f u l l y used as a d e t e c t o r .  T h i s new  of warming the specimens up r a t h e r r a p i d l y  technique consisted  (about 1 1 / 2  h o u r s ) from 4.2  by s l o w l y r a i s i n g them out of the h e l i u m b a t h i n t o a h e l i u m - f i l l e d bag kept a t room t e m p e r a t u r e .  Specimen J-5 was  K  plastic  then i n s t a l l e d i n the  " p u l s e h o l d e r " and the r e v e r s e p r o c e d u r e c a r r i e d o u t .  I n t h i s way the t u r n  around time was reduced by an o r d e r of magnitude and the time o f exposure to " h i g h " temperatures minimized. The mechanism by w h i c h 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 t o note t h a t f o l l o w i n g the i n i t i a l t h e r m a l  c y c l i n g , specimen J-5 was kept a t 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 w h i c h would seem t o e x o n e r a t e the t e s t i n g and o p e r a t i n g p r o c e d u r e s . the  Great c a r e was always t a k e n t o a v o i d  c o n d e n s a t i o n o f w a t e r 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  w h i c h cannot be r u l e d out however on the b a s i s of accumulated e v i d e n c e a r e : (1) S t r a i n i n the j u n c t i o n r e g i o n 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) C o n t a m i n a t i o n by water vapour d u r i n g the re-mounting t i m e , (3) C o n t a m i n a t i o n by vapour from remnants of s o l d e r f l u x l e f t on the "pulse holder" e l e c t r i c a l  terminals.  The o b v i o u s s o l u t i o n t o 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 t o a v o i d t h e r m a l c y c l i n g by b u i l d i n g a specimen h o l d e r i n which s e v e r a l specimens can be t e s t e d a t once and the b e s t ones s e l e c t e d f o r p u l s e measurements w i t h o u t ever b e i n g removed from the h e l i u m b a t h . 5.  M a g n e t i c F i e l d Dependence of Energy  Gap  The o b s e r v e d magnetic f i e l d dependence of t h e s u p e r c o n d u c t i n g energy gap i s shown i a f i g u r e 6-5.  I f the v a l u e of the gap i s chosen by  e x t r a p o l a t i o n t o t h e v o l t a g e a x i s , one  2A(B = 100) 2A(B = 0 )  =  finds  1.05 meV 1.11 meV  =  w h i c h 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 o t h e r workers (Douglass & F a l i c o v , 1964) f o r f i l m s a t t h e s e temperatures i n t h a t t h e i r e m p i r i c a l r e l a t i o n  -120-  F i g u r e 6-5:  M a g n e t i c F i e l d Dependence of Energy  Gap  -121( e q u a t l o n 2-7)  yields  A(H) ' H = 1 A(0) cf  Here, H  1  2 s  1  —  = 382 Oe i s the c r i t i c a l f i e l d  T c a l c u l a t e d from (Meservey & Douglass,  H  fioo'  .93  f o r a t h i n Sn f i l m a t temperature 1964)  (T) = H ( 0 ) ( 1 - (T/T ) ) ( 1 c c 2  ct  =  x — 382  (2X/d)tanh(d/2X))~i  with  T = 1.2 T H  = 3.72  c  c  K K (Critical  (0) = 306 Oe  temperature)  ( B u l k C r i t i c a l f i e l d a t I = 0 K)  X = 500 A ( P e n e t r a t i o n Depth) d = 2000 A ( F i l m T h i c k n e s s ) The magnetic f i e l d used w i t h the d e t e c t o r j u n c t i o n (J-5) was t h a t the d e c r e a s e i n the energy gap due be l e s s than 5% and may B.  o n l y 30 G so  to magnetic b i a s i n g may  be taken to  be s a f e l y n e g l e c t e d i n f u r t h e r a n a l y s i s .  O b s e r v a t i o n o f P u l s e s from A l p h a  Particles  F o l l o w i n g the dc t e s t s and re-mounting procedure d e s c r i b e d i n the p r e v i o u s s e c t i o n , sample J-5 was  c o o l e d t o 1.2  K and the optimum  o p e r a t i n g p o i n t determined as d i s c u s s e d p r e v i o u s l y . 1.  Counting  (see f i g u r e  6-6)  of P u l s e s  W i t h the e l e c t r o n i c s s e t up as i n f i g u r e 5-6,  the bandwidth  h a v i n g been a d j u s t e d f o r e q u a l r i s e and f a l l times of 100 n s e c , 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 p u l s e s were produced by  bombardment o f the j u n c t i o n a r e a i t s e l f was  e v i d e n t from the o b s e r v a t i o n  t h a t the p u l s e s c o u l d be c o n t r o l l e d by c l o s i n g the m e c h a n i c a l i n f i g u r e 4-3)  s h u t t e r (shown  o r by l o w e r i n g the s u b s t r a t e u n t i l the o v e r l a p a r e a  below the l e v e l o f the h e l i u m  bath.  was^just  -122-  I (mA)  ,  ol  0  1 0.2  ;  »  I  0.4  0.6 V(mV)  F i g u r e 6-6:  Detector  Biasing  Conditions  L  L  0.8  1.0  -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 s e t a t 5 times the rms n o i s e o u t p u t from the a m p l i f i e r s (33 mV f o r t h i s b a n d w i d t h ) , was 222 + 3 counts per minute  (cpm).  This  agrees r e a s o n a b l y w e l l w i t h the p a r t i c l e f l u x a t the j u n c t i o n p r e d i c t e d f r o m the s o u r c e s t r e n g t h and j u n c t i o n - s o u r c e geometry f i g u r e 6-7. N., J  If N  illustrated in  = 1.1 x 10^ i s the p a r t i c l e f l u x from the source then  the number of a l p h a 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 !  N. = N A COS(0)/4TT£ 3 o  2  =198.  A reduced number of c o u n t s , about 65 cpm, was o b s e r v e d when the s h u t t e r was i n the " c l o s e d " p o s i t i o n — a r e s u l t w h i c h f o r t u n a t e l y has a ready e x p l a n a t i o n .  also  disturbing  I n o r d e r t o maximize the  p a t h l e n g t h of the a l p h a 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 t o one s i d e of the s u b s t r a t e (see f i g u r e 6-7) so t h a t the s h u t t e r , w h i c h moved i n the p l a n e of the j u n c t i o n , when."closed" or p r e s s e d a g a i n s t the  s o u r c e l e f t about 35% of the source s t i l l exposed t o 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 w h i c h 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.  cpm  Some c o n t a m i n a t i o n  was d e t e c t e d on the s h u t t e r but the number of a l p h a p a r t i c l e s coming from -4 t h i s c o u r s e were a n e g l i g i b l y s m a l l f r a c t i o n , r o u g h l y 5 x 10  , of those  coming from the main s o u r c e . (b) I n f l u e n c e of Veto  System  Chapter 5 c o n t a i n s a d e s c r i p t i o n o f an i n t e r f e r e n c e v e t o i n g system which was d e s i g n e d t o p a r a l y z e 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 e l e c t r o - m a g n e t i c i n t e r f e r e n c e were p r e s e n t t h e r e b y p r e v e n t i n g t h e s e b e i n g counted as m e a n i n g f u l p u l s e s .  Thus, o n l y t h o s e  p u l s e s from the d e t e c t o r a m p l i f y i n g system were"counted whose a m p l i t u d e 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 disc 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 p u l s e . A l l o t h e r p u l s e s , whose a m p l i t u d e exceeded V,. , were counted on another s c a l e r — s e e f i g u r e 5 - 6 — disc ° c  and the count r a t e f o r t h e s e ranged from 13,000 t o 18,000 cpm.  Clearly,  w i t h such a h i g h i n t e r f e r e n c e count l e v e l , the s m a l l count r a t e of 200  cpm  due t o l e g i t i m a t e p u l s e s from the d e t e c t o r i t s e l f would have been u n d e t e c t a b l e w i t h o u t 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° © = 79 + 1° 2  Fig.  6-7:  Fig.  J u n c t i o n - S o u r c e Geometry ( S h u t t e r n o t shown f o r c l a r i t y )  6-8:  Pulses observed  a t optimum  V e r t i c a l - 1.75^iA/div.;  bandwidth; H o r i z . - 200 n s e c / d i v .  ( V e r t i c a l s c a l e c a l i b r a t e d f o r a square p u l s e as d e s c r i b e d  i n s e c t i o n B.2  input  (p.125) .)  -125Even with such a r e l a t i v e l y 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 n e g l i g i b l y small, being less than about 0.5%. 2.  Pulse Characteristics While i t i s interesting and s a t i s f y i n g to observe that the  count rate agrees with the predicted alpha p a r t i c l e f l u x at the junction, the key to understanding the physical processes taking place i n the detector l i e s i n the interpretation of the c h a r a c t e r i s t i c s of the actual pulses. The pulses were recorded photographically on a wide-band oscilloscope (2.4 nsec r i s e time) which was connected to the amplifier and vetoing systems as shown i n figure 5-6. Optimization of the amplifier system bandwidth  (x. ^ andx,.,., "out" on Ortec amplifier) led to the obserlnt diff  vation of pulses (shown i n figure 6-8) whose amplitudes were up to 19 times the rms noise l e v e l .  The upper and lower 3-db points of the amplifier  system at this bandwidth setting were l a t e r measured to be 0.1 MHz and 3.5 MHz respectively which i s consistent with the observed response (100 nsec r i s e 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 s e n s i t i v i t y , the amplifier system was used to measure pulses of known amplitude produced by a pulse generator i n a lumped constant model of the junction small signal equivalent c i r c u i t (see figures 6-9 and 3-5) V. (I  )  o step  V  iS_  o 3  =  -1  9  ES  _  (attenuation) (44ft)  For V = 110 mV and attenuation of 50 db, ( I ) ^ = 8pA which corresponds pg o step to an output voltage of ^ ^. 2.3 V making the amplifier system s e n s i t i v i t y r  =  ou  S = 1V/3.5 pA  (6-1)  The signal to noise r a t i o was a maximum for the operating point shown i n figure 6-6 at which r = (9V/8I)  was maximum.  Reducing r  -126-  (i ) o step  IF Pulse Generator  Attenuator  v  39Q. V.  10fi;  Lumped c o n s t a n t Model o f specimen (C = 3500 pF)  F i g u r e 6-9:  A —  T.  Current  Preamp.  Sensitivity  P u l s e A m p l i t u d e r a t i o as measured from p u l s e photographs  Calibration  'Normalization point  Theory V "\_ ( r = r .) out 1 V Cr = r ) out o J  a.  (\  ai load  10 r = (3V/3I) n T  F i g u r e 6-1Q:  P u l s e Amplitude vs Dynamic R e s i s t a n c e  out  -127by s h i f t i n g the b i a s v o l t a g e above or below 0.3 l e v e l t o i n c r e a s e , as shown i n f i g u r e 6-12,  mV  caused b o t h the n o i s e  and the p u l s e a m p l i t u d e  to  d e c r e a s e as r e c o r d e d by a s e r i e s of photographs taken a t v a r i o u s b i a s voltages.  T h i s v a r i a t i o n o f p u l s e a m p l i t u d e 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  a l o n g 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 e q u i v a l e n t c i r c u i t o f the j u n c t i o n d e p i c t e d i n the i n s e t .  ( c f . F i g u r e 3-5).  If r = (8V/9I)  T  and R = 10.3ft i s the i n p u t  impedance o f 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) i s r 61 1+r =  then  a n d  V  „(r = r . ) out i _ V . (r = r ) out o  R + r  r.  o  1  #  r  *  R + r.  o  1  w h i c h i s c o n s i s t e n t w i t h the measured v a l u e s . 3.  Pulse Height  Spectrum  F i g u r e 6-11  i s a h i s t o g r a m showing the number of  p u l s e s of a g i v e n a m p l i t u d e were observed  p l o t t e d against pulse  As no m u l t i - c h a n n e l p u l s e h e i g h t a n a l y z e r was spectrum was  amplitude.  a v a i l a b l e , t h i s pulse height  o b t a i n e d by examining a s e r i e s of p u l s e photographs  t o t h a t of f i g u r e 6-8)  times  and s i m p l y c o u n t i n g the number o f p u l s e s  (similar falling  w i t h i n the ranges denoted by the s m a l l 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 graticule.  ( A l l photographs so a n a l y z e d were t a k e n 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 b i a s e d at the optimum o p e r a t i n g p o i n t . ) sharp c u t - o f f a t the low a m p l i t u d e triggering  The  end of the spectrum denotes the p u l s e  level. At f i r s t g l a n c e , t h i s wide range i n p u l s e a m p l i t u d e s  is  s u r p r i s i n g f o r the a l p h a 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 a r e e s s e n t i a l l y monoenergetic (5.10-5.15 MeV).  I t t u r n s o u t , as d i s c u s s e d i n more d e t a i l  i n Chapter 7, t h a t the a m p l i t u d e v a r i a t i o n may  be a t t r i b u t e d to two  (1) the energy l o s t by the p a r t i c l e s i n p a s s i n g through from 0.5  t o 0.14  MeV  causes:  the f i l m s v a r i e d  depending on the angle o f i n c i d e n c e ( f i g u r e 6-7);  (2)  the j u n c t i o n i s t h e r m a l l y c o u p l e d t o the s u b s t r a t e and some of the energy expended i n the s u b s t r a t e 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 t o  -128-  w  + w 00  c u  20  CO  01 CO I—I  3  p-l  o z  10  j  - 1  P u l s e A m p l i t u d e (E) ( O s c i l l o g r a p h  F i g u r e 6-11:  P u l s e Height  divisions)  Spectrum  -129t h e j u n c t i o n i n s u f f i c i e n t time t o c o n t r i b u t e to the p u l s e . C.  Noise 1.  Observed N o i s e The n o i s e output from t h e d e t e c t o r - a m p l i f i e r system  measured w i t h a K e i t h l e y Model 120 wideband (10 Hz-100 MHz)  was  voltmeter—  c a l i b r a t e d t o r e a d the rms v a l u e of a t r u e s i n e w a v e — c o n n e c t e d as shown i n f i g u r e 5-6.  ( S t r i c t l y s p e a k i n g , because the meter i s c a l i b r a t e d to r e a d  t h e rms o f s i n u s o i d a l , not random, s i g n a l s , n o i s e v o l t a g e s i n d i c a t e d by the meter s h o u l d be m u l t i p l i e d by 1.13 1958).)  Of s p e c i a l i n t e r e s t was  to o b t a i n t h e i r t r u e rms v a l u e  (Partridge,  the observed v a r i a t i o n o f n o i s e 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 d a t a w i l l  be  c o n s i d e r e d a g a i n i n S e c t i o n D f o r purposes o f e s t i m a t i n g the j u n c t i o n capacitance^ N o i s e r e a d i n g s were a l s o taken w i t h a number o f c a p a c i t o r s (C  ) connected  i n p a r a l l e l to t h e j u n c t i o n .  As i t was not p o s s i b l e  d u r i n g t h e c o u r s e o f t h e r u n to p l a c e them 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 , t h e y were connected  a t room temperature  i n the " f i l t e r b o x " — s e e f i g u r e  5-3.  U n f o r t u n a t e l y , o n l y t h e nominal v a l u e s o f t h e s e c a p a c i t o r s a r e known as t h e 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 b e f o r e they c o u l d be calibrated.  Other c a p a c i t o r s o f s i m i l a r manufacture were, by measurement,  found t o be o f t y p i c a l l y ± 20% p r e c i s i o n a t 1 kHz but s e l f - r e s o n a t i n g a t about 9 MHz.  T h i s means t h a t i n the a m p l i f i e r bandpass (0.1 t o 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 p r o b a b l y e x h i b i t a r e a c t a n c e c o r r e s p o n d i n g t o much l e s s than t h e nominal c a p a c i t a n c e which makes t h e n o i s e measurements t a k e n as a f u n c t i o n of C  of l i t t l e v a l u e o t h e r t h a n a ext  c o n s i s t e n c y check (see f i g u r e 6-13). 2.  O r i g i n of N o i s e T h i s s e c t i o n w i l l show t h a t the dominant source o f n o i s e i n  t h e experiment was  the common base p r e a m p l i f i e r and not the  superconducting  tunnel junction p a r t i c l e detector. (a) R e p r e s e n t a t i o n o f D e t e c t o r as N o i s e  Generator  The e q u i v a l e n t c i r c u i t i n c l u d i n g n o i s e g e n e r a t o r s f o r t h e d e t e c t o r as b i a s e d and connected 5-3 and 5-6)  to the a m p l i f y i n g system  i s g i v e n i n f i g u r e 6-14(a) where  (see f i g u r e s  -130140 \  130  \  \ X \ \ \  120 E  'I a) w  110  •H  O  c  _L  100'  r = (8V/3I) Figurp  6-17:  T  ft  Output N o i s e vs j u n c t i o n Dynamic R e s i s t a n c e  300  X  200  CO  100  0)  w  •H  O C  ±  .01 Figure  6-13:  ,02  .03 C ( E x t ) uF(Nominal)  .04  Output N o i s e v s E x t e r n a l C a p a c i t a n c e  .05  -131-  T  = 1.2 K  T 1  I  ©  Kt)  "b  r = (3V/Sl) ,  = 295 K 1  2  ©K GX  To Preamp.  = Bias Resistance, I(t) = I  T  q  |  exp^-t/x) = s i g n a l current  i i 1  F i g u r e 6-14 ( a ) :  Detector  Equivaient c i r c u i t with Noise Generators  © .2 1  1 2  . 2 , . 2 \ B  =  +  v,  X  F i g u r e 6-14 ( b ) :  r  , e  input  r  e  v  =  .2  l  r  2  D e t e c t o r n o i s e Generator e q u i v a l e n t  i  e  b'  Lg>J  , = e m i t t e r r e s i s t a n c e , r ^ , = base r e s i s t a n c e , r , = c o l l e c t o r resistance c  e  Circuits  F i g u r e 6-14 ( c ) :  CB P r e a m p l i f i e r w i t h Noise  -1322 i^ i,  2 D  a  2 Johnson n o i s e i n b i a s r e s i s t a n c e (R^)=I ^=4kTB/R^ R  ~~2 = shot n o i s e on b i a s c u r r e n t ( I ) = I = 2eIB B  w i t h T t h e room temperature (295 K ) , k Boltzmann's c o n s t a n t and B t h e system bandwidth.  In practice,  = 20 Kft>>r, hence t h e e q u i v a l e n t  circuit  may be reduced t o t h e forms shown i n f i g u r e 6-14(b), where, b e i n g u n c o r r e l a t e d , the n o i s e c u r r e n t s add q u a d r a t i c a l l y .  v  2 d  = r  2  4kTB  r  = 4kTBr  Thus, over t h e bandwidth B,  + 2eIB  2elr 4kT  (6-2)  4kTBr (M)  70 uA w h i c h , a l o n g w i t h t h e o t h e r v a l u e s  At the o p e r a t i n g p o i n t I -2 makes M - 1.35 x 10  cited,  (b) E q u i v a l e n t 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 N o i s e The  "low f r e q u e n c y "  equivalent c i r c u i t for a transistor  o p e r a t i n g i n common base (CB) mode w i t h w h i t e n o i s e g e n e r a t o r s W o l l and H e r s c h e r (1962) i s shown i n f i g u r e 6-14(c). noise generators  as g i v e n by  I n t h i s case, t h e  are:  2 V  T  2 e  2 e  "  2 e I  E  B r  e'  kT el.  2kTBr' e  (6-3)  V  4kTBr' b  and v 2 where I e  2  c  = 2 e r , a ( l - a ) r B = 2 k T a ( l - a ) r B/r , E c c e 2  2  a i s t h e shot n o i s e on t h e e m i t t e r c u r r e n t I„ and H„„= ^ = 40 E FE 1-a  i s t h e measured common e m i t t e r f o r w a r d c u r r e n t g a i n . r e p r e s e n t a t i o n i s used as t h e upper frequency  (The "low f r e q u e n c y "  o f t h e a m p l i f y i n g system  -133bandpass i s about 3.5 MHz whereas t h e upper c u t o f f f r e q u e n c y  f ^ f o r the  16G2 t r a n s i s t o r used i n t h e p r e a m p l i f i e r i s quoted by t h e m a n u f a c t u r e r as 1.1 GHz.) (c) C a l c u l a t i o n o f N o i s e  Figure  To a p p r e c i a t e t h e r e l a t i v e importance o f t h e d e t e c t o r and p r e a m p l i f i e r as n o i s e s o u r c e s , i t i s c o n v e n i e n t  to define the f o l l o w i n g  noise figure  F =  t o t a l output n o i s e power output n o i s e power due t o n o i s e o f d e t e c t o r From t h e c i r c u i t f o r t h e p r e a m p l i f i e r and d e t e c t o r shown  i n f i g u r e 6-15 i t i s e v i d e n t t h a t t h e n o i s e power i s p r o p o r t i o n a l t o t h e 2 mean square v o l t a g e e simplicity  appearing  Q  a c r o s s R^.  T a k i n g t h e open loop case f o r  ( i e . R^ = °°), one f i n d s ; i  d  =  e'  b'  ., (6-4) I S  r , + r .)  (X +  e  v  and  •• - • e =-v, + v + i ( r , . + a r ) o b c e b' c  Substituting i  i n t o e^, c o l l e c t i n g terms and t a k i n g t h e mean square v a l u e  of both s i d e s y i e l d s  2  a  e =  2 2 71 , ~1 , ~1 . . ~2 .••• . c d b e' c < .- e ' ' b ^  r  ( v  +  V  +  V  }+  V  r  + r  .2  +  —  o  ( r  +  r  e'  +  r  b'  (6-5)  }  where i t has been assumed t h a t t h e n o i s e g e n e r a t o r s a r e u n c o r r e l a t e d , a r >>r, , and a r >>r . + r . c b c e' The o u t p u t n o i s e power due t o t h e n o i s e o f t h e d e t e c t o r 2 ~2 "2 i s found by s e t t i n g >j» « - 0 i n 6-5 w h i c h g i v e s —  v  =  v  =  e  v  c  ~2 , ~2 b ' e» F = 1 + — — V  +  *T v, d  V  V  +  ~2 . . c e' ( r  +  r  +  2 2 2 a r v, c d  r  , b' }  ,2  -134-  oii  e  F i g u r e 6-15:  E q u i v a l e n t c i r c u i t f o r D e t e c t o r and A m p l i f i e r w i t h n o i s e  ai  v = v  F i g u r e 6-16:  8  or V  s  E q u i v a l e n t c i r c u i t f o r S i g n a l to N o i s e R a t i o  estimate  -135U s i n g e q u a t i o n s 6-2  and 6-3  makes .2 (r + r , + r, , ) ' e* b 1  F  "  1  V  rM  =  +  ^e'  2 r  The  (6-6)  +  t r a n s i s t o r b i a s c u r r e n t ( I ) was £  e'  10 mA,  H  FE  so t h a t by 6-3  r  g  l  = 2.6 ft.  A n g e l l (1967) g i v e s the i n p u t impedance of a CB t r a n s i s t o r as R  i n  = r  g  l  + r , ( l - a ) making r ^ , = 216 ft u s i n g the measured (see c h a p t e r b  5)  i n p u t impedance o f a p p r o x i m a t e l y 8 ft. P u t t i n g t h e s e v a l u e s i n t o e q u a t i o n  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 noise. (d) E q u i v a l e n t Input N o i s e I t i s convenient  ( f o r C i r c u i t Used) to r e p r e s e n t the p r e a m p l i f i e r n o i s e  g e n e r a t o r s by an e q u i v a l e n t v o l t a g e g e n e r a t o r  (v ) a c r o s s t h e j u n c t i o n 8 2 2 (see f i g u r e 6-14(b) and 6-15). Thus F = v /v, o r g d  dynamic r e s i s t a n c e . v  = (4kTBrMF)* = 5.2  yV(rms),  r = 9.3ft  (e) S i g n a l to N o i s e R a t i o E s t i m a t e From the p u l s e photograph ( f i g u r e 6-8), t h e maximum s i g n a l i s seen t o c o r r e s p o n d  t o about 8.1 uA which can be r e p r e s e n t e d  a voltage generator V  uA • r = 75 uV a c r o s s the j u n c t i o n dynamic  resistance.  = 8.1  and V g s a r e thus l o c a t e d a t the same p o i 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 e q u i v a l e n t c i r c u i t so t h a t the q u a s i - t h e o r e t i c a l s i g n a l to n o i s e r a t i o (S/N) i s s i m p l y S/N  = V /v s  g  (see f i g u r e s 6-14(b) and 6-16).  = 14.5  The  by  two g e n e r a t o r s v  i n r e a s o n a b l e agreement w i t h the measured v a l u e of  19.  ( f ) N o i s e w i t h Lumped Constant Models of Specimens To c o n f i r m e x p e r i m e n t a l l y t h a t the observed  output  n o i s e depends e s s e n t i a l l y o n l y on the source impedance seen by the p r e a m p l i f i e r and not on n o i s e s o u r c e s 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 c o n s t a n t model of a t u n n e l 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 g a i n s unchanged from measurements w i t h the a c t u a l specimen, was junction.  indeed e q u a l to t h a t o b t a i n e d w i t h the o p e r a t i n g  To some e x t e n t , 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 n o i s e was  observed  (see f i g u r e 6-12,13) to v a r y w i t h j u n c t i o n  -136r e s i s t a n c e and p a r a l l e l c a p a c i t a n c e and y e t the lumped c o n s t a n t components of 9.6ft and 3000 pFs. were o n l y a p p r o x i m a t i o n s  to the a c t u a l v a l u e s .  N o n e t h e l e s s , because the absence o f the d e t e c t o r has not  significantly  a l t e r e d the n o i s e , i t i s r e a s o n a b l e to c o n c l u d e t h a t the p r e a m p l i f i e r was f a c t the dominant n o i s e D.  in  source.  D e t e r m i n a t i o n of J u n c t i o n C a p a c i t a n c e For m a t h e m a t i c a l  s i m p l i c i t y the presence o f C j was  t h e n o i s e i n v e s t i g a t i o n s of the p r e v i o u s s e c t i o n .  ignored i n  The experiments  mentioned  above i n w h i c h the n o i s e 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 t h a t 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 t h e a m p l i f i e r b e i n g t h e main n o i s e source) a r r i v e d a t t h e o r e t i c a l l y w i t h Cj = 0. I n o r d e r t o proceed w i t h the a n a l y s i s of the r e s u l t s out i n c h a p t e r 7, however, a r e a s o n a b l y c l o s e e s t i m a t e o f C The purpose o f 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 r e q u i r e m e n t ,  carried  i s required. first  by  c a l c u l a t i n g C j 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 o f t h e n o i s e measurements t a k i n g the e f f e c t of Cj  i n t o account.  out t h a t the range of v a l u e s found by the two approaches a r e  I t turns  reasonably  compatible. 1.  P a r a l l e l P l a t e Model A rough e s t i m a t e of the j u n c t i o n c a p a c i t a n c e may  be  o b t a i n e d by r e g a r d i n g the d e v i c e as a s i m p l e p a r a l l e l p l a t e c a p a c i t o r h a v i n g 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 S n 0 . 2  (a) D i e l e c t r i c Constant Van D a a l d i e l e c t r i c constant  of  SnO^  (1968) g i v e s the s t a t i c o r zero  (K = £/e ) Q  °f b u l k SnO^  depending on the c r y s t a l o r i e n t a t i o n .  frequency  as l y i n g between 9 and  14  (Other w o r k e r s , eg. Coon and F i s k e  (1965), have used K = 5 w i t h o u t 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  t o 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  h e r e ) quoted by some a u t h o r s , eg. Summitt and B o r r e l l i (1960), as b e i n g a p p r o x i m a t e l y 4.)  (1965) and  I t i s not r e a d i l y apparent  d i e l e c t r i c c o n s t a n t f o r b u l k m a t e r i a l may  Arai  t h a t the  be m e a n i n g f u l l y c a r r i e d over to  v e r y t h i n l a y e r s — a s i n a t u n n e l j u n c t i o n — a l t h o u g h t h e r e 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 The  Thickness t h i c k n e s s o f the i n s u l a t i n g l a y e r i s e s t i m a t e d  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 t h i c k n e s s ( S ) from f i g u r e 2-3, i t i s n e c e s s a r y t o know, i n a d d i t i o n to K, the mean b a r r i e r h e i g h t of the i n s u l a t o r d> and the -4 2 ° T  normalized lap  resistance a =  R n  * A where A = 7 x 10  cm  i s the j u n c t i o n o v e r -  area. The magnitude o f the f o r b i d d e n energy band i n Sn02  (E  ) i s g i v e n by A r a i (1960) and Summitt e t a l (1964) as 4 eV. Assuming gap t h a t the F e r m i s u r f a c e l i e s w i t h i n the band gap, <j> must be i n t h e range Q  0 < d> < E .A o gap  p r e c i s e v a l u e o f a) cannot be s p e c i f i e d because the 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 o f the F e r m i s u r f a c e i n the band gap T a k i n g o = 5.4  uncertain.  -3 2 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 o f 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 o f  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 c o n f u s e d w i t h some average t h i c k n e s s <S> A = ^ S ( x , y ) dx dy where S ( x , y ) i s the a c t u a l i n s u l a t o r thickness at  x,y.)  Chow (1963) and Hurych (1966) p o i n t o u t , however, t h a t the e f f e c t i v e t h i c k n e s s ( S ) t o be used f o r c a p a c i t a n c e c a l c u l a t i o n s i s c  1.5  t o 3 times g r e a t e r than S . T  T h i s 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  t h a t the t h i c k n e s s o f the i n s u l a t i n g l a y e r i s not c o n s t a n t s t o c h a s t i c a l l y from p o i n t t o p o i n t . a f f e c t the t u n n e l c u r r e n t — b e c a u s e  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 of  t h i c k n e s s — b u t o n l y weakly a f f e c t the For 9 $ K ^ 1 4  but v a r i e s  i t s e x p o n e n t i a l dependence upon the capacitance.  amd  10 $ S  $ 42 A, the c a l c u l a t e d v a l u e c, o f C j i s i n the range 1300 S Cj £ 8OO0 pF. S i n c e ^ < 1 eV i m p l i e s S > 42 A and C < 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 T  !  0 < Cj $ 8i000 pF  -138A more d i r e c t and more p r e c i s e e s t i m a t e was measurements as d e s c r i b e d i n the next 2.  o b t a i n e d from the n o i s e  section.  N o i s e Measurements In  s e c t i o n C o f t h i s c h a p t e r , the n o i s e output from the  j u n c t i o n - a m p l i f i e r system (V ) was  d i s c u s s e d a t some l e n 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 i n p u t l o o p e x e m p l i f i e d by the observed v a r i a t i o n of n o i s e 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 ) for  unknown c o n s t a n t j u n c t i o n c a p a c i t a n c e Cj  w i l l be denoted as ^ ( r , C j ^ , Meas.).  (actual) = C j ; this quantity a  From t h e s m a l l s i g n a l e q u i v a l e n t  c i r c u i t o f the d e t e c t o r - a m p l i f i e r system, the output n o i s e may c u l a t e d as a f u n c t i o n o f r and C^; V ( r , C , Thy.). XI  be  cal-  t h i s q u a n t i t y w i l l be denoted as  The purpose of t h i s s e c t i o n i s to e s t i m a t e the j u n c t i o n  J  c a p a c i t a n c e by d e t e r m i n i n g the range of C  f o r which V ( r , C , Thy.)  c o n s i s t e n t w i t h the e x p e r i m e n t a l l y measured n o i s e  v  ( > j » Meas.). r  n  G  a  The r e a d e r u n i n t e r e s t e d i n the f o l l o w i n g d e t a i l s may  is  mathematical  w i s h t o s k i p on to s e c t i o n E where the a c c e p t a b l e range o f  C.  v a l u e s o b t a i n e d from the n o i s e measurements, a l o n g w i t h the o t h e r p r i n c i p a l r e s u l t s f r o m t h i s c h a p t e r , are summarized. (a) C a l c u l a t i o n of V ^ r , C j , F i g u r e 6-17  Thy.)  shows the e q u i v a l e n t c i r c u i t o f the  t u n n e l 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 6-14  and 6-16)  ( e q u a t i o n 6-3)  w i t h the p r e a m p l i f i e r n o i s e g e n e r a t o r s v , , v^, and e  i n place.  6-15)  v , c  I n keeping w i t h the c o n c l u s i o n s o f s e c t i o n C,  t h e j u n c t i o n n o i s e g e n e r a t o r s have been n e g l e c t e d . i t may  Assuming r >>R^+r^,, c  be shown t h a t the mean square n o i s e v o l t a g e a c r o s s  (see f i g u r e  i s g i v e n by  R  2 e  °  =  2  a .4kTB 2  (r+r ,+r ,(l-a)) e  [  (r r , r ,) +  e  +  b  2  b  To extend t h i s r e s u l t t o the c i r c u i t of f i g u r e 6-17, e q u a t i o n 5-4  3-5,  FE e' r  one may  recall  and r e p l a c e r by the t r a n s m i s s i o n l i n e - d e t e c t o r impedance  Z = Z ( r , C ) = R + j X where  -139-  l.OtlF.  I  I  F i g u r e 6-17: •  L_  I  Junction Transmission Line  l.OuF  Filter Box  Preamplifier  E q u i v a l e n t 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 w i t h N o i s e Sources o n l y (no a l p h a p u l s e s ) -  -140-  1 + (corC )  R  + R (1.2 K,co).  Z  %  -r^uCjr) X =  j 1 + (torCj)  + juL;  S i n c e 1 Kft > 470ft >>  L = 1.63 uH.  r  g  I  + r , ( l - a ) , the f i l t e r b  box-CB t r a n s i s t o r c o m b i n a t i o n may be r e p l a c e d by a s e r i e s impedance Z.' in  =  R! + in  jX'. where ( c f . f i g u r e 5-5 and e q u a t i o n 5-5) in n  R' . = 7.9ft = r , + (1 - c t ) r m e' b' u l  X  'in  =  j ( a ) L  "  1 / a ) C )  w i t h L = 0.23 uH. and C = 0.5 uF.  ^  =  " c X  (Noise g e n e r a t o r s c o r r e s p o n d i n g t o t h e  Johnson n o i s e i n t h e p a r a l l e l r e s i s t o r s R  c  and R_ i n f i g u r e 6-17 a r e i g n o r e d  t  D  as t h e i r c o n t r i b u t i o n t o t h e n e t output n o i s e 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 d e t e c t o r impedances Z o f i n t e r e s t , r^,>>r ,+|Z|, e  w h i c h means t h a t over a bandwidth d f  e  o  d f  = 4  °  2  *  k T d f  <V  * e»  +  r  +  r  b? FE%»> / H  ( Z  +  Z  in  ) _ 2  (6-7) K d f (Z + Z! ) in  -  2  , Izl «  r, , b  The magnitude o f t h e t o t a l n o i s e v o l t a g e f o r p a r t i c u l a r v a l u e s ( r ' , C j ' ) o f t h e j u n c t i o n e q u i v a l e n t c i r c u i t parameters i s t h e r e f o r e  -141-  e  df = K  o  d f / | Z ( r ' , C.')  + Z!. ||2  xn.  «J  where the l i m i t s of i n t e g r a t i o n a r e 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 . = 3.5 MHz. MHz. Now 1 2 may form the n o i s e v o l t a g e r a t i o  V (r, C , n J T  2  (r, C ,  V n  Y(Thy.) = V  N  Thy.)  Thy.)  °=  e (r, C ) o J T  one  C  (6-8)  Thy.)  W i t h C j ' = C j and r ' = 9.3P., y(Thy.) was  e v a l u a t e d by Simpsons' r u l e  i n t e g r a t i o n f o r an a p p r o p r i a t e range o f v a l u e s f o r r , C j thereby the f a m i l y o f c u r v e s shown i n f i g u r e  so t h a t  ;;•<*. J>  e  J  ( r \ Cj',  y  generating  6-18.  (b) Computation o f ^ ( r , C j , Meas.) From e q u a t i o n 6-7  i t may  be seen t h a t the t h e o r e t i c a l  output n o i s e v o l t a g e depends upon the p r e a m p l i f i e r i n p u t l o o p impedance Z + Z.' . The measured o u t p u t n o i s e V (Meas.), however, i n c l u d e s c o n t r i b u m n t i o n s f r o m the p o s t a m p l i f i e r s whose n o i s e i s i n s e n s i t i v e t o Z + Z.' so t h a t in  i t i s convenient  to w r i t e V (Meas.) = [ V ( Z + Z.') n m  + V ] o  2  Here V(Z + Z ^ ) Z +  and V  q  2  (6-9)  2  i s the component o f output n o i s e v o l t a g e depending on i s the "open i n p u t l o o p " output n o i s e v o l t a g e .  The  quantity  of i n t e r e s t f o r comparison t o t h e o r y i s t h e r e f o r e  V  where, by experiment,  n  (r, C  , Meas.) = [ V (Meas.) - V ] n o 2  Ja  T  For comparison to y(T\iy.)  V ' = 36 mV. V  y(Meas.) = V  n  n  (r, C  Ja  T  2  t  (6-10)  the r a t i o  , Meas.) ?  (9.3, C , Meas.) ' Ja'  i s computed from the d a t a o f f i g u r e 6-12 f i g u r e 6-18.  2  T  and e q u a t i o n 6-9  and p l o t t e d i n  (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 a r e a measure  -142-  F i g u r e 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 N o i s e Output vs Dynamic R e s i s t a n c e of J u n c t i o n  -143of the experimental u n c e r t a i n t y . ) (c) " B e s t " v a l u e o f C j I n s p e c t i o n o f f i g u r e 6-18 shows r e a s o n a b l e agreement o f t h e o r y t o experiment  f o r a j u n c t i o n capacitance  1500 $ C j $ 4500 pF  w h i c h l i e s i n s i d e the range o f v a l u e s p r e d i c t e d from t h e p a r a l l e l p l a t e model i n s e c t i o n 1 o f t h i s  chapter.  (d) C o n s i s t e n c y Check W i t h i n e x p e r i m e n t a l e r r o r , t h i s v a l u e o f the j u n c t i o n c a p a c i t a n c e agrees w i t h t h a t o b t a i n e d g r a p h i c a l l y f r o m s e t t i n g  (r = 0 , V  C j = 0, Thy.)  V  ( r = 9.3, C , Thy.)  ( t r a n s m i s s i o n l i n e s h o r t e d , Meas.) ^ V  T  XI  r  J  (r =9.3, XI  C  T  ^  , Meas.)  J3.  The l e f t hand s i d e o f 6-11 was computed s i m i l a r l y t o e q u a t i o n 6-8; the r i g h t hand s i d e i s t h e r a t i o o f t h e o u t p u t n o i s e measured w i t h the t r a n s m i s s i o n l i n e t e r m i n a t e d w i t h a dc s h o r t ( a t 1.2 K) t o t h e n o i s e 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 T h i s c h a p t e r has d e s c r i b e d i n some d e t a i l t h e r e s u l t s o b t a i n e d  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 a r e 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 t h e o r y  and t h e r e s u l t s of o t h e r workers. r = (9V/3I)  T  From these c h a r a c t e r i s t i c s , t h e maximum  was found t o be 9.3ft. (2)  W i t h t h e j u n c t i o n b i a s e d a t i t s optimum o p e r a t i n g p o i n t ,  w h i c h 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 , t o c o r r e s p o n d  t o those v a l u e s  of I , V and B f o r w h i c h r was a maximum and the s u p e r c u r r e n t was minimum, p u l s e s were d e t e c t e d whose count r a t e agreed w i t h t h a t p r e d i c t e d from the s o u r c e 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 t h a t the p u l s e s c o u l d be t u r n e d 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 t h e source and j u n c t i o n , l e d t o the c o n c l u s i o n t h a t the  -Imp u l s e s were produced by a l p h a p a r t i c l e bombardment o f the j u n c t i o n i t s e l f . (3) (x  The p u l s e s o b s e r v e d ,  a t optimum a m p l i f i e r bandwidth  . = 100 n s e c , x = 1.7 p s e c ) , had a m p l i t u d e s rise fall r  r  up t o 19 times the r  p r e a m p l i f i e r - d o m i n a t e d rms n o i s e l e v e l o f r o u g h l y 100 keV r e f e r r e d t o t h e input. (4)  From n o i s e measurements, the j u n c t i o n c a p a c i t a n c e C j was  found t o l i e i n t h e range 1500 S Cj $ 4500 pF  w h i c h i s c o n s i s t e n t w i t h t h e range of v a l u e s 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 t o the r e a d e r b u t o f major  t o the e x p e r i m e n t e r  was the observed  d e t e r i o r a t e upon t h e r m a l  cycling.  concern  tendency o f t u n n e l i n g j u n c t i o n s t o  -145-  CHAPTER 7  ANALYSIS OF RESULTS A.  Introduction As d i s c u s s e d i n Chapter 1, the q u a n t i t y of most i n t e r e s t t o  be o b t a i n e d from the p r e s e n t experiment i s w — t h e average energy l o s s by a charged p a r t i c l e t o 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 c o r r e s p o n d i n g q u a n t i t y i n o t h e r t y p e s of d e t e c t o r  and, e v e n t u a l l y , f o r  comparison w i t h t h e o r y when i t i s s u f f i c i e n t l y w e l l d e v e l o p e d .  To determine  w from the measured p u l s e response of the s u p e r c o n d u c t i n g t u n n e l j u n c t i o n , i t i s n e c e s s a r y t o know the form of the c u r r e n t p u l s e i ( t ) w h i c h i s s u p e r imposed on the t h e r m a l e q u i l i b r i u m t u n n e l i n g c u r r e n t f o l l o w i n g each a l p h a 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  c r e a t e d by the a l p h a p a r t i c l e h a v i n g l o s t energy AE i n the j u n c t i o n may estimated, giving w =  AE/N  Q  be  .  I n t h i s e x p e r i m e n t , 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 t o d e t e r m i n e d i r e c t l y the form of i ( t ) .  T h e r e f o r e , i t i s assumed t h a t the  c u r r e n t p u l s e s have t h e form ( e q u a t i o n 3-11) where the peak a m p l i t u d e i  i(t) = i  e x p ( - t / t ) , ( t £ 0)  i s p r o p o r t i o n a l t o the magnitude o f the excess  d e n s i t y of q u a s i p a r t i c l e s g e n e r a t e d by the passage of the a l p h a 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 d u r i n g which t h i s excess d e n s i t y decays to zero.  A d m i t t e d l y 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 , b e i n g 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 y e t s u f f i c i e n t l y developed t o 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 t h e o r y i s b e i n g c a r r i e d out by Mr. George May of t h i s  laboratory.)  The purpose of t h i s c h a p t e r i s t o e s t i m a t e w by deducing the parameters x and i  from the a m p l i f i e r o u t p u t p u l s e d a t a .  In order to  deduce t h e s e 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 for  the e f f e c t s of n o i s e i n the system must be  ( s e c t i o n s C and  allowed  D).  H a v i n g determined x and of charge w h i c h a c t u a l l y t u n n e l e d .  i ,  one may  Q  e a s i l y f i n d the t o t a l amount  An e s t i m a t e of AE  then makes i t p o s s i b l e  to g i v e ( s e c t i o n E) a f i g u r e f o r w ( e x p ) , the average energy l o s s tunneling  q u a s i p a r t i c l e , and w the average energy l o s s per  pair excited.  A discussion  Derivation  quasiparticle  ( s e c t i o n F) of problems p e r t a i n i n g t o a  c a l c u l a t i o n of AE c o n c l u d e s the B.  of 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  i s the system t r a n s f e r f u n c t i o n H(s)  H(s)  where v ^ ( t ) and  v Q  Transfer Function  d e r i v e H(s)  1.  o  eg.  as  ±  output s i g n a l s r e s p e c t i v e l y ,  t i s the t i m e .  The  f r o m the known c h a r a c t e r i s t i c s of the  l i n e - a m p l i f i e r system and  defined  (see  = v (t)/v (t)  ( t ) are the i n p u t and  i s the complex f r e q u e n c y and  rigorous  chapter.  A fundamental concept of l i n e a r systems a n a l y s i s Bohn, 1963)  per  s=jui  aim of t h i s s e c t i o n i s to detector—transmission  f i n d V ( t ) u s i n g an assumed form of v ^ ( t ) . Q  Small Signal Equivalent C i r c u i t F i g u r e 7-1 (a) i s a schematic of the p u l s e  network and  amplifying  f i g u r e 7-1 (b) i s the corresponding' s m a l l s i g n a l e q u i v a l e n t  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 i n p u t v o l t a g e s i g n a l o b t a i n e d from the assumed form of i ( t ) and P-P  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  i n f i g u r e 7 - 1 ( a ) ; thus v . ( t ) = i ( t ) Z, = i 1 d o  Incorporating  exp(-t/x)  r —;—: — 1 + JtorCj  the r e s u l t s of c h a p t e r 5,  the e f f e c t i v e i n p u t impedance of the " c o l d " t r a n s m i s s i o n preamplifier resistance  may  (equation  (7-1)  5-3),  line-filter  box-  be r e p r e s e n t e d a p p r o x i m a t e l y as a s e r i e s i n d u c t a n c e L  R where L = 1.69  uH and  R = 10.3ft (From f i g u r e 5-9,  t o range from 9.5ft t o 11.6ft o v e r the system bandwidth b u t ,  R = R^  c  and i s seen  i n o r d e r 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 v a l u e of R=10.3ft  -148has 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 o f t h i s p a r t  of the network i s then  Y(s) - i ( t ) / v . ( t )  (7-2)  e  The CB t r a n s i s t o r p r e a m p l i f i e r t r a n s f o r m s the i n p u t signal current i ( t )  a t  l o w  e  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 t h a t  v ' ( t ) = «i (t)R e  where R' i s t a k e n t o be f r e q u e n c y  L  = i (t)R'  (7-3)  e  independent.  I t was p o i n t e d out i n c h a p t e r 6 t h a t the optimum a m p l i f i e r system bandwidth f o r p u l s e d e t e c t i o n corresponded t o a 10-90% r i s e time o f 100 nsec.  To t a k e t h i s time c o n s t a n t i n t o a c c o u n t , an i n t e g r a t i n g  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 . o o  a  circuit  For t h i s p o r t i o n o f t h e  circuit  W(s) = v " ( t ) / v ' ( t )  (7-4)  and, r e p r e s e n t i n g the g a i n of t h e a m p l i f y i n g system by v (t) = G v"(t)  G,  Q  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 e q u a t i o n s 7-2  to  (7-5)  thus be o b t a i n e d from  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 L a p l a c e t r a n s f o r m of v ( t ) w r i t t e n V(s) =  jT[v ( t ) ]  v(t) exp(-st)dt  = o  -149Thus,  V ( s ) = GR'V, ( s ) W ( s ) Y (s) o i and t h e d e s i r e d q u a n t i t y V ( t ) may be found by t a k i n g t h e i n v e r s e t r a n s f o r m Q  "^"V  v(t) =  A l l t h a t remains now i s t o determine 2.  L a p l a c e Transform  (s)]'  V ^ ( s ) , W(s) and Y ( s ) from f i g u r e  7-1(b),  Representation  (a) V . ( s ) W r i t i n g t h e impedances i n terms o f t h e i r  transforms  and a p p l y i n g t h e d e f i n i t i o n of V ( s ) t o e q u a t i o n 7-1 y i e l d s  v  where a = x \  i  ( s ) v  '  J»  Cj  .  JL  s+a  . JL  (7-7)  s+d  d = (rC.) 3  (b) W(s) I n s p e c t i o n o f t h e i n t e g r a t i n g c i r c u i t shows  W  w i t h b = (x ) a  _  1  < > " CT o S  '  R  o  A/C  s o  •  i+b  = (R C o o  (c) Y ( s ) Somewhat more c o m p l i c a t e d a l g e b r a i c a l l y than t h e o t h e r terms, Y ( s ) i s g i v e n by  ( 7  '  8 )  -150-  Y ( s ) = [R + Ls + r / ( l +  srCj)]"  1  (7-9) = (s + d ) / L ( s + a s + a ) 1 o 2  n  where a  1  = (R + L d ) / L , a  = (RC.d + 1 ) / L C . j J  o  T  Combining e q u a t i o n s 7-7 to 7-9 g i v e s  GR'bi V (S) = q  C L J T  Inverse  2. • -=— • s+a s+b  (7-10)  1 2  s  + a, s + a 1 o  Transform To o b t a i n t h e i n v e r s e t r a n s f o r m o f 7-10 i t i s c o n v e n i e n t  to set GR'bi /CjL o  = c o n s t a n t = A, wl where A i s independent o f t i m e , and t o  f a c t o r t h e q u a d r a t i c term such t h a t  3  (s+a) V  o  (s+b)  1  4  o  a  2  <  OR  a  (s+b)  *  (s+a )(s+a ) »  (s)  (s+a)  1  2 (s+ct)  2 ' + y  J « JO>  C  C  l  ( O R  * C  4  a  o  J >  So)  Both forms o f V ( s ) must be c o n s i d e r e d as C , the v a l u e of C which o JU J s a t i s f i e s a„ = 4 a , i s about 3300 pF w h i c h l i e s r i g h t i n t h e m i d d l e o f 1 o rn  t h e a c c e p t e d range of v a l u e s f o r Cj as determined i n c h a p t e r 6.  T  from n o i s e measurements  The i n v e r s e t r a n s f o r m s , o b t a i n e d from s t a n d a r d  Laplace  -151T r a n s f o r m T a b l e s [eg. E r d e l y i (1954) and McCollum and Brown  (1965)],  are as f o l l o w s :  o ^  V  _ _  exp (-at)  A  exp(-bt)  (a-b) ( a - a ^ (a-ct )  (a-b) (b-c^) (b-« )  2  2  exp(-a t)  exp(-a t)  1  ^ l~ 2^ ~ l^ "°'l^ a  a  a  a  Cj *  2  (a-a ) (b-a ) ( a - a )  b  2  2  1  C  J Q  t * 0  2  [7-11(a)]  V  Q  ( t )  exp (-bt) (a-b)[(a-b)  exp (-at) 2  + Y ]  (a-b)[(a-a)  2  exp(-at)cos(Yt Y  [(a-b)  2  + YV  + j>)  [(a-a)  2  +  2  +  y} 2  , Yr 2  C  5 C J ' JO t  Z  0  [7-ll(b)]  - - l Y where o> = t a n — — a-a t_  -  1  a  <_ - i a-b tan Y  fl = 2  R + Ld 2L  =  Y = (a -4 V o  A  A ]  = GR'bi /C L o J  -1524.  J u s t i f i c a t i o n o f Approximate T r a n s m i s s i o n L i n e Treatment I n c h a p t e r 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 t h e i n p u t  impedance o f the t r a n s m i s s i o n l i n e t e r m i n a t e d w i t h the p r e a m p l i f i e r as R^  c  + JX^  c  ( e q u a t i o n 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 o f t h i s  approximation  i n d e s c r i b i n g the impulse response o f the system, square p u l s e s from a p u l s e g e n e r a t o r were developed  i n a lumped c o n s t a n t model o f the specimen and  output waveform from t h e 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. f i g u r e 6-9  the  (The arrangement was  except t h a t f o u r d i f f e r e n t c a p a c i t a n c e s  s i m i l a r to  (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 t h e manner  d e s c r i b e d above u s i n g the average room temperature components of the l i n e  r e s i s t a n c e and i n d u c t i v e  (R = 13 P., L = 1.69uH, see f i g u r e 5-9).  In a l l four  c a s e s , the agreement between the t h e o r e t i c a l and e x p e r i m e n t a l 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 o r L)  t h a t t r e a t i n g the l i n e and i t s l o a d by the more p r e c i s e , but complicated s u c c e s s i v e - r e f l e c t i o n s technique C.  (Bohn, 1963)  algebraically  seemed unwarranted.  D e t e r m i n a t i o n o f C u r r e n t P u l s e Parameter T The o b j e c t of t h i s s e c t i o n i s t o e s t i m a t e T f r o m a l e a s t  f i t o f the t h e o r e t i c a l p u l s e shape ( e q u a t i o n s 7-11) observed  on photographs (see eg. F i g u r e 6-8).  squares  t o the p u l s e shape  T h i s s t a t i s t i c a l approach  made n e c e s s a r y by the n o i s e superimposed on the s i g n a l p u l s e s .  was  Thus,  f o l l o w i n g Orear (1958), i f x^ a r e the t h e o r e t i c a l v a l u e s c h a r a c t e r i z i n g the p u l s e shape and x^ a r e the c o r r e s p o n d i n g measured v a l u e s , assumed t o be G a u s s i a n d i s t r i b u t e d w i t h s t a n d a r d 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 m i n i m i z e d i s  M* =  N t ... i=l  (x. - x . ) / o . 2  1  X  (7-12)  2  1  2 I t t u r n s out  (Orear, 1958)  t h a t M*  i s merely  the x  d i s t r i b u t i o n of  (N-n)  degrees o f freedom where n i s the number o f parameters s o l v e d f o r and N i s the number of e x p e r i m e n t a l p o i n t s . w i t h t h e s t a n d a r d d e v i a t i o n AM*  The p r o b a b l e v a l u e o f M* i s M* =  = [2(N-n)] . z  N-n  -153( l ) E x t r a c t i o n o f t h e x^ and o\ from Photographs C o n s i d e r t h e t y p i c a l p u l s e photograph i n f i g u r e 6-8. Let  t = t  - 250 nsec be the time a t which a g i v e n p u l s e r e a c h e s i t s maximum  a m p l i t u d e , then t h e p u l s e a m p l i t u d e a ( t ) a t N o t h e r t i m e s ( t ^ ) , f o r 0 $ t ^ £ 2 t , may be measured d i r e c t l y from t h e photograph Q  ( f o r t h e same  p u l s e ) t o form the r a t i o  y  i  =  a  (  t  i  )  /  a  (  t  o  '  )  1  =  1  "-'«  N  Because t h e a m p l i f y i n g system i s l i n e a r , the p u l s e shape (as c h a r a c t e r i z e d by y^) depends o n l y on t h e time c o n s t a n t s o f t h e system and n o t on t h e 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 g r e a t e r t h a n t h e rms n o i s e v o l t a g e .  C o n s e q u e n t l y , maximum a m p l i t u d e p u l s e s  on 5 d i f f e r e n t photographs were chosen t o be a n a l y z e d i n t h i s f a s h i o n t o a r r i v e a t mean v a l u e s  x. = - 7 T y.. i 5 ... i i J=l  , 1 = 1,...,7  w i t h v a r i a n c e (Evans, 1955) 2  a. I  ? i 2 = 7- ) (y..-x.) 4 . , i i i J=l 1  —  v  .  ,  i = l,...,7  where y.. i s t h e measured r a t i o a ( t . ) / a ( t ) on t h e j t h p u l s e . 'IJ l o J  v  The r e m a i n i n g q u a n t i t i e s needed t o c a l c u l a t e M* a r e the  c o r r e s p o n d i n g 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 e q u a t i o n s 7-11,  x  i  =  v  o  < t  i  ) / v  o  ( t  o  '  )  w h i c h , as i t s h o u l d b e , i s independent o f i  Q  1  =  1  7  .  At t h i s j u n c t u r e , a word i s i n o r d e r c o n c e r n i n g t h e p r e c i s i o n w i t h which t h e times t . a r e known.  I n p r a c t i c e , s i n c e the zero o f time f o r  l  a p a r t i c u l a r p u l s e i s n o t c l e a r l y d i s c e r n i b l e from t h e photograph, t h e t ^ must be measured w i t h r e s p e c t t o t , which can o n l y be e s t i m a t e d from Q  photographs t o l i e w i t h i n a range o f v a l u e s t  Q  + At . Q  Hence,  -154t  = t + t ' ± At + A t ! where - t 4 t . t and A t ! 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 t a k e t h i s u n c e r t a i n t y i n t o X•  a c c o u n t , M* must be c a l c u l a t e d o v e r a range o f t . o (2) M*  Calculation I n s p e c t i o n of e q u a t i o n s 7-11  r e v e a l s t h a t x i s the o n l y  unknown v a r i a b l e i n the x^ but t h a t t h e v a l u e o f x which m i n i m i z e s M* depend upon the p a r t i c u l a r v a l u e s chosen f o r C  T  J  and t . o  I t was  will  shown i n  section B that 1500  and t  Q  may  * Cj  S? 4500 pF  be e s t i m a t e d from f i g u r e 6-8, by e x t r a p o l a t i o n o f the maximum  a m p l i t u d e p u l s e to t h e ac ground l e v e l , t o l i e i n the range  245 $ t  o  $ 265  nsec  A computer program was w r i t t e n to e v a l u a t e e q u a t i o n s o v e r the s t i p u l a t e d ranges of C  and t J  7-2.  7-11  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 o  As a measure o f the u n c e r t a i n t y i n the s o - c a l l e d most p r o b a b l e  t h a t v a l u e w h i c h m i n i m i z e s M* f o r a p a r t i c u l a r C  and t — i t J o T  x—i.e.  i s convenient,  a l t h o u g h a r b i t r a r y , to a c c e p t a l l x which f a l l w i t h i n a 90% c o n f i d e n c e l e v e l . I n t h i s c o n t e x t "90% c o n f i d e n c e l e v e l " means t h a t i f the experiment  were  r e p e a t e d 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 ) i s such t h a t the Q  p r o b a b i l i t y i s 90% t h a t the newly measured x. would g i v e M* £ Q where Q=10.6 2 i s the v a l u e 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 o f freedom 2 (see eg. x t a b l e s , Handbook of P h y s i c s and C h e m i s t r y ) . Thus, f o r example (see f i g u r e 7-2) a 90% c o n f i d e n c e l i m i t on x = *x (C = 3000 pF, t = 260 nsec) 1  J  o  g i v e s 123 £ x $ 167 nsec w i t h a most p r o b a b l e v a l u e of r o u g h l y 142 F i g u r e 7-3  summarizes t h e . r e s u l t s o f the M*  nsec. calculations.  The r e g i o n shown i n heavy b l a c k o u t l i n e i s the 790% c o n f i d e n c e volume" i n Cj,  t , x Q  space c o m p a t i b l e w i t h the u n c e r t a i n t y i n t  Q  and Cj d e s c r i b e d  above; t h e s u r f a c e of most p r o b a b l e v a l u e s of x l i e s r o u g h l y midway between t h e upper and lower  extremes. C l e a r l y , i t i s not m e a n i n g f u l to speak o f a unique  x  -155-  -157as g i v i n g t h e b e s t f i t of the t h e o r e t i c a l t o measured p u l s e shapes. purposes o f e s t i m a t i n g i , Q  For  however, i t i s c o n v e n i e n t t o t a k e a c e n t r a l  v a l u e (shown i n f i g u r e 7-3) of T = T ( 3 0 0 0 pF, 255 nsec) - 138 n s e c . (3) V a l i d i t y o f the Assumed Input C u r r e n t P u l s e The v a l u e s of x chosen by the l e a s t squares f i t a r e based on t h e assumption t h a t t h e c u r r e n t p u l s e 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 t o c a l c u l a t e the form of t h e observed p u l s e .  Some i n f o r m a t i o n about the  v a l i d i t y o f t h i s assumption may a l s o be o b t a i n e d from the M*  calculations.  I n p a r t i c u l a r , i f ( C j , t ^ , x') m i n i m i z e s M* then M*  (C' J  t ' , x*) s h o u l d l i e i n t h e range M* ±AM* = 6  ± 3.5 i f the t h e o r e t i c a l  o  p u l s e shape i s adequate  ( B r a d d i c k , 1966) whereas M*(C', t ' , x') < 2.5 would J o i m p l y x i s not determined by the measurement and M*(C', t ' , x') > 9.5 J  would i m p l y more parameters s h o u l d have been used. shows t h a t the c a l c u l a t e d M*  o  The p l o t o f f i g u r e  7-4  (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 o n l y t h a t x i s m e a n i n g f u l but t h a t a h i g h e r o r d e r parameter form of i ( t ) i s n o t j u s t i f i e d w i t h the p r e s e n t d a t a . D.  E s t i m a t e of C u r r e n t P u l s e A m p l i t u d e ( i ) Q  A r e s u l t w h i c h w i l l be needed l a t e r i s the c u r r e n t p u l s e amplitude i  c o r r e s p o n d i n g ( f o r reasons t o be d i s c u s s e d i n s e c t i o n E  f o l l o w i n g ) t o the maximum a m p l i t u d e observed p u l s e s .  The method o f o b t a i n i n g  t h i s parameter from the photographed p u l s e s ( 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. E q u a t i o n s 7-11 c o n t a i n i GR'  i b  2.  A = CjL  i m p l i c i t l y i n that v (t)  =  -2  :•:  f(t)  where v ( t ) i s the peak p u l s e a m p l i t u d e measured p h o t o g r a p h i c a l l y from o o t h e o s c i l l o s c o p e ( i n o u t p u t v o l t s V ( o u t p u t ) ) a t time t r i g h t hand s i d e o f e q u a t i o n s 7-11 e v a l u a t e d a t t = t . t o f i n d the a m p l i f i e r system t r a n s f e r c o n s t a n t  Q  and f ( t ) i s the A l l t h a t remains i s  GR'.  T h i s q u a n t i t y i s d e t e r m i n e d , as shown i n the next two p a r a g r a p h s , 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 d e s c r i b e d i n c h a p t e r 6 where v e r y  -158-  16  14  O  1500 pF  • X © &  2000 3000 4000 4500  S u b s c r i p t s 1-5 i n d i c a t e t = 245o 265 nsec r e s p e c t i v e l y  12k  6  3  e  •H  c  •H  100  120  140  160  180  i(nsec) F i g u r e 7-4:  M*(minimum') v/s R e l a x a t i o n time ( T )  200  -159-  fast rising  ( s t e p ) c u r r e n t p u l s e s o f known a m p l i t u d e were i n j e c t e d i n t o a  lumped c o n s t a n t model o f t h e s m a l l s i g n a l e q u i v a l e n t c i r c u i t o f the specimen (see f i g u r e 6 - 9 ) and the output p u l s e on an o s c i l l o s c o p e .  v o c  ( t ) was r e c o r d e d p h o t o g r a p h i c a l l y  (The g a i n and bandwidth s e t t i n g s o f the a m p l i f i e r s  were o f c o u r s e unchanged from the s i g n a l p u l s e measurements.) 1.  T r a n s f e r F u n c t i o n f o r Input C u r r e n t Step P u l s e The f a s t r i s i n g i n p u t c u r r e n t p u l s e may be r e p r e s e n t e d  a p p r o x i m a t e l y by i ( t ) = i ^ ^ - ) where 1  1  c  0 , t < = k , t = 1 , t >  u(t)  0 0 0  and i i s the measured maximum a m p l i t u d e i n p u t 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 - ( t ) i n t o e q u a t i o n 7 - 1 and t r a n s f o r m i n g g i v e s c  /  TT  V  ic  1  \  is)  '  v  =  1  1  • — •  C Jc  s  T  w h i c h i s the same as e q u a t i o n specimen v a l u e o f C j . )  oc  s+d  with a =  7-7  (C  0.  The forms o f b o t h W(s)  J c  =  pF i s the "dummy"  3500  and Y ( s ) a r e unchanged from  t h e d i s c u s s i o n i n s e c t i o n B so t h a t the a p p r o p r i a t e t r a n s f e r f u n c t i o n i s g i v e n by 7 - 1 1 ( b ) — s i n c e C  V  oc  J c  > C  ( t ) =  J Q  —with a = 0.  _  Thus  1  exp(-bt)  B  b[(a-b)  +Y ]  2  b(a  2  2  + y ) 2  (7-13)  exp(-ctt)  Yt(a-b)  cos ( y t + <ft)  + y J (a + Y V  Z  2  where ,  .  A = tan  and  —1  Y1  -- a  GR'i B =  r  b T  ^  -1  tan  a-b Y  -160Q u a n t i t i e s a and y,  as b e f o r e , a r e g i v e n 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 c o n s t a n t model v a l u e s of C = C J  T  , r  = r  Jc  = 10 Jl and R = R c  = 12 Q.  (R  c  i s the midband room  temperature  c  v a l u e of t h e 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 v a l u e s of a and y 7 - 1 7 - 1 p e r t i n e n t t o 7-13 a r e about 1.8 x 10 sec and 0.74 x 10 sec respectively. These v a l u e s , w h i c h make t h e denominators of a l l t h r e e terms of the e q u a t i o n the same o r d e r o f magnitude, combined w i t h the f a c t t h a t the  times  of i n t e r e s t f o r measured v  ( t ) are the o r d e r of 1 usee, d r a s t i c a l l y —6 Hence v (t - 10 sec)-* v ( t = °°) or oc oc  s i m p l i f y e q u a t i o n 7-13. n  v  (t ) oc  00  b(a  2  2 + Y )  Remembering t h e d e f i n i t i o n s of B and a  GR' , =_  v  oc i  From c h a p t e r 6, v  oc  (t ) °°  + r  yields  c  c  uA so t h a t  00  , .11 = YJ  c  r  oc  Q  given e a r l i e r R  (t ) / i = 1 V/3.5 oc  GR'  Q  ba  . 22 _ - , ^ • — = 0.628 0 Q  V(output) uA(input)  3.  E v a l u a t i o n of io From the d e f i n i t i o n of A g i v e n e a r l i e r , the s i g n a l c u r r e n t  p u l s e a m p l i t u d e 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 :  1  =  ° T a b l e 7-1  v ( t ) C. L o o J f(t ) o  GR'b  c o n t a i n s the v a l u e s of i  o b t a i n e d f o r t h e maximum amplitude o  pulse  -161of f i g u r e 6-8 ( c o r r e s p o n d i n g t o v ( t ) - 2.35 V ( o u t p u t ) ) and f ( t ) and C o o o J e v a l u a t e d f o r extreme, v a l u e s o f 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  CjfeF)  t  o  O  (nsec)  x (nsec)  i (yA) o  1500 1500 2600 3000 3500 3900 4000 4500 4500  245 245 265 265 245 249 265 250 265  153 120 160 171 114 110 169 105 160  21.2 24.8 20.6 19.7 24.9 25.5 19.6 26.2 20.3  *3000  255  138  22.3  T a b l e 7-1:  E s t i m a t e of i *  f o r maximum- A m p l i t u d e P u l s e .  C e n t r a l v a l u e s of C , t , T J o T  The u n c e r t a i n t y i n C , t J o T  and x i s seen t o l e a d t o a  spread i n i  so t h a t f o r t h e maximum a m p l i t u d e observed output p u l s e i n  question, i  may be s a i d t o l i e i n t h e 90% c o n f i d e n c e i n t e r v a l 20 £ i  $ 26uA  w i t h a most p r o b a b l e v a l u e o f 22pA. E.  Energy Loss p e r 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, t h e 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 t h e a l p h a p a r t i c l e may be found by r e c a l l i n g  t h e e x p r e s s i o n <n(t)>, t h e expected number o f q u a s i p a r t i c l e s t u n n e l i n g i n time t , from e q u a t i o n 3-15. S e t t i n g  oo  <( oo )> = w n  i exp ( - t / x ) d t  (7-14)  gives N  o  = i  o  /W e T T  (7-15)  -162where i ( t ) = i  exp  (-t/x) i s the form assumed f o r the c u r r e n t p u l s e  superimposed on t h e ambient t u n n e l i n g c u r r e n t . B e f o r e g o i n g on t o e v a l u a t e 7-15 estimate W  and d i s c u s s the s i g n i f i c a n c e o f W i  i t i s necessary  = W-W  .  R (a) T u n n e l i n g  T  Probability  From e q u a t i o n 3-17, t h e low temperature,  to  W  = G(e  T  2  -N (0)^ X A )  _ 1  where G i s  normal s t a t e conductance of t h e 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  p e r u n i t volume near the Fermi s u r f a c e , X i s the f i l m t h i c k n e s s and A the j u n c t i o n overlap area.  For t h e 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 A = 7.1 x 10  U s i n g e q u a t i o n s 3-18  to 3-23,  -4  (7-16)  cm  2  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 d e r i v e d from anomalous s k i n e f f e c t measurements ( W i l s o n , 1965), 22 3 one f i n d s N = 4.05 x 10 e l s / c m , E (Sn) = .4.4 eV making f  N (0) = 1.4 m  x 10  2 2  eV  1  cm"  (7-17)  3  The v a l u e f i n a l l y o b t a i n e d f o r the t u n n e l i n g p r o b a b i l i t y i s t h e r e f o r e  W  T  (b) Recombination  = 4.1 x 1 0  5  sec"  (7-18)  1  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  a l p h a 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 o t h e r s 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 c h a p t e r 2 ) .  This  r e c o m b i n a t i o n t a k e s p l a c e p r i m a r i l y by e m i s s i o n of phonons of energy  -163ntoi 2 A (T) r a t h e r t h a n photons, as the r a t e of the former exceeds t h a t of the l a t t e r by a f a c t o r of about 10^ (Rothwarf and Cohen, 1963).  Subsequently  t h e s e 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 t o the s u b s t r a t e  (Jones and Pennebaker, 1963) or t h e s u p e r f l u i d h e l i u m  ( W i l k s , 1967), may  c r e a t e 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 t h a t 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 b e i n g d i s s o c i a t e d by t h e r m a l phonons o f energy tuo£ 2A(T) and q u a s i p a r t i c l e s a r e c o n t i n u a l l y r e c o m b i n i n g i n t o p a i r s v i a the e m i s s i o n of phonons. Rothwarf and T a y l o r (1967) have shown t h a t f o r  quasipart-  i c l e s i n j e c t e d i n t o a s u p e r c o n d u c t i n g f i l m a t a c o n s t a n t r a t e , the observed r e c o m b i n a t i o n r a t e W'  i s dominated by the r a t e W = x y y are l o s t from the range tiu> 2A(T) and not by the r a t e W  a t which phonons _  7  R  x  = x 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 From t h e s e c o n s i d e r a t i o n s ,  at which R  pairs.  i t i s apparent t h a t the  r e c o m b i n a t i o n r a t e W'  deduced from the observed c h a r a c t e r i s t i c decay r a t e -1 W = W + W (W , W ) = x i s not s i m p l y t h a t c a l c u l a t e d from the t h e o r y of q u a s i p a r t i c l e r e c o m b i n a t i o n (W , see eg. Woo and Abrahams, 1968) but i s T  R  R  R  a l s o a f u n c t i o n of t h e phonon l o s s p r o c e s s e s which go a c c o r d i n g t o (c) E f f e c t o f t h e Energy Gap  W^.  Parameter  I t s h o u l d perhaps be noted t h a t as the temperature d i s t r i b u t i o n i n the t u n n e l j u n c t i o n s  r e l a x e s back t o the e q u i l i b r i u m  s i t u a t i o n , the magnitude o f the energy gap 2A(T) i n c r e a s e s and w i l l the manner i n w h i c h the excess q u a s i p a r t i c l e p o p u l a t i o n decays. appreciation  affect  Some  f o r t h e i n f l u e n c e o f A(T) may be. g a i n e d from t h e f o l l o w i n g  approximate c a l c u l a t i o n s . For equilibrium density  a s u p e r c o n d u c t o r a t temperature T, the t h e r m a l  o f q u a s i p a r t i c l e s n(T) i s g i v e n by e q u a t i o n 3-2:  n(T) = N (0) A(T) K... ( $ A ( T ) ) , m 1 « N (0) A(T) e x p ^ A C O / k ^ T ) m 15  T < T  c  (3-2)  ( 3 - 3)  (Here, N (0) i s the normal s t a t e d e n s i t y o f f r e e e l e c t r o n s t a t e s per u n i t  -164volume near t h e Fermi s u r f a c e (E = 0 ) , A(T) i s t h e energy gap parameter and k g i s Boltzmann's c o n s t a n t .  F p r convenience K ^ ( x ) , t h e f i r s t  m o d i f i e d B e s s e l f u n c t i o n o f t h e second k i n d , i s approximated  order  as ( c f . p.55)  K^(x) = oc(x) exp (-x) = exp(-x) as o ( x ) v a r i e s s l o w l y w i t h x and i s t h e o r d e r o f one.) Thus, as t h e temperature to  i n the tunnel j u n c t i o n  t h e e q u i l i b r i u m l e v e l (T ) , t h e change i n t h e excess  decreases  quasiparticle  d e n s i t y 6n = n(T) - n ( T ) , i g n o r i n g t h e r e c o m b i n a t i o n phonons, would be Q  g i v e n by ( f o r T o  < T < T ) c  3(6n)  9T  N (0) exp m  A(T) k  B  1- A(T)  9A(T)  V  T  A (T) 2  9T  k T  2  B  From c h a p t e r 2, i t may be r e c a l l e d t h a t f o r T < { T , 2A(T)=3.5 k„T c Be so t h a t i n t h i s range o f t e m p e r a t u r e s , Boltzmann f a c t o r exp (-1.75 TjT)  3(6n)/9T would be governed by a  s i m i l a r to that governing the d e n 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 . h o w e v e r , A(T) v a r i e s as i (1 - T / T ^ ) r e s u l t i n g i n c o n s i d e r a b l e d e p a r t u r e from t h e 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. c  2  (d) E v a l u a t i o n o f N o The.number o f 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  c r e a t e d by  the passage o f an a l p h a p a r t i c l e may now be c a l c u l a t e d from e q u a t i o n 7-15 and t h e e s t i m a t e t h a t  = 4.1 x 10^ sec  1  ( e q u a t i o n 7-18).  i n s e c t i o n D t h a t t h e most p r o b a b l e v a l u e o f i  I t was shown  c o r r e s p o n d i n g t o a maximum  a m p l i t u d e observed p u l s e was 22yA; t h e r e f o r e '  N 2.  o  = i /W^e = 3.4 x 10 o T  Energy Loss C o r r e s p o n d i n g  t o Maximum Amplitude  (7-19) Pulse  The r e m a i n i n g q u a n t i t y needed t o c a l c u l a t e w  =AE/N  q  is  AE,  t h e t o t a l amount o f energy d e p o s i t e d i n t h e s u p e r c o n d u c t i n g t u n n e l j u n c t i o n by t h e a l p h a p a r t i c l e . AE  I d e a l l y , f o r purposes  of nuclear spectroscopy,  s h o u l d be e q u a l t o t h e t o t a l a l p h a p a r t i c l e energy E^ = 5.1 MeV b u t  t h i s c o n d i t i o n c o u l d n o t be a c h i e v e d w i t h t h e t h i n f i l m j u n c t i o n s used.  -165The a l p h a p a r t i c l e p a t h l e n g t h t h r o u g h t h e f i l m o f t o t a l t h i c k n e s s 2X i s x = 2A /cos6 where 6 i s t h e a n g l e o f i n c i d e n c e .  From  f i g u r e 6-7 t h e most o b l i q u e a n g l e o f i n c i d e n c e was 80° w h i c h c o r r e s p o n d s -4 t o a p a t h l e n g t h o f 2.3 x 10  cm.  Thus, w i t h t h e use o f f i g u r e 3-3, t h e  maximum amount o f energy l o s t d i r e c t l y AE ( d i r e c t ) i n t h e t i n f i l m s i s seen t o be o n l y about 0.5 MeV. The remainder o f E^, w h i c h was d e p o s i t e d i n t h e g l a s s s u b s t r a t e a s t h e a l p h a p a r t i c l e comes t o r e s t , may n o t however, be n e g l e c t e d i n d e t e r m i n i n g w.  No attempt was made t o t h e r m a l l y d e c o u p l e t h e f i l m s  from t h e s u b s t r a t e w i t h t h e r e s u l t t h a t , perhaps as much as one h a l f o f t h e energy d e p o s i t e d i n t h e g l a s s may have d i f f u s e d back i n t o t h e t i n s u f f i c i e n t l y r a p i d l y to contribute to the pulse.  For purposes o f t h e p r e s e n t c a l c u l a t i o n  therefore,  AE = A E ( d i r e c t ) + h ( E - A E ( d i r e c t ) ) a  where 0 $ h $ 0.5 making AE $ 2.75 MeV.  (7-20)  The r e l a t i o n o f e q u a t i o n 7-20  t o t h e o b s e r v e d p u l s e spectrum ( f i g . 6-11) i s c o n s i d e r e d i n s e c t i o n F o f t h i s chapter. 3.  Estimate of w T a k i n g AE £ 2.75 MeV as t h e t o t a l energy d e p o s i t e d i n t h e  j u n c t i o n c o r r e s p o n d i n g t o t h e maximum a m p l i t u d e p u l s e and t h e r e s u l t f o r N  q  from e q u a t i o n 7-19 g i v e s an upper  w(tin) * ~  limit  = 8.2 x 10~ eV/q.p. p a i r 3  o  (7-21)  w h i c h i s r o u g h l y 7.5 times t h e v a l u e o f t h e energy gap i n s u p e r c o n d u c t i n g t i n a t 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  t h e average amount o f energy w h i c h must be d e p o s i t e d i n a superconductor t o c r e a t e 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 n o t t o be confused w i t h E  = 2A(T) d e r i v e d from m i c r o s c o p i c t h e o r y which i s t h e minimum amount o f g energy r e q u i r e d t o break up a Cooper p a i r a t temperature T. Another q u a n t i t y o f i n t e r e s t . f o r comparing t h e energy r e s o l u t i o n o f t h e t u n n e l j u n c t i o n d e t e c t o r t o c o n v e n t i o n a l semiconductor  -166s p e c t r o m e t e r s i s w ( e x p ) , the energy l o s s per tunneled.  Thus, f r o m e q u a t i o n  7-15  w(exp) $ <n(°°)> ' v  F.  Energy Loss and  q u a s i p a r t i c l e which a c t u a l l y  = — N  • —  - .145  eV  (7-22)  W T  o  m  Diffusion  T h i s s e c t i o n o u t l i n e s some c o n s i d e r a t i o n s  relevant  to  the  c a l c u l a t i o n of the manner i n w h i c h the energy l o s t by the a l p h a p a r t i c l e d i f f u s e s t h r o u g h the j u n c t i o n and 1.  substrate  g i v i n g r i s e to a heat  pulse.  Energy Loss P r o c e s s e s The  p r i m a r y means by w h i c h the a l p h a p a r t i c l e l o s e s  i s by the i o n i z a t i o n and atoms a l o n g i t s p a t h .  e x c i t a t i o n of the e l e c t r o n s (In m e t a l s , semiconductors and  p a r t i c l e g i v e s up o n l y about 0.1% c r e a t i n g phonons and ( S e i t z and  c o l l i s i o n i s generally  As  with  insulators,  of i t s energy d i r e c t l y to the  dislocations—with  K o e h l e r , 1956).)  associated  energy the  the  lattice—  the remainder g o i n g to the  electrons  the energy t r a n s f e r a t each i o n i z i n g  a s m a l l f r a c t i o n ( a p p r o x i m a t e l y the o r d e r of 10  of the p a r t i c l e energy (Dearnaley and  N o r t h r o p , 1966), the d e f l e c t i o n s  a s t r a i g h t l i n e p a t h are s l i g h t . -12 -11  p a r t i c l e comes to r e s t i n about  10  to 10  The  sec w i t h r o u g h l y 90% o f i t s p a t h (see f i g u r e 7-5)  glass substrate  making i t n e c e s s a r y t h e r e f o r e  p r o c e s s e s i n b o t h m e t a l s and  ) from  b e i n g i n the  to c o n s i d e r the energy l o s s  insulators.  I n a m e t a l , the p r i m a r y e l e c t r o n e x c i t a t i o n s a r e s u f f i c i e n t l y energetic  t h a t the u s u a l argument (Jones, 1956)  s t a t i n g that  electron-  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 e l e c t r o n - p h o n o n i n t e r a c t i o n s does not h o l d ; due  f i r s t because the It space r e s t r i c t i o n s on  t o the e x i s t e n c e o f the Fermi sphere are g r e a t l y reduced and  because the phonon d e n s i t y  scattering second  i s s m a l l as a r e s u l t of the f a c t o r of 1000  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  in  argument g i v e n  by Jones depends on t h e r e b e i n g a s t a t e of t h e r m a l e q u i l i b r i u m i n w h i c h energy i s shared e q u a l l y between e l e c t r o n and T h e r e f o r e the c o u p l i n g  between e l e c t r o n and  weaker t h a n e i t h e r the e l e c t r o n - e l e c t r o n t h a t the two  types of e x c i t a t i o n may  l a t t i c e degrees of freedom.)  phonon e x c i t a t i o n s i s much  o r phonon-phonon c o u p l i n g s ,  be c o n s i d e r e d to t h e r m a l i z e  so  -167-  F i g u r e 7-5:  P a r t i c l e Track Geometry  AE(direct) Figure  7-6:  (MeV)  T h e o r e t i c a l P u l s e Height Spectrum, h=0  (Un-normalized)  -168-12 i n d i v i d u a l l y i n t i m e s of t h e o r d e r o f 10  -11 - 10  sec (Dearnaley and  N o r t h r o p , 1966) w i t h the e l e c t r o n temperature a p p r o x i m a t e l y 200 t i m e s g r e a t e r than the l a t t i c e t e m p e r a t u r e .  F i n a l l y , t h e weak e l e c t r o n - p h o n o n  c o u p l i n g causes the two t y p e s 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 K o e h l e r , 1956). I n 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 c o n s i d e r e d free.  I n 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 t o r e l a x -12  t o some common temperature i s p r o b a b l y the o r d e r of 10  sec (as e s t i m a t e d  f o r s e m i c o n d u c t o r s by D e a r n a l e y and N o r t h r o p , 1966). I t seems r e a s o n a b l e to assume t h e r e f o r e t h a t w i t h i n a time the -9 o r d e r of 10  sec i t i s m e a n i n g f u l t o speak o f a l o c a l  electron-lattice  t e m p e r a t u r e , i n a narrow c y l i n d e r c o - a x i a l w i t h t h e p a r t i c l e t r a c k , which i s i n e x c e s s o f the j u n c t i o n e q u i l i b r i u m t e m p e r a t u r e .  The next paragraph  c o n s i d e r s the manner i n w h i c h t h i s heat energy d i f f u s e s outward from the track. 2.  Energy T r a n s f e r The heat energy i s s u b s e q u e n t l y t r a n s p o r t e d away from t h e  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 d e s c r i b e d by c l a s s i c a l h e a t d i f f u s i o n t h e o r y ( C r i t t e n d e n , 1968).  be  A complete  a n a l y t i c s o l u t i o n o f the d i f f u s i o n e q u a t i o n p c ( 9 T / 8 t ) = 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 t h e r m a l c o n d u c t i v i t y i s p r e c l u d e d >  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 o f 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 t h e t r a c k , t h e temperature dependence o f t h e s p e c i f i c heat and t h e r m a l c o n d u c t i v i t y may not be s a f e l y n e g l e c t e d .  (For example,  p u b l i s h e d d a t a f o r g l a s s (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 o r d e r s o f magnitude r e s p e c t i v e l y o v e r the temperature range from 1 K to 300 K.)  W i t h c = c ( T ) and K = K ( T ) , however, the d i f f u s i o n e q u a t i o n  i s n o n - l i n e a r making n u m e r i c a l methods i m p e r a t i v e . (b) T r a c k Geometry The d i s t a n c e which the a l p h a p a r t i c l e p e n e t r a t e s the g l a s s s u b s t r a t e i s g i v e n by R where R s a t i s f i e s  -169fR AE* = E a  a  - AE  a  (direct) =  dx.  ft)  Here, x i s the d i s t a n c e measured a l o n g 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 a l p h a p a r t i c l e energy  (5.1 MeV)  i s the energy l o s t by the a l p h a p a r t i c l e i n the t i n f i l m s . e s t i m a t e of R may t h a t R = AE^  and An  AE(direct)  approximate  be o b t a i n e d by t a k i n g (dE^ /dx) = c o n s t a n t i n g l a s s so  /OE^  /3x).  The r a t e o f a l p h a p a r t i c l e energy  loss  (8E  / 8 x ) , e s t i m a t e d from the atomic s t o p p i n g c r o s s s e c t i o n s o f the 3 c o n s t i t u e n t s o f soda l i m e g l a s s , i s found to be r o u g h l y 1.6 x 10 MeV/cm. a  Now  AE^ depends o n l y s l i g h t l y on the a n g l e a t w h i c h the a l p h a 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 t h a t R ranges from -3 2.9 to 3.1 x 10 cm w h i c h , f o r a l l p r a c t i c a l p u r p o s e s , may be taken as -3 R = 3 x 10  cm independent of a n g l e of i n c i d e n c e .  C o n s e q u e n t l y , as s k e t c h e d  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 C o n d i t i o n s W i t h r e f e r e n c e t o f i g u r e 7^5, i t i s c l e a r t h a t the heat  1  f l o w a c r o s s two i n t e r f a c e s s h o u l d be c o n s i d e r e d . A l t h o u g h the a c o u s t i c impedances o f t i n and g l a s s a r e q u i t e c l o s e (American I n s t i t u t e o f P h y s i c s 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 a t 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 t h a t a f i n i t e t h e r m a l impedance e x i s t s .  Little  large  (1959) has c a l c u l a t e d t h a t  a t low t e m p e r a t u r e s the heat f l o w between p y r e x and copper i s  dQ/dt = 3.7 x l O  5  ^  4  - T ) 4  2  erg sec cm" - 1  2  w h i c h , i n the l i m i t of s m a l l 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 a t the -3 2 interface.  T h i s c o r r e s p o n d s then to a t h e r m a l impedance Z^ = .67 T  w h i c h i s presumably not f a r d i f f e r e n t from a t i n - s o d a l i m e g l a s s  K cm  boundary.  The s i t u a t i o n a t the t i n - s u p e r f l u i d h e l i u m (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 t h e r m a l 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 a l o n e .  Other f a c t o r s w h i c h have been suggested ( L i t t l e ,  1961)  a r e the c o u p l i n g of the c o n d u c t i o n e l e c t r o n s i n the m e t a l t o the t o t a l l y  -1 W  -170-  r e f l e c t e d phonons of the l i q u i d and  the a d s o r p t i o n of a t h i n l a y e r of  h e l i u m atoms on the m e t a l i n t e r f a c e . provide  T h i s l a y e r of s o l i d h e l i u m would  an a c o u s t i c match between the l i q u i d and m e t a l because of i t s  intermediate  a c o u s t i c impedance.  E x p e r i m e n t a l measurements  (Gittleman  and B o z o w s k i , 1962) f o r the normal t i n - l i q u i d h e l i u m i n t e r f a c e g i v e -3 2-1 Z  T  = 5.48  T  K cm  W  h e l i u m i n t e r f a c e was  .  The v a l u e of Z^ f o r a s u p e r c o n d u c t i n g  some 10.6%  tin-liquid  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 t h a t  the  problem of the heat p u l s e d i f f u s i o n t h r o u g h 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 t o m e r i t d e t a i l e d c o n s i d e r a t i o n and, e a r l i e r , such work i s b e i n g c o n t i n u e d 3.  in this  E v i d e n c e f o r Heat C o n t r i b u t i o n from The  as mentioned  laboratory. Substrate  o b s e r v e d p u l s e h e i g h t spectrum i n f i g u r e 6-11  contains  some i n f o r m a t i o n c o n c e r n i n g 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 w h i c h w i l l now I t was  be  considered.  s t a t e d e a r l i e r t h a t the t o t a l energy c o n t r i b u t i n g  t o the p u l s e s h o u l d be of the form  AE * AE ( d i r e c t ) + h ( E * a where h i s an e x p e r i m e n t a l 0 £ h £  - AE(direct))  (7-20)  parameter assumed t o l i e i n the range  0.5. The  upper l i m i t o f 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 w h i c h shows t h a t the minimum time r e q u i r e d 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 , t o 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 g l a s s and —3 3 —1 —6 r e t u r n t o 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 p r o p a g a t i o n v e l o c i t y (Klemens, 1951)  and  the phonon mean  f r e e p a t h f o r g l a s s a t low t e m p e r a t u r e s , w h i c h i s known to be (Ziman, 1965  "long"  and K i t t e l , 1956), i s assumed t o be of the o r d e r of  s u b s t r a t e t h i c k n e s s s.)  From f i g u r e 6-8, —6  h e a t p u l s e must be l e s s than 0.5  x 10  the  i t can be seen t h a t the a c t u a l sec i n w i d t h so i t i s not  t h a t the " r e f l e c t e d " heat p u l s e , c o r r e s p o n d i n g to r o u g h l y  one-half  expected 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 p u l s e .  -171To see the e f f e c t o f t h e 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 c o n v e n i e n t t o c o n s i d e r two l i m i t i n g c a s e s : h =  h = 0 and  0.5. I f h = 0, w h i c h i m p l i e s t h e r e i s no feedback of heat energy  f r o m the s u b s t r a t e , the p u l s e h e i g h t spectrum s h o u l d be o f the form s k e t c h e d i n f i g u r e 7-6, f o r t h e energy A E ( d i r e c t ) d e p o s i t e d i n t h e 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 t o cos 6 where 6 i s the a n g l e of i n c i d e n c e . C o n t r a r y t o what was observed ( f i g u r e 6-11), t h i s spectrum p r e d i c t s t h a t l o w e r energy p u l s e s a r e v e r y much more p r o b a b l e (up t o 2 o r d e r of magnitude) t h a n h i g h e r energy ones.  I t must be c o n c l u d e d t h e r e f o r e t h a t  h e a t d i f f u s i n g back f r o m t h e s u b s t r a t e , the amount o f w h i c h i s l e s s s e n s i t i v e t o 9, was r e s p o n s i b l e t o a l a r g e e x t e n t f o r t h e observed p u l s e s . Thus h = 0 may be r u l e d o u t . On the o t h e r hand i f h = 0.5  and i s independent o f 0 so  t h a t the j u n c t i o n a c t s m a i n l y 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 p u l s e h e i g h t spectrum would be a " s p i k e " c e n t e r e d around the  2.3  MeV  to the  p o i n t on t h e energy a x i s — f a r removed from the 0.5 MeV  A E ( d i r e c t ) of f i g u r e 7-6.  p u l s e s due  Such a d i s t r i b u t i o n i s i n c o m p a t i b l e w i t h  o b s e r v e d spectrum w h i c h t h e r e f o r e r u l e s out t h e s i m p l e c a s e 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 e x p e c t e d t h a t h w i l l depend upon 0,  perhaps as t h e p r o j e c t i o n  ( s i n 0) o f the t r a c k on the g l a s s - t i n  so t h a t from t h e s e s i m p l e c o n s i d e r a t i o n s , h = h (0) and 0 < h  interface (0) $ 0.5.  I t might be argued t h a t the minimum v a l u e of h would be the  one y i e l d i n g AE such t h a t w = A E / N  Q  =  gap parameter a t the b a t h temperature T.  2 A ( T ) where A(T) i s the energy  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 a t p r e s e n t as a d e t a i l e d t h e o r y r e l a t i n g secondary i o n i z a t i o n and m i c r o p l a s m a phenomena i n s u p e r c o n d u c t o r s t o an energy gap model i s not yet  available. A l t h o u g h 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 t o  e s t a b l i s h n a r r o w e r 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 o f t h e p a u c i t y o f d a t a 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 ,  this  problem of heat feedback w i l l be o b v i a t e d when t h e experiment i s r e p e a t e d i n t h i s l a b o r a t o r y by Dr. B. White and Mr. G. May w i t h a phonon b a r r i e r between t h e s u b s t r a t e and t u n n e l j u n c t i o n .  -172G.  Summary For c o n v e n i e n c e ,  a resume''of the i m p o r t a n t r e s u l t s of t h i s  c h a p t e r i s g i v e n below. I t was preamplifier  shown t h a t the i n p u t impedance o f the t r a n s m i s s i o n l i n e -  system c o u l d be w r i t t e n as Z = R + jcoL where R = 10.3ft ,  L = 1.69uH and OJ i s the a n g u l a r f r e q u e n c y . c i r c u i t of the complete  From the s m a l l s i g n a l e q u i v a l e n t  system, the o u t p u t p u l s e , c o r r e s p o n d i n g to an  i n p u t c u r r e n t p u l s e o f the assumed form i ( t ) = i u s i n g Laplace Transform  e x p ( - t / t ) was d e r i v e d  techniques.  By l e a s t - s q u a r e s f i t t i n g the t h e o r e t i c a l output p u l s e shape t o the v a l u e of T was  t h a t observed on photographs,  deduced to be  x = 138 ± 33 nsec where the e r r o r s are those c o r r e s p o n d i n g t o a 90% confidence  limit. Knowing t h e a m p l i f i e r system t r a n s f e r c o n s t a n t GR'  s e n s i t i v i t y c a l i b r a t i o n , one c o u l d then 20 $ i  from a  calculate  $ 26uA  f o r maximum a m p l i t u d e 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 particle N  o  was  found t o be  N  where  by the a l p h a  o W  = 4.1 x 10^ sec  (7_15)  =  1  i s the e s t i m a t e d t u n n e l i n g 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.  T a k i n g i n t o account t h a t up to o n e - h a l f of the energy d e p o s i t e d i n the s u b s t r a t e by the a l p h a 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 c o r r e s p o n d i n g to the maximum a m p l i t u d e p u l s e s was  e s t i m a t e d t o 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 =  AE/N  q  $ 8.2  x 10~  eV/q.p.pair  (7-21)  -173Another q u a n t i t y (quasielectron  o f 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  o r q u a s i h o l e ) w h i c h a c t u a l l y t u n n e l e d w(exp), was found t o  be  w(exp) $ 0.145  eV  (7-22)  C o n c l u d i n g the c h a p t e r i s a s h o r t d i s c u s s i o n o f problems c o n c e r n i n g d e t a i l e d c a l c u l a t i o n o f the way  i n w h i c h the heat energy d e p o s i t e d  a l o n g t h e a l p h a 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 h e l i u m system.  -174-  CHAPTER 8  CONCLUSIONS T h i s c h a p t e r summarizes the i m p o r t a n t f i n d i n g s from t h e p r e s e n t work ( s e c t i o n A) and attempts t o p l a c e 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 o f the s u p e r c o n d u c t i n g t h i n f i l m t u n n e l j u n c t i o n as a n u c l e a r spectrometer  (section 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 a r e a s h o u l d proceed to improve b o t h the u n d e r s t a n d i n g and performance A.  of t h i s device.  Major R e s u l t s The experiment has demonstrated  f i l m t u n n e l j u n c t i o n may  t h a t the s u p e r c o n d u c t i n g t h i n  be used as a d e t e c t o r 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 t h a t the p u l s e o f t u n n e l i n g c u r r e n t 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  a l p h a p a r t i c l e s was  s u f f i c i e n t a m p l i t u d e and d u r a t i o n t o be r e a d i l y o b s e r v a b l e . r e f i n e m e n t s r e q u i r e d t o improve  of  Fabrication  the t u n n e 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  so s u r p a s s the measured s i g n a l t o n o i s e r a t i o of 19 a r e , 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 o f p r e s e n t t e c h n o l o g y . -3 An upper l i m i t of 8.2  x 10  eV has been p l a c e d 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 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 s u p e r c o n d u c t i n g t i n a t 1.2 K.  T h i s phenomenological q u a n t i t y ,  w h i c h h i t h e r t o had n o t been measured i n s u p e r c o n d u c t o r s , i s o f i n t e r e s t p r i m a r i l y t o n u c l e a r s p e c t r o s c o p i s t s f o r the comparing medium w i t h another 7.5  (see s e c t i o n B ) .  times the energy gap  o f one d e t e c t i n g  T h i s v a l u e of w(Sn)  i s approximately  (2A) i n s u p e r c o n d u c t i n g t i n a t 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 c r e a t e d i n a semiconductor  (eg. s i l i c o n ) where w ( S i ) =3.6  the energy gap i n s i l i c o n .  eV which i s 3.3  (Dearnaley and N o r t h r o p ,  1966)  times  -175B.  E v a l u a t i o n o f t h e D e v i c e as a N u c l e a r Spectrometer 1.  Energy  Resolution 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 N o r t h r o p , 1966)  R = a/q = N " = (w/E) " _2  2  _  where q = Ne i s t h e s i g n a l charge c o l l e c t e d , a/e = N  x 2  i s the standard  d e v i a t i o n i n N, ( t h e number o f c a r r i e r s produced by t h e charged p a r t i c l e ) , w i s t h e mean energy r e q u i r e d t o g e n e r a t e a c a r r i e r , E i s t h e p a r t i c l e energy and e i s t h e e l e c t r o n i c charge. To compare t h e t u n n e l j u n c t i o n d e t e c t o r t o a n o t h e r spect r o m e t e r t h e v a l u e o f w t o be used i s w(exp) = 0.145 eV where w(exp) i s t h e maximum v a l u e o f t h e energy l o s s p e r r q u a s i p a r t i c l e w h i c h t u n n e l e d (see s e c t i o n E, c h a p t e r 7 ) .  actually  Thus, f o r t h e same p a r t i c l e , i t  may be i n f e r r e d t h a t t h e 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 o f the  i p r e s e n t s u p e r c o n d u c t i n g t u n n e l j u n c t i o n i s a f a c t o r (0.145/3.6)  2  = 0.20  s m a l l e r than t h a t o b t a i n e d w i t h a s i l i c o n semiconductor d e t e c t o r ,  assuming  t h e same Fano f a c t o r f o r b o t h m a t e r i a l s . I t i s c o n c e i v a b l e , w i t h f o r s e e a b l e improvements i n j u n c t i o n f a b r i c a t i o n t e c h n o l o g y , t h a t t h i s r e s o l u t i o n f a c t o r c o u l d be improved t o  i converge perhaps on t h e approximate l i m i t  t g P ( s u p e r ) / E g a p ( s e m i ) ] = 0.03.  F o r example, t h e t u n n e l i n g p r o b a b i l i t y  may be improved by d e c r e a s i n g  E  a  2  t h e 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 t h e use o f i n s u l a t o r s w i t h s m a l l e r d i e l e c t r i c c o n s t a n t s 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 a l s o be d e c r e a s e d , o f course,, by m i n i m i z i n g t h e i n s u l a t o r o  t h i c k n e s s , b u t a p r a c t i c a l l o w e r l i m i t o f about 10 A p r e c l u d e s s i g n i f i c a n t advances from t h i s approach.) Furthermore, t h e r e c o m b i n a t i o n p r o b a b i l i t y W — d i s c u s s e d i n c h a p t e r 7 — c o u l d be decreased by s u r r o u n d i n g t h e 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 ) ,  t o reduce t h e l o s s o f r e c o m b i n a t i o n phonons t o t h e s u b s t r a t e and s u p e r f l u i d helium f i l m . The a c t u a l , r a t h e r than t h e s t a t i s t i c a l , l i m i t o f energy r e s o l u t i o n i s determined by o t h e r f a c t o r s b e s i d e w.  As d i s c u s s e d i n  -176c h a p t e r 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 t h i c k n e s s a c r o s s the j u n c t i o n overlap area imply that  i s not u n i f o r m a c r o s s 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 o c c u r i n the number of q u a s i p a r t i c l e s w h i c h t u n n e l f o r a g i v e n i n p u t o f energy.  At p r e s e n t , the p r e a m p l i f i e r n o i s e dominates  j u n c t i o n n o i s e 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 h i g h e r o u t p u t impedances w i l l improve t h i s 2.  Linearity  situation.  '  Another p o i n t of concern i s the dependence o f w upon the type o f p a r t i c l e , i t s energy  (E) and the geometry o f t h e j u n c t i o n .  I n t h e i d e a l s p e c t r o m e t e r , w(E) would be a c o n s t a n t k so t h a t 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 t o the energy  p a r t i c l e o f 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.  AE which  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 a l p h a 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 o f s e v e r a l gas and c o n d u c t o r d e t e c t o r s (see eg. Dearnaley and N o r t h r o p , 1966). f o r t h i s are t h a t 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 t h e gap i n the semiconductor  a  and  semi-  The  reasons  energy  remain e f f e c t i v e l y c o n s t a n t as the p r i m a r y  p a r t i c l e slows down and t h a t the c o u n t e r s are s u f f i c i e n t l y l a r g e t h a t t h e i n t e r a c t i o n of t h e i r b o u n d a r i e s w i t h p r o c e s s e s t a k i n g p l a c e near p a r t i c l e t r a c k may  be s a f e l y i g n o r e d .  the  As d i s c u s s e d below, n e i t h e r  c o n d i t i o n h o l d s i n the s u p e r c o n d u c t i n g t u n n e l j u n c t i o n d e t e c t o r . The energy gap i n the s u p e r c o n d u c t o r temperature  i s , of c o u r s e ,  dependent such t h a t i t v a n i s h e s i n the hot r e g i o n near  the  p a r t i c l e t r a c k c e n t e r , f i n a l l y r e a p p e a r i n g and i n c r e a s i n g i n magnitude as the temperature the b a t h .  of the s u p e r c o n d u c t i n g f i l m s r e l a x e s back to t h a t of  I n a sense, the s i t u a t i o n i s analogous  to t h a t i n semi-  c o n d u c t o r d e t e c t o r s when the d i s o r d e r l e f t i n the wake o f a d e n s e l y 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 d e p a r t u r e from l i n e a r i t y o f t e n r e f e r r e d t o as the " f i s s i o n d e f e c t " . I n c h a p t e r 7, a t t e n t i o n was r a t e a t which phonons escaped  drawn t o the f a c t t h a t the  from the t u n n e l j u n c t i o n p l a y e d an  r o l e i n d e t e r m i n i n g the temperature  important  r e l a x a t i o n time T and c o n s e q u e n t l y  w.  I t would seem t h e r e f o r e t h a t c o n d i t i o n s a t the b o u n d a r i e s of the t u n n e l j u n c t i o n d e t e c t o r 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 v a l u e o f  w.  -177C l e a r l y , c o n s i d e r a b l y more t h e o r e t i c a l and  experimental  e v i d e n c e i s r e q u i r e d t o e s t a b l i s h the v a r i a t i o n o f w w i t h p a r t i c l e type and energy i n s u p e r c o n d u c t i n g  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  Efficiency  I f t h e j u n c t i o n i s to be u s e f u l as a s p e c t r o m e t e r , i t s h o u l d be s u f f i c i e n t l y massive t o s t o p i o n i z i n g r a d i a t i o n so t h a t a l l i t s energy i s d e p o s i t e d i n t h e j u n c t i o n .  The s t o p p i n g e f f i c i e n c y of a s i n g l e  t u n n e l j u n c t i o n i s l i m i t e d by i t s mass pXA w h i c h , f o r t h e p r e s e n t specimen, —7 8 i s a p p r o x i m a t e l y 2 x 10 g — a f a c t o r 10 s m a l l e r than f o r contemporary semiconductor  detectors.  The a r e a A cannot be made much l a r g e r than a t  p r e s e n t because the j u n c t i o n c a p a c i t a n c e would become s u f f i c i e n t l y l a r g e t o p r e v e n t the c u r r e n t p u l s e from b e i n g observed; o n l y a t the expense of d e c r e a s i n g  X c o u l d be i n c r e a s e d but  ( e q u a t i o n 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 w h i c h t u n n e l and the speed of response of the d e t e c t o r system; p c o u l d be i n c r e a s e d , but o n l y some 55%, by u s i n g l e a d i n s t e a d of t i n . I t i s c o n c e i v a b l e however, w i t h the p r e s e n t r a p i d growth o f 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 t e c h n i q u e s , t h a t i t may  be p o s s i b l e to overcome t h i s handicap  by m a n u f a c t u r i n g 4.  (Schroen,  1968)  of t h i n f i l m j u n c t i o n s  them i n t h r e e - d i m e n s i o n a l a r r a y s .  Advantages Over C o n v e n t i o n a l  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 c o n v e n t i o n a l semiconductor  spectrometers  i n the measurement of n u c l e a r and X r a d i a t i o n o  w h i c h 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 r e p o r t e d 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 g o a l w i t h a r e a s o n a b l e amount of f u r t h e r developmental  effort,  (see s e c t i o n C) As mentioned i n paragraph  2, s t o p p i n g e f f i c i e n c y might be  i n c r e a s e d so t o be comparable to s e m i c o n d u c t o r s , j u n c t i o n i n three-dimensional arrays. however, how  by f a b r i c a t i n g  I t remains to be  the  calculated,  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 t h i c k n e s s a c r o s s the  many j u n c t i o n s r e q u i r e d to s t o p an e n e r g e t i c 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 r e s o l u t i o n . - -  -1785.  Disadvantages  f o r O p e r a t i o n as a  An o b v i o u s shortcoming  Spectrometer  i s the o p e r a t i n g temperature  1.2 K w h i c h n e c e s s i t a t e s a l i q u i d h e l i u m c r y o s t a t .  of  Even i n l a b o r a t o r i e s  where a s u p p l y of h e l i u m i s a v a i l a b l e , the b u l k and c o m p l e x i t y of a c r y o g e n i c system would p r o b a b l y d e t e r a l l but the most s t r o n g l y m o t i v a t e d p o t e n t i a l users. To p l a c e a s u p e r c o n d u c t i n g d e t e c t o r on l i n e w i t h an a c c e l e r a t o r - p r o d u c e d beam i n v o l v e s s t i l l o t h e r d i f f i c u l t i e s . the t i n y j u n c t i o n a r e a , t h e s o l i d a n g l e subtended e x t e r n a l source i s v e r y s m a l l i n d e e d .  Because of  by the j u n c t i o n s to an  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 o n l y a t the expense o f 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 t o m i n i m i z e  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 c o u l d w e l l prove t o be a c h a l l e n g i n g c r y o g e n i c and n u c l e a r e n g i n e e r i n g problem.  Too h i g h a  p a r t i c l e f l u x would tend t o o v e r h e a t 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 r e n d e r i t i n o p e r a t i v e . A f u r t h e r h i n d r a n c e to r o u t i n e use i s the tendency  of t h i n  f i l m t u n n e l 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 s t o r a g e o r t h e r m a l c y c l i n g ( c f . Chapter  6).  Workers i n the a r e a of development of t u n n e l  j u n c t i o n a r r a y s 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 a r e a w i t h p h o t o - r e s i s t and growing  the o x i d e l a y e r w i t h an oxygen glow d i s c h a r g e .  A l t h o u g h none seems i n s u p e r a b l e , these o b s t a c l e s w i l l doubt impede the development o f a p r a c t i c a b l e s u p e r c o n d u c t i n g C.  no  spectrometer.  F u t u r e Work 1.  Use of J u n c t i o n s w i t h L a r g e r Dynamic R e s i s t a n c e ( r = 3V/3I) The  s i g n a l to n o i s e 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 o f j u n c t i o n s w i t h r e a s o n a b l y l a r g e (500-1000°-) output' impedances, f o r not o n l y 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 n o i s e i s a minimum ( W o l l and 1962).  I n the p r e s e n t work, the magnitude o f r was  Herscher,  l i m i t e d by the  r e s t r i c t i o n o f w o r k i n g w i t h t i n and t h e d e t e r i o r a t i o n of t u n n e l i n g 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 e q u a t i o n 3- 9 , i t may  be seen t h a t r <*  exp(A(T)/kT)  -179which immediately  s u g g e s t s two ways i n w h i c h h i g h e r dynamic r e s i s t a n c e  j u n c t i o n s c o u l d be r e a l i z e d i n f u t u r e :  t h e use o f m a t e r i a l s w i t h l a r g e r  energy gaps (2A(T)) and t h e use o f lower o p e r a t i n g t e m p e r a t u r e s .  Lead  and n i o b i u m , f o r i n 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 t u n n e l 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. R o w e l l but Nb l e s s so ( G i a e v e r , 1 9 6 3 ) ) .  Lower o p e r a t i n g temperatures t h a n 1 K 3  are r e a d i l y o b t a i n e d w i t h l i q u i d He  b u t , of c o u r s e , a t t h e expense o f  considerable complication i n the a n c i l l a r y The  (1963),  cryogenics.  problems a s s o c i a t e d w i t h t h e r m a l c y c l i n g a r e e a s i l y  o b v i a t e d by d e s i g n o f a system i n which t h e specimens a r e kept a t h e l i u m t e m p e r a t u r e s from t h e 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 a r e no l o n g e r o f i n t e r e s t .  I n f a c t , s h o u l d i t be n e c e s s a r y ,  fundamental r e a s o n why an e v a p o r a t o r  t h e r e i s no  c o u l d n o t be b u i l t i n t o a c r y o s t a t  so t h a t a s e t o f j u n c t i o n s need n e v e r be exposed t o a n y t h i n g b u t vacuum or i n e r t helium 2.  atmosphere.  Thermal D e c o u p l i n g  o f J u n c t i o n s from  Substrate  Two b e n e f i t s would accrue from t h e r m a l l y d e c o u p l i n g t h e tunnel j u n c t i o n from the s u b s t r a t e :  the recombination  probability W  would be d e c r e a s e d and t h e a m b i g u i t y  i n w caused by t h e d i f f u s i o n o f  h e a t from t h e s u b s t r a t e t o t h e j u n c t i o n would be e l i m i n a t e d . w i t h e x p e r i m e n t s done on s i n g l e s u p e r c o n d u c t i n g  In connection  s t r i p p a r t i c l e detectors  ( S p i e l e t 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 t h e r m a l l y i s o l a t e d a t h i n f i l m s t r i p from t h e s u b s t r a t e 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 o f about 300 A o f 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 t h e work o f C h r i s t y (1960), c o n s i s t s o f bombarding t h e s u b s t r a t e ( p r i o r t o f i l m e v a p o r a t i o n ) w i t h 100-200 eV e l e c t r o n s i n t h e presence o f 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 t h e d e t a i l e d a n a l y s i s o f c h a p t e r  r e l a x a t i o n time T was e x t r a c t e d from p h o t o g r a p h i c  7 i n which t h e p u l s e  pulse data, i t i s  e v i d e n t much o f t h e u n c e r t a i n t y c o u l d be reduced by t h e use o f a w i d e r 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 o f t h e c u r r e n t p u l s e waveform t o much s h o r t e r t i m e s . the dominant n o i s e source  Furthermorej  t h e p r e a m p l i f i e r used was  so t h a t any i n n o v a t i o n r e d u c i n g t h i s f a c t o r  would, o f c o u r s e , be b e n e f i c i a l as w e l l .  -180-  APPENDIX A  JUNCTION PREPARATION A.  Substrate The  Preparation s u b s t r a t e s used were o r d i n a r y m i c r o s c o p e s l i d e s cut on  s i d e t o have o v e r a l l dimensions 2 cm x 7.6  one  cm and s c o r e d t o d i v i d e i t i n t o  f i v e a r e a s of 1 cm x 2 cm as shown i n f i g u r e A - l ( a ) .  I n t h i s way,  a l l five  j u n c t i o n s c o u l d be e v a p o r a t e d and dc t e s t e d a t once and then the s u b s t r a t e c o u l d r e a d i l y be b r o k e n up i n t o the f i v e s m a l l e r s u b s t r a t e s f o r p u l s e  testing  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 " p r o c e s s the s l i d e s had  c l e a n e d " by the m a n u f a c t u r e r  (Corning),  t o be c l e a n e d a g a i n because of the h a n d l i n g 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 . "Teepol" detergent  C l e a n i n g was  done w i t h  a f t e r which the s u b s t r a t e s were r i n s e d s e q u e n t i a l l y i n  h o t , c o l d and d i s t i l l e d water f o l l o w e d by hot methanol.  Fairly  thick,  o  probably  5,000-10,000 A, s i l v e r c o n t a c t s were then evaporated through an  a p p r o p r i a t e 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 s u b s t r a t e s were s t o r e d 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 b e i n g 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 s u b s t r a t e was hot methanol and mounted i n the B.  Evaporation 1.  r i n s e d again i n  evaporator.  Procedure  Base F i l m ( f i g u r e A - l ( b ) ) Evaporation  took p l a c e i n a CVC  (Model CVE-15) o i l d i f f u s i o n -7 o pump e v a p o r a t o r w i t h a base p r e s s u r e of l e s s than 5 x 10 T o r r . The 2000 A —6 t h i c k bottom f i l m was evaporated a t a p r e s s u r e of 2-5 x 10 T o r r and a r a t e P  of a p p r o x i m a t e l y  2000 A/min.  A s h u t t e r p l a c e d between the vapour source and  the s u b s t r a t e masked the s u b s t r a t e from the evaporant u n t i l the source  was  u n i f o r m l y h o t , i m p u r i t i e s were d r i v e n o f f and e v a p o r a t i o n was w e l l under  way.  -181-  Microscope S l i d e  666  i o  (a)  •  t-i l n  *  • i n  i ! n  -7.6 cm— Contact  i !  t  n  ...  2  1  *  Arrangement Vapour D e p o s i t e d Ag c o n t a c t s  (b)  -\ rh n rh h r •1 P P Pi Mi i i L  n  n  1  •n  ! n  1 n  Base F i l m E v a p o r a t i o n .  •  •  rH J - I,! 1  (c)  i "! Lpl  yr~i  4  i i i r-i i r-i  r-i  I  I  rL  rh Lpl  i i ! r-l !  n  Oxidation 0.2  mm  -ilr-rh  r - i T »/' ™T "V U i I  (d)  rh  J  i -I  i  Cross F i l m  i  1  r n.  i  1  V  1  r  r ~t  T l i r l  Evaporation  Figure A - l Junction Preparation  1 T  0.2  mm  -1822.  Oxidation Some 30 minutes a f t e r t h e base f i l m e v a p o r a t i o n was complete,  the d i f f u s i o n pump was v a l v e d o f f and d r y oxygen was s l o w l y l e a k e d i n t o t h e b e l l j a r u n t i l t h e p r e s s u r e was about 1/3 atmosphere.  The s u b s t r a t e 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 h e a t e r , was t h e n warmed s l o w l y ( i n about one hour) up t o 110°C and k e p t a t t h a t temperature f o r 15-20 h o u r s .  A f t e r c o o l i n g t o room temperature i n t h e oxygen  atmosphere,  the specimens were exposed t o ambient room a i r f o r about one minute f o l l o w i n g w h i c h t h e b e l l j a r was e v a c u a t e d .  Presumably t h e b r i e f exposure t o 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 h u m i d i t y , p r o v i d e d t h e c a t a l y t i c a c t i o n r e q u i r e d t o make t h e 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 o f s u c c e s s w i t h t i n j u n c t i o n s n o t s u b m i t t e d t o t h i s phase o f t h e p r o c e d u r e emphasizes i t s i m p o r t a n c e . ( I t s h o u l d be p o i n t e d o u t t h a t t h e b e n e f i c i a l e f f e c t o f a b r i e f exposure t o room a i r was n o t d i s c o v e r e d u n t i l r a t h e r l a t e i n t h e e x p e r i m e n t a l program so t h a t i t s r o l e was n o t 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 t h e v a g a r i e s o f  t h i n f i l m s w i l l a p p r e c i a t e t h a t one h e s i t a t e s t o tamper w i t h a r e c i p e t h a t works.) 3.  Top F i l m E v a p o r a t i o n  A f t e r t h e p r e s s u r e had been reduced t o t h e u s u a l base l e v e l -7 of l e s s than 5 x 10 T o r r , t h e 2000 A t h i c k top f i l m was e v a p o r a t e d a t t h e 0  same p r e s s u r e and r a t e as t h e l o w e r f i l m i n t h e geometry shown i n f i g u r e A-l(d).  The s u b s t r a t e was a t room temperature throughout b o t h e v a p o r a t i o n s .  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 r e a s o n a b l y c o n t r o l l e d environment from t h e time t h e s u b s t r a t e was i n s t a l l e d u n t i l t h e completed j u n c t i o n s were removed f r o m t h e e v a p o r a t o r t o have t h e e l e c t r i c a l l e a d s attached. A p h o t o m i c r o g r a p h o f 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. 4.  (The w h i t e speck on t h e m i d d l e r i g h t i s a dust p a r t i c l e ) .  Attachment o f E l e c t r i c a l Leads and Mounting E l e c t r i c a l c o n n e c t i o n t o t h e t h i n f i l m s was made w i t h f i n e  (0.003 i n . ) g o l d w i r e s i n d i u m - s o l d e r e d t o t h e s i l v e r c o n t a c t s .  (None o f  t h e s e j o i n t s , e a s i l y made w i t h a m o d e r a t e l y warm, v e r y c l e a n s o l d e r i n g i r o n ( f r e e o f any f l u x ) , was ever observed t o f a i l ) .  The g o l d l e a d s were then  i n d i u m - s o l d e r e d t o t h e t e r m i n a l s o f t h e dc o r p u l s e t e s t h o l d e r (see f i g u r e s 4-2 and 4-3) a f t e r w h i c h t h e j u n c t i o n s were mounted i n t h e h e l i u m dewar  -183ready 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 d r y n i t r o g e n o r h e l i u m gas atmosphere.  T y p i c a l l y , t h e time r e q u i r e d t o e f f e c t t h e t r a n s f e r o f  specimens f r o m e v a p o r a t o r  t o h e l i u m dewar w i t h a l l l e a d s a t t a c h e d 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 m i n u t e s ; throughout t h i s p e r i o d the j u n c t i o n s were, o f n e c e s s i t y , exposed t o room a i r . C.  Evaporation 1.  Apparatus  Masks The masks used t o d e l i n e a t e t h e evaporated s t r i p s were  made f r o m 0.003 i n . B e r y l l i u m - C o p p e r  shim s t o c k w h i c h combines ease o f  p h o t o e t c h i n g w i t h t h e d e s i r a b l e p r o p e r t i e s o f low t h e r m a l e x p a n s i o n , 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 t o c r e a s i n g .  low  To ensure t h a t t h e  masks would f i t c l o s e l y and u n i f o r m l y a g a i n s t t h e s u b s t r a t e s , t h e masks r e 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 t o + 0.001 i n . ) and t h e s u b s t r a t e h o l d e r 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 t h e mask, h e l d i n p l a c e by t h e w e i g h t o f t h e h o l d e r , (see f i g u r e A-2)  The s u b s t r a t e h o l d e r moved o n l y i n a v e r t i c a l  c h a n g i n g o f masks w h i l e under vacuum was a c h i e v e d  plane;  by r o t a t i o n o f t h e mask  table 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 d e s i g n was the d e s i r e t o m i n i m i z e t h e penumbra r e g i o n on t h e f i l m s w h i c h , a c c o r d i n g t o Rowell  (1968), may g i v e r i s e t o v a r i o u s u n f a v o u r a b l e  junction characteristics.  I n s p e c t i o n o f photomicrographs o f 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 g o a l was l a r g e l y 2.  Substrate  achieved.  Holder  The s u b s t r a t e h o l d e r s e r v e d a t w o - f o l d purpose:  to a l i g n  and f i r m l y p r e s s t h e g l a s s s l i d e s on top of t h e masks and t o heat the s u b s t r a t e u n i f o r m l y t o r o u g h l y 100°C i n vacuum.  As c o n s t r u c t e d , see f i g u r e A-2,  the h o l d e r was machined f r o m copper w i t h an i n t e r n a l nichrome w i r e (.4°,/ft) heater.  M i c a sheets p r o v i d e d t h e r e q u i r e d e l e c t r i c a l i n s u l a t i o n .  measurement was e f f e c t e d w i t h a c o p p e r - c o n s t a n t a n  Temperature  thermocouple w i t h room  temperature r e f e r e n c e , mounted d i r e c t l y on the h o l d e r body ( v i s i b l e on t h e photograph i n f i g u r e A - 2 ) . E n e r g i z e d by a f i l t e r e d dc power s u p p l y , t h e h e a t e r was c a p a b l e o f m a i n t a i n i n g 110 ± 2°C f o r p e r i o d s of 16-20 h o u r s .  -184-  Fig. A-2:  F i g . A-3:  Evaporation  Apparatus  P h o t o m i c r o g r a p h o f Sn-Sn T u n n e l J u n c t i o n ( M a g n i f i c a t i o n : X100)  -1853.  E v a p o r a t i o n Sources Commercial e v a p o r a t i o n s o u r c e s o r " b o a t s " , o b t a i n e d from  R. D. M a t h i s Co., 1345 G a y l o r d 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 o f W, Mo or Ta s t o c k , the p r e f e r a b l e  f o r m of s o u r c e was t h e s o - c a l l e d "open" type (No. S5) which c o n s i s t e d o f a s i n g l e p i e c e of m a t e r i a l ( a p p r o x i m a t e l y 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 m i d d l e as i n the p l a n v i e w s k e t c h e d below.  A l l t h r e e r e f r a c t o r y m e t a l s were used w i t h l i t t l e t o choose among them. Because o f 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 t o bend i n t o a shape c o m p a t i b l e w i t h the e v a p o r a t o r e l e c t r o d e s b u t , a p a r t from t h a t , performed q u i t e a c c e p t a b l y . A second t y p e of s o u r c e found t o 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 " t y p e (No. S17A, see s k e t c h )  w h i c h tended t o g i v e n o n - u n i f o r m 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  t h a t the "hot s p o t " of e v a p o r a t i o n wanders p e r i o d i c a l l y a l o n g the l e n g t h of the b o a t s e v e r e l y 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 w h i c h the vapour is ideally distributed. 4.  F i l m T h i c k n e s s Measurement A q u a r t z c r y s t a l o s c i l l a t o r was used t o measure f i l m t h i c k n e s s  and r a t e o f e v a p o r a t i o n . The p r i n c i p l e o f o p e r a t i o n and p r a c t i c a l d e t a i l s a r e d i s c u s s e d i n a r e v i e w by Behrndt (1966).  There, i t i s shown t h a t the  s h i f t i n t h e fundamental f r e q u e n c y f of a c r y s t a l on w h i c h a f i l m of t h i c k n e s s 6s has been d e p o s i t e d i s g i v e n by  -186where p i s the d e n s i t y of the m a t e r i a l b e i n g d e p o s i t e d , p ^ i s the d e n s i t y of q u a r t z and N i s the f r e q u e n c y c o n s t a n t w h i c h , 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 t h e a p p r o p r i a t e v a l u e s f o r the c o n s t a n t s i n A - l  y i e l d s the u s e f u l r e s u l t  Sf=  f o r p i n g cm  -3  Hz  (A-2)  ° and 6s i n Angstroms. The  commercial  u n i t u s e d — a Sloan Instruments C o r p o r a t i o n  D e p o s i t T h i c k n e s s M o n i t o r , Model D T M - 3 — i n c l u d e d c r y s t a l , c r y s t a l  oscillator  h e r m e t i c a l l y s e a l e d i n sensor head (not shown i n f i g u r e A - 2 ) , v a r i a b l e f r e q u e n c y o s c i l l a t o r and f r e q u e n c y meter. I n p r a c t i c e , because g e o m e t r i c a l 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 c o n v e n i e n t to c a l i b r a t e crystal 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) , ^ substrate t  (A-3)  where a i s a c o n s t a n t which depends on p and the c r y s t a l - s u b s t r a t e e v a p o r a t i o n source geometry. To e v a l u a t e a ( P b ) , two independent were used.  methods of c a l i b r a t i o n  One method c o n s i s t e d of w e i g h i n g the Pb f i l m s w i t h a m i c r o -  b a l a n c e and d e t e r m i n i n g t h e i r t h i c k n e s s from the f i l m a r e a (as measured w i t h a t r a v e l l i n g microscope)  and the b u l k d e n s i t y ; t h e o t h e r method i n v o l v e d  the d i r e c t measurement of the f i l m t h i c k n e s s w i t h a Na l i g h t i n t e r f e r o m e t e r (a S l o a n I n s t r u m e n t s "Angstrometer",  Model M100).  The r e s u l t s are  summarized below:  Method  ct(Pb)Hz/A  Gravimetric  3.1  Interferometrie  3.2  o  A p p l y i n g approximate  g e o m e t r i c a l c o r r e c t i o n s t o a(Pb) y i e l d s 7.3  Hz/A  -187w h i c h i s c o n s i s t e n t , w i t h i n e x p e r i m e n t a l e r r o r , w i t h the v a l u e o  p/1.78 = 6.4  Hz/A  p r e d i c t e d from e q u a t i o n  The v a l u e of a(Sn) was e q u a t i o n A-2,  A-2.  d e r i v e d f r o m ot(Pb) by u s i n g  giving a(Sn) = a(Pb) p. /p  = 2.02  Hz/A  -188-  APPENDIX B  dc CHARACTERISTICS OF LEAKY  A.  SPECIMENS  Introduction T h i s appendix i s a summary o f work c a r r i e d out i n a t t e m p t i n g t o  u n d e r s t a n d and s y s t e m a t i z e "leaky" superconducting  the experimental  tunnel junctions.  r e s u l t s obtained w i t h s o - c a l l e d (In t h i s context, "leaky" i s  used t o d e s 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 i n c o m p a t i b l e w i t h t h e 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 s c r i b e d i n Chapter 2 ) . I f such j u n c t i o n s a r e mentioned a t a l l i n t h e l i t e r a t u r e , i t i s u s u a l l y o n l y i n p a s s i n g so t h a t t h e new worker i n t h e f i e l d , who p r e p a r e s a s e t o f 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 t o d i f f e r markedly from t h e " t y p i c a l " r e s u l t s g i v e n i n most a r t i c l e s ; i s l e f t t o wonder what went wrong. fill  The p r i m a r y  f u n c t i o n o f t h i s s e c t i o n then i s t o attempt t o  t h i s gap i n t h e l i t e r a t u r e f o r t h e newcomer by (1) d e s c r i b i n g some o f  the l i t t l e - p u b l i c i z e d "bad" t u n n e l j u n c t i o n r e s u l t s and (2) g i v i n g a g u i d e t o p l a c e s i n t h e 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 a r e d i s c u s s e d . P a r t B i s devoted t o a p r e s e n t a t i o n o f t h e r e s u l t s and p a r t C to  their analysis.  Some o f t h e u n r e s o l v e d problems i n a n a l y z i n g t h e  l e a k y j u n c t i o n c h a r a c t e r i s t i c s a r e o u t l i n e d i n p a r t 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 o f a l e a k y Sn-Sn02~Sn  tunnel 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 o b t a i n e d f o r  the Pb-Pb 0 -Pb specimens d i s c u s s e d i n Appendix C.) x y  The 4 t e r m i n a l  network by w h i c h these X-Y p l o t s were o b t a i n e d i s s i m i l a r t o t h a t a l r e a d y d e s c r i b e d i n Chapter 5.  I t s h o u l d be noted t h a t a l l these curves a r e taken  i n z e r o a p p l i e d magnetic f i e l d b u t no e f f o r t was made t o s h i e l d t h e specimens from t h e e a r t h ' s f i e l d o f a p p r o x i m a t e l y  0.2 G.  -189-  — •  Figure B - l :  Typical  1.22 2. 01 2.97  I-V  K K K  C h a r a c t e r i s t i c s f o r "Leaky" Specimens  -1902.  M a g n e t i c F i e l d Dependence of S u p e r c u r r e n t The  supercurrent,  I-V  c h a r a c t e r i s t i c s of most specimens showed some  i e . the j u n c t i o n c o u l d pass a f i n i t e c u r r e n t w i t h o u t  developing  a v o l t a g e between the f i l m s .  Because b o t h the dc Josephson  e f f e c t and  s u p e r c o n d u c t i n g m e t a l l i c f i l a m e n t s t h r o u g h the i n s u l a t i n g  l a y e r c o u l d 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 d e t e r m i n e the magnetic f i e l d dependence of the maximum s u p e r c u r r e n t (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  e x c e p t t h a t the i n p u t t o the X a x i s was current passing  l i k e t h a t of f i g u r e  a voltage p r o p o r t i o n a l to  magnetic f i e l d was  5-1,  the  t h r o u g h the H e l m h o l t z c o i l s p r o d u c i n g the magnetic  p a r a l l e l t o the p l a n e of the j u n c t i o n . )  t o z e r o ; I was  (I ) c  field  To produce the ^ ^ ~ ^ p l o t , the c r  t  s e t a t some l e v e l B^ w i t h the j u n c t i o n c u r r e n t I e q u a l  then s l o w l y i n c r e a s e d u n t i l the v o l t m e t e r was  observed t o  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 t r a c e d out the recorder increased  then b e i n g p r o p o r t i o n a l t o  t o B^  and  +  the curve of f i g u r e C.  A n a l y s i s of  I  i ( ^)' B  c  r  t  T  he  f i e l d was  1.  then  the o p e r a t i o n r e p e a t e d t h e r e b y e v e n t u a l l y  generating  B-2.  Results  F o r purposes of d i s c u s s i o n i t i s c o n v e n i e n t to c o n s i d e r of f i g u r e B-1  by  i n two  p a r t s : (a) 0.1  mV  < V and  I-V C h a r a c t e r i s t i c f o r V > 0.1 Theory p r e d i c t s and  (b) 0 $ V $ 0.1  the c u r v e s mV.  mV  experiments show (see Chapters 2 and  6)  t h a t 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 c u r r e n t f o r b i a s v o l t a g e s V<2A/e should evident  d e c r e a s e markedly w i t h d e c r e a s i n g from a comparison of f i g u r e s B-1  not t r u e f o r the l e a k y specimens.  temperature. and  6-1  I t i s immediately  t h a t such b e h a v i o u r i s  T h i s s u g g e s t s t h a t the c u r r e n t i n t h i s  r e g i o n of the c h a r a c t e r i s t i c does not a r i s e from t u n n e l i n g 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 c o n s i d e r a b l y  l e s s than t h a t of  the i n s u l a t i n g l a y e r when b o t h 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  d i o d e i n p a r a l l e l w i t h numerous f i 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 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.  R^. It  would seem t h a t such a model e x p l a i n s the g e n e r a l shape of the observed  -191-  -192-  F i g u r e 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 with Metallic. Filaments  -193c h a r a c t e r i s t i c but t h e r e s t i l l e x i s t s an excess c u r r e n t the o r i g i n of w h i c h , as d i s c u s s e d i n s e c t i o n D, i s not f u l l y  understood.  That some of the c u r r e n t p a s s i n g between the f i l m s was to  due  t u n n e l i n g i s i n d i c a t e d by t h e sharp i n c r e a s e i n c u r r e n t a t b i a s v o l t a g e s  c o r r e s p o n d i n g t o t w i c e the s u p e r c o n d u c t i n g energy gap The temperature was  2).  dependence o f the energy gap e x h i b i t e d by t h e s e specimens  checked by p l o t t i n g A(T)/A(0) a g a i n s t T/T  energy gap a t temperature the v o l t a g e a x i s .  , where A(T) i s o n e - h a l f  t h e o r e t i c a l curve  (figure  Supercurrent  the  T o b t a i n e d e x p e r i m e n t a l l y be e x t r a p o l a t i n g t o  U s i n g the g e n e r a l l y accepted v a l u e s of A(0) = 1 . 1  and t r a n s i t i o n temperature  2.  (see Chapter  T  £  = 3.72  meV  K, gave f a i r agreement w i t h t h e  2-2).  (V = 0, F i g u r e B ^ l )  As mentioned e a r l i e r , I  . -B p l o t s l i k e t h a t o f f i g u r e crit °  B-2  r  were made i n an attempt  t o e s t i m a t e the r e l a t i v e magnitudes o f the dc  Josephson and m e t a l l i c f i l a m e n t s u p e r c u r r e n t s .  The r e a s o n i n g behind  t e s t i s t h a t the dc Josephson c u r r e n t i s h i g h l y magnetic f i e l d (see Chapter  2) and may  this  dependent  be quenched f o r v e r y s m a l l v a l u e s o f B.  On  the  o t h e r hand, t h e 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 i l a m e n t s u p e r c u r r e n t s h o u l d be v e r y h i g h (Anderson,  1964)  as the f l u x quanta a r e expected t o pass  t h r o u g h 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 superconducting s h o r t s .  Upon t h i s p r e m i s e , t h e magnitude of the s u p e r c u r r e n t  due t o s u p e r c o n d u c t i n g 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 t a k e n t o be t h a t s u p e r c u r r e n t p r e s e n t a t t h e l o w e s t minimum d e t e c t e d i n the I . -B p l o t and t h e remainder was assumed t o be dc Josephson c u r r e n t I crxt J  T  (a) Josephson C u r r e n t To c o n f i r m the assumption t h a t dc Josephson c u r r e n t was  indeed b e i n g o b s e r v e d ,  f i e l d current I of I  T  = I  T  zero  and the p e r i o d i c i t y of the magnetic f i e l d dependence  were compared w i t h t h e o r y . The  by  n  the magnitude o f the maximum observed  ( c f . Chapter  t h e o r e t i c a l maximum dc Josephson c u r r e n t i s g i v e n  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 h e r e , the low temperature normal tunneling resistance R  was  estimated  by decomposing the a c t u a l I-V -1 and s e t t i n g (R ) e q u a l t o the n  n  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  slope  w h i c h 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).  r a t i o y = I ^ / I ^ was  The  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 o t h e r e x p e r i m e n t s . (Langenberg e t 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 crit r  l i k e f i g u r e B-2  may  be e s t i m a t e d  from the number of minima N  an i n t e r v a l of magnetic f i e l d AB so t h a t T' = AB/N*.  1  occurring i n  From Chapter 2,  the  t h e o r e t i c a l p e r i o d between s u c c e s s i v e minima of the Josephson c u r r e n t 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  t h e 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  d i m e n s i o n of the j u n c t i o n normal to B (see i n s e t , F i g u r e B-4). 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  f o r the t h r e e specimens c o n s i d e r e d ,  Both T'  i t i s evident  that,  the agreement between T' and T i s q u i t e  p o o r . I t must be remembered, however, t h a t the s a m p l i n g 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 — crit c  6  c o r r e s p o n d e d t o r e l a t i v e l y l a r g e magnetic f i e l d i n c r e m e n t s o f b - 0.23 so t h a t one  G  does not expect t o be a b l e 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 c l o s e ^ b . the s k e t c h below. )*- T-H  T h i s p o i n t i s i l l u s t r a t e d f o r a v e r y s i m p l e case i n There, i t i s assumed t h a t b = 2T/3  and  i t is  observed form, N' = 3 •actual form, N = 6  quite evident  t h a t N',  the a c t u a l number N.  t h e number of minima o b s e r v e d , would be about When the d a t a of f i g u r e B-4  are corrected  approximately  t o t a k e t h i s e f f e c t i n t o a c c o u n t , the agreement between experiment theory i s q u i t e  1/2  and  reasonable. I n the l i g h t of t h e s e f i n d i n g s , i t may  dc Josephson s u p e r c u r r e n t  was  be concluded t h a t  p r e s e n t a l t h o u g h i t s p r e c i s e magnitude remains  somewhat o b s c u r e d . (b) M e t a l l i c F i l a m e n t C u r r e n t The  a r e a A^ of m e t a l l i c f i l a m e n t s or  "superleaks"  -195-  I  1 0  Fig.B-5: S t r u c t u r e i n I-V  / / — J 0 C h a r a c t e r i s t i c Near V  v =0  -196r e q u i r e d t o account f o r t h e o b s e r v e d s u p e r c u r r e n t  i s an e x t r e m e l y  small  f r a c t i o n o f t h e g e o m e t r i c a l j u n c t i o n a r e a and i t i s e n t i r e l y p l a u s i b l e t h a t i m p e r f e c t i o n s o f t h i s e x t e n t c o u l d occur i n t h e i n s u l a t i n g l a y e r . Consider, B-3.  According  f o r example, t h e j u n c t i o n o f f i g u r e B-2 and  t o Anderson and R o w e l l  (1963), t h e c r i t i c a l c u r r e n t s o f 7 -2 f i n e l y d i v i d e d specimens n e v e r exceeds 10 A cm w h i c h , f o r I = 1.7 mA -10 2 (see f i g u r e B-3) p l a c e s a lower bound o f about 1.7 x 10 cm on A . -10 2 T h i s i s c o m p a t i b l e w i t h t h e v a l u e A = L/aR = 1.8 x 10 cm where ° 4 - 1 - 1 L = 10 A i s t h e assumed l e n g t h o f t h e f i l a m e n t s , a = 10 ft cm i s the f  f  f  f  assumed f i l a m e n t c o n d u c t i v i t y (Anderson and R o w e l l , 1963) and R^ i s t h e f i 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 o f t h e 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 g e o m e t r i c a l a r e a A o f t h a t 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 t h a t A^/A^IO and t h e r e a r e 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 junctions studied. Bardeen (1962) f i n d s t h a t t h e c r i t i c a l c u r r e n t o f 2 3/2 s m a l l specimens a t t e m p e r a t u r e T i s p r o p o r t i o n a l t o (1 - t ) t = T/T^.  where  I t might be expected t h a t 1^ s h o u l d behave s i m i l a r l y as a f u n c t i o n  of t e m p e r a t u r e b u t u n f o r t u n a t e l y l ^ ~ c r  B  t  p l o t s were made a t o n l y one  temperature (1.2 K) making i t i m p o s s i b l e t o check t h a t h y p o t h e s i s . Further evidence of the i n f l u e n c e of m e t a l l i c filaments i s the constant  c u r r e n t jump, shown i n f i g u r e B-1 a t about 1.0 mA, w h i c h i s  examined i n t h e n e x t 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 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 Q  the dc Josephson e f f e c t , t h e t r a n s i t i o n from V = 0 t o t h e o r d 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 s h o u l d be a s i n g l e d i s c o n t i n u o u s as shown i n f i g u r e 2- 7.  Such b e h a v i o u r  j u n c t i o n s as summarized i n f i g u r e B-5.  step  was n e v e r observed i n l e a k y  The manner i n w h i c h t h e v o l t a g e  i n c r e a s e d f r o m zero as t h e c r i t i c a l c u r r e n t was r e a c h e d , f e l l i n t o t h e two categories labelled (a),  (a) and (b) i n f i g u r e B-5.  w i l l be c o n s i d e r e d b r i e f l y h e r e ; type  The s m a l l jump, c a l l e d  type  ( b ) , i n which no d i s c r e t e jump i s  d e t e c t e d , i s d i s c u s s e d 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  -197v 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  s u p e r c o n d u c t i n g w i r e which i s d r i v e n normal by a c u r r e n t i n excess of i t s c r i t i c a l c u r r e n t 1^.  London (1950) shows t h a t the v o l t a g e drop V per u n i t  l e n g t h of an i n f i n i t e l y l o n g c y l i n d r i c a l w i r e c a r r y i n g c u r r e n t i £ i  is c  v = (i/2)n  i[i +  o  (i-(i /i) )£] 2  c  2 where Sl  Q  = 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 . jump of ( l / 2 ) f t i Q  c  T h i s r e l a t i o n p r e d i c t s t h a t when i = i  o c c u r s and  the s t r a i g h t l i n e V = ift « 0  a voltage  t h a t 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  p o i n t of t h i s d i s c u s s i o n i s t h a t 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 t y p e ( 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 a s s e s s i n v i e w 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  a r e 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 .  e v i d e n c e , however, one constant-current  Lacking  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 t h a t the  v o l t a g e jump, w h i c h appears o n l y a t the lowest  better small  temperatures,  i s the r e s u l t of s w i t c h i n g a l o n g the m e t a l l i c f i l a m e n t i n t e r m e d i a t e load D.  state  line. U n r e s o l v e d Problems i n the A n a l y s i s T h i s 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 l e a k y j u n c t i o n s which are not y e t f u l l y understood. S p e c i f i c a l l y , t h e s e 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 the s t r u c t u r e near V = 0 i n f i g u r e 1.  Excess  and  B-5.  Currents  I n the l e a k y specimens a c u r r e n t was  observed which exceeded  t h a t w h i c h c o u l d 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 with m e t a l l i c filaments. C a r e f u l e x a m i n a t i o n 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 0.15<V<2A/e  region  shows t h a t i n a d d i t i o n to the main c u r r e n t jump a t 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 Comparison of c u r v e s t a k e n at 1.2  been seen by o t h e r workers as w e l l and they have proposed.  a t 2A/n,  n = 2,3,4,5.  K and 2 K shows l i t t l e dependence of  a m p l i t u d e of the s i n g u l a r i t i e s upon temperature. explanations  i n the  the  These s i n g u l a r i t i e s have  t h i s s e c t i o n i s a s y n o p s i s of  the  -198T a y l o r and B u r s t e i n (1963) f i r s t n o t i c e d t h e 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 o f second o r d e r t u n n e l i n g 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 o r r e c o m b i n a t i o n .  The  r a t i o o f t h e magnitude o f t h e d i s c o n t i n u i t y i n t h e c u r r e n t ( J ^ ) a t eV = A f o r t h e double p a r t i c l e p r o c e s s e s t o t h e magnitude o f t h e c u r r e n t discontinuity  a  t  e  V  =  ( S c h r i e f f e r and W i l k i n s ,  J  1963)  A  J  In B - l ,  ^A f o r t h e s i n g l e q u a s i p a r t i c l e p r o c e s s e s i s  2 A  exp (-4<k >d) x  exp (-2<kj.>d)  <k > i s an average component o f wave number k p e r p e n d i c u l a r t o J>  t h e b a r r i e r and d i s t h e b a r r i e r t h i c k n e s s .  F o r an i d e a l , u n i f o r m b a r r i e r  J"A~ J^^/IO, b u t , as t h e a u t h o r s p o i n t o u t , t h e e f f e c t i v e d f o r t h e two p r o c e s s e s need n o t be e q u a l f o r a patchy o x i d e l a y e r .  The t h i n r e g i o n s o f  the f i l m would be emphasized more by t h e double p a r t i c l e than t h e s i n g l e p a r t i c l e p r o c e s s because of t h e e x p o n e n t i a l f a c t o r s i n  and J^A*  ^  e  double p a r t i c l e p r o c e s s e s a r e e s s e n t i a l l y independent o f t h e q u a s i p a r t i c l e d e n s i t i e s and a r e c o n s e q u e n t l y , i n agreement w i t h experiment, e s s e n t i a l l y independent o f t e m p e r a t u r e . Yanson e t a l (1965) r e p o r t e d t h e h i g h e r o r d e r s i n g u l a r i t i e s 2A/n, to  f o r n up t o 6, i n Sn-Sn j u n c t i o n s a t 1.61 K i n a f i e l d o f 33 Oe a p p l i e d  suppress t h e dc Josephson c u r r e n t .  S t r o n g e v i d e n c e was a l s o p r e s e n t e d  c l a i m i n g t h e i r j u n c t i o n s t o be f r e e o f m e t a l l i c f i l a m e n t s a c r o s s t h e i n s u l a t o r implying t h e r e f o r e that the s i n g u l a r i t i e s a r e 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 t h e b a r r i e r and n o t w i t h p r o c e s s e s i n v o l v i n g t h e presence o f s u p e r l e a k s . The magnetic f i e l d dependence o f t h e 2A/n s i n g u l a r i t i e s i n Pb-Pb t u n n e l j u n c t i o n s was s t u d i e d by Marcus (1966, A,B) whose  specimens  e x h i b i t e d s u p e r c u r r e n t s due t o b o t h Josephson t u n n e l i n g and m e t a l l i c filaments.  He found t h a t f o r a magnetic f i e l d normal t o t h e p l a n e o f t h e  j u n c t i o n ( B j . ) , t h e a m p l i t u d e and sharpness o f t h e s i n g u l a r i t i e s decreased  -199i n d i s c r e t e s t e p s w i t h i n c r e a s i n g Bj. i n the range 10<B <100 G.  The  <1  c o n c l u s i o n t h a t Marcus draws from t h i s e v i d e n c e , et a l , i s t h a t the 2A/n  s i n g u l a r i t i e s are due  i n c o n t r a d i c t i o n to Yanson  to a t u n n e l i n g p r o c e s s t a k i n g  p l a c e 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 o t h e r hand, i n t e r p r e t s the 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  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 b e i n g due  data  2A/n  t o 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 h i g h e r o r d e r quantum m e c h a n i c a l p r o c e s s e s w i t h a v o l t a g e t h r e s h o l d 2AP/n.  T a k i n g a p p r o p r i a t e v a l u e s of P and n, he was  t o 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  able  Douglass.  S t i l l a n o t h e r e x p l a n a t i o n of the subharmonic t u n n e l i n g phenomenon i s suggested by Zawadowski i n the same paper. t h a t t u n n e l i n g j u n c t i o n s a c t as a microwave g e n e r a t o r  I t i s w e l l known  at f i n i t e applied  v o l t a g e because of the ac Josephson c u r r e n t ; i t i s c o n c e i v a b l e t h a t the s i n g u l a r i t i e s c o u l d be c o m p l e t e l y microwave p h o t o n - a s s i s t e d  therefore  or p a r t i a l l y the r e s u l t of  quasiparticle tunneling.  I n a l a t e r paper, Zawadowski (1967) c o n s i d e r s b r i e f l y p r o b l e m of h i g h e r o r d e r t u n n e l i n g p r o c e s s e s c o n c l u d i n g t h i s type may  the  t h a t p r o c e s s e s of  be much more i n t e n s e than expected i f the t r a n s i t i o n s o f  the  q u a s i p a r t i c l e s t h r o u g h the b a r r i e r o c c u r a t some d e f e c t i n the b a r r i e r . S i n c e Yanson e t 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 l i n g 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 t h a t the s u p e r l e a k s  the samples of Marcus and those c o n s i d e r e d  h e r e are important  sense of b e i n g d e f e c t s i n the b a r r i e r a t which h i g h e r - o r d e r  in  o n l y i n the  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 b e h a v i o u r 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 l e a k y j u n c t i o n s ; i t appears t o f a l l i n t o the  g e n e r a l c a t e g o r i e s of f i g u r e B-5. approximately supercurrent was  two  R e t r a c e l o o p s , whose a r e a v a r i e d  i n v e r s e l y w i t h t e m p e r a t u r e , were always p r e s e n t but i n (a) was  r e s t o r e d at I = 1^ < I ^ »  restored at I = I  cr  t  a n  d  i n (b) the  supercurrent  . . S i m i l a r s t r u c t u r e t o case (a) has been observed crit by Yanson e t 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 a t 2 K  -200i n z e r o magnetic f i e l d .  The same workers a l s o found s t r u c t u r e r e s e m b l i n g  t h a t o f case (b) when they f i r s t a p p l i e d a magnetic  f i e l d o f tens o f  o e r s t e d s , t u r n e d i t o f f and r a n an I-V curve a t z e r o f i e l d . made whether t h e change i s permanent. to magnetic  No mention i s  Yanson e t a l a t t r i b u t e t h e i r r e s u l t s  f l u x b e i n g p a r t i a l l y t r a p p e d by i n h o m o g e n e i t i e s a t the edges o f  the s u p e r c o n d u c t i n g f i l m s ; t h i s e x p l a n a t i o n can p r o b a b l y be extended t o the p r e s e n t j u n c t i o n s f o r t h e m e t a l l i c f i l a m e n t s would s e r v e m a i n l y t o increase the f l u x trapping.  The i n f l u e n c e o f f l u x t r a p p i n g on t h e d e t a i l e d  b e h a v i o u r o f t u n n e l j u n c t i o n s has been f e l t by o t h e r workers as w e l l . J a k e l e v i c e t a l (1965), f o r i n s t a n c e , d i s c l o s e t h a t they found i t n e c e s s a r y to  s h i e l d t h e i r j u n c t i o n s from even the e a r t h ' s f i e l d w h i l e c o o l i n g them t o  p r e v e n t s e r i o u s a t t e n u a t i o n o f t h e 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 a t  eV = 2A/n w i t h n = 1,2,3 and as h i g h as 5 i n some specimens.  At b i a s voltages  eV<2A/5-0.22 mV, however, t h e s e s i n g u l a r i t i e s were obscured by a much more c o m p l i c a t e d 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  c o n s t a n t - v o l t a g e and c o n s t a n t - c u r r e n t s t e p s ; a g a i n , s i m i l a r  has been r e p o r t e d by Yanson e t a l (1965).  behaviour  I t seems q u i t e r e a s o n a b l e t h a t  these s t e p s a r e e v i d e n c e o f t h e ac Josephson  current c o u p l i n g w i t h the  r e s o n a n t modes o f t h e j u n c t i o n which would a c t as an open ended p a r a l l e l plate resonator. Eck et  T h i s 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  e t a l (1964, 1965); i t i s a l s o d i s c u s s e d a t some l e n g t h by Langenberg, a l (1966).  (Note added i n p r o o f :  S i n c e t h i s appendix was w r i t t e n , t h e author has become  aware o f a r e c e n t r e v i e w ( W i l k i n s , 1969) of t h e s u b j e c t o f m u l t i p a r t i c l e t u n n e l i n g which d e a l s w i t h some of t h e 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, t h a t t h e most d e s i r a b l e j u n c t i o n s f o r use as a charged p a r t i c l e d e t e c t o r on each s i d e o f t h e i n s u l a t i n g l a y e r (Pb-Pb).  tunnel  i s one w i t h l e a d f i l m s  T h i s appendix o u t l i n e s some  of t h e o b s t a c l e s encountered i n t h e c o u r s e of a t t e m p t i n g  t o 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 n o t p o s s i b l e to use t h e s e j u n c t i o n s i n the detector  experiment.  S e c t i o n A i s a summary o f t h e low temperature r e s u l t s o b t a i n e d w i t h Pb-Pb specimens and t h e c o n c l u s i o n s describes  derived therefrom.  Section B  t h e phenomenon o f nodule growth on the l e a d f i l m s and r e c o u n t s  the s t e p s t a k e n t o understand and e l i m i n a t e t h i s u n d e s i r a b l e Section C contains  side e f f e c t .  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 l e a d i n p a r t i c u l a r ; t h e plasma  o x i d a t i o n t e c h n i q u e 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 t o t h e convent 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 1.  Junction  Results Preparation  The r e c i p e f o l l o w e d i n p r e p a r i n g  these j u n c t i o n s , derived  from t h a t used by w o r k e r s i n t h e B e l l Telephone L a b o r a t o r i e s  ( R o w e l l , 1968),  i s e s s e n t i a l l y t h e same as t h a t g i v e n i n Appendix A f o r Sn-Sn specimens.  In  t h i s c a s e , t h e i n s u l a t i n g l a y e r i s a t h e r m a l l y grown o x i d e o f l e a d — p r o b a b l y PbO, see s e c t i o n C. of e v a p o r a t i n g  (The t e c h n i q u e o f T a y l o r and B u r s t e i n  (1963)  a v e r y t h i n l a y e r o f aluminum on top o f t h e base f i l m and  a l l o w i n g i t t o o x i d i z e t o f o r m t h e t u 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 o f i n c o m p l e t e c o n v e r s i o n . ) V a r i a t i o n s on t h e r e c i p e t h a t were t r i e d i n an e f f o r t t o make r e l i a b l e j u n c t i o n s w i l l be p o i n t e d out i n t h e d i s c u s s i o n o f s e c t i o n B.  -2022.  Low temperature  dc C h a r a c t e r i s t i c s  Without e x c e p t i o n , a l l Pb-Pb t u n n e l j u n c t i o n s t e s t e d a t l i q u i d helium temperatures—about d e f i n e d i n Appendix B.  50 a l t o g e t h e r — w e r e l e a k y i n t h e sense  I n many cases t h e Pb-Pb j u n c t i o n s were worse than  the l e a k y Sn-Sn j u n c t i o n s i n t h a t 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 l i n g was e v i d e n t , as denoted by t h e absence o f a sharp i n c r e a s e i n t h e c u r r e n t a t V = 2A/e. Furthermore,  t h e l a r g e s u p e r c u r r e n t e x h i b i t e d by t h e Pb-Pb  specimens was almost a l l due t o s u p e r c o n d u c t i n g m e t a l l i c f i l a m e n t s ( o r s u p e r l e a k s ) as i t was i n excess o f t h e expected maximum t h e o r e t i c a l dc J o s e p h s o n c r i t i c a l c u r r e n t and, u n l i k e t h e dc Josephson c u r r e n t , showed no s i g n i f i c a n t dependence upon t h e 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 m A — t h e maximum c u r r e n t output a v a i l a b l e from t h e dc s u p p l y used, see c h a p t e r 5 — a n d i n s e v e r a l j u n c t i o n s , t h e t h i n f i l m s s e r v i n g as c u r r e n t l e a d s were observed t o go normal and literally  "burn o u t " b e f o r e t h e j u n c t i o n would s w i t c h from t h e s u p e r c u r r e n t  characteristic.) D u r i n g an e x t e n s i v e e x p e r i m e n t a l program aimed a t i m p r o v i n g the t u n n e l j u n c t i o n performance^ t h e e f f e c t s o f v a r y i n g a l l t h e obvious parameters  such as e v a p o r a t i o n 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 s t u d i e d b u t w i t h o u t s u c c e s s . 3.  C o n c l u s i o n s from Pb-Pb J u n c t i o n S t u d i e s The o b v i o u s c o n c l u s i o n , o f c o u r s e , i s t h a t t h e l e a d o x i d e  formed on t h e base f i l m was v e r y patchy so t h a t , when t h e t o p f i l m was e v a p o r a t e d t o 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  a r e a c o n s i s t e d o f two l e a d f i l m s i n i n t i m a t e c o n t a c t f o r m i n g an e s s e n t i a l l y continuous superconducting s t r i p .  I n s p i t e o f t h e enormous e f f o r t p u t i n t o  growing a s u i t a b l e o x i d e l a y e r and t h e a s s e r t i o n s o f o t h e r workers  concerning  t h e ease o f d o i n g s o , t h e dc measurements i n d i c a t e t h i s g o a l was never reached.  T h e r e f o r e , because no Pb-Pb j u n c t i o n s c o u l d be prepared i n which  t h e s u p e r c u r r e n t c o u l d be suppressed nor a 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 c h a r a c t e r i s t i c o f 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  u s i n g Pb-Pb j u n c t i o n s i n t h e a l p h a p a r t i c l e d e t e c t i o n experiment had t o  be abandoned. One avenue o f 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 f o l l o w e d i n t h i s s t u d y was t h e e f f e c t on j u n c t i o n performance o f nodules observed t o  -203have d e v e l o p e d on the s u r f a c e o f the t h i n f i l m s . r e c o u n t s the d e t a i l s o f t h i s B.  The next s e c t i o n  investigation.  I n v e s t i g a t i o n o f Nodule Growth on Lead F i l m s M i c r o s c o p i c e x a m i n a t i o n o f Pb-Pb t u n n e l j u n c t i o n s , see  C - l and C-2,  r e v e a l e d t h a t n o d u l e s , h a v i n g 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 . A-3, n e v e r formed t h e s e b l e m i s h e s ) .  Now  ( T i n f i l m s , see eg.  but, i f  they form on the base f i l m b e f o r e the top f i l m i s evaporated and p r o j e c t through i t ,  figure  i f these growths are merely s u r f a c e  f e a t u r e s o f the completed j u n c t i o n they are not v e r y i m p o r t a n t  suggested.  figures  consequently  a mechanism f o r the f o r m a t i o n o f s u p e r l e a k s i s c l e a r l y  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 t h a t the nodules d i d  i n d e e d form on t h e base f i l m but when i t l a t e r became p o s s i b l e t o f a b r i c a t e Pb-Pb j u n c t i o n s which were a p p a r e n t l y n o d u l e - f r e e , they proved to be j u s t as l e a k y as t h e i r p r e d e c e s s o r s .  I t seems, t h e r e f o r e , t h a t the presence of  n o d u l e s must c o n t r i b u t e to the f o r m a t i o n o f s u p e r l e a k s but i t i s not the fundamental  reason why  the o x i d e p a t c h i n e s s p e r s i s t s .  films i s discussed b r i e f l y i n section 1.  D e s c r i p t i o n of  (Oxide growth on t h i n  C.)  Nodules  That the w h i t e specks o f f i g u r e C - l , photographed  with  r e f l e c t e d l i g h t , were i n d e e d growths on the s u r f a c e of the f i l m s and not p i n h o l e s i n the f i l m s was m a g n i f i c a t i o n ( f i g u r e C-2)  e s t a b l i s h e d by the o b s e r v a t i o n a t h i g h e r o f the shadows they c a s t and by the f a c t t h a t  the specks were opaque t o t r a n s m i t t e d l i g h t . e l e c t r o n microscope been squeezed  An attempt was made w i t h an  t o determine whether the nodules were m e t a l t h a t had  from s u r f a c e f i s s u r e s o r l a r g e i s l a n d s o f m e t a l o x i d e which  had formed on the s u r f a c e 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 d e t a c h i n g the l e a d f i l m s from the g l a s s s u b s t r a t e w i t h o u t 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 , p r o b a b l y no fundamental  under-  s t a n d i n g o f these growths can be r e a l i z e d ; the purpose of the p r e s e n t study was  s i m p l y t o l e a r n how 2.  t o p r e p a r e f i l m s f r e e of these d e f e c t s .  O r i g i n of Nodules-The R e s u l t of Thermal Treatment S u r p r i s i n g l y , the nodules were found to form whether a l e a d  f i l m was h e a t e d t o 395 K i n about 500 T o r r o f oxygen or c o o l e d to 1.2 K i n a h e l i u m atmosphere.  To observe t h i s , a s e t of j u n c t i o n s was  prepared  a c c o r d i n g t o the s c h e d u l e 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 c o m p l e t i o n o f s t a g e s b, c, d and a l i q u i d h e l i u m  -204-  (a) F i g . C-1:  (b)  P h o t o m i c r o g r a p h s o f Pb-Pb T u n n e l J u n c t i o n (a)  Before exposure  (b) A f t e r  F i g . C-2:  exposure  to L i q u i d to L i q u i d  Helium Helium  Photomicrograph showing ( M a g n i f i c a t i o n X1220)  Nodules  (X100)  -205exposure.  No n o d u l e s were v i s i b l e on the r e c e n t l y formed base  film  ( s t a g e b) b u t t h e y were a f t e r the base f i l m was heated t o e x p e d i t e o x i d e formation (stage c ) .  The n o d u l e s p e r s i s t , b e i n g v i s i b l e a l l a l o n g the base  f i l m , and appear t o p e n e t r a t e the c r o s s e d f i l m when i t i s e v a p o r a t e d (stage d ) .  (Photomicrograph C - l ( a ) was t a k e n a t t h i s p o i n t ; the v e r t i c a l  f i l m i s t h e 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 t o l i q u i d  h e l i u m t e m p e r a t u r e s caused f u r t h e r growth o f the n o d u l e s on b o t h f i l m s as i l l u s t r a t e d i n figure C-l(b).  As an a d d i t i o n a l check, f i v e o t h e r specimens  were p r e p a r e d s i m u l t a n e o u s l y on one s u b s t r a t e and then s e p a r a t e d such t h a t t h r e e were heated and two were c o o l e d ( t o 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 r e g a r d t o the s u r f a c e d e n s i t y o f n o d u l e s formed, was  discernible. T h i s 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 v i e w p o i n t ,  a s u p e r c o n d u c t i n g t u n n e l 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 v i e w p o i n t , a c h e m i c a l p r o c e s s t h a t goes e q u a l l y w e l l a t r o u g h l y 1 and 400 K i s h a r d t o e n v i s a g e . One e x p l a n a t i o n suggested by Young (1968) i s t h a t the e v a p o r a t e d f i l m s a r e i n compression and t h a t the nodules r e s u l t from the d i f f e r e n t i a l e x p a n s i o n and c o n t r a c t i o n o f the g l a s s and l e a d upon t h e r m a l cycling.  A l t h o u g h t h e argument i s weakened somewhat by t h e 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 o t h e r workers who have a l s o used g l a s s s u b s t r a t e s 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 o f the next paragraph where i t i s seen t h a t the n o d u l e s d i d not form i n f i l m s of t h i c k n e s s l e s s o  t h a n 1000 A.  Such f i l m s ^ presumably, would f l o w more f r e e l y w i t h the g l a s s  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 t o r u p t u r e s which c o u l d g i v e r i s e to nodules. 3.  T e s t s Made t o I s o l a t e N o d u l e - P r o d u c i n g Parameters There a r e , of c o u r s e , a h o s t of parameters w h i c h a f f e c t a  g i v e n e v a p o r a t i o n . No c l a i m i s made t h a t the l i s t of parameters t e s t e d h e r e i s e x h a u s t i v e f o r one i s concerned o n l y w i t h t h o s e w h i c h c o u l d c o n c e i v a b l y g i v e r i s e t o the n o d u l e growth.  A " t e s t " o f the e f f e c t o f a parameter change  c o n s i s t e d o f p r e p a r i n g a s e t of j u n c t i o n s ( u s u a l l y f i v e ) under the a p p r o p r i a t e c o n d i t i o n s , examining t h e f i l m s m i c r o s c o p i c a l l y f o r t h e presence of nodules i m m e d i a t e l y f o l l o w i n g e v a p o r a t i o n , c o o l i n g the j u n c t i o n s t o 77 K i n a h e l i u m atmosphere and r e - e x a m i n i n g them f o r nodule growth.  (Once the t h e r m a l -  -206t r e a t m e n t 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  means o f p r o d u c i n g nodule growth f o r t e s t purposes,  chosen as the  as the h e a t i n g c y c l e  d u r i n g the o x i d a t i o n s t a g e c o u l d c o n c e i v a b l y be o m i t t e d 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 n e v e r t h e c o o l i n g c y c l e i n u s i n g them. words, a t h i n f i l m t h a t i s s u b j e c t t o nodule p r o d u c t i o n a t low i s u s e l e s s whether o r not nodules First,  are produced a t e l e v a t e d  In other temperatures  temperatures.)  the parameter v a r i a t i o n s o f i n t e r e s t a r e l i s t e d  e x p l a i n e d ; the r e s u l t s are summarized i n the f i n a l paragraph. e x p e r i m e n t a l l i m i t a t i o n s o n l y one parameter was  and  (Within  changed a t a  time.)  (a) Evaporant P u r i t y Two  grades and forms of Pb were used as  evaporant:  99.999% pure b a r s t o c k f r o m 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 e v a p o r a t i o n s o u r c e s were used i n each i n s t a n c e . (b) E v a p o r a t i o n Source E v a p o r a t i o n s were performed w i t h " b o a t s " o f s i m i l a r  geometry ( t y p e S5, Appendix A) but made of Ta, Mo o r  W.  (c) Order of F i l m D e p o s i t i o n To t e s t the h y p o t h e s i s t h a t the nodules were due c o n t a m i n a t i o n e v o l v e d from the i n i t i a l h e a t i n g of a new  charge of  the u s u a l o r d e r o f e v a p o r a t i n g the f i l m s , see f i g u r e A - l , was  to  evaporant,  reversed.  (d) S u b s t r a t e 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 t e c h n i q u e s ,  a g l a s s s u b s t r a t e of the type d e s c r i b e d i n Appendix A was 1/3  prepared  i n which  of the s c o r e d a r e a 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  very d i l u t e h y d r o f l u o r i c acid.  with  The r e m a i n i n g 1/3 had been c l e a n e d i n the  u s u a l 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 e v a p o r a t i o n r a t e on nodule growth i s somewhat obscured by the u n a v o i d a b l e v a r i a t i o n s i n r a t e w h i c h occur d u r i n g t h e course of a g i v e n e v a p o r a t i o n .  I n g e n e r a l , an average d e p o s i t i o n r a t e  o  o f about 1000 A/min  was  achieved.  (f) F i l m Thickness The  importance  o f f i l m t h i c k n e s s t o nodule  formation  -207-  was  s t u d i e d f i r s t of a l l w i t h a " p i n - h o l e " s o u r c e (see Appendix A) w h i c h  gave f i l m s whose t h i c k n e s s v a r i e d c o n t i n u o u s l y  a l o n g the l e n g t h of  the  s u b s t r a t e but w h i c h o t h e r w i s e were formed under i d e n t i c a l c o n d i t i o n s . j u n c t i o n s were made i n w h i c h the t h i c k n e s s of one  f i l m was  o  1000  greater  Later,  than  o  A and  the t h i c k n e s s of the o t h e r l e s s than 1000  A.  (g) Summary The negative  r e s u l t s of a l l the t e s t s except the l a s t were  i n the sense t h a t the nodule p o p u l a t i o n produced was  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 parameter.  I t was  those changes of  o  1000 A. F i l m s t h i c k e r than 1000 A, prepared s i m u l t a n e o u s l y ones, formed n o d u l e s i n the u s u a l manner. Conclusions  eventual  form,  h e a t e d o r c o o l e d , i n a f i l m of t h i c k n e s s l e s s than  °  The  evaporation  found, as s t a t e d e a r l i e r , t h a t the n o d u l e s d i d not  e i t h e r when the f i l m was  4.  apparently  t o the  thinner  from T e s t s  consequences of t h i s d i s c o v e r y w i t h r e g a r d t o  the  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  serious.  H i g h d e t e c t i o n e f f i c i e n c y demands t h a t the f i l m s of the t u n n e l 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 s e v e r e h a n d i c a p . I t should  a l s o be mentioned t h a t these tests do not r u l e out  the p o s s i b i l i t y t h a t the n o d u l e s are an h y d r o x i d e of l e a d formed when the j u n c t i o n s are t r a n s p o r t e d w a t e r has  i n room a i r .  P e r s o n a l e x p e r i e n c e has  shown t h a t  a v e r y 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 o t h e r workers  (eg. R o w e l l (1968) and  Schroen (1968)) have emphasized the need of  exposure of l e a d f i l m s to water vapour.  avoiding  As t h e r e i s no apparent r e a s o n  why  o  t h i s p r o c e s s would d i s c r i m i n a t e a g a i n s t f i l m s t h i n n e r than 1000 be t h a t 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  A, i t may  expansion-contraction  mechanism suggested e a r l i e r , the p o i n t s of r u p t u r e b e i n g more s u s c e p t i b l e t o c o r r o s i o n than the s u r r o u n d i n g The  "oxidized" surface.  c o r r o s i v e e f f e c t o f mercury on m e t a l s l i k e l e a d 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 ,  t h a t mercury c o n t a m i n a t i o n i n the e v a p o r a t o r may respect.  (Although  not be n e g l i g i b l e i n t h i s  the e v a p o r a t o r 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 temporarily  i n s t a l l e d i n i t a t an e a r l i e r time, and  been  the p o s s i b i l i t y of  -208r e s i d u a l mercury vapour c o n t a m i n a t i o n e x i s t s . ) C.  Oxide Growth on Lead One  preceding  Films  q u e s t i o n which i s not y e t u n d e r s t o o d and upon w h i c h the  work sheds no l i g h t i s the reason why  i t proved i m p o s s i b l e  grow a c o n t i n u o u s l a y e r of o x i d e on the l e a d f i l m s .  to  Compounding the problem  i s the f a c t t h a t few d a t a are a v a i l a b l e on the o x i d a t i o n of l e a d a t 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  p u b l i c a t i o n s w h i c h are a v a i l a b l e and  g i v e a v e r y b r i e f summary of  the  their  f i n d i n g s w h i c h are r e l e v a n t t o t u n n e l j u n c t i o n f a b r i c a t i o n . o  Anderson and Tare (1964) p r e p a r e d t h e i r 2500 A t h i c k l e a d f i l m s by e v a p o r a t i o n  from a w e l l outgassed 99.999% pure l e a d charge at  the  o  r e l a t i v e l y slow r a t e of about 250 A/min. i n j e c t e d i n t o the e v a p o r a t i n g the f i l m s was  v e s s e l and  Doses of oxygen were s u b s e q u e n t l y the i n i t i a l uptake of the oxygen by  found t o be v e r y r a p i d , (complete i n l e s s t h a n one m i n u t e ) .  i m p o r t a n t r e s u l t was  An  t h a t 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 o x i d e s t r u c t u r e w h i c h , 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 t e c h n i q u e s , was  found t o be o r t h o r h o m b i c l e a d monoxide (PbO).  A p a r t from the slow o  evaporation  r a t e used by these workers compared t o the 1000  p r e s e n t e x p e r i m e n t , no s p e c i a l p r e c a u t i o n s ensure the growth of PbO  A/min r a t e of  appear t o have been t a k e n t o  t h a t have been o m i t t e d  i n work h e r e .  A l a t e r paper by the same a u t h o r s (1965) extends the work on o x i d a t i o n r a t e of pure l e a d f i l m s and  the  the  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 l e a d i m p u r i t i e s such as bismuth and t h a l l i u m . S e v e r a l papers d e a l i n g g e n e r a l l y and problems i n m e t a l s are a v a i l a b l e as w e l l . T. N. Rhodin e t a l , J . F. C h i t t u m , D. W.  t h e o r e t i c a l l y with oxidation  (see eg. the f o u r a r t i c l e s P a s h l e y and H. F i s c h m e i s t e r  c o n f e r e n c e r e p o r t a r r a n g e d by Benard, 1965).  in a  Of p a r t i c u l a r u s e f u l n e s s  r e v i e w by U h l i g (1967) which i n c l u d e s a f a i r l y e x t e n s i v e f o l l o w s the sequence (1) p h y s i c a l a d s o r p t i o n of 0^ oxides layer.  chemisorption  at d i s c r e t e s i t e s and The  is a  bibliography.  B r i e f l y , he s t a t e s t h a t the i n t e r a c t i o n of oxygen w i t h a c l e a n m e t a l by d i s s o c i a t i o n of 0^ and  by  f o l l o w e d i n most  surface instances  of 0 atoms, (2) n u c l e a t i o n of m e t a l  (3) f o r m a t i o n  and growth of a c o n t i n u o u s  time r e q u i r e d f o r t h i s to happen i s q u i t e s h o r t .  oxide  For example,  -209a c o n t i n u o u s Cu^O l a y e r i s formed i n about 6 sec i n 1 T o r r o f 0^ a t 550°C; even s h o r t e r times a r e r e q u i r e d a t h i g h e r p r e s s u r e s  or lower temperatures.  C e r t a i n l y t h e 15-20 hours d u r i n g w h i c h t h e l e a d f i l m s o f t h i s experiment were h e a t e d t o 100°C and exposed t o 500 T o r r o f 0^ would seem t o be ample t i m e t o form a c o n t i n u o u s o x i d e l a y e r . The g l o w - d i s c h a r g e o x i d a t i o n t e c h n i q u e  d e s c r i b e d by Schroen (1968)  i s perhaps t h e b e s t way t o a v o i d t h e u n c e r t a i n t i e s i n h e r e n t i n t h e r m a l l y grown o x i d e s .  While admitting that present  t h e o r e t i c a l understanding  the p r o c e s s e s i n v o l v e d i s based upon t e n t a t i v e models, Schroen c l a i m s  of that  t h i s plasma method has been r e f i n e d t o the p o i n t t h a t r e p r o d u c i b l e and s t a b l e t u n n e l j u n c t i o n s a r e r o u t i n e l y made.  -210-  APPENDIX D  EFFECT OF FINITE FILM RESISTANCE ON TUNNEL JUNCTION CHARACTERISTICS A.  Introduction T h i s a p p e n d i x , of c o n c e r n p r i m a r i l y t o those who  are  interested  i n problems connected w i t h t h i n f i l m t u n n e l j u n c t i o n f a b r i c a t i o n , d e s c r i b e s two  l i t t l e - p u b l i c i z e d e f f e c t s w h i c h 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 e x p e r i m e n t a l s t e p s t a k e n i n a t t e m p t i n g t o u n d e r s t a n d these e f f e c t s . discussed  Though an i n t e r e s t i n g s i d e l i g h t , the problems  h e r e are not germane t o an u n d e r s t a n d i n g of the t u n n e l  junction  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 ( S e c t i o n B) was  samples, when t e s t e d w i t h the c o n v e n t i o n a l i n f i g u r e D - l ( a ) d i s p l a y e d an I-V  t h a t many c r o s s e d - f i l m  4-terminal  technique sketched  curve w i t h n e g a t i v e s l o p e .  Pedersen  and  Vernon (1967) observed s i m i l a r b e h a v i o u r i n the p a r a l l e l - f i l m type o f j u n c t i o n (Sn-Sn) s k e t c h e d i n f i g u r e D - l ( b ) . The  second e f f e c t d i s c o v e r e d  was  t h a t the s l o p e of the  I-V  curve depended on <|>, the a n g l e of i n t e r s e c t i o n of the f i l m s . Theoretical  ( S e c t i o n C) and e x p e r i m e n t a l model s t u d i e s  D and E) o f these phenomena show t h a t they may  (Sections  be a t t r i b u t e d to the  finite  r e s i s t a n c e o f 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.  E x p e r i m e n t a l O b s e r v a t i o n s of E f f e c t s w i t h Tunnel 1.  "Negative  Junctions  Resistance"  F i g u r e D - l ( c ) shows the I-V c h a r a c t e r i s t i c s o b t a i n e d two  with  t y p i c a l Sn-Sn c r o s s e d - f i l m t u n n e l 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 A l - P b and Pb-Pb j u n c t i o n s .  temperatures the s l o p e of the I-V  I t i s evident  curve i s n e g a t i v e ,  that at c e r t a i n  ( w i t h I and V measured  as shown i n f i g u r e D - l ( a ) ) , so t h a t the j u n c t i o n e x h i b i t s " n e g a t i v e r e s i s t a n c e . " I n a d d i t i o n , the magnitude of the s l o p e i s temperature dependent so 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 n e g a t i v e v a l u e a t 296 K,  that through  -211-  #1  Figure D-l(c):  T y p i c a l I-V  V  (mV)  #11  C h a r a c t e r i s t i c s of 2 Sn T u n n e l i n g  V  n T  >V)  Junctions  |  -212z e r o around 4 K t o a s m a l l p o s i t i v e average v a l u e a t 3 K and below. These r e s u l t s a r e i n c o m p a t i b l e w i t h the s i m p l e  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 t o d e s c r i b e f o u r t e r m i n a l measurement o f t u n n e l j u n c t i o n s as shown i n f i g u r e D-2. I  A  -vV^/-  ?R  I n the f i g u r e ,  a  r  e  2  the  D  -AVvV-  ~1  -WW-  -o< B  iR  2  2  and R  F i g u r e D-2 F o u r - T e r m i n a l E q u i v a l e n t C i r c u i t o f Tunnel J u n c t i o n  r e s i s t a n c e o f 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 resistance.  tunneling  F o r the p o l a r i t i e s shown, w h i c h are i d e n t i c a l w i t h those o f  f i g u r e D - l ( a ) , i t i s c l e a r t h a t 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  transition  temperature T^, t h e I-V c h a r a c t e r i s t i c s o f t u n n e l j u n c t i o n s composed o f s u p e r c o n d u c t i n g f i l m s have s m a l l r e g i o n s o f n e g a t i v e  s l o p e w h i c h a r i s e from  the r e l a t i v e magnitudes o f the energy gaps o f the s u p e r c o n d u c t o r s i n v o l v e d (see eg. F i g u r e 2-6); discussion).  such e f f e c t s are not o f i n t e r e s t i n the  present  As t h e r e i s no way i n which the c i r c u i t o f 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 required:  t h e t h e o r e t i c a l approach i s d i s c u s s e d i n s e c t i o n C and t h e  experimental  approach, c o n s i s t i n g o f 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  d i s c u s s e d i n s e c t i o n D. S e v e r a l o t h e r f e a t u r e s i n the I-V curves o f f i g u r e D - l ( c ) s h o u l d be n o t e d . 1.1  Breaks a p p e a r i n g i n the I-V curve a t 3 K between 0.6 and  mV are caused by t h e f i l m s b e i n g t e m p o r a r i l y d r i v e n normal by t h e  measuring c u r r e n t .  The r e v e r s a l o f s l o p e shown t o o c c u r a t about 1.2 mV  f o r sample I I i s caused by the measuring c u r r e n t h a v i n g d r i v e n the f i l m s completely  normal.  The f a c t t h a t t h e r e s i s t a n c e i s a p p r o x i m a t e l y  the same  a t 1.25 K and 3.0 K f o r b o t h 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 t h a t o n l y a v e r y s m a l l component o f the c u r r e n t f l o w i n g between the f i l m s was due t o t u n n e l i n g .  I t must be  emphasized, however, t h a t w h i l e the s o - c a l l e d " n e g a t i v e " r e s i s t a n c e i s  -213a 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 n o t 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  o f t h i s comes from t h e r e s u l t s o f Pedersen and Vernon (1967) whose j u n c t i o n s displayed conventional 2.  t u n n e l i n g b e h a v i o u r below t h e t r a n s i t i o n temperature.  S t r i p I n t e r s e c t i o n Angle-Dependence o f S l o p e L e t <K^ 90°) be t h e a n g l e o f i n t e r s e c t i o n o f t h e two t h i n  f i l m s t r i p s s e r v i n g as c u r r e n t t e r m i n a l s  (see f i g u r e D - l ( a ) ) .  Thus,  depending on t h e c h o i c e o f t e r m i n a l s used i n the measurements, a g i v e n 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 , with reference  t o f i g u r e s D - l ( a ) and D-2, ^ ( 0 ) = R..(180-<|>) as t h e  e f f e c t i v e r e s i s t a n c e s h o u l d be t h e t u n n e l i n g r e s i s t a n c e R^, w h i c h depends on t h e o v e r l a p a r e a and i s independent o f t h e c h o i c e o f c u r r e n t Experimentally, t o R.. (180-<j>).  terminals.  i t was found t h a t a t room t e m p e r a t u r e , R. (<)>) was not e q u a l T a b l e D - l summarizes t h e e x p e r i m e n t a l  r e s u l t s where  p = Rj (<J) l a r g e ) / R j (<}> s m a l l ) i s introduced  as a c o n v e n i e n t measure o f the j u n c t i o n r e s i s t a n c e asymmetry.  T h i s a n g l e dependence o f 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 model s t u d y , t h e r e s u l t s o f which a r e d i s c u s s e d  Sample ( P b - A l ) No.  i n s e c t i o n E.  Rjft) $ = 55°  R.Ofr = 125°) P  -.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  + .03  .53  .3+130  1.0  others  +.016 .3+130  T a b l e D - l : Dependence o f J u n c t i o n R e s i s t a n c e C.  R. (<j> = 55°) J  66-3  66-29 16  R. (ft) J <j> = 125°  experimental  on <j>  Theoretical Investigations The s i m p l e s t j u n c t i o n t o c o n s i d e r  i n d e t a i l i s the p a r a l l e l - f i l m  t y p e f o r , i f u n i f o r m t h i c k n e s s and w i d t h a r e assumed f o r t h e f i l m s and  -214i n s u l a t o r , the a n a l y s i s i s o n e - d i m e n s i o n a l . of t h i s j u n c t i o n can be d e s c r i b e d  I t t u r n s o u t t h a t the b e h a v i o u r  a n a l y t i c a l l y whereas t h e b e h a v i o u r o f 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  numerically.  As shown i n f i g u r e D - l ( b ) , the c u r r e n t  I i s fed into terminal 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 t h e j u n c t i o n V  b e i n g measured f r o m t e r m i n a l D t o B.  <}> = 180°  voltage  T h i s c a s e , which corresponds t o  ( c f . f i g u r e s D - l ( a ) and ( b ) ) , has been a n a l y z e d by Pedersen and  Vernon (1967) who f i n d t h a t R_. = V j / I i s g i v e n by R..(180 ) =a£R [csch(a£) + 2 R R ( R  + R )~ tanh  o  r  ;[  2  2  1  2  (aA/aJ-R^/ ( R ^ R ^ ; ( D - l )  where A i s t h e l e n g t h of t h e j u n c t i o n and 2 a  = g (r + r ) x  2  R-. = r , A 1 1 R  0  2  = r„A 2 .  . -1  Here, g i s t h e t u n n e l conductance per u n i t l e n g t h and r ^ , r  2  a r e the  r e s i s t a n c e s p e r u n i t l e n g t h o f 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 tunneling  resistance.  F i r s t o f a l l , i t i s c l e a r t h a t when b o t h f i l m s a r e s u p e r c o n d u c t i n g , R^ = R  2  = 0 = a, t h e measured r e s i s t a n c e R. becomes t h e a c t u a l  r e s i s t a n c e R^,.  O b v i o u s l y , f o r f i n i t e v a l u e s o f R^ and R  2>  R  tunneling ^ Rp and  f u r t h e r m o r e , because o f the n e g a t i v e term, t h e r e w i l l e x i s t v a l u e s o f R^ and  R  2  f o r which R  i s negative.  The e f f e c t o f 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 e q u a t i o n D-l i s s i m p l i f i e d t o t h e case of a symmetric j u n c t i o n , ie. R  = R  2  = R.  Then otA =  (R/2R ) T  and Rj(sym) = (RR /2)£ c o t h T  (R/2Rj,)^  - R/2,  <j>=180°  (D-2)  -215The  p l o t of equation  D-2, i n f i g u r e D-3 shows t h a t 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 o f R / R ^ , R^ (sym) i s negative. The  v a l i d i t y o f 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 b e i n g o b t a i n e d  between t h e e x p e r i m e n t a l l y  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 t h a t c a l c u l a t e d f r o m e q u a t i o n experimental  values  D-l using  f o r R ^ , R^ and R^'  A n o t h e r j u n c t i o n c o n f i g u r a t i o n t h a t may be t r e a t e d as a one p r o b l e m i s t h e cj> = 0° case ( c f . f i g u r e D - l ( b ) ) when t h e  dimensional  c u r r e n t I i s f e d 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 B on f i l m 2 w i t h t h e j u n c t i o n v o l t a g e V_. b e i n g measured from t e r m i n a l D t o C.  U s i n g t h e same e q u i v a l e n t  c i r c u i t as Pedersen and Vernon (1967) and  a p p l y i n g t h e a p p r o p r i a t e 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 , 1 2 * J  V°  0)  =  V  1 = (2R  V*  c s c h (2R/  V*  (D_3)  E q u a t i o n D-3, p l o t t e d i n f i g u r e D-3, shows t h a t the measured r e s i s t a n c e R^ becomes t h e t u n n e l i n g r e s i s t a n c e as R-H) and t h a t R . - K ) i n t h e l i m i t o f l a r g e f i l m resistance. Unfortunately,  i t i s not p o s s i b l e to deal with the crossed-film  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 t h e lumped e q u i v a l e n t i s a three-dimensional  network.  circuit  C o n s e q u e n t l y , t h e v o l t a g e V ( x , y ) between  the f i l m s , cannot be e x p r e s s e d i n a s i m p l e , c l o s e d form and i t would be n e c e s s a r y t o r e s o r t t o some form o f n u m e r i c a l D i f f e r e n c e o r t h e Monte C a r l o method.  a n a l y s i s such as t h e F i n i t e  ( B i n n i s and Lawrenson, 1963).  W h i l e d i f f e r i n g i n d e t a i l from those o f t h e 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 o f t h e <J> = 180° p a r a l l e l - f i l m case l e a d t o some g e n e r a l conclusions  f o r l i m i t i n g cases c o n c e r n i n g t h e e f f e c t o f f i n i t e f i l m r e s i s t a n c e  on t h e measured c r o s s e d  film junction resistance.  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 a r e e f f e c t i v e l y i n s u l a t e d from one a n o t h e r so t h a t t e r m i n a l s D and B have e s s e n t i a l l y t h e 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 ) / I > 0; f o r R >> R „ , f i l m s 1 j u a l and 2 a r e v i r t u a l l y a s i n g l e sheet w i t h , as i s c o n f i r m e d e x p e r i m e n t a l l y i n D  -216-  -217t h e 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 b e i n g n e g a t i v e w i t h r e s p e c t to B such that R D.  = (V  D  - V ) / I < 0. B  E x p e r i m e n t a l S i m u l a t i o n of C r d s s e d - F i l m J u n c t i o n s 1.  G r a p h i t e Coated  Paper  To s i m u l a t e 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 c o n d u c t i n g on one s i d e , was i n t o a shape r e s e m b l i n g two f i l m s " c r o s s i n g " a t an a c u t e a n g l e . f i g u r e D-4;  t h i s model corresponds  i n f a c t to R^ = 0 ) .  cut  (see  Equipotential lines  were p l o t t e d by s u p p l y i n g a s t e a d y c u r r e n t as shown and measuring  the v o l t a g e  a t every i n t e r s e c t i o n of a r e a s o n a b l y f i n e g r i d l a i d out on the g r a p h i t e surface.  The d i s t a n c e from the i n t e r s e c t i o n t o the end of t h e " f i l m s "  was  about 10 times t h e " f i l m " w i d t h so t h a t by t a k i n g c a r e to p l a c e the c u r r e n t e l e c t r o d e s c e n t r a l l y on t e r m i n a l s A and C a u n i f o r m c u r r e n t f l o w was V o l t a g e was  measured w i t h a F l u k e D i f f e r e n t i a l v o l t m e t e r , Model 881A,  e s s e n t i a l l y i n f i n i t e i n p u t impedance a t the n u l l p o i n t . f i g u r e D-4  obtained.  t h a t R. = (V_ - V_,)/I. < 0. j u a A—L  I t i s clear  having from  A r e s i s t a n c e of t h e same s i g n but  g r e a t e r magnitude was o b t a i n e d by p a s s i n g the c u r r e n t from A t o B and t e r m i n a l C.  For t h i s  grounding  "junction"  = P  R(102°) R(78°)  =  -135ft -99ft  =  ' *  T h i s s i m p l e t e s t p r o v i d e s f u r t h e r c o n f i r m a t i o n t h a t the r e a s o n t h a t a c t u a l t u n n e l j u n c t i o n s d i s p l a y e d an a n g l e dependent n e g a t i v e r e s i s t a n c e (Table D-l) was  because t h e i r t u n n e l r e s i s t a n c e was  sufficiently  s m a l l as t o make them l o o k e l e c t r i c a l l y l i k e a s i n g l e c o n d u c t i n g s h e e t . 2.  S o l d e r e d Manganin S t r i p s As a c l o s e r a p p r o x i m a t i o n 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 s o l d e r e d  t o g e t h e r a t t h e i r m i d d l e r e g i o n w i t h c a r e b e i n g taken t o make the s o l d e r l a y e r as u n i f o r m as p o s s i b l e .  (A rough e s t i m a t e shows R/R  T  = 10  for this  "junction") ' '•  E q u i p o t e n t i a l l i n e s mapped out u s i n g the t e c h n i q u e o u t l i n e d above were s i m i l a r to those o f the c o n d u c t i n g s h e e t .  Thus, the p r e d i c t i o n s  o f the p a r a l l e l - f i l m s t r u c t u r e i n t h e r e g i o n o f v e r y s m a l l t u n n e l i n g r e s i s t a n c e a r e borne o u t 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-  -2193.  Compressed Nichrome S t r i p s In  for  o r d e r t o study the b e h a v i o u r of c r o s s e d - f i l m j u n c t i o n s  o t h e r v a l u e s o f R/R^,, a l u c i t e j i g was  strips  (1/8 x 4 x .021  prepared  i n w h i c h two nichrome  i n . ) c o u l d be c o n s t r a i n e d to l i e one on top o f the  o t h e r and i n t e r s e c t a t 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 c o u l d then be v a r i e d by e x e r t i n g p r e s s u r e n o r m a l l y on the i n t e r s e c t i o n w i t h a small b a k e l i t e plunger. S t r a i n I n d i c a t o r , Model P-350, was  (see f i g u r e D-5)  (A Budd  used t o measure the l o a d developed  d e p a r t m e n t a l workshop h y d r a u l i c p r e s s . )  by  B e f o r e b e i n g p l a c e d i n the j i g ,  s u r f a c e s of the s t r i p s w h i c h were t o be superimposed were c l e a n e d p o l i s h e d w i t h f i n e carborundum paper (No.  the the  and  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 t h e 4 - t e r m i n a l 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 t h a t 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 l o a d s on the p l u n g e r and n e g a t i v e f o r l a r g e r l o a d s i n b o t h c o n f i g u r a tions.  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 t h a t 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 t o the l o a d p r e s s i n g the s u r f a c e s t o g e t h e r 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 p r e s s u r e than the b u l k r e s i s t a n c e R of the nichrome strips.  For l a r g e l o a d s ( s m a l l R ) , the s t r i p s a r e e s s e n t i a l l y a T  continuous  p i e c e l i k e the g r a p h i t e coated paper and R.. i s n e g a t i v e ; f o r s m a l l l o a d s ( l a r g e R ^ ) , t h e s t r i p s a r e i n s u l a t e d from each o t h e r making R^  positive.  The adequacy of t h e p a r a l l e l f i l m model to e x p l a i n these c r o s s e d 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 manner. R/R  T  F i g u r e D-3  e x p l o r e d i n the f o l l o w i n g  i n d i c a t e s t h a t i n the l i m i t o f cj> = 0, j / r p > 0 f o r a l l R  and i n the l i m i t <j> = 180°,  R  j /  R T  <  0  f  °r  R  j /  R  > T  2.88.  R  I t seems  r e a s o n a b l e t o assume t h e r e f o r e t h a t f o r s t r i p s c r o s s i n g a t 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 a l i n e a r sum of the two l i m i t i n g cases simplified  ( e q u a t i o n s D-2  be expressed  and D-3).  as  Thus, i n  form,  B = ( l - g ( c b ) ) [ ( A / 2 ) * c o t h ( A / 2 ) * -A/2]  where A = R(<b)/R  T>  B =  R  j /  R  a n c T  + B($)(2A)* csch(2A)* ,  (D-4)  * the m i x i n g 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 v a l u e s g(0) = 1 and  0(180) = 0.  The  e f f e c t i v e b u l k s t r i p r e s i s t a n c e R - (cb) i s taken t o be a f r e e parameter R  -220-  Press  B a k e l i t e Plunger Nichrome S t r i p  Lucite J i g Load  Figure U-V.  P r e s s e d Nirhrnme S r r i n " J n n r r i rms"  cell  -221because t h e c u r r e n t d i s t r i b u t i o n i n t h e s t r i p s i n t h e o v e r l a p r e g i o n a t a r b i t r a r y a) i s n o t known; i n t h e extreme c a s e s , R(0) = R(180) = 1.9 mfi/sq. f o r t h e s t r i p s used.  To r e l a t e e q u a t i o n D-4 t o t h e s i m u l a t e d j u n c t i o n  measurements i t i s n e c e s s a r y t o make some assumption between Rj, and t h e compressing  l o a d L.  c o n c e r n i n g the r e l a t i o n  The form chosen, which i s s i m p l e s t  t o t r e a t m a t h e m a t i c a l l y and y e t has t h e c o r r e c t a s y m p t o t i c b e h a v i o u r , i s Rj, = k(o))/L where k=k(<j>) i s a parameter t o be determined e x p e r i m e n t a l l y . (The <f> dependence o f k, l i k e t h a t o f R(<J>), a r i s e s from t h e unknown a n g u l a r dependence o f t h e c u r r e n t d e n s i t y i n t h e " j u n c t i o n " r e g i o n . )  For a given  l o a d L., t h e r e f o r e 1  A  ±  = R(q )L /k(q)) and B )  i  ±  = R  (cfOl^/k^).  W i t h e q u a l weight assumed f o r t h e e x p e r i m e n t a l v a l u e s B^(expt) - R ^ ( e x p t ,  <f>)L^/k(a>), e q u a t i o n D-4 was l e a s t squares f i t t e d t o  the d a t a o b t a i n e d f o r t h e 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 . v a l u e s o f t h e parameters  8, R and k w h i c h  minimized  ^[B (expt) - B ( t h e o r y ) ] ±  The  2  ±  a r e g i v e n 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  T a b l e D-2 L e a s t Squares F i t Parameters F i g u r e D-6 summarizes,the r e s u l t s .  The s o l i d c u r v e s r e p r e s e n t  e q u a t i o n D-4 e v a l u a t e d f o r t h e v a l u e s o f 3 g i v e n i n t a b l e D-2. (For r e f e r e n c e p u r p o s e s , t h e l i m i t i n g b e h a v i o u r o f e q u a t i o n D-4 i s shown by t h e dashed c u r v e s . )  The e x p e r i m e n t a l r e s u l t s have been reduced, as shown on t h e  graph, by t h e a p p r o p r i a t e b e s t f i t v a l u e s o f  R(tf>)  and k(a>).  -222-  ^  = (1-8)[(R/2R ) T  F i g u r e D-6:  J  coth(R/2R ) -R/2R ] + 3 W ^ ) * 2  T  T  csch(2R/R )  Comparison of P a r a l l e l f i l m t h e o r y w i t h Crossed S t r i p "Junction" data.  J  T  Nichrome  -223I t i s evident  t h a t t h i s s i m p l e 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) t h a t 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 v a l u e 1.9  c o n f i g u r a t i o n s , b e i n g r e a s o n a b l y c l o s e to  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  a s m a l l change i n B from the cb = 0 and r e q u i r e d t o f i t the 53° The  and  127°  important conclusion  cb = 180°  the  (2) t h a t  only  configuration values i s  data. t o be drawn from these s t u d i e s i s t h a t  the  nichrome s t r i p " j u n c t i o n " can s i m u l a t e , the b e h a v i o u r of t h i n f i l m  junctions  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  tunneling  r e s i s t a n c e s i n d e t e r m i n i n g the s i g n and magnitude of the measured j u n c t i o n resistance f o r a given terminal configuration.  The  a n g u l a r dependence of  the j u n c t i o n r e s i s t a n c e i s m e r e l y a n o t h e r m a n i f e s t a t i o n E.  A n g u l a r Dependence of the T h i n F i l m J u n c t i o n  of the same e f f e c t .  Resistance  T h i s s e c t i o n demonstrates t h a t the observed cb dependence of 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 s t o o d f r o m the nichrome s t r i p  (see T a b l e D-l) may  the  be under-  simulations.  A c o n v e n i e n t 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 s e c t i o n B and of R << R^,  to a n g l e dependence i s the q u a n t i t y p which i s d e f i n e d  appears i n the r i g h t hand column of T a b l e D - l .  ( I n the  in limit  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 t h a t R.. i s expected  t o be independent of the c h o i c e of c u r r e n t  and v o l t a g e  t e r m i n a l s and  s h o u l d tend t o 1 as i t does f o r 16 of the specimens of T a b l e D - l .  p  In  the  l i m i t of R >> R j , the c o n t i n u o u s sheet a p p r o x i m a t i o n , i t i s expected t h a t 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 v a l u e depending on the geometry and  tunneling  junction  conductance.)  By drawing smooth c u r v e s through the d a t a o b t a i n e d d i f f e r e n t nichrome s t r i p " j u n c t i o n s "  from  two  (samples a and b ) , i t i s p o s s i b l e  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. e v i d e n c e , depending on the s i g n of Rj(cb). r e s u l t i s t h a t , as s e t out i n T a b l e D-3,  The  Three r e g i o n s  are i n  s i g n i f i c a n t p a r t of  a l l the v a r i o u s  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 s i m u l a t e d  this  asymmetries by the nichrome s t r i p  model. I t may  be c o n c l u d e d t h e r e f o r e  to  t h a t , as i n the nichrome s t r i p  -224-  -225" j u n c t i o n s " , the a n g u l a r dependence of R. i n t h i n f i l m j u n c t i o n s i s s i m p l y  Region  Tunnel J u n c t i o n s E x h i b i t i n g Same Asymmetry  I  66-17,-21,-29  II  66-15,-19  III  66-3,-11,-18  T a b l e D-3'  J u n c t i o n R e s i s t a n c e Asymmetry Observed.  a n o t h e r d e m o n s t r a t i o n of the i n t e r r e l a t i o n between f i l m and t u n n e l i n g resistances.  -226-  BIBLIOGRAPHY  ABELES, B. & GOLDSTEIN, Y., (1965), Phys. Rev. L e t t e r s , 14, 595. AJZENBERG-SELOVE, F., e d . , (1960), N u c l e a r S p e c t r o s c o p y , P a r t A, P r e s s , New Y o r k . )  (Academic  AMBEGAOKAR, V. & BARATOFF, A., (1963), Phys. Rev. L e t t e r s , 10, 486; Phys. Rev. L e t t e r s , 11, 104. ANDERSON, J . R. & TARE, V.B., (1964), J . Phys. Chem., 68, 1482. 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