UBC Theses and Dissertations

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UBC Theses and Dissertations

Dielectric properties of thin insulating films Wilcox , Philip Stanley 1968

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DIELECTRIC PROPERTIES OF THIN INSULATING FILMS  by  PHILIP STANLEY WILCOX B.Sc,  U n i v e r s i t y of B r i t i s h Columbia, 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the_ Department of Electrical  Engineering  We accept t h i s t h e s i s as conforming required Research  t o the  standard  Supervisor,  Members of Committee  Head o f Department Members of t h e Department of E l e c t r i c a l  Engineering  THE UNIVERSITY OP BRITISH COLUMBIA December, 1968  In p r e s e n t i n g  t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s  an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e I further agree that permission f o r s c h o l a r l y p u r p o s e s may by h i s r e p r e s e n t a t i v e s .  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Columbia  thesis or  publication  g a i n s h a l l n o t be a l l o w e d w i t h o u t  of  that  Study.  Department  It i s u n d e r s t o o d t h a t c o p y i n g o r  permission.  Department  and  copying of t h i s  be g r a n t e d b y t h e Head o f my  of t h i s t h e s i s f o r f i n a n c i a l written  for extensive  I agree  for  my  ABSTRACT  The d i e l e c t r i c p o l a r i z a t i o n p r o c e s s e s , c o n d u c t i o n mechanisms and space charge e f f e c t s o c c u r i n g i n t a n t a l u m / t a n t a l u m p e n t o x i d e / metal d e v i c e s a r e i n v e s t i g a t e d .  The d i e l e c t r i c p r o -  p e r t i e s a r e a n a l y z e d on t h e b a s i s of an i o n i c r e l a x a t i o n  process  with a nearly f l a t d i s t r i b u t i o n of a c t i v a t i o n energies.  This d i s -  t r i b u t i o n l e a d s t o s t e p response  p o l a r i z a t i o n currents  following  an i n v e r s e time l a w . The e f f e c t o f an i n j e c t e d e l e c t r o n i c charge on t h e response  space  of t h e d e v i c e due t o the removal of a s t e p  v o l t a g e i s - a n a l y z e d and r e s u l t s a r e g i v e n d e m o n s t r a t i n g The d e v i c e s used e x h i b i t '  this effect.  a r e c t i f i c a t i o n behaviour.  F o r t a n t a l u m p o s i t i v e the c u r r e n t s f o l l o w a S c h o t t k y law and f o r tantalum negative, the.bulk Poole-Frenkel law. H y s t e r e s i s e f f e c t s a r e observed  as w e l l as t h e e f f e c t s o f a space charge on.the  S c h o t t k y law c u r r e n t s .  On one sample, s u f f i c i e n t l y h i g h f i e l d s 3  causes an i n c r e a s e i n t h e conductance by a f a c t o r o f 10  5 t o 10 .  For t h i s deformed sample, no h y s t e r e s i s . , o r r e c t i f i c a t i o n i s obs e r v e d and t h e c u r r e n t s f o l l o w a S c h o t t k y law f o r both p o l a r i t i e s . The v a l i d i t y of the i o n i c r e l a x a t i o n model i s d i s c u s s e d i n l i g h t of t h e observed The  d i e l e c t r i c losses at  low.temperatures.  e x p e r i m e n t a l r e s u l t s i n d i c a t e t h a t an e l e c t r o n i c r a t h e r than  an i o n i c process c o u l d be r e s p o n s i b l e f o r the d i e l e c t r i c l o s s e s .  i  TABLE OF CONTENTS ABSTRACT  i  TABLE OF CONTENTS  i i  LIST OF ILLUSTRATIONS  . .'v,  LIST OF SYMBOLS  .. v i i  ACKNOWLEDGEMENT. 1.  INTRODUCTION  2.  THEORY  4.  1  "  3  '.  2.1  D i e l e c t r i c P r o p e r t i e s of Amorphous F i l m s  3  2.2  Conduction i n T h i n I n s u l a t i n g F i l m s  7  2.3  3.  :ix  2.2.1  Schottky I n j e c t i o n  11  2.2.2  Poole-Frenkel Effect  11  2.2.3  F i e l d E m i s s i o n or T u n n e l l i n g .  12  2.2.4  Space Charge L i m i t i n g  14  Space Charge E f f e c t s i n Thin F i l m s  16  2.3«1  B u i l d up of a Space Charge  16  2.3.2  Decay of a Space Charge  17  2.3-3  E f f e c t on C o n d u c t i o n C u r r e n t s  22  2.3«4  E f f e c t on D i e l e c t r i c P r o p e r t i e s  22  EXPERIMENTAL PROCEDURES  '. .  23  3.1  Sample P r e p a r a t i o n  23  3-2  Counterelectrodes  25  3.3  E x p e r i m e n t a l Setup  26  3.4  Series Resistance  29  RESULTS  31  4.1  C u r r e n t Measurements  31  4.1.1  Charge and D i s c h a r g e C u r r e n t s  31  4.1.2  D i s c h a r g e C u r r e n t s as a F u n c t i o n of Voltage  36  Page 4.1.3 4.2  D.C. C o n d u c t i o n C u r r e n t s  43  B r i d g e Measurements  48  4.2.1  C a p a c i t a n c e and Loss Measurements  48  4.2.2  Temperature Dependence of C a p a c i t a n c e and Losses  51  4.2.3  Comparison o f B r i d g e and Step Response Values of t h e Losses  5.  6.  DISCUSSION  55 '  59  5.1  Step Response Measurements  59  5.2  D.C. C o n d u c t i o n C u r r e n t s  60  5.3 5.4  Comparison o f Samples #2 and #3 Frequency and Temperature Dependence o f the D i e l e c t r i c Properties..  62 63  5.5  The I o n i c R e l a x a t i o n Model  64  CONCLUSIONS  66  APPENDIX I  .  Debye E q u a t i o n APPENDIX I I  68 ,.  I o n i c R e l a x a t i o n Model  69 69  APPENDIX I I I  71  D i e l e c t r i c Response f o r a Uniform D i s t r i b u t i o n of A c t i v a t i o n Energies APPENDIX IV  71 75  S c h o t t k y Law, P o o l e - F r e n k e l E f f e c t and Simmons' D e f e c t Model  75  APPENDIX V  79  Zero F i e l d P o i n t and C a l c u l a t i o n o f Space Charge Decay Currents .'. APPENDIX V I  68  79 .  82  C a l c u l a t i o n of Zero F i e l d P o i n t and Space Charge D e n s i t y  82  Page APPENDIX V I I .  .  Space Charge L i m i t e d Currents. REFERENCES  85 85 86  iv  LIST OF ILLUSTRATIONS Figure  Page  2-1  Energy l e v e l diagram of a m e t a l / i n s u l a t o r / m e t a l d e v i c e . .  10  2-2  Energy l e v e l diagram f o r t u n n e l l i n g p r o c e s s e s  10  2-3  Energy l e v e l diagram f o r model  17  2- 4  E l e c t r o n i c space charge d i s t r i b u t i o n  19  3-1  A n o d i z a t i o n setup  24  3-2  C u r r e n t measurements  3-3  Test chamber f o r low temperature measurements  28  3-4  Test chamber f o r measurements  28  3- 5  S e r i e s r e s i s t a n c e example  30  4- 1  Charge c u r r e n t s f o r T a - n e g a t i v e v o l t a g e s  32  4-2  Charge c u r r e n t s f o r T a - p o s i t i v e v o l t a g e s  33  4-3  Discharge currents f o r Ta-negative voltages  34  4-4  Discharge currents f o r Ta-positive voltages  35  4-5  Log J v s . l o g t f o r d i f f e r e n t e l e c t r o d e s on sample # 3 . . .  37  4-6  D i s c h a r g e c u r r e n t s as a f u n c t i o n of v o l t a g e - sample # 3 .  38  4-7  D i s c h a r g e c u r r e n t s as a f u n c t i o n of v o l t a g e - sample # 2 .  40  4-8  Log J v s . l o g t f o r d i f f e r e n t e l e c t r o d e s on sample # 2 . . .  41  4-9  Leakage c u r r e n t as a f u n c t i o n of f i e l d - sample # 3  44  26  circuit  i n vacuum  4 - 1 0 Leakage c u r r e n t as a f u n c t i o n of f i e l d - sample # 3 . .  45  4-11  Temperature dependence of l e a k a g e c u r r e n t s  46  4-12  Leakage c u r r e n t s as a f a n e t i o n of f i e l d - sample # 2  47  4 - 1 3  Frequency dependence of  4-14  Temperature dependence of C K ' and C K " - sample # 1  52  4-15  Temperature dependence of C K ' and C K " - sample # 2  53  4-16  Frequency dependence of  C K  and  1  O  Q  Q  295°K  O  Q  C K'  at  C K"  O  and  at  C K" O  82°K  49  54  4 - 1 7 D i e l e c t r i c l o s s e s v s . r e l a x a t i o n time  56  4 - 1 8 Temperature dependence of C K ' and C K " - sample # 3  58  Q  v  O  Figure  Page  II-l  I o n i c r e l a x a t i o n model...  69  III-l  A p p r o x i m a t i o n of Debye e q u a t i o n - r e a l p a r t  72  III-2  A p p r o x i m a t i o n of Debye e q u a t i o n - i m a g i n a r y part....  72  III- 3  Step response curve  IV- 1  Image f o r c e i n S c h o t t k y e f f e c t  IV-2  B a r r i e r l o w e r i n g due t o S c h o t t k y e f f e c t  76  IV-3  Insulator defect structure  76  IV-4  Simmons' i n s u l a t o r d e f e c t s t r u c t u r e  77  . 73  vi  (  75  LIST OF SYMBOLS Angstroms Richardson's  constant  o n e - h a l f d i s t a n c e between p o t e n t i a l  minima  .slope of S h o t t k y law s l o p e of P o o l e - F r e n k e l law dielectric electric  displacement  field  capacitance c a p a c i t a n c e of an a i r f i l l e d c a p a c i t o r r a t i o of f r e e t o trapped charge c o n d u c t i o n band energy v a l e n c e band energy equilibrium fermi l e v e l quasi fermi l e v e l a c c e p t o r energy l e v e l donor energy l e v e l  t r a p energy l e v e l p e r m i t t i v i t y of a i r r e a l p a r t of d i e l e c t r i c p e r m i t t i v i t y i m a g i n a r y p a r t of d i e l e c t r i c p e r m i t t i v i t y static  permittivity  p e r m i t t i v i t y at. h i g h f r e q u e n c i e s Fermi-Dirac d i s t r i b u t i o n f u n c t i o n d i s t r i b u t i o n of a c t i v a t i o n  energies  current polarization  c u r r e n t / u n i t area  space charge c u r r e n t / u n i t a r e a  vii  J  leakage c u r r e n t / u n i t  T  = e/e  o  = dielectric  area  constant  k  Boltzmann's c o n s t a n t  L  t h i c k n e s s of f i l m acceptor state.density/unit trap density/unit  energy  donor s t a t e d e n s i t y / u n i t n  c  free electron  energy  energy  density  attempt t o escape f r e q u e n c y n^  trapped e l e c t r o n  n^'  trapped e l e c t r o n d e n s i t y / u n i t  P  polarization  P  s  static polarization *  Q  charge/unit area  q  a c t i v a t i o n energy  <T~  conductivity  \T~' o t  low f i e l d  t  r e l a x a t i o n time  Zo  c o n s t a n t = 1 /' 1 ) o^  T  conductivity J  time  temperature  u  mobility  V  volts  w  angular  T  density  c  frequency  c h a r a c t e r i s t i c temperature  0  electron a f f i n i t y  x  d i s t a n c e through  x*  zero f i e l d point  film  viii  energy  ACKNOWLEDGEMENT  The a u t h o r would l i k e t o express h i s a p p r e c i a t i o n t o Dr. L. Young f o r h i s p a t i e n t h e l p and guidance throughout t h e course of t h i s i n v e s t i g a t i o n . .The a u t h o r . i s a l s o i n d e b t e d t o Dr. D. P u l f r e y f o r r e a d i n g t h e manuscript and f o r h i s many h e l p f u l suggestions. The a u t h o r i s g r a t e f u l t o Mr. C.J. D e l l ' O c a .for h i s a s s i s t a n c e i n making t h e s p e c t r o p h o t o m e t r y measurements and f o r many v a l u a b l e d i s c u s s i o n s .  G r a t e f u l acknowledgement i s g i v e n t o  Messrs. G. Yan and A. Torrens f o r s e v e r a l h e l p f u l d i s c u s s i o n s ; t o Messrs. N. Taneja, S. Matheson, C. D e l l ' O c a , G. Yan and G. O l i v e f o r p r o o f r e a d i n g t h e t h e s i s ; t o Mr. A. MacKenzie  for his technical  h e l p ; and t o Miss B. Harasymchuk and M i s s A. Hopkins f o r t y p i n g the t h e s i s . G r a t e f u l acknowledgement f o r f i n a n c i a l support i s g i v e n . t o t h e N a t i o n a l Research C o u n c i l (Grant A 3392), t h e Department of Defence P r o d u c t i o n ( C o n t r a c t No. PX69-600019) and t h e Sprague E l e c t r i c Co. ( C o n t r a c t 65-7275).  ix  1  1.  INTRODUCTION  The purpose of t h e work d e s c r i b e d i n t h i s t h e s i s was t o study t h e d i e l e c t r i c p r o c e s s e s , t h e c o n d u c t i v i t y the  e f f e c t of space charge on t h e e x t e r n a l  mechanisms and  currents i n thin i n -  s u l a t i n g amorphous f i l m s . T h i n i n s u l a t i n g amorphous f i l m s a r e used i n s e v e r a l types of d i s c r e t e  e l e c t r o n i c d e v i c e s , e.g., t h e MOS f i e l d  effect  t r a n s i s t o r , t h e t h i n f i l m t r a n s i s t o r , the e l e c t r o l y t i c c a p a c i t o r and  the t h i n f i l m capacitor.  In monolithic s i l i c o n  integrated  c i r c u i t s , i n s u l a t i n g f i l m s a r e used f o r s u r f a c e p a s s i v a t i o n  and  e l e c t r i c a l i n s u l a t i o n between t h e s i l i c o n and m e t a l f i l m i n t e r c o n n e c t i o n s , as w e l l as f o r t h i n f i l m c a p a c i t o r s and MOS  field  e f f e c t d i o d e s and t r a n s i s t o r s . Various factors a particular application,  e n t e r i n t o t h e c h o i c e of a t h i n f i l m f o r t h e most obvious b e i n g t h e e l e c t r i c a l  i n s u l a t i n g p r o p e r t i e s of t h e f i l m .  For capacitor  applications,  a t h i n f i l m , b e s i d e s p r o v i d i n g adequate i n s u l a t i o n , s h o u l d have s u f f i c i e n t l y low d i e l e c t r i c l o s s e s and r e s i d u a l v o l t a g e  effects.  R e s i d u a l v o l t a g e s may be due t o p o l a r i z a t i o n p r o c e s s e s w i t h l o n g decay t i m e s , o r t o e l e c t r o n i c  space charge e f f e c t s .  e f f e c t t r a n s i s t o r s , such e f f e c t s can cause d r i f t characteristics, i n addition  I n MOS  field  i n the device  t o e f f e c t s due t o sodium i o n d r i f t .  The f i l m s used i n t h i s study were anodic t a n t a l u m pentoxide.  Device a p p l i c a t i o n s  o f these f i l m s i n c l u d e the e l e c t r o -  l y t i c c a p a c i t o r and t h i n f i l m RC c i r c u i t s .  2  The p e r m i t t i v i t y * was measured as a f u n c t i o n of f r e quency and temperature u s i n g a c o n v e n t i o n a l A.C.  bridge.  In  a d d i t i o n , the l o s s (e") measurements were extended t o v e r y low f r e q u e n c i e s ( 0.01 Hz) by measuring the c u r r e n t response t o a (l step voltage  9) '  .  S i n c e l e a k a g e c u r r e n t s may  o f t e n be l a r g e  enough t o e f f e c t i v e l y cover up the p o l a r i z a t i o n c u r r e n t s , the .method used was t o measure the c u r r e n t s upon removal of a v o l t a g e . In a d d i t i o n may  to d i e l e c t r i c displacement c u r r e n t s , there  a l s o be c u r r e n t s due t o the decay of a t r a p p e d space charge.  A c c o r d i n g l y , the charge and d i s c h a r g e t r a n s i e n t c u r r e n t s were s t u d i e d as a f u n c t i o n of time and a p p l i e d v o l t a g e , and v a r i o u s t e c h n i q u e s were used t o i d e n t i f y the t r a n s i e n t c u r r e n t components. C o n d u c t i o n c u r r e n t s through t h i n f i l m s are sometimes (2 3) a f f e c t e d by the development  of a space charge i n the f i l m  '^ .  The c o n d u c t i v i t y mechaniam was i n v e s t i g a t e d by measuring the steady s t a t e l e a k a g e c u r r e n t s as a f u n c t i o n of v o l t a g e and  temperature  and r e l a t i n g the curves o b t a i n e d t o the absence or presence of a space charge as determined by the t r a n s i e n t c u r r e n t measurements. Chapter 2 i s a r e v i e w of the t h e o r i e s of the d i e l e c t r i c p r o c e s s e s , c o n d u c t i o n mechanisms and space charge e f f e c t s i n t h i n amorphous f i l m s .  The e x p e r i m e n t a l procedures are d e s c r i b e d i n  Chapter 3, and the r e s u l t s o b t a i n e d are p r e s e n t e d i n Chapter 4. I n Chapter 5 the r e s u l t s are d i s c u s s e d i n the l i g h t of e x i s t i n g knowledge and the c o n c l u s i o n s t o be drawn from the work are g i v e n i n Chapter 6. * The d i e l e c t r i c p r o p e r t i e s are d e s c r i b e d by the f r e q u e n c y and temperature dependent complex d i e l e c t r i c p e r m i t t i v i t y , e(u),T) = e' (a),T) - j e " (w,T) 1-1 where e" i s u s u a l l y n e a r l y independent of f r e q u e n c y f o r amorphous f i l m s .  3 2. 2.1  THEORY  D i e l e c t r i c P r o p e r t i e s of Amorphous F i l m s A number of p h y s i c a l processes  t i o n of i o n i c s o l i d s .  Each process may  c u l a r c h a r a c t e r i s t i c frequency  c o n t r i b u t e t o the  polariza-  be c h a r a c t e r i z e d by a p a r t i -  or range of c h a r a c t e r i s t i c  frequencies,  and responds t o o n l y those a p p l i e d f r e q u e n c i e s which are l e s s t h a n the c h a r a c t e r i s t i c  frequency.  I n a p e r f e c t i o n i c c r y s t a l , i . e . , one which i s f r e e from a l l d e f e c t s , i t i s u s u a l t o s e p a r a t e the p o l a r i z a t i o n i n t o e l e c t r o n i c and  i o n i c components.  displacement  E l e c t r o n i c p o l a r i z a t i o n , which i s due  t o the  of e l e c t r o n s w i t h r e s p e c t to the i o n s , responds t o the  h i g h e s t frequency  and has a c h a r a c t e r i s t i c frequency  or u l t r a - v i o l e t r e g i o n s . r e l a t i v e displacement  i n the  optical  I o n i c (or atomic) p o l a r i z a t i o n i s due t o t h e  of o p p o s i t e l y charged i o n s and has  character-  i s t i c f r e q u e n c i e s g i v e n by the l a t t i c e v i b r a t i o n f r e q u e n c i e s , which are i n the i n f r a - r e d r e g i o n .  However, e l e c t r o n i c and atomic p o l a r i -  z a t i o n components cannot be c o m p l e t e l y ment induces  an e l e c t r o n i c displacement  separated ( r e f . 4,  as the i o n i c d i s p l a c e P.  151).  I n the case of a c r y s t a l which, c o n t a i n s d e f e c t s  (e.g.,  i m p u r i t y i o n s , l a t t i c e d e f e c t s ) t h e r e are a d d i t i o n a l p o l a r i z a t i o n components due t o processes some processes  a s s o c i a t e d w i t h the d e f e c t s .  Examples of  are d i p o l e o r i e n t a t i o n , i o n s moving between p o t e n t i a l  minima s i t e s , e l e c t r o n s hopping or t u n n e l i n g between l o c a l i z e d i t y s t a t e s , and  impur-  o r i e n t a t i o n of i m p u r i t y p o l a r m o l e c u l e s (e.g., H^O).'  Another type of process which can c o n t r i b u t e to the low  frequency  l o s s e s i n v o l v e s the d i f f u s i o n of i o n s up to and away from a b a r r i e r . Such i n t e r f a c i a l p o l a r i z a t i o n has, f o r example, been p o s t u l a t e d to exist i n electroluminescent  phosphors.^)  4 The p o l a r i z a t i o n p r o c e s s e s a s s o c i a t e d w i t h d e f e c t s are r e l a x a t i o n p r o c e s s e s and have c h a r a c t e r i s t i c f r e q u e n c i e s below the infra-red.  These processes may  be f o r m a l l y described, by a s u p e r p o s i -  t i o n of Debye type p r o c e s s e s ^ w i t h a d i s t r i b u t i o n of times.  A s i n g l e Debye process i s g i v e n by (Appendix e (to)  where  T e  s  relaxation  I)  -e  e  =  + j js ^ f r  =  relaxation  time  =  static  2-1  permittivity  Eoo =  p e r m i t t i v i t y at high frequencies  to =  a p p l i e d a n g u l a r frequency  P o l a r i z a t i o n p r o c e s s e s i n a n o d i c T&^O^  f i l m s are complicated  by the amorphous n a t u r e of the f i l m . T h e r e i s no l o n g range o r d e r . The i o n i c l a t t i c e i s a p e r i o d i c . I t i s r e a s o n a b l e t o assume t h a t the  elec-  t r o n i c d i s p l a c e m e n t s and e l a s t i c i o n d i s p l a c e m e n t s are not e s s e n t i a l l y changed from t h o s e i n r e g u l a r c r y s t a l s , except t h a t i n an amorphous substance t h e r e w i l l be a spread i n the c h a r a c t e r i s t i c f r e q u e n c i e s so t h a t , f o r example, a broader a b s o r p t i o n band w i l l be o b s e r v a b l e i n the infra-red.  Below i n f r a - r e d f r e q u e n c i e s , Ta^O^  are o f t e n almost independent  of frequency over a v e r y wide range of  8)  (l  frequencies  d i s p l a y s l o s s e s which are  '  .  T h i s s i t u a t i o n r e q u i r e s a d i s t r i b u t i o n of  t i o n times of the form  relaxa-  l / j ^ ^ .  The f a c t t h a t a l / x d i s t r i b u t i o n g i v e s l o s s e s independent of frequency can be e a s i l y shown.  S e t t i n g e q u a t i o n ( l - l ) e q u a l to  e q u a t i o n (2-1) t h e n 8  i  2 + M  x  2  5  I n t e g r a t i n g over a l / x d i s t r i b u t i o n then oo  e"(w) = (e  Jo  s  " T o l+co -^ 2  — ^  2  This d i s t r i b u t i o n of r e l a x a t i o n of p h y s i c a l with L=  (e -e«) s  5 2  -• c o n s t a n t  times may be produced by a v a r i e t y  models, e.g., a Maxwell l a y e r model ( r e f . 8, P. 161)  f>(x) = f  thickness  models  =  > o  e  where  X / / L  of  the  involving  jO ( x ) ^ r e s i s t i v i t y ,  film,  ions  and x  = distance  electrons  or  film.  of  the  through  hopping between  film, the  film;  sites  in  or the  .. I  The i o n i c  relaxation-  model w i t h a r e l a t i v e l y f l a t  distri-  b u t i o n o f a c t i v a t i o n e n e r g i e s has been c l a i m e d ' ' ^ t o a d e q u a t e l y desc r i b e t h e f r e q u e n c y and temperature dependent p e r m i t t i v i t y amorphous i n s u l a t o r s .  o f many  The e x p e r i m e n t a l r e s u l t s p r e s e n t e d i n t h i s  t h e s i s w i l l be d i s c u s s e d on t h e b a s i s o f t h i s model.  The i o n i c  relax-  a t i o n model g i v e s an e x p o n e n t i a l approach t o t h e steady s t a t e and thus can be d e s c r i b e d by t h e Debye e q u a t i o n (Appendix I I ) .  The r e l a x a t i o n  time o f t h i s p r o c e s s i s g i v e n by ~ X  where  q  =  _  ~  %o  q/ T  Z  K  Z  Z  activation  energy or b a r r i e r h e i g h t between  potential  minima X"  =  c o n s t a n t ( i n v e r s e o f jump f r e q u e n c y )  T  =  temperature i n °K  k  = Boltzmann's  constant  Assuming an almost f l a t d i s t r i b u t i o n o f a c t i v a t i o n e n e r g i e s , G-(q), then t h e i n t e g r a t i o n  '  o f t h e Debye e q u a t i o n over t h i s d i s t r i b u t i o n  6  ( i n Appendix I I I ) g i v e s  where  e'  =  e"  -  q  Q  £  o  +  o  J  ( S g - e ^ )  ( e - e - » ) f  kT G ( q  s  (2-3)  G(q)dq  Q  )  (2-4)  i s d e f i n e d a c c o r d i n g t o e q u a t i o n (2-2) as £  =^f e ° q  /  (2-5)  M  E q u a t i o n s (2-3) and (2-4), which were f i r s t d e r i v e d by Severs and Du P r e ^ , can be s i m p l y i n t e r p r e t e d .  I n e q u a t i o n ( 2 - 3 ) , o n l y those  w i t h a c t i v a t i o n energies l e s s t h a n q w where co i s r e l a t e d t o q  by ( 2 - 5 ) .  Q  Q  processes  respond t o an a p p l i e d f r e q u e n c y I n ( 2 - 4 ) , l o s s e s occur o n l y f o r  those a c t i v a t i o n e n e r g i e s which have t h e c h a r a c t e r i s t i c a n g u l a r f r e qviency co. (Note:  from t h e r e l a t i o n i n E q u a t i o n ( 2 - 2 ) , a f l a t  dis-  t r i b u t i o n o f a c t i v a t i o n e n e r g i e s G-(q), corresponds t o a l/x d i s t r i b u t i o n of r e l a x a t i o n times). The p o l a r i z a t i o n response  to a step voltage f o r a n e a r l y  f l a t d i s t r i b u t i o n o f a c t i v a t i o n e n e r g i e s has a l s o been c a l c u l a t e d i n Appendix I I I , and i s J where  p  =  (£ - )kT G(q') | s  £o0  J  = p o l a r i z a t i o n c u r r e n t / u n i t area  E  = applied f i e l d  (2-6)  step  and q' i s d e f i n e d a c c o r d i n g t o e q u a t i o n (2-2) as  t ro =  q , / e  kr  ( 2  _  7 )  T h i s d e f i n i t i o n of q' means t h a t a t some time t , t h e p o l a r -  i z a t i o n c u r r e n t i s dominated by a process w i t h r e l a x a t i o n time  ^-  t g i v e n by X = ^  which i s e q u i v a l e n t t o the a c t i v a t i o n energy  E q u a t i o n (2-6)  may  q'.  be used t o extend d i e l e c t r i c l o s s mea-  surements t o v e r y low f r e q u e n c i e s as f o l l o w s : Combining e q u a t i o n s (2-6) and (2-4) g i v e s ~ E , ,  <  = 2  u )  V  G(q  )  ~E~  cTi^7  The a n g u l a r frequency to i s determined 2jt  co  ( 2  by s e t t i n g q. = q 1  ~  8 )  which g i v e s  _ t • 2  Thus e»( ) u  = |  with  co = | S  (2-9)  Measurements of p o l a r i z a t i o n c u r r e n t s have been used by Hamon^^ t o extend l o s s measurements t o f r e q u e n c i e s l e s s t h a n 0.1 E x c e l l e n t agreement was  found between t h e measured v a l u e s and  o b t a i n e d by u s i n g a b r i d g e a t v e r y low f r e q u e n c i e s .  to assume  and then c a l c u l a t e what the  l o s s e s would be a t a c o r r e s p o n d i n g f r e q u e n c y . i n E q u a t i o n (2-9) so l o n g as 0.3<  those  The method of  a n a l y s i s used by Hamon, and l a t e r reviewed by B a i r d ^ ^ was —n p o l a r i z a t i o n c u r r e n t s of the form t  Hz.  n<1.2.  The r e s u l t s are as g i v e n  I n the present  (Appendix I I I ) , these numbers would correspond t o how  analysis  much the d i s t r i -  b u t i o n G-(q) would be allowed t o v a r y b e f o r e i n v a l i d a t i n g t h e assumpt i o n s used c o n c e r n i n g the f l a t n e s s of G-(q). 2.2  Conduction i n T h i n I n s u l a t i n g F i l m s . The leakage c u r r e n t s through t h i n i n s u l a t i n g f i l m s may  due to movement of e l e c t r o n s , h o l e s or i o n s .  be  8  For t h e p a r t i c a l a r f i l m T a 2 0 ^ , experiments by V e r m i l y e a  '  and S t a n d l e y & M a i s s e l ^"Siave been claimed t o i n d i c a t e t h a t h o l e s carry a n e g l i g i b l e part of the current. and M a i s s e l , u s i n g a  Both V e r m i l y e a and S t a n d l e y  Ta/Ta20,-/metal system found a rough c o r r e l a t i o n  between m e t a l work f u n c t i o n and c u r r e n t magnitudes.  Large c u r r e n t s  were o b t a i n e d f o r low work f u n c t i o n m e t a l s , thus i n d i c a t i n g h o l e i n j e c t i o n i s not an important f a c t o r i n the c o n d u c t i o n p r o c e s s . V e r m i l y e a a l s o a r r i v e d a t a s i m i l a r c o n c l u s i o n f o r t h e Ta/Ta20^/electrolyte  system. Ta^O^ i s grown a n o d i c a l l y by p l a c i n g a p i e c e of t a n t a l u m  i n a n e l e c t r o l y t e and a p p l y i n g a p o s i t i v e p o t e n t i a l r e l a t i v e t o another e l e c t r o d e .  The f i l m grows by i o n movement and growth i s  a p p r e c i a b l e o n l y f o r f i e l d s o f the o r d e r 6 x 10^ v/cm. A t f i e l d s much below t h i s v a l u e the c u r r e n t i s predominantly e l e c t r o n i c i n n a t u r e a n d  f i l m growth i s n e g l i g i b l e .  I n the f o l l o w i n g r e v i e w of c o n d u c t i o n t h e o r i e s t h e r e f o r e , i t w i l l be assumed t h a t t h e c u r r e n t s a r e e l e c t r o n i c i n n a t u r e .  Some  of t h e p o s s i b l e c o n d u c t i v i t y mechanisms i n c l u d e : 1.  Schottky i n j e c t i o n (13)  2.  P o o l e - F r e n k e l e f f e c t '( 1 4 )  3.  F i e l d emission or t u n n e l i n g  4.  Space charge l i m i t e d ^ '  L  (15) 2  ^^  The leakage c u r r e n t d e n s i t y may be l i m i t e d by one o r more o f the above mechanisms a c t i n g s i m u l t a n e o u s l y . An energy l e v e l diagram  f o r the m e t a l / i n s u l a t o r / m . e t a l system . (17)  w i t h one r e c t i f y i n g and one ohmic c o n t a c t appears i n F i g u r e 12-1;. I n t h i s f isrure  9  0^ and 0^ = work f u n c t i o n s of metals 1 and 2 9 - e l e c t r o n a f f i n i t y o f the i n s u l a t o r E E Ep  c  = c o n d u c t i o n hand edge ° = v a l e n c e hand edge = equilibrium Fermi-level  T h i s diagram i s a r r i v e d a t by u s i n g t h e . p r i n c i p l e t h a t i n e q u i l i b r i u m (17) the Fermi l e v e l i s t h e same throughout the system  .  I f one con-  s i d e r s the j u n c t i o n b e i n g made by b r i n g i n g t h e m e t a l and i n s u l a t o r i n t o c o n t a c t , then when t h e m e t a l and i n s u l a t o r a r e w i d e l y  separated  n e a r l y the t o t a l c o n t a c t p o t e n t i a l w i l l f a l l a c r o s s t h e s e p a r a t i o n distance.  When brought i n t o c o n t a c t , the c o n t a c t p o t e n t i a l  a c r o s s the i n s u l a t o r .  With two metal  falls  c o n t a c t s and no space charge i n  the i n s u l a t o r , t h e n the f i e l d i n the i n s u l a t o r i s g i v e n by — —  .  I f t h e r e i s s u f f i c i e n t f r e e charge i n the i n s u l a t o r t h e n the c o n t a c t p o t e n t i a l may f a l l a c r o s s space charge r e g i o n s near the i n t e r f a c e s as shown i n F i g u r e ( 2 - 1 ) .  E l e c t r o n d e p l e t i o n causes the bands t o  bend up, g i v i n g a r e c t i f y i n g c o n t a c t .  E l e c t r o n accumulation  the bands down, g i v i n g an ohmic c o n t a c t .  bends  S u r f a c e s t a t e s on the i n s u -  l a t o r can cause band bending b e f o r e a m e t a l c o n t a c t i s a p p l i e d and can make the b a r r i e r h e i g h t n e a r l y independent of m e t a l work f u n c t i o n . Referring to Figure  ( 2 - 1 ) , the ohmic c o n t a c t a t the m e t a l  2 i n t e r f a c e a l l o w s e l e c t r o n s t o be drawn f r e e l y i n t o the i n s u l a t o r under an a p p l i e d f i e l d .  A t the metal 1 i n t e r f a c e , c o n d u c t i o n can  take p l a c e only' by the e l e c t r o n s e i t h e r g a i n i n g enough thermal energy to pass over the b a r r i e r of h e i g h t 0^-9 barrier.  or by t u n n e l l i n g through the  F i g . 2-2  Energy l e v e l diagram f o r t u n n e l l i n g p r o c e s s e s  11  2.2.1  Schottky  Injection  Schottky i n j e c t i o n i s t h e f i e l d a s s i s t e d t h e r m a l of e l e c t r o n s over a p o t e n t i a l h a r r i e r .  emission  The c u r r e n t i s g i v e n by  (Appendix I V ) , J  (2-10)  = AT exp-(0,-<9)/kT'exp°cE*(O)/kT 2  L  (18) where  A = Richardson's  constant  e3  a = ZZZ f  e  =  h i g h frequency d i e l e c t r i c  permittivity  E(0) = f i e l d a t x = 0 e = electronic  charge.  E x p e r i m e n t a l l y , t h e f i e l d E ( 0 ) a t t h e i n t e r f a c e cannot be o b t a i n e d d i r e c t l y and so i t i s u s u a l l y assumed t h a t E ( o ) = ^ applied voltage.  where V i s t h e  The h i g h frequency d i e l e c t r i c c o n s t a n t i s used, as  an e l e c t r o n w i t h s u f f i c i e n t energy t o pass over t h e b a r r i e r w i l l be moving v e r y f a s t and t h e i o n s w i l l not be a b l e t o respond 2.2.2  t o i t s motion.  Poole - F r e n k e l E f f e c t T h i s i s a b u l k p r o c e s s i n which t h e c o n d u c t i v i t y o f t h e  i n s u l a t o r i s c o n t r o l l e d by f i e l d - e n h a n c e d t h e r m a l e m i s s i o n coulombic  traps.  The c a l c u l a t i o n of t h i s e f f e c t  t h a t o f the S c h o t t k y b y  law (Appendix  IV).  from  i s analagous t o  The c o n d u c t i v i t y i s g i v e n  (25,35) I.  a- (x) = <j- exp j9E '(x)/kT  (2-11)  P  where  0  = low f i e l d . c o n d u c t i v i t y , and  The leakage c u r r e n t i s then g i v e n by  £3= 2«c .  12  J  = ( r ( x ) E ( x ) = a- E(x)exp'flE*(x)/kT:  T  O  IJ  ^  2  R  As f o r the S c h o t t k y law, E ( x ) i s u s u a l l y assumed t o he V/L.  This  assumption i s dubious because t o observe a p u r e l y P o o l e - F r e n k e l  effect  would n e c e s s i t a t e the c o n d i t i o n of no b a r r i e r l i m i t a t i o n on the c u r rent.  Lack of any b a r r i e r l i m i t a t i o n means t h a t charge c o u l d be  i n j e c t e d i n t o the i n s u l a t o r w i t h a consequent b u i l d u p of a space charge., i n excess of any e q u i l i b r i u m space charge due t o c o n t a c t potentials.  The exact r e l a t i o n s h i p between E ( x ) and the a p p l i e d v o l -  tage then becomes c o m p l i c a t e d . E r a n k and Simmons  have computed I n J - V  f i l m s where they have c o n s i d e r e d  2  curves f o r t h i n  S c h o t t k y i n j e c t i o n at a b a r r i e r , the  b a r r i e r h e i g h t modulated by a space c h a r g e . i n the b u l k , and the b u l k c o n d u c t i v i t y determined by the P o o l e - F r e n k e l e f f e c t on a s i n g l e trapping l e v e l .  At h i g h e r v o l t a g e s t h e i r curves a r e a s y m p t o t i c t o  the S c h o t t k y i n j e c t i o n curve.  At low v o l t a g e s the e f f e c t s o f the  space charge a r e more pronounced and t h o curves d e v i a t e from the S c h o t t k y law.  A P o o l e - F r e n k e l c h a r a c t e r i s t i c , i . e . , l n Jjy-V  w i t h s l o p e g i v e n by  (20,21,22)  p l o t s have been observed w i t h A l ^ O ^ ^ ^ ^ ,  2  ( 2 3 ) ^ T a 0 ' ' , and o r g a n i c i n s u l a t o r s ^ ^ (  >  Z n 0  curve  i s not r e v e a l e d a t any t i m e .  Linear. I n J^-V S i 0  2  2  12  )  5  2 4  with  good agreement between the observed and c a l c u l a t e d v a l u e s o f the s l o p e a „ _ ,, (12,20,21,22) , . . . , — j — = TJn* However, some of these authors ' ' ' claim that d E ^ the c o n d u c t i o n was i n f a c t b u l k l i m i t e d d e s p i t e the observed S c h o t t k y J IHJL T  v a l u e of  j~ . A p o s s i b l e e x p l a n a t i o n f o r t h i s was p o i n t e d out d E by Simmons(2.5) and i s d i s c u s s e d i n Appendix IV. 2  2.2.3  F i e l d E m i s s i o n or T u n n e l i n g The energy l e v e l diagram i n F i g u r e  (2-2) shows s e v e r a l d i f -  13  f e r e n t tunnelling processes ( C h y n o w e t h ^ ) . 1.  T u n n e l l i n g from m e t a l t o c o n d u c t i o n hand effect  ( 2 6 )  These a r e : (Fowler-Nordheim  ).  2.  T u n n e l l i n g from t r a p l e v e l i n t o c o n d u c t i o n band.  3.  T u n n e l l i n g from v a l e n c e band i n t o c o n d u c t i o n band.  4.  T u n n e l l i n g from v a l e n c e band i n t o m e t a l . For  the  >  i n s u l a t o r s w i t h a wide band gap p r o c e s s 3, known as  Zener e f f e c t , i s n e g l i g i b l e i n comparison w i t h processes. 1 and 2.  Process 4 i s e s s e n t i a l l y p r o c e s s 1.  h o l e t u n n e l l i n g and i s analagous t o  A c c o r d i n g l y o n l y the f i r s t two p r o c e s s e s w i l l be con-  sidered here. (15) Similar expressions a r e o b t a i n e d f o r b o t h , namely, J where  = BE e n  L  K n < 2  E  (2-13)  0 = b a r r i e r h e i g h t f o r 1 and t r a p depth f o r 2. B, C = c o n s t a n t s , n e a r l y independent o f ' t e m p e r a t u r e .  One t u n n e l l i n g p r o c e s s which.has not been i n c l u d e d i n the diagram i s t h a t from "metal t o m e t a l " . o  Such a p r o c e s s r e q u i r e s f i l m s l e s s than .  100A i n t h i c k n e s s which i s much t h i n n e r than those used i n the p r e s e n t work. Tunnel c u r r e n t s a r e r e l a t i v e l y temperature independent. fll has been observed l e s s dependent  It  12) '  f o r Ha^O^  t h a t the l e a k a g e c u r r e n t s "become  on temperature as the temperature d e c r e a s e s .  This  i n d i c a t e s t h a t t u n n e l c u r r e n t s at room temperature a r e masked by t h e r (12) m a l l y produced c u r r e n t s .  Mead's  curves f o r T a 0 ^ i n d i c a t e a b u l k 2  t u n n e l l i n g c o n d u c t i o n p r o c e s s which he a t t r i b u t e d t o p r o c e s s 2.  14 2.2.4  Space Charge L i m i t e d I t was p o i n t e d  Currents  out by Mott and Gurney  (27)  that i t i s theor-  e t i c a l l y p o s s i b l e t o have a l a r g e amount of "space charge l i m i t e d " c u r r e n t f l o w i n g through a t h i n i n s u l a t i n g f i l m . an ohmic c o n t a c t contact  To make t h i s  and a t r a p f r e e i n s u l a t o r are r e q u i r e d .  possible,  The ohmic  i n s u r e s a p o o l of e l e c t r o n s a t t h e cathode, f r e e t o be drawn  i n t o t h e i n s u l a t o r by an a p p l i e d f i e l d .  For a perfect c r y s t a l , a l l  the space charge w i l l be f r e e charge and the c u r r e n t  i s g i v e n by  (Appendix V I I ) 2 J  where  L  =  8 ^  J>  (2-14)  u. = e l e c t r o n m o b i l i t y .  Traps p r e s e n t i n t h e c r y s t a l , o r amorphous f i l m w i l l i m m o b i l i z e  some  of the c a r r i e r s and t h e c u r r e n t w i l l be reduced by a f a c t o r A the r a t i o o f f r e e t o t o t a l charge n *  where  n  C  n c  %  +  n  t  ~  0  n  t  (2-15)  and n, a r e t h e f r e e and trapped charge d e n s i t i e s , X  respectively. For most i n s u l a t o r s t h e t r a p p e d charge i s u s u a l l y much g r e a t e r  than  the f r e e charge. (2) Rose for  , u s i n g a v e r y s i m p l i f i e d a n a l y s i s , has c a l c u l a t e d A  a v a r i e t y o f t r a p d i s t r i b u t i o n s i n t h e band gap.A s i n g l e  t r a p p i n g l e v e l . l e a v e s the V lowers the current  shallow  l a w o f e q u a t i o n (2-14) unchanged, but  magnitude.  A u n i f o r m d i s t r i b u t i o n of t r a p s i n t h e band gap g i v e s a value  15  A  a  h V  e  a  I  e*  if = c o n s t a n t  Y v  (2-16)  Then J  L  (2-17)  v  Ef/kT  For a t r a p d i s t r i b u t i o n of t h e form .  J  a  L  V  (V  T  n  a e  then  U  +  (2-18)  A c h a r a c t e r i s t i c s i m i l a r t o E q u a t i o n (2-18) has been (28) observed by Walker  i n Ta 0^. 2  Rose c o n s i d e r e d o n l y t h e s i m p l e s t  c o n d i t i o n s , i . e . , no double i n j e c t i o n , no r e c o m b i n a t i o n .  When these  e f f e c t s a r e c o n s i d e r e d the J - V c h a r a c t e r i s t i c s a r e q u i t e d i f f e r e n t from above Ohmic c u r r e n t s do not n e c e s s a r i l y mean c u r r e n t s l i m i t e d by the t h e r m a l p r o d u c t i o n of f r e e c a r r i e r s i n t h e b u l k .  A sufficiently  s m a l l s i g n a l w i l l g i v e a l i n e a r J-V curve i r r e s p e c t i v e o f t h e conduct i o n mechanism. G e n e r a l l y s p e a k i n g , t h e most commonly observed c o n d u c t i o n mechanism a t room temperatures and h i g h f i e l d s , i n t h i n f i l m s , has been o f t h e S c h o t t k y o r P o o l e - F r e n k e l t y p e .  At high, f i e l d s and v e r y  low temperatures t u n n e l l i n g i s observed and a t low f i e l d s an ohmic c h a r a c t e r i s t i c i s observed. No mention has been made, so f a r , o f t h e p o s s i b l e time dependence o f t h e leakage c u r r e n t s .  I f the leakage c u r r e n t s were t o  depend i n some manner on a space charge, and s i n c e t h e b u i l d u p o f a space charge i s time dependent, t h e n t h e l e a k a g e c u r r e n t s would be time dependent.  The e q u a t i o n s d e r i v e d so f a r a r e f o r t h e steady s t a t e ,  and p o s s i b l e t r a n s i e n t e f f e c t s a r e c o n s i d e r e d i n t h e next s e c t i o n .  16 2.3  Space Charge E f f e c t s As i n d i c a t e d p r e v i o u s l y , space charge r e g i o n s may  i n a f i l m due  to contact p o t e n t i a l s .  a d d i t i o n a l charge may  Under an a p p l i e d  exist  voltage,  he i n j e c t e d i n t o the i n s u l a t o r and  trapped.  I t i s l i k e l y t h a t an amorphous f i l m has a h i g h d e n s i t y of e l e c t r o n i c t r a p s , and  t h a t these t r a p s w i l l p l a y an important r o l e i n the e l e c -  t r o n i c p r o c e s s e s o c c u r i n g i n the  film.  An e l e c t r o n i c t r a p i s a l o c a l i z e d s t a t e w i t h an energy l e v e l such t h a t the e l e c t r o n s commute between the t r a p and the band.  Such a s t a t e may  be due  t o an i m p u r i t y  conduction  ( i . e . , doping i n a  semi-conductor) or a s t r u c t u r a l d e f e c t or s i m p l y the amorphous n a t u r e of some i n s u l a t o r s . I t i s o f t e n assumed t h a t , f o r amorphous i n s u l a t o r s , i n t h e . steady s t a t e most of any  space charge r e s i d e s i n t r a p s , and a n e g l i -  g i b l e amount i s f r e e charge. A space charge has a number of e f f e c t s , namely: 1)  The b u i l d u p of a space charge may charging  current J  r e s u l t i n an e x t e r n a l l y obseived  , sc  2)  under s h o r t c i r c u i t c o n d i t i o n s , an e x t e r n a l l y observed c u r r e n t may  3)  4)  be observed due t o the decay of a trapped space charge,  the space charge may s i e n t and  a f f e c t the l e a k a g e c u r r e n t s under b o t h t r a n -  steady s t a t e c o n d i t i o n s ,  the space charge may l o s s e s and  have an e f f e c t on the c a p a c i t a n c e ,  Buildup The  area) i s given  dielectric  p o l a r i z a t i o n currents.  Each of these p o s s i b i l i t i e s w i l l now 2.3.1  discharge  be c o n s i d e r e d  in detail.  of a Space Charge t o t a l i n j e c t e d space charge i n an i n s u l a t o r (per u n i t by  L (n (x,t). + n . ( x , t ) ) dx C  n  C  and n, a r e t h e f r e e and trapped "U  brium v a l u e s .  (2-19)  "U  charge d e n s i t i e s above t h e e q u i l i -  The c u r r e n t d e n s i t y due t o the b u i l d u p of a space  charge i s then J  sc  =  e  dt 1  ^c^' ^ 1  +  n  ( ' X  t  t ) )  (2-20)  d x  0  Any  attempt t o d e r i v e J  i s complicated  by the f a c t t h a t an i n j e c t i n g  sc c o n t a c t l i m i t s t h e r a t e a t which charge i s i n j e c t e d , and the space charge b u i l d u p a f f e c t s t h e v a l u e s o f t h e f i e l d at t h e i n j e c t i n g cont a c t , thus the r a t e o f i n j e c t i o n . The  t r a p p i n g k i n e t i c s a l s o become v e r y c o m p l i c a t e d  when  one c o n s i d e r s t h e p o s s i b i l i t y o f a d i s t r i b u t i o n of t r a p p i n g l e v e l s with v a r y i n g capture  cross sections trap d e n s i t i e s varying with d i s -  tance through the i n s u l a t o r , e f f e c t s o f the f i e l d  on the t r a p p i n g k i -  n e t i c s , and d i f f e r e n t types o f t r a p s . 2.3.2  Decay of a Space Charge I n c o n s i d e r i n g the e x t e r n a l l y observed c u r r e n t due t o t h e  decay o f space charge i n a m e t a l / i n s u l a t o r / m e t a l d e v i c e a s i m p l i f i e d model w i l l be taken i n which b o t h b a r r i e r h e i g h t s a r e e q u a l and t h e r e i s no space charge i n the i n s u l a t o r a t e q u i l i b r i u m . METAL  INSULATOR  E^  METAL  J  $ -6 >  E  x == 0  Fig.  2-3  v X  Energy l e v e l diagram f o r  = L  model  18  0 = m e t a l work f u n c t i o n 0 = e l e c t r o n a f f i n i t y of i n s u l a t o r E-^? = e q u i l i b r i u m Permi l e v e l L = w i d t h of i n s u l a t o r E E The  = c o n d u c t i o n band edge  c  = v a l e n c e band edge  v  &  above assumptions n e c e s s a r i l y r e s t r i c t the v a l u e of the  l e v e l Ej, t o the range e(0-0) = E - E p . 0  0  c  I t i s assumed t h a t a u n i -  form d i s t r i b u t i o n of t r a p p i n g s t a t e s through the band gap and  Fermi  exists  t h a t i n e q u i l i b r i u m these are f i l l e d t o about the F e r m i l e v e l . To a c h i e v e the assumed zero space charge i t i s n e c e s s a r y  t o p o s t u l a t e the e x i s t e n c e  of donor l e v e l s .  D e n o t i n g the p r o b a b i l i t y  of o c c u p a t i o n of a s t a t e by the F e r m i f a c t o r f , t h e n the zero space charge d e n s i t y c o n d i t i o n E  E  c  0 = e f  C  (l-f)N dE - e C d  JE . V  where  gives f N  d  t  dE  t  (2-21)  \ .  - d e n s i t y of donor s t a t e s per u n i t energy = d e n s i t y of t r a p p i n g s t a t e s per u n i t energy E  d  = energy l e v e l of donor s t a t e = energy l e v e l of t r a p p i n g  Under an a p p l i e d v o l t a g e ,  state  i t i s assumed t h a t e l e c t r o n i n j e c t i o n t a k e s  p l a c e , and t h a t the r e s u l t a n t n e g a t i v e space charge r e s i d e s i n the traps.  At time t = 0, when a s t e a d y s t a t e i s reached, the d e v i c e i s  s h o r t c i r c u i t e d . Assuming no i n s t a n t a n e o u s change i n the trapped space charge,then a r e d i s t r i b u t i o n of a p o s i t i v e charge on the  plates  t a k e s p l a c e such t h a t at a ' p o i n t x * i n the f i l m the f i e l d i s zero  19 (shown i n Appendix V).  .—  —  !  -  • -  -V ^  -  I  t  _  t t  —  'ext  + X  :  X*  = 0  Fig.  2  "t  •  1  Q  2-4  X  = L  E l e c t r o n i c space charge d i s t r i b u t i o n  When an e l e c t r o n i s r e l e a s e d from a t r a p i t w i l l move under t h e i n f l u ence o f t h e space charge f i e l d , of x* i t f i n d s i t s e l f .  i n a d i r e c t i o n depending on which s i d e  I n a d d i t i o n t o t h e e x t e r n a l c u r r e n t due t o  the  space charge decay, t h e r e w i l l be a d e p o l a r i z a t i o n c u r r e n t due t o  the  i o n i c r e l a x a t i o n processes considered e a r l i e r .  The p o l a r i z a t i o n  c u r r e n t s w i l l be a l t e r e d because o f t h e a l t e r e d f i e l d d i s t r i b u t i o n i n the  i n s u l a t o r due t o t h e space c h a r g e .  As d e r i v e d i n Appendix V, t h e  e x t e r n a l c u r r e n t i s g i v e n by  • ext  e n  t  £  (x*,t) § f  s"  £ c a y  , * , N dx* e n ( x * , t ) —-• dt t  - (e -e~)kT G(q' ) f;  J  -t/x G(q)dq  0 (x*,t<0)  (2-22)  where E ( x * , t < 0) - -  §_  e  (x,t<0)dx  +  E(0,t<0)  (2-23)  ^0  The f i r s t term i n (2-22) i s due t o space charge decay a l o n e .  The  l a t t e r term i s t h e d e p o l a r i z a t i o n c u r r e n t , s l i g h t l y a l t e r e d from equation (2-6).  The second term i s a m i x t u r e o f both.  The i n t e g r a -  t i o n i n t h e second term cannot be done u n l e s s t h e f u n c t i o n a l form o f G(q) i s known over t h e whole range o f a c t i v a t i o n e n e r g i e s .  However,  20 by i n t e g r a t i n g over a range o f q such t h a t t h e r e l a x a t i o n t i m e s go from much l e s s t h a n t t o much g r e a t e r than t , and assuming G(q) i s f l a t i n t h i s range, t h e n *2 e  t / / r  G(q)dq  =  -kT G(q) E i ( - t / 2 " ) 2  and E i ( - a ) = e x p o n e n t i a l i n t e g r a l which i s d e f i n e d on t h e f o l l o w i n g page. To determine J and t h e z e r o f i e l d  Q X  ^ r e q u i r e s knowing t h e space charge d e n s i t y  point x * ( t ) .  N e g l e c t i n g any r e t r a p p i n g , the decay k i n e t i c s o f n^. a r e governed by  an ' • ~ i where  (  x > t  )  =  -v n '(x,t)exp-(E o  t  c  - E )/kT  , (2-24)  t  n ^ ' ( x , t ) = space charge d e n s i t y / u n i t energy c e n t e r e d at E^ 3> = c o n s t a n t = jump f r e q u e n c y Q  Assuming a u n i f o r m d e n s i t y of t r a p s i n t h e band gap w i t h t h e i r o c c u p a t i o n governed by F e r m i s t a t i s t i c s t h e n , as c a l c u l a t e d i n Appendix V I , _^ n (x,t) = - k T N ^ E . t - ^ t t  Ei(-^t(e-  e  ( E  -  ( E  o- F E  c- F)/ E  k T  ) / k T  ;-  e"  t e  ~ (E -E (x))/kT c  F  0  + -(E -E (x))/kT , J e  c  F  }  }  (2-25) where  E p ( x ) i s t h e q u a s i F e r m i l e v e l a t t = CU  C~ da = —\ — a  and  E i ( - a ) i s the exponential i n t e g r a l  21 Some t a b u l a t e d v a l u e s f o r E^(-oc) are g i v e n bel'ow _  - E i (- °c )  OC  0.0  e  -  0 6  E i (-«*=)  CMS  H.  0.01 0.10 1.0 10.0  O  if  I.  € z.  o.  zz  Hi  JL •  a. I O '  K  6  ( 3 .  O  o  6  O?  I t i s e v i d e n t t h a t f o r l a r g e space charge d e n s i t i e s that E  c -  »  V  E  c "  such  V > x  then n ^ ( x , t ) becomes n e a r l y independent of x. The z e r o f i e l d p o i n t x* i s d e r i v e d by u s i n g the s h o r t c i r cuit condition.  Thus, L  V = 0  E ( x , t )dx  §  (2-26)  0 where  E ( x , t ) i s as g i v e n i n Appendix V I , E q u a t i o n s  (VI-9)  and  (VI-10). Equations  (2-25) and  (2-26) are v e r y d i f f i c u l t  to solve,  even by making v e r y s i m p l i f y i n g assumptions c o n c e r n i n g n ^ ( x , t ) .  Thus,  i t i s not c l e a r what form the space charge d i s c h a r g e c u r r e n t s i n E q u a t i o n (2-22) w i l l  take.  Lindmayer (34) has c l a i m e d t h a t a space charge w i l l g i v e a -r; d i s c h a r g e c u r r e n t .  H i s model was  s i m i l a r t o t h a t used h e r e ,  t h a t he n e g l e c t e d the i o n i c p o l a r i z a t i o n e f f e c t s . (2-22), J  e  x  t  = en (x*,t) ~ t  Thus, from  and from E q u a t i o n (2-26), 0 = £  He f u r t h e r assumed Ep(x)>> Ep°,  except Equation dx^n^.  thus making n ^ ( x , t ) independent of x.  From e q u a t i o n (2-26) t h i s makes x* independent of t , and hence J , = 0 ' ext ) c o n t r a r y t o what Lindmayer concluded. What Lindmayer a c t u a l l y  22  c a l c u l a t e d was — £ However, J ^ ex  2.3.3  which from E q u a t i o n (2-2'S) i s p r o p o r t i o n a l t o ^. dn+i s not g i v e n by  The E f f e c t on C o n d u c t i o n C u r r e n t s The e f f e c t of space charge on the l e a k a g e c u r r e n t s  been covered i n S e c t i o n 2.2.. Space charge a f f e c t s t h e f i e l d  has distri-  b u t i o n i n the i n s u l a t o r and hence the l e a k a g e c u r r e n t s f o r any cond u c t i o n mechanism.  A space charge would a l s o be expected t o cause  h y s t e r e s i s i n the J-V c h a r a c t e r i s t i c . 2.3.4  The E f f e c t on D i e l e c t r i c  Properties  The e f f e c t of a space charge on the p o l a r i z a t i o n of a d i e l e c t r i c was covered i n S e c t i o n 2.3.2 charge may ting film.  I t i s not c l e a r how a space  e f f e c t the measured c a p a c i t a n c e and l o s s e s i n an i n s u l a I t i s p o s s i b l e t h a t l o s s e s may  r e s u l t from e l e c t r o n s  hopping between t r a p s i t e s i n a manner s i m i l a r t o t h a t c o n s i d e r e d for  the i o n i c r e l a x a t i o n l o s s e s .  23  33•1  EXPERIMENTAL PROCEDURE  Sample P r e p a r a t i o n Both b u l k tantalum  s l i d e s * were t r i e d .  and tantalum  s p u t t e r e d on to g l a s s  The b u l k tantalum s u r f a c e was  prepared  by  c l e a n i n g i n t r i c h l o r e t h y l e n e and then c h e m i c a l l y p o l i s h i n g . * * L i t t l e success was  a c h i e v e d u s i n g the b u l k t a n t a l u m ,  however,  as most of the Ta/Tap^Op./ m e t a l d e v i c e s c o n s t r u c t e d broke down a t r e l a t i v e l y low a p p l i e d f i e l d s  ( 5 x 10^v/cm.).  Microscope exam-  i n a t i o n showed t h a t sharp s t e p s at the g r a i n b o u n d a r i e s developed d u r i n g the chemical p o l i s h i n g .  An oxide would develop c r a c k s a t  these sharp edges, thus l e a d i n g t o low breakdown v o l t a g e s . T h i s d i f f i c u l t y was  not p r e s e n t when u s i n g the  sputtered  " tantalum f i l m s as they had a m i r r o r s u r f a c e as r e c e i v e d and hence r e q u i r e d no p o l i s h i n g .  They c o u l d thus be used d i r e c t l y a f t e r a  simple c l e a n i n g process. The  s p u t t e r e d tantalum  s i o n i n s u l p h u r i c a c i d (96% chromate.  samples were degreased by Immer-  Reagent) s a t u r a t e d w i t h potassium d i -  The m e t a l s u r f a c e s cleaned by t h i s treatment  were hy-  d r o p h i l i c , i . e . , they c o u l d be wetted by a.uniform t h i n f i l m water and would dry i n a s i m i l a r u n i f o r m manner.  The  of  samples were  then r i n s e d i n d i s t i l l e d water and mounted i n a t e f l o n h o l d e r . f o r the a n o d i z a t i o n . *' S u p p l i e d by D. M i l l s , N o r t h e r n Ottawa.  E l e c t r i c Research L a b o r a t o r i e s  ** Chemical p o l i s h was made up of 5 p a r t s by volume 36% HpSO., 2 p a r t s 70% HNO and 2 p a r t s 48/ HP. ^4 0  24  Ta WIRE t »  Pt  t  t 1 ll  1  7~  ^- PLEXIGLASS LID  TEFLON  WIRE  GLASS SLIDE WITH SPUTTERED Ta  ELECROLYTE  BEAKER  Fig.  3-1  Anodization  An oxide i s grown on tantalum  setup by p l a c i n g t h e m e t a l i n  an e l e c t r o l y t e and a p p l y i n g a v o l t a g e t o i t which i s p o s i t i v e w i t h respect t o another e l e c t r o d e  (Platinum i n t h i s case).  Oxide  growth occurs by t h e movement o f t a n t a l u m and oxygen i o n s , which at room temperatures r e q u i r e s f i e l d s i n the f i l m o f t h e o r d e r 6 x 10^ v/cm. a constant  The f i l m s used i n t h i s study were grown by a p p l y i n g  c u r r e n t u n t i l t h e d e s i r e d t h i c k n e s s was reached  (which  i s r o u g h l y p r o p o r t i o n a l t o t h e v o l t a g e a p p l i e d t o the c e l l  during  growth).  The t e r m i n a l v o l t a g e was t h e n l e f t on t h e sample f o r  a number of hours so t h a t weaker spots i n t h e f i l m would be a l l o w e d o to grow t h i c k e r .  F i l m t h i c k n e s s e s were kept below 1000. A , as  experiments by V e r m i l y e a i n d i c a t e d  that the conduction  proper-  t i e s o f t h i c k e r f i l m s were dominated by f l a w s . R e s u l t s a r e presented Sample 1 was anodized  f o r three d i f f e r e n t samples.  i n 40% s u l p h u r i c a c i d a t 100-105°C a t app-  r o x i r n a t e l y 25ma/cm t o 30v. The sample was l e f t a t ^ O v . f o r two hours a t which time t h e c u r r e n t had f a l l e n t o a p p r o x i m a t e l y G^a/cm'  25  Sample 2 was anodized ature at approximately  i n 0.5% s u l p h u r i c a c i d a t room temper22 ma/cm  t o 30 v o l t s .  I t was l e f t on con-  s t a n t v o l t a g e f o r two hours a t which time t h e c u r r e n t was about 5[ia/cm  .  Some o f t h i s c u r r e n t may have been due t o leakage  the e l e c t r o l y t e and t h e t a n t a l u m  wire c o n t a c t  between  (see F i g . 3-1)• F o r  sample 3> t h e procedure was t h e same as f o r sample 2 except t h a t 2 the c o n s t a n t  c u r r e n t used was 3 ma/cm .  F i l m t h i c k n e s s e s were measured u s i n g a model 14 Cary spectrophotometer and t h e curves 3.2  on page 80 o f r e f e r e n c e 8.  Counterelectrodes A f t e r t h e a n o d i z a t i o n was completed t h e f i l m s were r i n s e d  i n d i s t i l l e d water, a i r d r i e d and p l a c e d i n a Veeco vacuum system for  evaporation of the counterelectrodes.  were used.  Aluminum, g o l d and indium  The aluminum and g o l d w i r e s were wound on t u n g s t e n  h e l i x w i r e and then cleaned agitation.  i n trichlorethylene using ultrasonic  The indium was p l a c e d i n a tantalum  in a similar fashion. _5 imatelylO  boat and cleaned  The e v a p o r a t i o n was c a r r i e d out a t approx-  mm of Hg, and i n a l l cases t h e e v a p o r a t i o n was  s t a r t e d b e f o r e opening a s h u t t e r and exposing the oxide f i l m t o the source.  The d i s t a n c e between s o u r c e  and f i l m was a p p r o x i m a t e l y  s i x inches.  The masks used produced c i r c u l a r e l e c t r o d e s w i t h  diameters from 0.2 mm t o 2 mm. The aluminum c o u n t e r e l e c t r o d e s gave v e r y u n s a t i s f a c t o r y d e v i c e s w i t h low breakdown s t r e n g t h .  G o l d * and indium were found  to g i v e good r e c t i f y i n g c o n t a c t s . *  S i l c o x and M a i s s e l ( 3 3 ^ found g o l d e l e c t r o d e s gave h i g h breakdown s t r e n g t h s . They a t t r i b u t e d t h i s t o a tendency of g o l d t o b r i d g e f i n e c r a c k s i n t h e oxide s u r f a c e whereas aluminum tends to p e n e t r a t e these c r a c k s thus l e a d i n g t o low breakdown v o l t a g e s .  26 The sample # 2 used i n t h i s work . i n i t i a l l y had aluminum electrodes.  These were l a t e r removed w i t h s u l p h u r i c and n i t r i c  a c i d s and r e p l a c e d by indium 3.3  Experimental  electrodes.  Setup  The e a r l i e r measurements were c a r r i e d out i n d r y a i r w i t h the sample mounted i n an e l e c t r i c a l l y screened which c o u l d be m a i n t a i n e d 8CTK t o 40CTK.  s e a l e d chamber ( P i g . 3.3)  a t d i f f e r e n t temperatures i n the range  Contact was made t o t h e c o u n t e r e l e c t r o d e w i t h a  s m a l l g o l d probe.  E f f o r t s were made t o m i n i m i z e t h e c o n t a c t  pre-  ssure and no apparent damage t o t h e f i l m was observed due t o t h e contact.  S h i e l d e d , low n o i s e c a b l e s were used f o r a l l c o n n e c t i o n s .  C o n n e c t i o n p o i n t s had t o be f u l l y s h i e l d e d as t h e c u r r e n t s  being  measured were i n t h e picoamp range. The c u r r e n t s were measured u s i n g a K e i t h l e y 417 picoammeter i n t h e following  circuit.  PICOAMMETER  VOLTAGE I  DIVIDER  J  a  CHAMBER SAMPLE  X-Y RECORDER  F i g . 3-2  Current measurements c i r c u i t  27  The x - y r e c o r d e r was used f o r mapping c u r r e n t t r a n s i e n t s . L a t e r experiments chamber ( P i g . 3-4) Hg.  which c o u l d be evacuated  The temperature  Statham SD6 +150°C.  were c a r r i e d out i n a d i f f e r e n t  was  t o about 300u.  c o n t r o l l e d by p l a c i n g the whole u n i t i n a  oven which a l l o w e d f o r a temperature  range -40°C t o  C o a x i a l l e a d s w i t h t e f l o n i n s u l a t i o n were used i n ,the  c i r c u i t i n P i g . 3-2 w i t h t h i s Capacitance General Radio 1615  setup.  and d i e l e c t r i c l o s s e s were measured on a  - A t r a n s f o r m e r b r i d g e i n the t h r e e t e r m i n a l  mode (which e l i m i n a t e s c a p a c i t a n c e s between the l e a d s and  ground).  The frequency range used was 100 Hz t o 100 KHz w i t h s i g n a l a m p l i tudes between lOmv and 100 mv peak t o peak.  The measured v a l u e s  of c a p a c i t a n c e and l o s s e s d i d not depend on the s i g n a l  amplitude  'in t h i s range.  on  Measurements were extended t o 500 KHz  sample u s i n g a Boonton Model-75c  one  capacitance bridge.  The c i r c u i t i n P i g . 3-2 was  checked f o r e l e c t r i c a l  leakage by removing the sample, a p p l y i n g a v o l t a g e g r e a t e r , t h a n t h a t used i n the experiments  and then e n s u r i n g t h a t any  resultant  c u r r e n t was much s m a l l e r than.the c u r r e n t s b e i n g measured i n the experiments. The s p e c i f i e d a c c u r a c y of the picoammeter was a l l ranges except the l o w e s t , where i t was -12 c u r r e n t v a l u e s l e s s than 10 measured v a l u e s w i l l be w i t h i n The G.R.  +5%.  +37°  for  Thus except f o r  amps, the o v e r a l l a c c u r a c y of the +57°.  b r i d g e accuracy i s dependent on the measurement  frequency and on the magnitude of c a p a c i t a n c e b e i n g measured. P e r c e n t v a l u e s are g i v e n i n the r e s u l t s , where a p p r o p r i a t e .  The  Fig. 3 - 4  Test chamber f o r measurements i n vacuum  29  Boonton b r i d g e a c c u r a c y a l s o depends on the parameters of the sample b e i n g t e s t e d , and v a l u e s a r e g i v e n where  appropriate  i n the r e s u l t s . 3•4  Series  Resistance  The measured v a l u e s f o r the l o s s e s have t o be c o r r e c t e d f o r s e r i e s r e s i s t a n c e due t o c o n t a c t s , l e a d r e s i s t a n c e , e t c . C o r r e c t i o n f o r t h i s can be most e a s i l y made u s i n g the s e r i e s R-C model f o r the c a p a c i t o r .  Thus  where R-^ i s the e x t r a s e r i e s r e s i s t a n c e due t o l e a d s , e t c . tan  5 =  (R-^+RQ)  wC  r e c t t a n / once R  g  and i t i s a simple m a t t e r t o determine the cori s known.  T  Thus  R  T  L  i s more e a s i l y d e r i v e d from the  p a r a l l e l R-C model,  r^WV A W —  From t h i s ,  RT (1 + co R 2  tan At h i g h  /  =  C )  2  2  + R,  coR C P P 2  frequencies tan $  - coRL C p T  tan S a g a i n s t l/co e x t r a p o l a t e d t o i n f i n i t e f r e q u e n c y co C g i v e s R ^ . F o r m a t e r i a l s w i t h t a n -4.0.1, C = C^ t o w i t h i n 1% and  A p l o t of  s  so  —— s  may be p l o t t e d a g a i n s t l/co. The example i l l u s t r a t e d below i s f o r the b r i d g e measure-  ments on sample #3, e l e c t r o d e a , shown i n F i g . 4-17.  30  F i g . 3-5  S e r i e s r e s i s t a n c e example  31  4. 4-1  Current  4-1-1  RESULTS  Measurements  Charge and Discharge  Currents  P i g s . 4-1 t o 4-4 a r e a l l concerned w i t h the step r e s ponse of a p a r t i c u l a r sample (#3) w i t h a g o l d e l e c t r o d e "a" of -?  -?  a r e a 3-62 +0.05 x 10 " cm  0  .  P i l m t h i c k n e s s was 644 +5 A.  Meas-  urements were c a r r i e d out w i t h the sample under a vacuum of 300 u.Hg. P r e v i o u s t o these measurements the sample had been annealed at 140 + 10°C f o r a p p r o x i m a t e l y c i r c u i t e d f o r approximately made.  40 hours.  The sample was s h o r t  8 hours b e f o r e any measurements were  P i g s . 4-1 and 4-2 show the c h a r g i n g c u r r e n t d e n s i t i e s as a  f u n c t i o n of time a f t e r a p p l y i n g s t e p v o l t a g e s of i n c r e a s i n g magnitude.  P o l a r i t i e s were a l t e r n a t e d every one or two t e s t s a t the  lower v o l t a g e s .  The s t r a i g h t l i n e s drawn through the p o i n t s f o r  the lower v o l t a g e s have a s l o p e of -0.90.  At the lowest  voltages  (0.25, 0.50, and l.Ov) the c h a r g i n g c u r r e n t s became too s m a l l t o be measured a f t e r a p p r o x i m a t e l y  500 s e c .  F i g s . 4-3 and 4-4 show the d i s c h a r g e c u r r e n t d e n s i t i e s as a f u n c t i o n of time a f t e r removal of the a p p l i e d v o l t a g e s .  The  c h a r g i n g v o l t a g e s were l e f t on f o r a t l e a s t 1000 s e c , i . e . a t l e a s t f i v e times the l e n g t h of time i n which the d i s c h a r g e were observed.  currents  The l i n e s have been drawn w i t h a s l o p e of -0.90  and the c l o s e f i t between the p o i n t s and the l i n e s i n d i c a t e s the d i s c h a r g e c u r r e n t s can be r e p r e s e n t e d  by a law J <*. t ^'90^  Y±g  4-5  shows the 2v Ta- p o s i t i v e d i s c h a r g e curve f o r f o u r d i f f e r e n t g o l d e l e c t r o d e s , a l l on the same sample (#3)• w i t h a s l o p e of -0.90.  The l i n e s a r e a g a i n drawn  The f o u r s e t s of p o i n t s , r o u g h l y  superimposed  1  0  9]o ft/1  lo  F i g . 4-1  2  sec)  Charge c u r r e n t s f o r Ta- n e g a t i v e  voltages  F i g . 4-3  Discharge c u r r e n t s f o r Ta-negative v o l t a g e s  F i g . 4-4•  Discharge currents f o r T a - p o s i t i v e  voltages  36 on t h e . l o w e r l i n e were taken at d i f f e r e n t s t a g e s i n the a n n e a l i n g ( i . e . from a p p r o x i m a t e l y  5 hours to 40 h o u r s ) .  The  one upper s e t  of p o i n t s i s f o r the same e l e c t r o d e as one of the lower s e t of p o i n t s , but was  taken p r i o r t o any a n n e a l i n g .  F i g . 4-5 i n d i c a t e s  t h a t the r e s u l t s are r e p r o d u c i b l e f o r d i f f e r e n t e l e c t r o d e s on the same sample a f t e r the sample has been annealed.  There does not  appear t o be a s t r o n g dependence on the a n n e a l i n g time. 4.1.2  Discharge C u r r e n t s as a F u n c t i o n of V o l t a g e On the b a s i s of the t h e o r y presented i n s e c t i o n 2.1 f o r  p o l a r i z a t i o n c u r r e n t s , the d e v i a t i o n of J  from an exact =jr law  can be a t t r i b u t e d to the d i s t r i b u t i o n f u n c t i o n G-(q) b e i n g not q u i t e flat..  A c c o r d i n g to e q u a t i o n ( 2 - 6 ) , J  s h o u l d be l i n e a r w i t h v o l t a g Jt i n the absence of a space charge. Thus, i n F i g . 4-6, — y i s p l o t t e d as a f u n c t i o n of v o l t a g e where J i s taken from F i g s . 4-3 P  0 and 4-4.  The v a l u e of J t  0  ,  9  0  90 i s o b v i o u s l y independent of time.  The o t h e r t h r e e e l e c t r o d e s t e s t e d on t h i s sample and r e f e r r e d t o i n F i g . 4-5 d i s p l a y e d a s i m i l a r dependence of J t ^ ' on V. v a l u e of J t ^ ' ^ i n c r e a s e d w i t h i n c r e a s i n g magnitude of TaV  The ne-  g a t i v e v o l t a g e s , went through a minimum at low Ta- p o s i t i v e v o l tages and then i n c r e a s e d a g a i n at l a r g e Ta- p o s i t i v e v o l t a g e s . F i g . 4-6  p o s s i b l y i n d i c a t e s space charge e f f e c t s .  l a r i z a t i o n c u r r e n t s , than f o r a l i n e a r response,  For p u r e l y poJ t ^ ' ^ should V  be independent of V and, i f a p o l a r i z a t i o n s a t u r a t i o n phenomenon o c c u r s , should be a d e c r e a s i n g f u n c t i o n of V ( r e f e r t o equations I I - 8 and I I - 9 ) • F i g . 4-6 may  Thus the curve a t low Ta- p o s i t i v e v o l t a g e s i n  indicate a polarization saturation.  p o r t i o n s of the curve may  The i n c r e a s i n g  be due t o a space charge decay c u r r e n t  component or to an i n c r e a s e d p o l a r i z a t i o n c u r r e n t due to the e f f e c t  -8h  SAMPLE TEST  5  No 3 VOLTAGE  GOLD ELECTRODES = 2'0V  Tc -  TESTED  IN  300/J . VACUUM  POSITIVE  -5  o  -W  0  2  t°9j  0  ft/  1  sec)  F i g . 4-5 Log J v s . l o g t f o r d i f f e r e n t e l e c t r o d e s on sample #3  F i g . 4-6  D i s c h a r g e c u r r e n t s as a f u n c t i o n of v o l t a g e - sample  #3  39  of the space charge.  R e f e r r i n g t o e q u a t i o n 2-22,  decay c u r r e n t i s of the form n ^ ( x * , t ) ^  the space charge  . . The p o l a r i z a t i o n c u r r e n t  c h a r a c t e r i s t i c i s unchanged, though, the magnitude of the i s changed (from V/L t o E(x*) c h a r a c t e r i s t i c (t and 4-4  ).  The f a c t t h a t the  current  discharge  ±s the same f o r a l l v o l t a g e s i n P i g s .  suggests the c u r r e n t i s due  o n l y t o one mechanism.  low v o l t a g e c h a r a c t e r i s t i c i n F i g s . 4-1  t o 4-4  4-3  The  indicate a polari-  z a t i o n c u r r e n t because the charge and d i s c h a r g e c u r r e n t s are  equal  and independent of p o l a r i t y (up t o 1 v o l t w i t h a maximum d e v i a t i o n of 20% at t = 0 s e c ) .  I t can a l s o be seen from F i g s . 4-1  t h a t d i s c h a r g e and charge c u r r e n t s are equal f o r Ta- n e g a t i v e v o l t a g e s =2v  (for t ^ 1 0  to  4-4  sec.)  and f o r T a - p o s i t i v e v o l t a g e s = 5v.  T h i s would seem t o i n d i c a t e p o l a r i z a t i o n c u r r e n t s even though the d i s c h a r g e c u r r e n t s are of d i f f e r e n t magnitudes f o r o p p o s i t e  pol-  a r i t i e s at the same v o l t a g e . S i m i l a r experiments to those above are now  discussed f o r  a d i f f e r e n t sample (#2) which had indium r a t h e r than g o l d e l e c t r o d e s . o The f i l m t h i c k n e s s was  the same (644+5 A) but t h i s speciman  t e s t e d i n dry a i r r a t h e r than i n low vacuum and a l s o was  was  annealed  at a maximum of 70°C.  The r e s u l t s were c o n s i d e r a b l y d i f f e r e n t Jt * from those f o r sample #3F i g . 4-7 shows a p l o t of — y — v s . V, analagous t o the J t ^ * ^ p l o t i n F i g . 4-6 f o r sample #3. t"*"*^ i s V 1  used because f i g u r e 4-8 The  shows a s l o p e of  0  -1.0.  curve decreases f o r both p o l a r i t i e s and has a peak  v a l u e about f o u r times l a r g e r than t h a t f o r sample #3erent e l e c t r o d e s on sample #2 gave s i m i l a r c u r v e s .  Three d i f f -  F i g . 4-8  shows the  d i s c h a r g e c u r r e n t f o r the t h r e e e l e c t r o d e s f o r f i v e v o l t s TaFrom e q u a t i o n  (2-9), ^  i s a measure of the l o s s e s at v e r y  positive.  low  40  ~5  Jt  V /10 SAMPLE  ,  ?  COUL/cm No 2  INDIUM  VOLT ELECTRODE  © 2\  5 Ta-NEGATIVE  F i g . 4-7  0 VOLTAGE  ®  ©  5  Ta-  D i s c h a r g e c u r r e n t s as a f u n c t i o n of v o l t a g e  POSITIVE  - sample #2  "c'  I  t  t  I  ;  Fig. 4 - 8  >  log  1Q  ft/1  I  sec)  I  Log J vs l o g t f o r d i f f e r e n t e l e c t r o d e s on sample # 2  I  2  t  42  frequencies,  assuming t h a t J i s p u r e l y p o l a r i z a t i o n c u r r e n t .  The  above r e s u l t s appear t o show t h a t sample #2 i s t h r e e or f o u r times as l o s s y as sample #3 a t v e r y low f r e q u e n c i e s .  However, a t 1 KHz,  the l o s s e s o f t h e t h r e e e l e c t r o d e s on sample #2 v a r i e d  between  t a n / = .0046 and t a n S = .0052 w h i l e t h e f o u r e l e c t r o d e s on sample #3 had l o s s e s v a r y i n g between t a n <f = .0061 and t a n S- .0081.  The  s t e p response b e h a v i o u r o f sample #2, as demonstrated i n F i g . 4-7 c o u l d n o t be s a t i s f a c t o r i l y e x p l a i n e d .  Some o t h e r samples t e s t e d  d u r i n g the course of t h e i n v e s t i g a t i o n showed a s i m i l a r l a c k of c o r r e l a t i o n between t h e b r i d g e measurements and t h e s t e p response measurements.  Data f o r these o t h e r samples i s l i s t e d below and a  s i n g l e p o i n t f o r each sample i s p l o t t e d i n F i g . 4-7 t o show t h e s c a t t e r . The p l o t t e d v a l u e s have been c o r r e c t e d f o r t h e d i f f e r e n t film  thicknesses. 1.  F i l m formed t o lOOv on b u l k Ta i n 0.5% H S 0 , a t 2  4  4 ma/cm , A l e l e c t r o d e s , t a n / ( l KHz) = 0.028, t e s t e d in air.  At 5v. Ta- p o s i t i v e , ^  = 3-3 x 1 0 ~  9  coul../cm.^ v o l t . 2.  F i l m formed t o 50v on b u l k Ta i n 0.5%  H  10 ma/cm . 2  S 2  °4  a  t  A l e l e c t r o d e s , t a n / ( l KHz) = 0.008,  tested i n a i r .  A t 3v Ta- p o s i t i v e , ^  = 2 x 10~ coul./cm 9  2  volt. 3.  F i l m formed t o 40v on b u l k Ta i n 0.5% H^PO^ a t 2 22 ma/cm . Gold e l e c t r o d e s , t a n /= 0.01 t e s t e d i n a i r . At 2v Ta- p o s i t i v e ,  4.  ~ 2 x 1 0 ~ coul./cm 9  F i l m formed on b u l k Ta t o 40v i n 4 0 % H S 0 2  a t 20 ma/cm . in air.  4  2  volt. a t 85°C  Indium e l e c t r o d e s , t a n /= 0.005; t e s t e d  At 5v Ta- p o s i t i v e , ^  = 2 x 10 ^ c o u l . / c m  2  volt.  43  S i n c e g o l d e l e c t r o d e s appear t o g i v e b e t t e r r e s u l t s , i t may  be t h a t some of the poor r e s u l t s o b t a i n e d w i t h the other e l e c t r o d e  m a t e r i a l s were due to oxide d e p o s i t e d on t h e f i l m when the e l e c t r o d e m a t e r i a l s were evaporated. why  However i t i s not e x a c t l y c l e a r  t h i s s h o u l d l e a d t o a d i f f e r e n t s t e p response.  The use  of  g o l d e l e c t r o d e s appears t o e l i m i n a t e the problem. 4.1.3  D.C.  Conduction F i g . 4-9  i s a Schottky p l o t ( l o g J  conduction currents. F i g s . 4-1  and 4-2  Currents L  vs  The leakage current. J-^ was  (j) ^ ) 1  of the  e s t i m a t e d from  by e x t r a p o l a t i n g t o l a r g e r times the l i n e s drawn  through the measured p o i n t s . are not drawn i n F i g s . 4-1 and labeled /kT- and 02  6/kT  The h i g h e r v o l t a g e s i n F i g . 4-9  and 4-2.  The s t r a i g h t l i n e s drawn  .correspond t o the c a l c u l a t e d v a l u e s of  the S c h o t t k y and P o o l e - F r e n k e l s l o p e s r e s p e c t i v e l y . c a l c u l a t e d from e q u a t i o n s 2-10 of 5.  2  These were  and 2.-11 u s i n g a d i e l e c t r i c  constant  The Ta- p o s i t i v e v o l t a g e curve appears s i m i l a r t o those (12)  observed  by Mead  and t o the curves c a l c u l a t e d by Prank and  (3)  Simmons  .  Namely,at h i g h e r v o l t a g e s the curve i s a s y m p t o t i c  the S c h o t t k y s l o p e .  The change i n s l o p e at 5v may  when charge i n j e c t i o n o c c u r s .  Prom P i g . 4-6,  a t the minimum i n the curve and may  correspond  correspond  to to  5v Ta- p o s i t i v e i s to when space charge  e f f e c t s b e g i n t o show up i n the Ta- p o s i t i v e d i s c h a r g e c u r v e s . I t was n o t i c e d t h a t s u f f i c i e n t l y h i g h v o l t a g e s  (from  10 - 15 v o l t s of e i t h e r p o l a r i t y ) caused a change i n the p r o p e r t i e s of sample #3. ature.  Such changes are c a l l e d ' " d e f o r m a t i o n " i n the  liter-  The conductance of the sample i n c r e a s e d by an amount depending  on the magnitude of the a p p l i e d v o l t a g e . P i g . 4-10  which i s f o r sample #3,  This i s i l l u s t r a t e d i n  e l e c t r o d e C of a r e a  Flo . 1  A—Q  T,OQVQ  mi  v  v  o  n  +.  oo  a  - F n  V-I  /-> -f S  n-p f  i  oi d  _  som^l  n  41 "z,  F i g . 4-10  Leakage c u r r e n t as a f u n c t i o n of f i e l d - sample #3  F i g . 4-11  Temperature dependence of l e a k a g e c u r r e n t s  F i g . 4-12  Leakage c u r r e n t s as a f u n c t i o n of f i e l d - sample #2  48 -2 3.50 + 0.05 x 10 i n a 3 0 0 | i vacuum.  2 cm , measurements b e i n g t a k e n w i t h t h e sample P i g . 4-10 shows t h e J-V c h a r a c t e r i s t i c f o r a  f i l m deformed t o a h i g h degree.  B e f o r e t h e f i l m was deformed, t h e  l e a k a g e c u r r e n t s were t h e same order of magnitude as those i n P i g . 4 3 i . e . a f a c t o r of between 10 film.  5 t o 10  s m a l l e r than i n t h e deformed  A l s o t h e r e c t i f i c a t i o n b e h a v i o u r d i s a p p e a r s and h y s t e r e s i s  e f f e c t s were reduced. the c a l c u l a t e d  The s t r a i g h t l i n e drawn i n P i g . 4-10 i s  Schottky slope.  Steady s t a t e l e a k a g e c u r r e n t s were measured as a f u n c t i o n of temperature  and v o l t a g e on sample #2, e l e c t r o d e  —3 + 0.10 x 10 t i o n process.  ( a r e a 4.25  b  2 cm ) t o determine  t h e a c t i v a t i o n energy of the conduc-  P i g . 4 - 1 1 i s a p l o t o f l o g J-^ v s . ^ where t h e mea-  surements were t a k e n f o r d e c r e a s i n g temperatures. energy i s determined  The a c t i v a t i o n  from e q u a t i o n 2-10 f o r a S c h o t t k y  process  and from a s i m i l a r e q u a t i o n w i t h ec r e p l a c e d by (3 f o r a P o o l e Prenkel process.  P i g . 4-12 i s a S c h o t t k y p l o t of t h e leakage  w i t h t h e drawn l i n e s c o r r e s p o n d i n g t o the c a l c u l a t e d P o o l e - F r e n k e l s l o p e s as b e f o r e .  current  S c h o t t k y and  The r e a s o n a b l e f i t between t h e  p o i n t s and t h e l i n e s j u s t i f i e s c a l c u l a t i n g t h e a c t i v a t i o n  energies  shown i n F i g . 4 - 1 1 by u s i n g t h e t h e o r e t i c a l v a l u e s of ©c and |3 f o r t h e Ta- p o s i t i v e and Ta- n e g a t i v e c u r r e n t s r e s p e c t i v e l y . Ta- p o s i t i v e v o l t a g e a c t i v a t i o n energy was 0 . 6 4 ev.  The  and t h e Ta-  n e g a t i v e v o l t a g e a c t i v a t i o n energy 0.82 ev. 4.2 B r i d g e Measurements 4.2-..1  Capacitance Fig.  and Loss Measurements  4 - 1 3 i s a p l o t of c a p a c i t a n c e and l o s s e s v s . f r e -  quency f o r sample #3, g o l d e l e c t r o d e b of a r e a 6.3+0.2x10  —4  2 cm".  50  The p l o t t e d v a l u e s are CoK"  = C t a n £ and CoK'  where K' - jK" i s  the complex d i e l e c t r i c c o n s t a n t and Co i s the c a p a c i t a n c e of the air filled  capacitor.  t o be a c c u r a t e t o 0.01% CoK"  The G.R.  b r i d g e v a l u e s f o r CoK'  f o r f < 10 KHz  i s a c c u r a t e t o 0.1%.  and t o 0.2%  at 100  specified  KHz.  The Boonton b r i d g e measurements of  are s p e c i f i e d t o be a c c u r a t e t o 0.25% t o o n l y 10%.  are  + 0.2pf and CoK"  CoK'  i s accurate  The d i f f e r e n c e i n the l o s s measurements shown f o r  the two b r i d g e s i s about 10%.  The v a l u e s of CoK'  above 100  have been c o r r e c t e d f o r s e r i e s r e s i s t a n c e i n the l e a d s ,  KHz  connections  e t c . , and approximate c o r r e c t i o n s have . been made f o r l e a d i n d u c t a n c e s u s i n g the t a b l e s p r o v i d e d w i t h the Boonton b r i d g e . 500 KHz was  The decrease  i n CoK"  at f r e q u e n c i e s l e s s  than  unusual as f o r most samples t e s t e d the l o s s e s c o n t i n u e d  t o i n c r e a s e a t low f r e q u e n c i e s .  A s m a l l e l e c t r o d e was  required  f o r these h i g h frequency measurements as the Boonton can measure a maximum c a p a c i t a n c e of o n l y 1000 external  pf ( w i t h o u t  range-extending  capacitors). Por sample # 3 , measurements i n a 300|i vacuum on two g o l d  e l e c t r o d e s of a r e a Two  3-5 x 10  g o l d e l e c t r o d e s of a r e a  t o 27.1  + .8.  -2  2 cm  gave a v a l u e of K' = 25-9 + —3 2  4 . 9 x 10  cm  gave a v a l u e of K'  up t o 140°C i n vacuum.  t h a t c a p a c i t a n c e and l o s s v a l u e s tended a f r e s h l y prepared  sample.  about a day, the observed  In g e n e r a l , i t was t o decrease  found  w i t h time f o r  Once the samples had been s h o r t e d f o r v a l u e s of the c a p a c i t a n c e would change  l e s s than 0 . 3 % over a p e r i o d of 2-15  days.  P u t t i n g a sample i n a  vacuum (300|i) would cause a drop of about 0.5% i n Cok"  =26.5  The p r e v i o u s h i s t o r y of t h i s sample i n c l u d e d a n n e a l i n g  at temperatures  7-15%  0.8.'  i n a few minutes,  i n CoK'  and  and then a slow decrease  from  as mentioned  51  above.  H e a t i n g a sample and then c o o l i n g i t would r e s u l t i n the  same e f f e c t as s h o r t c i r c u i t i n g t h e sample f o r a day.  This pro-  cedure was c a r r i e d out on most samples. 4.2.2  Temperature Dependence of C a p a c i t a n c e and Losses P i g . 4-14 shows a h y s t e r e s i s e f f e c t i n t h e v a l u e s . o f  CoK'  and CoK" when a sample i s heated. o  T h i s f i g u r e i s f o r sample  #1, o f t h i c k n e s s 718 + 5 A w i t h a g o l d c o u n t e r e l e c t r o d e o f a r e a -3  2  4 x 10 ^ cm . Measurements were made a t 1 KHz i n d r y a i r .  A  s i m i l a r e f f e c t was observed f o r a l l samples, though i n most cases not so pronounced.  The v e r y r a p i d i n c r e a s e i n CoK' and CoK" as  the temperature was i n c r e a s e d s l i g h t l y may have been a s p e c i a l f e a t u r e w i t h t h i s sample due t o i t s p r e v i o u s h i s t o r y , i n t h a t i m m e d i a t e l y p r i o r t o t h i s experiment, f i e l d s up t o 3«5 x 10^ v/cm w i t h Ta- p o s i t i v e , had been a p p l i e d to..the sample.  The l o s s e s  are p l o t t e d as CoK" , because a c c o r d i n g t o e q u a t i o n 2-4, t h i s can T be i n t e r p r e t e d as a p l o t o f t h e d i s t r i b u t i o n f u n c t i o n G-(q) . At the p o i n t marked "steady s t a t e " t h e sample was h e l d a t c o n s t a n t temperature f o r t h r e e hours and Cok' and CoK" changed l e s s than T 0.4% and 1.5% r e s p e c t i v e l y i n t h i s i n t e r v a l .  The sample was then  s l o w l y c o o l e d t o l i q u i d n i t r o g e n temperatures. Fig. CoK"  4-15 shows t h e temperature dependence o f CoK' and.  o f sample #2 e l e c t r o d e a between 295°K and 85°K a t 1 KHz, i n  dry a i r .  The a r e a of t h e indium e l e c t r o d e was 3-7 + .1 x 10  T h i s f i g u r e , as d i d P i g . 4-14, demonstrated temperature  cm  t h a t below a c e r t a i n  ( a p p r o x i m a t e l y 250°K here) t h e l o s s e s became l i n e a r  w i t h temperature.  I f t h e l o s s e s a r e due to t h e i o n i c r e l a x a t i o n  p r o c e s s c o n s i d e r e d i n s e c t i o n 2.1 then a c c o r d i n g t o e q u a t i o n (2-4) the d i s t r i b u t i o n G-(q) i s f l a t over t h i s l o w e r range of a c t i v a t i o n  10501  950h  850\-  0-20Y  0-161  0-12Y  O'OSl  004 h  100 TEMPERATURE T K 200 300 F i g . 4-14 Temperature dependence o f C^K' and C K" - sample #1 ' n  400  o 5 3  1235  SAMPLE O  No 2  ELECTRODE  f=  "a"  kHz  /  -C K Q  Ta/Ta  2  0  5  /  INDIUM  i  o O  145  TEMPERATURE  205 o T K  265  P i g . 4-15 Temperature dependence of C K' and C K" - sample #2 o  o  55  energies. CoK"  F i g . 4-16 shows t h e f r e q u e n c y dependence of CoK' and  of t h e same sample a t 82°K.  The l o s s e s a r e more independent  of f r e q u e n c y than t h e l o s s e s a t room' temperature.  T h i s would be  expected, as F i g . 4-15 shows t h a t t h e d i s t r i b u t i o n G(q) i s n o t f l a t near room temperature.  The v a l u e of K' c a l c u l a t e d a t 1 KHz  and a t room temperature was 24.3 + 0.8 t o 24.1 + 0.8.  The + 0.8  e r r o r i s due m a i n l y t o u n c e r t a i n t y i n the a r e a of t h e e l e c t r o d e . T h i s sample had been heated t o a maximum of about 70°C i n i t s h i s t ory. An e s t i m a t e o f t h e a c t i v a t i o n energy of t h e l o s s p r o c e s s e s may be a c c u r a t e l y o b t a i n e d from t h e r e s u l t s i n F i g . 4-15. e q u a t i o n 111-10 and n e g l e c t i n g and Du Pre) g i v e s l n  —  (  a  = 3 4 . 8 + 1.5.  Applying  s was done by Gever's Therefore the a c t i v a t i o n  o e n e r g i e s a r e q = 0.25 + .01 ev. a t 85 K and 0.88 + .05 ev. a t 295°K.  By e x t r a p o l a t i n g t h e curves i n F i g . 4-15 t o 0°K, then  from e q u a t i o n ( 2 - 3 ) t h e intercept„of CoK' a t T = 0°K g i v e s CoK«o . The r e s u l t i s t h a t K ^ > 0 . 9 K ' where K' i s taken a t 295°K.  Thus  the p o l a r i z a t i o n p r o c e s s e s w i t h c h a r a c t e r i s t i c f r e q u e n c i e s belcw the i n f r a r e d c o n t r i b u t e l e s s t h a n 1 0 % t o t h e p o l a r i z a t i o n of the dielectric. 4.2.3  Comparison of B r i d g e and Step Response Values f o r t h e l o s s Fig-. 4-17 i s a p l o t of t h e d i e l e c t r i c l o s s e s f o r sample  # 3 , g o l d e l e c t r o d e .a, o b t a i n e d by both b r i d g e measurements and s t e p response measurements, as a f u n c t i o n of t h e r e l a x a t i o n t i m e . F o r t h e b r i d g e measurements, t h e r e l a x a t i o n time ^ i s g i v e n by 2%/co and f o r t h e s t e p response measurements, e q u a t i o n 2-9 was used with  = t/2.  The s t e p response v a l u e s were taken from the one  ON  log.  F i g . 4-17  Q  (T/lsec)  D i e l e c t r i c l o s s e s vs. r e l a x a t i o n time  57 v o l t anodic curve i n F i g . 4-4.  The smooth j o i n i n g of the two  s e t s of p o i n t s i n d i c a t e s t h a t the c u r r e n t s measured a t one v o l t were p o l a r i z a t i o n c u r r e n t s .  The b r i d g e measurements i n F i g . 4-17  have been c o r r e c t e d f o r s e r i e s r e s i s t a n c e due t o the l e a d s , e t c . The d e v i a t i o n s i n the p o i n t s a t 60 KHz and 100 KHz may be due t o small inaccuracies i n determining  the s e r i e s r e s i s t a n c e .  The  d i e l e c t r i c l o s s e s c o u l d not be c a l c u l a t e d a c c u r a t e l y , f o r example , as a t 100 KHz the t r u e s e r i e s r e s i s t a n c e accounted f o r l e s s than 20% of the measured l o s s e s . To o b t a i n a complete curve of the l o s s e s v s . r e l a x a t i o n time i n F i g . 4-17 would r e q u i r e e i t h e r measuring the l o s s e s a t v e r y low f r e q u e n c i e s , very short times,  or measuring the p o l a r i z a t i o n c u r r e n t s a t  or by u s i n g a temperature sweep t e c h n i q u e .  l a t t e r method was attempted.  F i g . 4-18 shows CoK' and CoK" T  a f u n c t i o n of temperature between 300°K and 365 K.  then jt"  Q  = kT In 4 ^  .  Using equation  may be determined from F i g . 4-18.  Q  This procedure was not  independent of temperature.  values  temperature  T a k i n g CoK' t o be l i n e a r l y dependent on CoK"  temperature, t h e n , i n order t o have  F i g . 4-18.  TT i s needed  111-10 as before  too s u c c e s s f u l as the r e s u l t s i n d i c a t e d an a p p a r e n t l y dependent v a l u e of ^ -  as  To r e l a t e a  temperature change t o a f r e q u e n c y change, the constant i n equation. 2-5, q  The  constant,  This i s not t h e case as shown i n  Using the l a r g e s t value  of X ,  f o r the l o s s e s d i d f a l l r o u g h l y  s e t s of p o i n t s i n F i g . 4-17.  must be  Q  however, the c a l c u l a t e d  on the l i n e j o i n i n g the two  12,900  SAMPLE  No 3  Ta/Tq 0 2  5  ELECTRODE  /60LD  " a"  58  f = 1kH  o  o  z  i2,eoo  12,700  12,6 00\  12,5 00'I ©  300  320  F i g . 4-18  TEMPERATURE  T  340  Temperature dependence o f C^K  360  °K 1  and C K" 0  •sample #3  ©  59 5.  5•1  DISCUSSION  Step Response Measurements Measurements of low frequency d i e l e c t r i c l o s s e s u s i n g a  step response t e c h n i q u e are c o m p l i c a t e d and p o l a r i z a t i o n s a t u r a t i o n e f f e c t s . s e c t i o n 2.3«2 showed how the e x t e r n a l l y  by space charge  effects  The t h e o r e t i c a l a n a l y s i s i n  a space charge decay may  contribute to  observed d i s c h a r g e c u r r e n t s and how  the c u r r e n t due t o d i e l e c t r i c p o l a r i z a t i o n . F i g . 4-6 i n d i c a t e a space charge e f f e c t .  i t may  affect  The r e s u l t s shown i n  A polarization satura-  t i o n phenomenon might be expected at the f i e l d s used h e r e , as can be seen by comparing e q u a t i o n s I I - 8 and I I - 9 . F i g . 4-6 d i s p l a y s what might be such an e f f e c t .  Thus, t o measure low f r e q u e n c y  d i e l e c t r i c l o s s e s a c c u r a t e l y r e q u i r e s u s i n g s u f f i c i e n t l y low f i e l d s i f s a t u r a t i o n phenomenon are t o be a v o i d e d and a r e c t i f y i n g c o n t a c t t o reduce any charge i n j e c t i o n . (7) C h e r k i . et a l ,  c a r r i e d out f r e q u e n c y response and  response experiments on a Ta/Ta 0^/ g o l d d e v i c e w i t h r e s u l t s 2  d i f f e r e n t from those r e p o r t e d .here.  step slightly  T h e i r measured c u r r e n t s f o l l o w e d  an exact — law a t temperatures from 4°K t o 295°K, were p o l a r i t y independent and were l i n e a r w i t h f i e l d up t o 5 x 10 corresponds t o about 3 v o l t s i n F i g . 4-5)-  v/cm  (which  The l o s s e s were f r e q u -  ency independent over the same temperature range.  I n t h i s range  of f i e l d s they observed no space charge or p o l a r i z a t i o n s a t u r a t i o n effects.  The r e s u l t s i n t h i s work i n d i c a t e d  frequency-independent  l o s s e s below about 250 K. At 295 K the 1 osses were not q u i t e i n d e (29) / / pendent of f r e q u e n c y . D r e i n e r , u s i n g a Ta/TapO^/ e l e c t r o l y t e W  system, observed T- law d i s c h a r g e c u r r e n t s l i n e a r w i t h a Ta-  positive  60 f i e l d s up t o 2 x 10^ v/cm.  Above t h i s f i e l d the d i s c h a r g e c u r r e n t s  i n c r e a s e d more r a p i d l y w i t h f i e l d .  I t i s not c l e a r why  a satura-  t i o n phenomena does not appear i n D r e i n e r ' s and C h e r k i ' s r e s u l t s . I t i s a l s o not c l e a r why  Cherki  et a l . d i d not observe a space  charge e f f e c t f o r Ta- n e g a t i v e v o l t a g e s .  P o s s i b l y t h e i r sample  p r e p a r a t i o n t e c h n i q u e r e s u l t e d i n the Ta/TapO^ b a r r i e r of t h e i r  de-  v i c e s b e i n g h i g h e r than the b a r r i e r s of the d e v i c e s used i n t h i s work.  However s i n c e no d e t a i l s of t h e i r sample p r e p a r a t i o n t e c h n i q u e  were g i v e n a comparison i s not p o s s i b l e . 5•2  D.C.  Conduction  Currents  The i n v e s t i g a t i o n of d.c. e l e c t r o n i c c o n d u c t i o n difficult  was  because of marked time dependence of the c u r r e n t s .  For  sample #2 at h i g h Ta- p o s i t i v e v o l t a g e s the c u r r e n t s were s t i l l d r i f t i n g s l o w l y a f t e r twelve hours. tended  The Ta- n e g a t i v e c u r r e n t s  t o go through a minimum at between 1 and 30  minutes a f t e r  a p p l y i n g the v o l t a g e , and then s l o w l y i n c r e a s e d . P l o t t e d v a l u e s as i n F i g . 4-12  were t a k e n at the minimum c u r r e n t l e v e l .  s i e n t c h a r g i n g c u r r e n t s shown i n F i g . 4-1  and 4-2  v o l t a g e s are much l a r g e r than can be accounted currents.  The  tran-  f o r the l a r g e r  f o r by p o l a r i z a t i o n  They are p r o b a b l y due t o e i t h e r one or both of two  effects;  l ) The t r a n s i e n t i s due t o a space charge b u i l d u p a c c o r d i n g to e q u a t i o n (2-20), or 2) the c u r r e n t i s due t o l a t e d leakage c u r r e n t s , which may  space charge  modu-  be .given by an e q u a t i o n l i k e  (2-10)  where the f i e l d E(0) i s c o n t r o l l e d by the development of a space charge. The r e s u l t s i n F i g . 4-9 for  and 4-12  i n d i c a t e a Schottky  Ta- p o s i t i v e v o l t a g e and a P o o l e - F r e n k e l process f o r Ta-  voltage.  In F i g . 4-9  process  negative  the Ta- p o s i t i v e c u r r e n t s at the lower v o l t a g e s  are l e s s than p r e d i c t e d by the S c h o t t k y law due t o a space  charge.  The curve has been drawn w i t h a sudden change between t h r e e and f i v e v o l t s , s i n c e P i g . 4-6 appears t o i n d i c a t e t h a t space charge develops  i n this interval.  The asymmetry i n c o n d u c t i o n  i s marked by a pronounced r e c t i f i c a t i o n  processes  characteristic.  The a c t i v a t i o n e n e r g i e s f o r the c o n d u c t i o n c u r r e n t s noted (12) i n P i g . 4-11 a r e c o n s i d e r a b l y h i g h e r than those r e p o r t e d by Mead ( i . e . 0.4 e v ) . However i f Mead's r e s u l t s were f o r a deformed f i l m (32) then i t i s p r o b a b l y meaningless  t o compare v a l u e s .  u s i n g a Ta/TapO^ e l e c t r o l y t e system,reported  Cherki  et a l . ,  a v a l u e of 0.72 ev f o r  Ta- p o s i t i v e c u r r e n t s . w h i c h i s h i g h e r than t h e 0.64 ev r e p o r t e d here. I f these v a l u e s a r e due t o the m e t a l / o x i d e and e l e c t r o l y t e / o x i d e i n t e r f a c e s , i t would be expected t h a t they be d i f f e r e n t . Sample #3 was the o n l y one which deformed a t h i g h e r v o l t a ges.  Deformation  commenced a t about 10 - 15 v o l t s of e i t h e r p o l a r -  ity.  The r e s u l t s shown i n P i g . 4-10 f o r a h i g h l y deformed f i l m (12)  are s i m i l a r t o the curves p u b l i s h e d by Mead device.  f o r a Ta/TapO^/Au  That i s , there i s l i t t l e or no r e c t i f i c a t i o n , t h e c u r r e n t s  obey a S c h o t t k y law, and t h e c u r r e n t s a r e about the same order of magnitude f o r the same f i e l d .  Thus i t seems p o s s i b l e t h a t Mead's  r e s u l t s were f o r a deformed f i l m .  He r e p o r t e d t h a t he c o u l d not g e t  r e p r o d u c i b l e r e s u l t s due t o d r i f t . Such deformations  of oxide f i l m s due t o an a p p l i e d f i e l d (30 3 l ) have been r e p o r t e d p r e v i o u s l y ' . I t i s a necessary c o n d i t i o n (31) for  o b s e r v i n g the n e g a t i v e r e s i s t a n c e d i s p l a y e d by t h i n f i l m s  .  To observe a d i f f e r e n t i a l n e g a t i v e r e s i s t a n c e , r e s e a r c h e r s speak of a p p l y i n g a "forming" v o l t a g e t o t h e f i l m which causes a l a r g e i n crease i n f i l m c o n d u c t i v i t y . The p r o c e s s e s o c c u r i n g i n the f i l m s when they a r e deformed a r e not f u l l y understood.  62 5.3  Comparison•of Samples #2 and  #3  Samples //2 and #3 were prepared under i d e n t i c a l c o n d i t i o n s except f o r d i f f e r e n t f o r m a t i o n c u r r e n t d e n s i t i e s but d i s p l a y e d considerably different properties.  V o l t a g e s up to 25 v o l t s Ta-  positive  c o u l d be a p p l i e d to sample #2 w i t h o u t c a u s i n g breakdown or any f o r m a t i o n whereas 10-15 breakdown. and 10-15  v o l t s Ta- n e g a t i v e was  de-  s u f f i c i e n t t o cause  Sample #3 would breakdown a t about 15 v o l t s Ta-  positive  v o l t s n e g a t i v e and b e f o r e b r e a k i n g down would be c o n s i d -  e r a b l y deformed,as a l r e a d y d e s c r i b e d .  The s t e p response  behaviour  of the two f i l m s a l s o gave d i f f e r i n g r e s u l t s ; t h a t i s , the s t e p response  of sample #2 c o u l d not be c o r r e l a t e d w i t h the l o s s e s as  measured w i t h a b r i d g e .  The d i e l e c t r i c c o n s t a n t of sample #3  was h i g h e r than t h a t of sample #2. be a t t r i b u t e d t o :  These d i f f e r e n c e s c o u l d a p r i o r i  the d i f f e r e n t e l e c t r o d e m e t a l s , the d i f f e r e n t  f o r m a t i o n c u r r e n t d e n s i t y , or the d i f f e r e n t h i s t o r y of the samples and e x p e r i m e n t a l c o n d i t i o n s , ( i . e . sample #2 was measured i n a i r , sample #3  i n vacuum).  It.is p r o b a b l y t h a t the f i r s t and  reasons are the most i m p o r t a n t . sample #3 may  The h i g h temperature  a n n e a l i n g of  have a l t e r e d the s t o i c h i o m e t r y of the o x i d e .  iments by Smythe et a l .  Exper-  have i n d i c a t e d t h a t a n n e a l i n g causes  an i n c r e a s e i n the c a p a c i t a n c e , which may K' noted here.  third  e x p l a i n the d i f f e r e n c e i n  The d e f o r m a t i o n p r o p e r t i e s of sample #3 may  also  (37)  be caused by the vacuum a n n e a l i n g .  Hickmott  n e g a t i v e c o n d u c t i v i t y ( i . e . deformation)  w  has noted  that  i s a c h i e v e d more e a s i l y  under vacuum, though t h i s i s not a n e c e s s a r y c o n d i t i o n .  Schwartz  et a l [ ^ ^ , u s i n g Ta/TapO^/Au d e v i c e s , n o t e d t h a t the presence  of  water vapour caused h i g h l y asymmetric breakdown c h a r a c t e r i s t i c s . A f t e r vacuum-baking t h e i r samples, the Ta- p o s i t i v e breakdown  63  v o l t a g e had decreased t o a p p r o x i m a t e l y the same as the Ta-negative breakdown v o l t a g e .  T h i s perhaps e x p l a i n s why  sample #2 here showed  such h i g h Ta- p o s i t i v e breakdown s t r e n g t h when compared w i t h sample #3. As mentioned  p r e v i o u s l y , no s a t i s f a c t o r y reason c o u l d be  found f o r the d i f f e r e n t s t e p response 5•4  Frequency and Temperature Fig.  4-13  results.  Dependence of the D i e l e c t r i c P r o p e r t i e s  shows t h a t the l o s s e s CoK"  become f l a t at 500  KHz.  T h i s would be expected i f the sample had a temperature b e h a v i o u r s i m i l a r t o t h a t shown i n F i g . 4-15-  From e q u a t i o n (2-5), an i n c r e a s e  i n f r e q u e n c y has the same e f f e c t as a decrease i n temperature, and a c c o r d i n g t o F i g s . 4-14  and 4-15  the l o s s e s become f l a t as the.  temperature d e c r e a s e s . The r e a s o n f o r the r a p i d i n c r e a s e of CoK' . and CoK"  with  an i n c r e a s e i n temperature,as. shown i n F i g . 4-14, i s • n o t c l e a r . It: may  be due t o a p r o c e s s i n v o l v i n g the r e l e a s e of e l e c t r o n s from  t r a p s or t o some p r o c e s s i n which the_temperature i n c r e a s e a change i n the number of p o l a r i z a b l e p a r t i c l e s . b i l i t y may  The l a t t e r p o s s i -  a l s o e x p l a i n the apparent nonconstant v a l u e of  determined from F i g . 4-18.  causes  7T  o  I f the l o s s e s are due t o i o n s not bound  to any p a r t i c u l a r s i t e i n the f i l m , then the number of such, " i n t e r stitial"  i o n s c o u l d be temperature dependent. The a n a l y s i s i n  Appendix  I I considered  o n l y a f i x e d number of p o l a r i z a b l e p a r t i c l e s .  I t was n o t i c e d t h a t by h e a t i n g some samples, and then c o o l ing  them, the l o s s e s c o u l d be reduced by as much .as 50%.  A similar  e f f e c t was n o t i c e d a f t e r s i m p l y s h o r t i n g a f r e s h l y prepared sample for  a l o n g time.  Under vacuum c o n d i t i o n s , t h e l o s s e s decreased  i d l y at f i r s t and then s l o w l y , a s above.  This d r i f t i n the l o s s  rap-  64 v a l u e s may be due t o two e f f e c t s : "interstitial"  i o n s , then the l o s s e s w i l l decrease as t h e s e i o n s a r e  trapped i n l a t t i c e s i t e s . the' p r o c e s s .  l ) I f t h e l o s s e s a r e due t o  A n n e a l i n g a t h i g h temperatures w i l l  speed  T h i s argument' i s supported by the dependence of t h e  l o s s e s , on f o r m a t i o n c u r r e n t d e n s i t y and on time a f t e r c e a s i n g formation  ( r e f e r e n c e ( 8 ) , p. 163). 2) I f t h e l o s s e s are due t o  absorbed i m p u r i t i e s such as water t h e n t h e l o s s e s w i l l as t h e f i l m i s " d r i e d " .  decrease  Experiments by Schwartz and G r e s h ^ ^  have i n d i c a t e d t h a t t h e presence of water vapour can cause l a r g e increases i n the losses i n  Ta 0r-. o  2 5 The s t r o n g temperature dependence of the d i e l e c t r i c l o s s e s (7) i s i n d i c a t i v e of an a c t i v a t i o n energy p r o c e s s .  C h e r k i et a l .  measured l o s s e s down t o 4<.2°K on a TapO^ specimen, where they foundthe  d i s t r i b u t i o n of a c t i v a t i o n e n e r g i e s t o be f l a t .  However t h e i r  r e s u l t s i n d i c a t e t h a t the d i s t r i b u t i o n has a d i f f e r e n t v a l u e a t 4»2°K than a t 77°K and a t 295°K. of  From e q u a t i o n ( 2 - 4 ) , a measure CoK"  t h e d i s t r i b u t i o n G(q) i s g i v e n by — — - .  then,at- 295°K,  ^r~=  0.0407 pf/°K, a t 77 K., Q  and a t 4 . 2 ° K , ^ p - = 0.35 pf/°K.  Prom C h e r k i ' s paper . =' 0 . 0 4 0 3 p f / ° K  Th e i r d i s t r i b u t i o n i s f l a t  from  295 K t o 77 K wher eas t h a t shown i n Pigs. 4-14 and 4-15 i s f l a t a p p r o x i m a t e l y 250° t o 80°K.  from  I t i s p o s s i b l e t o c a l c u l a t e the a c t i -  v a t i o n e n e r g i e s of t h e p r o c e s s e s o c c u r i n g a t 4.2°K i n the paper of C h e r k i et a l . U s i n g e q u a t i o n 111-10 and t h e i r v a l u e s of CoK' and CoK" between 295° and 77°K, f e = 31.3- From q = kT I n o o then q = 0.011 ev a t 4.2°K. 5•5 The I o n i c R e l a x a t i o n Model t  h  e  n  l  n  I t seems improbable t h a t i o n s w i t h v e r y low b i n d i n g e n e r g i are  t h e cause of t h e l o s s e s a t 4.2°K, s i n c e , i f such s h a l l o w l e v e l s  65  e x i s t e d , t h e n a t room temperatures the i o n s would he f r e e t o move u n t i l trapped.  C o o l i n g t h e f i l m s would r e s u l t i n t h e r e b e i n g no  ions i n the shallow l e v e l s .  I f a t room temperature the i o n s were  not t r a p p e d , then i o n i c c o n d u c t i o n would o c c u r a t low f i e l d s .  This  perhaps s u g g e s t s t h a t t h e l o s s e s may be due t o an e l e c t r o n i c p r o c e s s , such as e l e c t r o n s h o p p i n g between i m p u r i t y s t a t e s a t h i g h e r temp e r a t u r e s and perhaps t u n n e l i n g a t low t e m p e r a t u r e s .  Cherki's  r e s u l t s show t h a t CoK" becomes much l e s s temperature dependent below 77°K, perhaps i n d i c a t i n g a t u n n e l i n g p r o c e s s .  One f u r t h e r f a c t  which c o n f l i c t s w i t h an i o n i c l o s s p r o c e s s i s t h a t t h e a c t i v a t i o n energy of t h e b u l k e l e c t r o n i c c o n d u c t i o n c u r r e n t s (~ 0.82 ev) i s about t h e same as the a c t i v a t i o n energy of t h e l o s s e s a t room temp e r a t u r e (~ 0.88 e v ) . I f t h e l o s s e s were i o n i c , t h e n t h e c o n d u c t i o n s h o u l d be a l s o . A p o s s i b l e model f o r an e l e c t r o n i c p r o c e s s would be an E -E e x p o n e n t i a l t r a p d i s t r i b u t i o n i n t h e bandgap of t h e form N^ct exp-( ^ Above t h e Eermi l e v e l , t h e t r a p o c c u p a t i o n would be g i v e n by a E^-Ep Boltzmann d i s t r i b u t i o n , e x p - ( — ^ — ) .  Thus t h e number o f e l e c t r o n s a t  a p a r t i c u l a r t r a p l e v e l w i l l be independent of t h a t t r a p l e v e l , which i s e s s e n t i a l l y t h e same s i t u a t i o n as was assumed i n c a l c u l a t i n g the i o n i c r e l a x a t i o n l o s s e s . The r a t h e r l a r g e a c t i v a t i o n energy (0.88ev) of t h e l o s s e s a t room temperature s u g g e s t s a r e a s o n f o r t h e apparent i n c r e a s e i n t h e d i s t r i b u t i o n n e a r room temperature shown i n P i g s . 4-14, 4-15 and 4-18.  At t h e s e a c t i v a t i o n e n e r g i e s , some i o n movement  may be expected ^and s o ~ t h e l o s s e s c o u l d be composed of an e l e c t r o n i c component and an a d d i t i o n a l i o n i c component.  The a n a l y s i s p r e s e n t e d  i n Appendix I I I would have t o be m o d i f i e d t o account f o r t h e changing number of p o l a r i z a b l e  particles.  66 6.  CONCLUSIONS  In s t u d y i n g the p r o p e r t i e s of a Ta/TapO^/ metal d e v i c e , the d i e l e c t r i c l o s s e s were found t o be n e a r l y independent of f r e q u e n c y over a v e r y wide range of f r e q u e n c i e s . temperature dependence of the energy type p r o c e s s .  losses indicated  The s t r o n g an a c t i v a t i o n  B r i d g e measurements at temperatures  below  250 K i n d i c a t e d ..that the d i s t r i b u t i o n o f a c t i v a t i o n e n e r g i e s became U  f l a t between a p p r o x i m a t e l y 0.25  ev and 0.75  ev.  Measurements at  500 KHz at room temperature a l s o showed the l o s s e s becoming frequency independent, as i n d i c a t e d by the low temperature measurements. The t h e o r y suggested t h a t space charge e f f e c t s w i l l show up i n the d i s c h a r g e c u r r e n t t r a n s i e n t s a f t e r removal of an a p p l i e d voltage.  The e f f e c t s were observed e x p e r i m e n t a l l y and were found t o  be p o l a r i t y dependent; oxide i n t e r f a c e s  t h a t i s , d i f f e r e n t b a r r i e r h e i g h t s at the m e t a l /  e i t h e r f a c i l i t a t e d or i n h i b i t e d charge i n j e c t i o n  i n t o the i n s u l a t o r .  The space charge can cause a d i s c h a r g e c u r r e n t  due t o the decay of the space charge and can cause a change i n t h e magnitude  of the p o l a r i z a t i o n c u r r e n t s .  The experiments  indicated  t h a t the main e f f e c t of the space charge was t o a l t e r the p o l a r i z a t i o n current  magnitude.  A p o l a r i z a t i o n s a t u r a t i o n e f f e c t was observed i n the s t e p response experiments when the space charge e f f e c t s were reduced by using a r e c t i f y i n g contact.  The p o l a r i z a t i o n s a t u r a t i o n e f f e c t and  the space charge e f f e c t s must be t a k e n i n t o c o n s i d e r a t i o n when measuring technique.  the low f r e q u e n c y d i e l e c t r i c l o s s e s by a s t e p response  67  The l o s s e s were found t o be v e r y dependent on the sample history.  P o r example, an a n n e a l i n g process reduced  t h e l o s s e s , by  as much as 50% i n some cases. The  c o n d u c t i o n c u r r e n t s were found t o be t i m e dependent,  which was a t t r i b u t e d t o t h e development of a space charge.  The Ta-  p o s i t i v e c u r r e n t s obeyed a S c h o t t k y law and t h e Ta- n e g a t i v e c u r r e n t s obeyed a P o o l e - F r e n k e l law.  T h i s asymmetry i n c o n d u c t i o n mechanisms  was marked by a pronounced r e c t i f i c a t i o n c h a r a c t e r i s t i c .  The S c h o t t k y  law c u r r e n t s showed t h e o r e t i c a l l y p r e d i c t e d space charge e f f e c t s , and the onset of these space charge e f f e c t s was c o r r e l a t e d w i t h t h e presence  of space charge as i n d i c a t e d i n t h e s t e p response  iments.  On one sample t e s t e d , h i g h v o l t a g e s caused a d e f o r m a t i o n  of t h e f i l m , c h a r a c t e r i z e d by an i n c r e a s e d conductance.  exper-  F o r these  deformed f i l m s , t h e r e c t i f i c a t i o n behaviour d i s a p p e a r s and t h e c u r r e n t s f o l l o w a S c h o t t k y lav/. The a c t i v a t i o n energy of the b u l k e l e c t r o n i c  conduction  c u r r e n t s was found t o be about t h e same as t h a t c a l c u l a t e d f o r t h e i o n i c r e l a x a t i o n l o s s e s a t room temperature.  This f a c t , c o u p l e d w i t h  the observed h i g h l o s s e s a t low t e m p e r a t u r e s , i n d i c a t e s t h a t t h e l o s s e s may be due to an e l e c t r o n i c process r a t h e r than an i o n i c process.  I f t h e l o s s e s were due t o i o n s , then an i o n i c  conduction  c u r r e n t would be expected. One of t h e o b j e c t i v e s of f u t u r e work on these f i l m s should be t o determine observed  what causes the l a r g e i n c r e a s e i n f i l m conductance,  i n some samples, when a l a r g e enough v o l t a g e i s a p p l i e d .  Such changes i n f i l m p r o p e r t i e s can be d e t r i m e n t a l i n d e v i c e applications.  68 APPENDIX I Debye  Equation The p o l a r i z a t i o n response t o a s t e p f i e l d f o r a r e l a x a -  t i o n process i s ( r e f . 4, p. 72) dP _ 1 (P dt - %  - P)  I - l  s  where P = p o l a r i z a t i o n f o r the process w i t h r e l a x a t i o n time P  s  = static  r  2:  polarization  I n t e g r a t i n g I - l from t = 0 g i v e s P(t) = P  (1 - e -  t / Y  )  1-2  C o n s i d e r i n g D = eE = E E + P , then,at f r e q u e n c i e s h i g h e r than  those  Q  at which the r e l a x a t i o n p r o c e s s e s w i l l respond, E = e E + P^  1-3  Q  At i n t e r m e d i a t e f r e q u e n c i e s . s(co)E = e E + P^  + P(co)  Q  1-4  and a t v e r y low f r e q u e n c i e s such t h a t a l l processes  f o l l o w the f i e l d  e x a c t l y ( i . e . co = 0) e E = s E s  where P  g  o  +  ^  P  +  .  s  1-5  i s t h e s t a t i c p o l a r i z a t i o n due o n l y t o the r e l a x a t i o n  processes L e t t i n g E be s i n u s o i d a l , i . e . E = E e'^'*',' then s u b t r a c t i n g 1-3 from 1-4 and 1-5 g i v e s P(co) = (e(co) - Zc, ) E e  J w t  0  1-6  and Ps = (es - r°° ) Eo e Substituting  1-7  j w t  1-6 and 1-7 i n t o I - l g i v e s 1 3w(e(co) - z^ ) = -  - z^ ) - (e(co) - ej)  (e  A-  b  which, when s o l v e d f o r e(o;), g i v e s t h e DeLye e(w) = z  00  + s" °° c  £  1+jco-r  equation 1-9  I-  APPENDIX I I I o n i c R e l a x a t i o n Model (4) A charged p a r t i c l e i s assumed t o possess two  equilibrium  p o s i t i o n s a d i s t a n c e 2a a p a r t , s e p a r a t e d by a p o t e n t i a l b a r r i e r of h e i g h t q. *  Pig.  I I - l I o n i c r e l a x a t i o n model  Assuming q « k T , then, i n t h e r m a l e q u i l i b r i u m , o n l y  a fraction  of the p a r t i c l e s , g i v e n by the Boltzmann f a c t o r e ^/^T^  w i l l have  enough energy a t any g i v e n time t o go oer the b a r r i e r .  Application  of a f i e l d r a i s e s the b a r r i e r f o r p a r t i c l e s i n p o s i t i o n 2 by eaE' and l o w e r s i t by the same amount f o r p a r t i c l e s i n p o s i t i o n 1. ting 9  Q  Let-  be the f r e q u e n c y of o s c i l l a t i o n ( i . e . attempt t o escape  frequency) of the p a r t i c l e , t h e n , under an a p p l i e d f i e l d ,the proba b i l i t y per second f o r a p a r t i c l e t r a n s f e r from 1 t o 2 i s co^  =  )  i  o  e  -(q-eaE)/kT  II-l  and from 2 t o 1 i s _  w  e  21 ~  -(q.*eaE)/kT  II-2  I f at any i n s t a n t of time t h e r e are N^ p a r t i c l e s at 1 and N^ a t 2, the r a t e s of change of N^ and N^ are g i v e n are independent of each o t h e r ) by dN. dt dN  1  = -N  l (  ^  +  2  N LO 2  2  IT  = Vl2  - V 21 J  2 1  (assuming the p a r t i c l e s  II-3  II-4  The t o t a l number of p a r t i c l e s i s f i x e d N  =  N  ±  +  N  2  H-5  70 S u b t r a c t i n g I I - 3 from I I - 4 and u s i n g I I - 5 g i v e s |^ ( N - N ) = - ( w 2  1  1 2  + io )(N 2 1  Assuming t h a t a t t = 0 ,  N - N = 12 " "21 12 21 N(co  2  x  W  }  +  W  - N ) + (0^2 - co )N  2  x  II-6  21  = N , then 2  (1 - e~ K2 21 ) t  +w  )  II-7  The induced p o l a r i z a t i o n i s p r o p o r t i o n a l t o ^ - N ^ and 2  thus approaches t h e s t e a d y s t a t e e x p o n e n t i a l l y . t i o n ( i . e . at  t=o*=>)  i s proportional to  e and  The s t a t i c p o l a r i z a -  eaE kT e eaE kT  -eaE kT - e _eaE kT + e  II-8 T T  Q  f o r low f i e l d s i s u s u a l l y approximated by .  eaE kT-  II-9  The r e l a x a t i o n time T of t h e p r o c e s s f o r low f i e l d s i s g i v e n by  1 10^  + co  21  That i s x  =  e  q  A  •  T  I  I  -  1  0  o E q u a t i o n I I - 7 can be w r i t t e n i n t h e same form as 1-2. That i s P = P (1 - e ^) s Throughout t h e above a n a l y s i s , t h e v e c t o r n a t u r e of a _t//  —>  —»  and E has been n e g l e c t e d . When "a and E a r e n o t p a r a l l e l then "a • E should r e p l a c e aE i n a l l t h e above e x p r e s s i o n s .  71  APPENDIX I I I D i e l e c t r i c Response f o r a Uniform D i s t r i b u t i o n of A c t i v a t i o n  Energies  A n e a r l y u n i f o r m d i s t r i b u t i o n of a c t i v a t i o n e n e r g i e s , a s c o n s i d e r e d i n Appendix I I , i s assumed.  The d i s t r i b u t i o n , G(q), i s  normalized G(q)dq  = 1  III-l  The Debye e q u a t i o n , s e p a r a t e d i n t o r e a l and i m a g i n a r y i s integrated  over t h i s d i s t r i b u t i o n where the r e l a x a t i o n time , TT ,  of t h e p r o c e s s i s g i v e n by e q u a t i o n 11-10. 1-1  and 2-1 (e  )  -  f  = U  Defining time  B  G  J  s  e  G  -2" = —^p  d  %  o  =  Substituting  III-2  ^ \ ^  an a c t i v a t i o n energy q  T  equations  1+1/7:  C  - e~)  ^  Q  III-3 corresponding to a  , then .from e q u a t i o n 2K = r p _ to 2  where  Thus from  e»  =  .  parts,  relaxation  11-10 qo/kT  III-4  6  l/V  ' o  e q u a t i o n s I I - 4 and 11-10 C  G(q)dq  J  s  into III-2 gives  1 + (2TC) e x p 2 ( q - q J / k T  o  X  5  O  The denominator i n the i n t e g r a n d i s of the form shown i n P i g . ( I I I - l ) . T h i s curve can be approximated e  =  + (e  g  to-that  - £<« )  \  of the d o t t e d l i n e , G(q)dq  giving III-6  S i m i l a r l y , equation I I I - 3 i s e  "  (  £  £  s  )  \  G(q)2Jt  exp  (q-q^)/kT  dq  1 + (2at) exp 2 ( q - q ) / k T 2  Q  m _ 7  72 A  /1 (zrrf'e <vr H~ 7°)  ->  Fig. I I I - l  Approximation  of Debye e q u a t i o n - r e a l p a r t  The i n t e g r a n d appears as i n P i g . I I I - 2 /r-  ZTT G  l+  KT  (z7rY lk(rh) e  Pig. III-2  Approximation  of Debye e q u a t i o n - imaginary  part  Assuming t h a t G(q) i s f l a t enough so t h a t i t doesn't change much i n t h e r e g i o n centered a t q , then G-(q) may be e v a l u a t e d at q Q  and taken o u t s i d e the i n t e g r a l . e"  =  (  Eg  -  £ c 0  )  J  Q  Thus  • kT  GCq ) Q  Two e x p e r i m e n t a l l y u s e f u l r e l a t i o n s can be d e r i v e d from  III-8 equations  73  I I I - 6 and I I I - 8 , namely -2  3 lnco  III-9  and 3 e'  = e__ 2  3  4_n_  n  £ o 0  III-IO  where a temperature independent s t a t i c p e r m i t t i v i t y has been assumed in  III-IO. The above e q u a t i o n s , f i r s t d e r i v e d by Gevers and. Du P r e ^ ^  may be extended t o i n c l u d e s t e p r e s p o n s e . The d i s p l a c e m e n t c u r r e n t d e n s i t y i s g i v e n by dD III-ll p dt which becomes, f o r a f i x e d f i e l d , J P  = dP dt  111-12  Por a s t e p a p p l i e d a t t=0, then ,from e q u a t i o n 1-2^ = P " 111-13 P s I n t e g r a t i n g e q u a t i o n I.TI-13 f o r d i s t r i b u t i o n o f a c t i v a t i o n e n e r g i e s J  t  G(q) , and u s i n g 11-10, then P  /  r  e  = P  o  S  ^ kT \ G(q)  S  ^  111-14  The i n t e g r a n d appears as i n P i g . I I I - 3  V*.  Pig. III-3  Step response curve  ->  74  Assuming the G-(q) changes v e r y l i t t l e  over the range of ac-  t i v a t i o n e n e r g i e s which span themajor p a r t of the above c u r v e , G-(q) may  be e v a l u a t e d at the peak ( i . e . T =  and t a k e n o u t s i d e the  t/2 =  then  e^  integral.  Thus J,. = j£T p sp  G(q' )  t  where  q' = kT In  t/  To  III-l5  75 APPENDIX IV S c h o t t k y Law, P o o l e - F r e n k I E f f e c t , and Simmons' Defect Model 1)  Schottky E f f e c t The maximum number of e l e c t r o n s p e r second  approaching  a b a r r i e r w i t h e n e r g i e s h i g h enough t o get over the b a r r i e r , or what would be t h e s a t u r a t i o n c u r r e n t d e n s i t y i n t h e absence of a f i e l d i s (18) g i v e n by the Richardson-Dushman e q u a t i o n P (0-9) • J  L  = AT^ e~  kT •  IV-1  where 0-^-9 = b a r r i e r h e i g h t as i n P i g . 2-1 A = Richardson's  constant  The S c h o t t k y e f f e c t i s the f i e l d l o w e r i n g of the m e t a l / i n s u l a t o r b a r r i e r and i s c a l c u l a t e d by c o n s i d e r i n g t h e "image f o r c e " a s s o c i a t e d w i t h an e l e c t r o n l e a v i n g a metal s u rTOR face. / M 6 VL A  —  tV\£T A L  ®< Pig.  IV-1  >E  Image f o r c e i n S c h o t t k y e f f e c t  '•  Under an a p p l i e d f i e l d E, the p o t e n t i a l of an e l e c t r o n p a s s i n g over t h e b a r r i e r i s W  I  =  -e  2  ~ 16TC  x  E  where E = h i g h frequency  - eEx  (0-9)  +  IV-2  permittivity.  A p o t e n t i a l maxima occurs when  <J  = 0  d x  Then from IV-2,' t h e maxima,' x m i s x = m  (l6steE j  The p o t e n t i a l a t the maxima i s from IV-2 E'  = (0  X  - 9)  -  1  ocE^  2  IV_4  ?6 The b a r r i e r h e i g h t  i s lowered by  o&E  METAL  as shown i n P i g . I V - 2  INSULATOR  Fig. IV-2 2)  2  Poole-Frenkel  B a r r i e r l o w e r i n g due t o S c h o t t k y  effect  Effect  This i s analagous t o the S c h o t t k y  e f f e c t , except t h a t the  coulombic a t t r a c t i v e f o r c e i s s u p p l i e d by a t r a p p i n g c e n t e r , than an image f o r c e . a Poole-Frenkel  rather  Thus only p a r t i c u l a r k i n d s of d e f e c t s w i l l  e f f e c t ; i . e . one t h a t has a charge when empty.  show The  e l e c t r o n and the p o s i t i v e l y charged t r a p are s e p a r a t e d by a d i s t a n c e x r a t h e r than the 2x shown i n F i g . I V - 1 .  C a r r y i n g through the c a l c u -  l a t i o n as b e f o r e r e s u l t s i n the t r a p b a r r i e r b e i n g lowered by the p E ( x ) where (3 = 2a .  amount  2  C a l c u l a t i o n of the c o n d u c t i v i t y r e q u i r e s assuming some (35)  defect s t r u c t u r e f o r the i n s u l a t o r . posed a c c e p t o r  Mark and Hartman  pro-  and donor l e v e l s w i t h p a r t i a l i o n i z a t i o n of the  donors E :  donor energy l e v e l  Q  E, E-F  E, E = acceptor a ^ = acceptor  E E F i g . IV-3  N.  v  Insulator defect  level state density  donor s t a t e d e n s i t y  structure  Assuming t h a t most of the i o n i z e d donor e l e c t r o n s r e s i d e i n the acceptor N  / a  l e v e l s , then 1  1 + exp-(Ep-E )/kT  Nd  a  1 + exp-(E -E )/kT F  d  77 which, f o r N,> B g i v e s . a a to  V  E  L.  i  = L  p  m  The c o n d u c t i v i t y i s g i v e n by CT-  where N  c  N , - N  n  + kT i n _o__a a .  d  I V  = n eu. = N en e x p - ( E - E ) / k T c ' c = constant c  _  5  IV-6  p  u = electron mobility The P o o l e - F r e n k e l e f f e c t w i l l operate  on the donor c e n t e r s  (which  are p o s i t i v e l y charged when empty) t o g i v e a c o n d u c t i v i t y  cr  (x) =  cr  exp 0 E * ( x ) / k T  o  I V _ 7  where 0  IV-8  = N" eu. ( d ~ a ) e x p - ( E - E ) / k T N  N  c  0  .  0  d  N  a (25) Simmons  has proposed a d e f e c t s t r u c t u r e i n which the  Poole-Frenkel e f f e c t d i s p l a y s a Schottky '  characteristic.  E  .  c  E Ep  E^ = t r a p . e n e r g y l e v  Ed  = trap density  t  E F i g u r e IV-4  Simmons' Defect  v  Model  He assumes t h a t most of the i o n i z e d donor e l e c t r o n s r e s i d e i n the shallow t r a p p i n g l e v e l .  N  d  ( l  -  Thus  1 . \ 1 + exp-(E -E )/kT / F  d  N  t  1 + exp-(E -E )/kT F  t  78 N  d  e  xp-(E -E )/kT F  ^  d  N  t  exp ( E - E ) / k T F  t  Thus E  = i(E  + E,) + i k T I n  \  IV-9  S u b s t i t u t i n g t h i s i n t o e q u a t i o n IV-6, and a l l o w i n g f o r t h e l o w e r i n g of t h e donor b a r r i e r by t h e P o o l e - F r e n k e l e f f e c t Q~ ( )  = Q-^  X  ex TO ^ E (x)/kT = <rg e x p x E (x) /kT * * IV-10 2  0  8  where °~o = c ^ I H N  8  (  then  exp-(E -*(E +E ))AT c  d  t  7 9  APPENDIX V Zero F i e l d P o i n t and C a l c u l a t i o n o f Space Charge Decay C u r r e n t s I t can be s i m p l y shown t h a t when t h e r e i s a t r a p p e d n e g a t i v e space charge i n t h e f i l m and t h e sample i s s h o r t c i r c u i t e d , the f i e l d  2-4).  i s z e r o a t some p o i n t x* i n t h e f i l m (see F i g . P o i s s o n ' s e q u a t i o n i n one dimension i s dE (x) -e / \ dx" = 7 t n  whereen^(x) = t r a p p e d space charge I n t e g r a t i n g from x = 0,  (  x  r  - ee \\nn ( x ) 7e From t h e s h o r t c i r c u i t c o n d i t i o n V = 0, t h e n  -I'  =  E(0)  X  V = - J E(x)dx = |  Jdx ^ n ( x ) d x -  Therefore  " = |n  E(0)  j  +  V"  dx  E(0)L = 0  t  \ d dx x  ~, 1  density.  then  E(x)  n  )  V-2  V-3  °  5^  \ n (x)dx  0  >  t  V-4  Now i t need o n l y be shown t h a t E(l)<0 and thus the f i e l d w i l l have to be z e r o a t some p o i n t x* where 0 < x * < L . Since  ^ n ^ ( x ) d x i s an i n c r e a s i n g f u n c t i o n o f x then Jdx  ^  n (x)dx t  <  I  J  n (x)dx t  V-5  From (V-2) E(L)  =  -  E(0)  |  ^  n (x)dx t  V-6  From e q u a t i o n V-4 and V-5 E(0)< T h e r e f o r e , from V-6 In  | §  n (x)dx t  E(L) < 0 .  QED.  c a l c u l a t i n g the-.space charge d i s c h a r g e c u r r e n t s , t h e  p o l a r i z a t i o n e f f e c t s o f t h e space charge w i l l be c o n s i d e r e d . to F i g . 2-5, plates.  Q-|_ and Q  2  Referring  a r e t h e f r e e charges p e r u n i t a r e a on t h e  80  Then = ext. " < J  dt  J  0 )  V  -  7  where J(0) i s t h e c u r r e n t d e n s i t y a t the m e t a l . / i n s u l a t o r i n t e r face.  This c u r r e n t w i l l be due t o e l e c t r o n s b e i n g r e l e a s e d from  t r a p s and b e i n g swept t o t h e e l e c t r o d e by t h e space charge f i e l d , Q-j^ i s g i v e n by t h e d i s p l a c e m e n t v e c t o r . Q  1  = D(0) = £ E(0) + P(0)  V-8  Q  D i s r e l a t e d t o t h e space charge by V *D = - e n ( x , t )  V-9  t  I n t e g r a t i n g i n one dimension g i v e s D ( x , t ) = -e  j^n (x,t)dx  + P(x*, t )  t  V-10  where D ( x * , t ) = P ( x * , t ) because E(x*, t ) = 0.  The p o l a r i z a t i o n  at t h e zero f i e l d p o i n t may not be zero because of t h e p o l a r i z a t i o n caused by f i e l d s a p p l i e d b e f o r e (i.e.  t< 0).  s h o r t c i r c u i t i n g t h e sample,  C o n s i d e r i n g t h e p o l a r i z a t i o n mechanism d i s c u s s e d  i n s e c t i o n (2.1) then  0 0  P(x* t ) = P (x*) B  where  J e~  G(q)dq  V-ll  ) E(x*,t<0)  V-12  t / / r  ° P (x*) = ( e Q  E(x*, t < 0 ) r e f e r s t o t h e s t a t i c f i e l d a t x* before  short  cir-  c u i t i n g t h e sample and i s g i v e n by E(x*, t < 0) =  ( n (x, t  t<0)dx  +  E(0, t < 0 )  V-13  To determine J(0) i n e q u a t i o n V-7,the c o n t i n u i t y e q u a t i o n Neglecting recombination  i s used  ( i . e . when an e l e c t r o n i s r e l e a s e d from  a t r a p i t does not get r e t r a p p e d ) then qj(x,t) dx Thus  = e ^jt dt  (x,t)  V v  T .  X 4  Thus J(x,t) = e  |  ( x , t ) dx  J  d  where  V-15  t  ^ J(x*, t ) = 0  as  E(x*,t) = 0  From e q u a t i o n s V-7 and V-8 vt  =  i  t  D  (  0  )  -  +  j  < ° >  S u b s t i t u t i n g f o r D ( 0 ) and J ( 0 ) from e q u a t i o n s V - 1 0 and V - 1 5 and u s i n g V - l l and V-12 then ext. \(x*>V - (Es"eo J  =  e  o )  G ( q , )  ""'  E(x  t  Cl-"b  - ^ s " e £  £ o  °^  oo  en. ( x * , t ) dx*. dt x  \  J  < 0 ) k T  "t  e" ' G(q)dq t//  r  V-16  82 APPENDIX VI C a l c u l a t i o n of Zero F i e l d P o i n t and. Space Charge D e n s i t y The decay of a t r a p p e d space charge, n e g l e c t i n g  recom-  b i n a t i o n i s g i v e n by dn|/  ,\  dt  ^ t ^ ' ^  =  exp-(E -E )/kT c  t  '  VI-1  w h e r e e n ( x , t ) i s the space charge d e n s i t y / u n i t energy, c e n t e r e d at an energy l e v e l E, and x> = jump f r e q u e n c y . u  O  I n t e g r a t i n g from z e r o t i m e , then - t / n '(x,t) =n '(x,0) e ^(E ) X / /  "t  t  x  ~  "'  r  ~  "t t  t  VI-2  where X ( E ) = ^- exp o  (E -E )/kT  t  c  t  Assuming t h a t a t t = 0 the s t e a d y s t a t e charge d i s t r i b u t i o n n^'(x,0) through the band gap can be g i v e n by F e r m i - D i r a c s t a t i s t i c s , then  , .s ( n. ( x , t ) = N. \  yf  w  0  J  t  c  e"  dE, t  t / / x  VI-3  1 + exp-(E (x)-E )AT F  •where N^ = t r a p d e n s i t y / u n i t  t  energy  Ep(x) = q u a s i - f e r m i - l e v e l The i n t e g r a t i o n i s c a r r i e d out o n l y above the e q u i l i b r i u m f e r m i l e v e l as t h i s i s the r e g i o n i n which the i n j e c t e d space charge r e s i d e s .  L e t t i n g a = exp- ( E (x) - E ) ATF  n. ( x , t ) = -N.kT x  1  c  . \  ^  then  §  -t/r,  ?(l+a/» Z 0  )  vi-4  83 t e  n (x,t) +  = kTN {  Ei(-t/zr)  +  - e  -(E U)-E )/kT F  l/p  c  Ei(-t/z - -£-)  0  where E i ( - a ) i s t h e e x p o n e n t i a l i n t e g r a l g i v e n by  J  Ei(-oc) Assuming E^/kT » E ( x ) / k T p  e  a  da a  then  f /IT F°)/kT n ( x , t ) = -kTN yEi(-» t e - c - F ) _ ( E  t  t  E  J / k l  Q  E i ( - V t (e- (E -Eg) /kT Q  The  c  Y> + - ( E - E ( x ) )/kT ^ (, t  e  e  o  0  F  •-. ( E - E p (x ) ) A T  +  0  }  }  j  VI-7  z e r o f i e l d p o i n t , x*, i s d e r i v e d u s i n g t h e s h o r t  c i r c u i t c o n d i t i o n , V = 0.  J-  Thus  V = 0 = - ( E(x,t)dx .  VI-8  ^  Jo  E ( x , t ) i s g i v e n by  -  E(x,t)•= i - (D(x,t) - P ( x , t ) ) o where D ( x , t ) i s g i v e n by e q u a t i o n V-10.  VI-9  P ( x , t ) may be d i v i d e d i n t o components due t o t h e decay of p o l a r i z a t i o n f o r f i e l d s a p p l i e d b e f o r e s h o r t c i r c u i t i n g ( i . e . Appendix I I I ) , t h e b u i l d up of p o l a r i z a t i o n o f t h e type  considered  i n Appendix I I I due t o t h e space charge f i e l d , and t o t h e h i g h f r e q u e n c y p o l a r i z a t i o n component due t o t h e space charge f i e l d . Thus P ( x , t ) = (e-e^) +( s  s  E ( x , t < 0) \ e ~ E(x, t ) J ~  -ec*)  t / / r  G(q)dq  (1 - e "  t/2:  ) G(q)dq  '+(t* - t' ) E ( x , t )  VI-10  Q  E ( x , t ) may be determined  i n terms o f n ^ ( x , t ) from VI-10 and V I - 9 ,  u s i n g V - l l , V-12, V-13 and V-2, however, i t g e t s v e r y  complicated  84 and no attempt has been made t o d e r i v e the time dependence of x*. I f i t i s assumed t h e r e i s no p o l a r i z a t i o n due t o i o n movement then e q u a t i o n V I - S has the s i m p l e form L  0 =  PC  dx  (  n (x,t)dx t  VI-11  APPENDIX V I I Space Charge L i m i t e d • Neglecting given  Currents d i f f u s i o n e f f e c t s , the c u r r e n t d e n s i t y i s  by J  where n c = f r e e e l e c t r o n  Jj = n c eu^ E  VII-1  T  density J  u = electron mobility Por a p e r f e c t c r y s t a l , the net charge i n the i n s u l a t o r w i l l be charge and  from P o i s s o n ' s e q u a t i o n dE / \  dl  Thus J  =  ( x )  L = " ^  _e  1  n  c  / \ (  x  VII-2  )  o l ^  E(x)  VI1  I n t e g r a t i n g from x = 0 and assuming E(0) J  Thus  L  x = — ^  / 2  =  /  3  ° J  L  =  3  = 0 then VII-4  "  2  L  V=JE(X)CIX and  E (x)  -  \ fe (  ~  l  " I^  V  2  h  V I I  -5  V  I  1  -  6  free  86  REFERENCES 1.  B a i r d , M.,"Rev. o f Mod. P h y s . , " v o l . 40, p. 219, 1968.  2.  Rose, A.,"Phys. 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