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Electronic conduction and dielectric properties of thin insulating films Shousha, Abdel Halim Mahmoud 1969

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ELECTRONIC CONDUCTION AND DIELECTRIC PROPERTIES OF THIN INSULATING FILMS 'by ABDEL HALIM MAHMOUD SHOUSHA B . S c , Ca i ro U n i v e r s i t y , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of v. E l e c t r i c a l Eng inee r ing We accept t h i s t h e s i s as conforming to the r equ i r ed s tandard Research Superv i sor . . . . Members of Committee. A c t i n g Head of Department Members of• the Department of E l e c t r i c a l Engineer ing THE UNIVERSITY OF BRITISH COLUMBIA November,.1969 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a D a t e 3>e,c. 2 ? \ C ^ < j ABSTRACT The work contained i n t h i s thes i s i s concerned mainly w i t h conduct ion mechanisms and p o l a r i z a t i o n processes i n t h i n amorphous i n s u l a t i n g f i l m s . A model has-been proposed a n d . i t s d . c . conduct ion c h a r a c t e r i s t i c s computed. The numer ica l r e s u l t s show the p o s s i b i l i t y of o b t a i n i n g e i t h e r space charge, S c h o t t k y , o r 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 s depending on the model parameters. The t r a n s i e n t e l e c t r o n i c d ischarge current has been analysed and the r e s u l t s show that t h i s e l e c t r o n i c current i s approximately independent of the p reapp l i ed vo l t age i n con t ras t to the i o n i c d ischarge cur ren t •which i s l i n e a r l y dependent on p reapp l i ed v o l t a g e . This r e s u l t , together w i t h the exper imental r e s u l t s obtained on Ta/TagO^/Au d iodes , suggests that the c a l c u l a t i o n s of low frequency d i e l e c t r i c l o s s e s u s ing step response measure-ments are complicated by space charge e f fec t s only when the p re -a p p l i e d f i e l d i s r e l a t i v e l y low ( t ^ l M V / c m fo r l ^ O ^ f i l m s ) . Ta/TagOp./Au diodes were prepared by s o l u t i o n anod iza -t i o n or plasma a n o d i z a t i o n . A l l prepared diodes e x h i b i t e d a r e c -t i f i c a t i o n behaviour . Over the frequency range 100 Hz -100 kHz capaci tance and l o s s tangent were found to decrease s l i g h t l y w i t h increasing" f r e q u e n c y wh i l e the equ iva len t s e r i e s r e s i s t a n c e was -1 05 found to be approximately p r o p o r t i o n a l to or . A l l prepared o d iodes , w i t h gold counter e l ec t rodes less, than 1000 A t h i c k , were found to w i th s t and , under a s l o w l y a p p l i e d f i e l d , f i e l d , s t rengths approaching the formation f i e l d v a l u e . TABLE OF CONTENTS Page ABSTRACT v i i TABLE OP CONTENTS.' i i i LIST OF ILLUSTRATIONS v LIST OF . SYMBOLS '. . . . . v i i ACKNOWLEDGEMENT ' x 1. INTRODUCTION ' 1 2. THEORETICAL REVIEW ' 3 2.1 E l e c t r o n i c Conduction i n Thin I n s u l a t i n g F i l m s . . . 3 2 .1 .1 P o s s i b l e Conduction Mechanisms 3 2 .1 .2 Schottky Emission 4 2.1.3 Poole-Frenkel Emission ' 5 2 .1 .4 Space Charge E f f e c t s 7 2.2 D i e l e c t r i c P o l a r i z a t i o n Processes 8 2.3 D i e l e c t r i c Breakdown.... 10 2 .3-1 Types of D i e l e c t r i c Breakdown ' 10 a. Thermal Breakdown '. . . 10 b. E l e c t r i c Breakdown 10 c. Heterogeneous Breakdown . 11 2.3-2 S e l f - H e a l i n g Breakdown 11 3. ELECTRONIC CONDUCTION CURRENT CHARACTERISTICS OF THIN . AMORPHOUS FILMS • 1 3 3-1 Proposed Model 13 3-2 B a s i c Equations 13 3-3 Computational Procedure 17 3-4 Numerical Results and Discussion 18 • 3-5 Transient E l e c t r o n i c Discharge current 26 i i i Page 4. EXPERIMENTAL PROCEDURES AND RESULTS 29 4-1 Sample P repa ra t i on 29 4.1.1 Tantalum Surface P r e p a r a t i o n 29 4.1 . 2 F i l m Growth 29 a. E l e c t r o l y t e S o l u t i o n Grown F i l m s . 29 b . Plasma Grown F i lms 29 4.1-3 Counterelectrod.es 30 4 . 2 E l e c t r i c a l Measurements ' 30 4.3 D . C . Conduction C h a r a c t e r i s t i c s 31 4.4 Step Response Measurements.' . 36 4- 5 A.C' . Br idge Measurements 39 4.6 Breakdown Tests 47 5. DISCUSSION 53 5.1 D . C . E l e c t r o n i c Conduction 53 5- 2 Step Response Measurements 54 5.3 Breakdown St rength 56 6. CONCLUSIONS , . 59 APPENDIX Ion ic P o l a r i z a t i o n Current fo r 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 61 REFERENCES ' 64 i v LIST OF ILLUSTRATIONS F igure Page 1.1 A n o d i z a t i o n Arrangement 1 2.1 E l e c t r o n i c Band S t ruc tu re of C r y s t a l l i n e and amorphous • d i e l e c t r i c 3 2.2 Image Force Reduct ion of M e t a l - I n s u l a t o r B a r r i e r (Schot tky e f f ec t ) 5 2.3 Image Force Reduct ion of Donor State B a r r i e r (Poole -F r e n k e l e f f e c t ) . . . 6 3.1 Band Diagram of Proposed Model 1 3 3-2 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of • Changing N Q d 19 3-3 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Negat ive Space Charge 21 3.4 D . C . Conduction Mechanisms. 2 3 3.5 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Changing and C^ 24 3- 6 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Changing N 25 3.7 Band Diagram f o r M e t a l / l n s u l a . t o r / M e t a l du r ing D i s -charge . 26 4 .1 D . C . Conduction C h a r a c t e r i s t i c s fo r T a - P o s i t i v e Vol tages 32 4.2 Vol tage-Thickness R e l a t i o n fo r Constant C u r r e n t . . . . 33 4.3 D . C . Conduction C h a r a c t e r i s t i c s f o r Ta-Negative Vol tages 34 4 . 4 D . C . Conduction C h a r a c t e r i s t i c s fo r Fi lms. Prepared by D i f f e r e n t Procedures 35 4- 5 Discharge Current as Func t ion of F i e l d f o r F i l m s of D i f f e r e n t Thickness ( l a - p o s i t i v e ) . 37 4 . 6 Discharge Current as Func t ion of F i e l d fo r F i l m s Prepared by D i f f e r e n t P r o c e d u r e s . ( T a - p o s i t i v e ) 38 4-7 Frequency Dependence of Capaci tance f o r F i l m s of D i f f e r e n t Thickness • 40 v P i g u r e ' Pa.ge 4 . 8 Frequency Dependence of Equ iva l en t 'Ser ies Res is tance f o r F i l m s of D i f f e r e n t T h i c k n e s s . . . . . 41 4.9 Frequency Dependence of Capacitance fo r F i lms p re -pared by D i f f e r e n t Procedures 42 4.10 Frequency Dependence of Equ iva len t Se r i e s Res is tance fo r F i l m s prepared by d i f f e r e n t Procedures 4 3 4.11 Capaci tance Dependence on d . c . Bias ( T a - n e g a t i v e ) . . .45 4.12 Loss Tangent Dependence on d . c . B ia s ( T a - n e g a t i v e ) . . 46 4.13 Apparatus f o r Measuring Capacitance and Loss Tangent dependence on d . c . B ias 47 4.14 Breakdown Measurement Apparatus 48 4.15 E f f e c t of V/eak Spots and Countere lec t rode Evapora t ion on I -V C h a r a c t e r i s t i c 50 4.16 A T y p i c a l F i n a l I - V C h a r a c t e r i s t i c 50" 4.17 Breakdown St rength as a f u n c t i o n of Countere lec t rode area f o r F i l m s of D i f f e r e n t Thickness Grown on S i n g l e C r y s t a l Ta ( T a - p o s i t i v e ) 51 4.18 Breakdown St rength as a Func t ion of Countere lec t rode area f o r F i lms of D i f f e r e n t Thickness Prepared on Sput tered Ta ( T a - p o s i t i v e ) 51 5.1 V a r i a t i o n of D i e l e c t r i c Loss w i t h Frequency: Compari-son of Bridge- and Step Response Resu l t s 55 NOTE: Unless otherwise s t a t e d , a l l drawn v a r i a b l e s are i n M . K . S . u n i t s . v i LIST 0"F SYMBOLS o A Angstrom A c o u n t e r e l e c t r o d e a r e a A^ . a c o n s t a n t C s diode c a p a c i t a n c e ^°(l000) diode c a p a c i t a n c e a t 1 kHz. ,C 0,C, c o n s t a n t s L Z j d f i l m t h i c k n e s s E e l e c t r i c f i e l d s t r e n g t h E^ "breakdown s t r e n g t h e e l e c t r o n i c charge f • p r o b a b i l i t y of o c c u p a t i o n of a t r a p f f r e q u e n c y a t which d i s s i p a t i o n goes through maximum I e l e c t r o n i c d i s c h a r g e c u r r e n t e 0 1-^  l e a k a g e c u r r e n t J d.c. c o n d u c t i o n c u r r e n t d e n s i t y J g e l e c t r o n i c d i s c h a r g e c u r r e n t d e n s i t y J e x ^ . t o t a l e x t e r n a l d i s c h a r g e c u r r e n t d e n s i t y J i o n i c d i s c h a r g e cur-rent d e n s i t y K ' Boltzmann's c o n s t a n t m,N c o n s t a n t s N. a c c e p t o r s t a t e d e n s i t y t r a p d e n s i t y N .^ t r a p d e n s i t y / u n i t energy n s u b s c r i p t denotes norma.liza.tion nc f r e e e l e c t r o n d e n s i t y n^ t r a p p e d e l e c t r o n d e n s i t y V I 1 p o l a r i z a t i o n s t a t i c p o l ar3_za t i o n c h a r g e o n c o u n t e r e l e c t r o d e e q u i v a l e n t s e r i e s r e s i s t a n c e e q u i v a l e n t s e r i e s r e s i s t a n c e a t one kHz r a t e o f e l e c t r o n c a p t u r e r a t e o f e l e c t r o n r e l e a s e f u n c t i o n o f e l e c t r i c f i e l d t e m p e r a t u r e t i m e a p p l i e d v o l t a g e . f i n a l a p p l i e d v o l t a g e e l e c t r o n t h e r m a l v e l o c i t y e n e r g y o f t h e d e e p e s t t r a p a c t i v a t i o n e n e r g y -d i s t a n c e t h r o u g h f i l m z e r o f i e l d p o i n t c o n s t a n t s P o o l e - F r e n k e l s l o p e S c h o t t k y s l o p e l o s s a n g l e . max. v a l u e f o r l o s s a n g l e d i e1e c t r i c p e rmi11 i v i t y r e a l p a r t o f 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 par"!: o f d i e l e c t r i c p e r m i t t i v i t y s t a t i c d i e l e c t r i c p e r m i t t i v i t y d i e 1 e c t r i c perrni11ivity a t h i g h f r e q u e n c i e p e r m i t t i v i t y o f a i r . trap energy J + t m . -n ext  L E ' u e l e c t r o n m o b i l i t y $ attempt to escape frequency 4 Jext/V ^ t o t a l charge <5~ capture cross s e c t i o n low f i e l d c o n d u c t i v i t y X r e l a x a t i o n time •'V constant = — L- O V 0 metal work fu n c t i o n e l e c t r o n a f f i n i t y of the i n s u l a t o r •Y, s t a t i c s u s c e p t i b i l i t y co angular frequency i x ACKNOWLEDGEMENT I would l i k e to express my g r a t e f u l a p p r e c i a t i o n to Dr. L. Young f o r h i s competent and i n s p i r i n g s u p e r v i s i o n and generous assistance throughout the duration of t h i s p r o j e c t . I am al s o indebted to Dr. D. P u l f r e y f o r many h e l p f u l discussions and f o r reading the manuscript. G r a t e f u l acknowledgement i s due to Messrs. .0. Dell'Oca, G. O l i v e , N . Taneja and G. Yan f o r many valuable d i s c u s s i o n s , -to Messrs. G. Anderson, A. MacKenzie and J.H. Stuber f o r t h e i r t e c h n i c a l help; and to Miss B. Harasymchuk f o r t y p i n g the t h e s i s . Many thanks f o r f i n a n c i a l support are given to the Na t i o n a l Research Co u n c i l (operating grant A 3392) and the B r i t i s h Columbia Telephone Company (B.C. T e l . Graduate f e l l o w s h i p ) . 1. INTRODUCTION The s tud ie s of t h i n ' i n s u l a t i n g amorphous f i l m s l e s s than 1 micron t h i c k have made cons iderab le advances i n the l a s t decade, p r i m a r i l y because of the i n d u s t r i a l demand f o r r e l i a b l e t h i n - f i l m m i c r o e l e c t r o n i c d e v i c e s . - This progress has brought much s c i e n t i f i c confidence i n the use of t h i n f i l m s . Thin i n s u l a t i n g amorphous f i l m s are now used e x t e n s i v e l y i n m i c r o e l e c t r o n i c s and in t eg ra t ed c i r c u i t s . The i r a p p l i c a t i o n i n M O S . F . E . T . , t h i n - f i l m F . E . T . , e l e c t r o l y t i c c a p a c i t o r , t h i n f i l m capac i t o r and R-C in t eg ra t ed c i r c u i t s i s of i n c r e a s i n g importance. The f i l m s i n v e s t i g a t e d i n t h i s study were anodic t a n t a -lum pentoxide which were grown as shown s c h e m a t i c a l l y i n F i g . 1.1. METAL (ANODE) OXIDE EL ECTROLY TE SOLUTION OR OXYGEN PLASMA OA THODE F i g . 1.1 A n o d i z a t i o n Arrangement The main aim i n the p ro j ec t was to i n v e s t i g a t e d i e l e c -t r i c processes , conduct ion mechanisms and the e f f ec t of space charge on the e l e c t r i c a l performance of t h i n f i l m s . The work to be reported here c o n s i s t s of two main p a r t s . In the f i r s t par t a model fo r a t h i n i n s u l a t i n g f i l m w i l l be discussed and the d . c . conduct ion c h a r a c t e r i s t i c - w i l l be computed us ing the proposed model. The second par t i s an exper imenta l study of t h i n ' f ^ O ^ f i l m s grown on tantalum s i n g l e c r y s t a l or on sput tered tantalum • film!; Chapter 2 i s a shor t review of the theo r i e s of the d i e l e c t r i c processes , conduct ion mechanisms and space charge ' e f f ee i n t h i n amorphous f i lms . . The proposed model and the computed d . c . conduct ion c h a r a c t e r i s t i c s are presented i n chapter 3« The e x p e r i -mental procedures are descr ibed and the r e s u l t s obtained are p re -sented i n chapter 4. In chapter 3 the r e s u l t s are d i scussed and the conc lus ions to be drawn from the \vork are g iven i n chapter 6. 2. THEORETICAL REVIEW 2•1 E l e c t r o n i c Conduction i n Thin I n s u l a t i n g F i lms 2 .1 .1 P o s s i b l e Conduction Mechanisms Since the bas i c fea tures of the band s t r uc tu r e are determined p r i m a r i l y by short range order , or more p r e c i s e l y by the a c t u a l chemical bonds between atoms the genera l nature of the band s t r uc tu r e i s preserved i n t r a n s i t i o n from the c r y s t a l l i n e to amorphous s t a t e . The disappearance of l o n g range orde does however change the d e t a i l e d s t r u c t u r e ; e . g . l e a d i n g to a b l u r r i n g of the edges of conduct ion and valence bands and a l s o g i v i n g r i s e to deep l o c a l i z e d l e v e l s F i g . 2 . 1 . Conduction Band M 2 Donor Leve l s a Trapping Levels ^ •Conduction Band , ' 'Quasi L o c a l i z e d Leve l s ' Deep Traps t r ^ ^ T ^ u * ^ ^ p ^ " w t y o f s t a t e F i g . 2.1 E l e c t r o n i c Band S t ruc ture of C r y s t a l l i n e and Amorphous D i e l e c t r i c s Assuming that the normal band s t r uc tu r e can be a.pplied to an amorphous f i l m , there would, appear to be f i v e p o s s i b l e con-duc t i on mechanisms through i n s u l a t i n g f i l m s . 1) I on i c conduct ion . 2) Space charge l i m i t e d c u r r e n t s . 3) T u n n e l l i n g and i n t e r n a l f i e l d emis s ion . 4) Schot tky emiss ion and P o o l e - F r e n k e l e f f e c t s . 5) Impuri ty conduct ion , e i t h e r through impur i t y bands or by hopping. The presence of flaws provides a p a r a l l e l conduction mechanism. The mechanism of such conduction and the'chemical and ( ~Z A \ p h y s i c a l nature of the flaws are mostly unknown ' . The most commonly observed e l e c t r o n i c conduction charac-(5) t e r i s t i c , at room temperature has the form : 1 J = C exp (3 E 2 (2.1) where J i s the current d e n s i t y . E i s the average e l e c t r i c f i e l d . G and 8 are temperature dependent constants. Both Schottky emission' and Poole-Erenkel emission have the- above form but w i t h d i f f e r e n t values f o r C and 8.. • Although they are often a p p l i e d to e x p l a i n experimental r e s u l t s , i t i s not always p o s s i b l e to obtain a s a t i s f a c t o r y agreement w i t h e i t h e r mechanism, t h e r e f o r e , i t i s worth examining each i n some d e t a i l to determine whether i n f a c t they are a p p l i c a b l e . 2.1.2 Schottky Emission Schottky emission i s thermionic emission over a f i e l d -lowered p o t e n t i a l b a r r i e r ( f i g . 2 . 2 ) . The current density i s given (1) by where 1 J = A T 2 e x p - ( | ^ ) exp B s E 2(o) (2.2) A i s Richardson'• s constant 8 i s the Schottky slope (2-3) E (0 ) i s the f i e l d strength at' the e m i t t i n g cathode 0 i s the metal work f u n c t i o n . % i s the e l e c t r o n a f f i n i t y of the oxide e i s the h igh f r e q . d i e l e c t r i c p e r m i t t i v i t y , e i s the magnitude of e l e c t r o n i c charge' K i s Boltzmannls ' cons tant . T i s the temperature IJ^LATOR_ BOTTOM OF N s V - -f:) CONDUCT/ON BAND 'A F i g . 2.2 Image Force Reduction of Meta l - Insu la to r "Barrier (Schot tky e f f ec t ) I t i s , w o r t h p o i n t i n g out some of the assumptions and l i m i t a t i o n s of t h i s cathode emiss ion l i m i t e d approach. 1. Since the emiss ion process i s fas t compared to i o n i c motion o p t i c a l va lues of the r e l a t i v e p e r m i t t i v i t y should be used. 2. E l e c t r o n s are assumed t o ' t r a v e r s e the b a r r i e r wi thout s c a t t e r i n g . A l s o t r app ing could mask the emiss ion process : t h i s i s expected to be more Important fo r r e l a t i v e l y t h i c k amorphous ' f i l m s . 3- Space charge e f f ec t s are neg lec t ed . 2 . 1 . '3 P o o l e - F r e n k e l Emiss ion P o o l e - F r e n k e l emiss ion i s ' f i e l d enhanced thermal emiss ion of e l ec t rons from coulombic . traps ( f i g . 2 .3) - Since the charge on the t raps i s f i x e d ' i n p o s i t i o n , t h i s a t t r a c t i v e force i s p r o p o r t i o n a l to ( X i s the d i s t ance between the e l e c t r o n and t rap centre) x 6 i n s t ead of ~ 2 i n the case of Schot tky emiss ion . This g ives r i s e . t 0 Py.-p == 2P and current dens i t y i s g iven by P . F ' s (6) (2.4) J = 6^ E(x) exp S p > F E2 ( x ) (2.5) where r j^ i s the low f i e l d c o n d u c t i v i t y BOTTOM OF CON DUCT ION BAND P i g . 2.3 Image Force Reduct ion of Donor State B a r r i e r (Poo le -F renke l e f f e c t ) . Some of the assumptions and l i m i t a t i o n s of the above- semi-c l a s s i c a l P o o l e - F r e n k e l approach are : '1. The P o o l e - F r e n k e l e f f ec t desc r ibed by equat ion (2.5) can occur only when the' bu lk m a t e r i a l conta ins s ta tes that are n e u t r a l when occupied; that i s , donor s t a tes fo r an n- type m a t e r i a l , because only then i s there a coulombic a t t r a c t i o n centre a p r e r e q u i -(6) s i t e fo r d e r i v i n g equat ion 2.5 . 2 . The p o t e n t i a l b a r r i e r i s assumed coulombic which i s t rue only f o r the i d e a l i s e d case of an hydrogenic impur i t y i n a c r y s t a l l a t t i c e . The p o t e n t i a l b a r r i e r corresponding to deep l e v e l s i n semiconductors and i n s u l a t o r s i s not known but one c e r t a i n l y expects d e v i a t i o n from a coulombic form. I t i s known that a l e v e l that i s s teeper than coulombic would experience l e s s p o t e n t i a l l ower ing 7 1 (2) than p p F E 2 3- Re trapping of free e l e c t r o n s and space charge e f f e c t s are neglected'. 2.1.4 Space Charge E f f e c t s The presence of space charge changes the f i e l d d i s t r i b u -t i o n i n s i d e the f i l m and hence the conduction c h a r a c t e r i s t i c s . Space.charges trapped i n deep traps with l o n g • r e l a x a t i o n time would be expected to cause h y s t e r s i s i n the J-V c h a r a c t e r i s t i c . e r i s t i c (I (7) 0'Dwyer analyzed the current-voltage c h a r a c t e r i s t i c w ith the help of a d i e l e c t r i c model o r i g i n a l l y proposed by E r b h l i c h and assuming emission processes at the electrode i n s u l a t o r i n t e r f a c e as being e i t h e r Schottky emission or the simple Fowler-Nordheim emission. E s s e n t i a l l y , the conclusions of 0'Dwyer were that'Schottky emission d i d not give r i s e to an i d e a l Schottky p l o t due to the presence of space charge b u i l d up. However, Powler-Nordheim emis-s i o n d i d give an almost i d e a l Schottky p l o t . ^ (9) 2 Frank and Simmons have computed lnJ-V curves f o r t h i n f i l m s where they have considered Schottky i n j e c t i o n at the metal/ i n s u l a t o r i n t e r f a c e , the bulk c o n d u c t i v i t y being determined by the Poole-Frenkel e f f e c t on a s i n g l e trap l e v e l . At high voltages t h e i r curves are'asymptotic to the Schottky i n j e c t i o n curve; at low voltages the e f f e c t s of space charge are more pronounced and the curves deviate from the Schottky law. A Poole-Frenkel c h a r a c 1 t e r i s t i c i s not revealed at any time. This was the case because Frank and Simmons assumed that the i n j e c t e d charge was i n excess of e l e c t r i c a l n e u t r a l i t y . 8 2.2 ' D i e l e c t r i c P o l a r i z a t i o n Processes A number of p h y s i c a l processes con t r i bu t e to the p o l a r i -z a t i o n of i o n i c c r y s t a l s . Each process may be c h a r a c t e r i s e d by a p a r t i c u l a r c h a r a c t e r i s t i c frequency or range of c h a r a c t e r i s t i c f r equenc ies . I t i s usua l to separate 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 p o l a r i z a t i o n (deformation of e l e c t r o n s h e l l s ) , i o n i c p o l a r i -z a t i o n (displacement of ions w . r . t . each o the r ) ' and p o l a r i z a b i l i t y such.as ions moving between p o t e n t i a l minima, e l ec t rons hopping or t u n n e l l i n g between l o c a l i s e d s t a t e s . The fu r the r mechanism of o r i e n t a t i o n of molecules w i t h permanent d i p o l e s i s u n l i k e l y t o a r i s e i n amorphous S in f i l m s s ince i t i s improbable that molecular groups having permanent d i p o l e moments could be incorpora ted i n the f i l m i n such a way as to r o t a t e w i t h the f i e l d . Since e l e c t r o n i c and i o n i c p o l a r i z a t i o n cannot be com-p l e t e l y separated, as i o n i c displacement induces e l e c t r o n i c d i s -placement, i t i s b e t t e r ^ ^ ' ^ " ^ to c h a r a c t e r i z e p o l a r i z a t i o n as e i t h e r u l t r a v i o l e t ( s o l e l y e l e c t r o n i c displacement) o r . i n f r a red (to which the displacements of n u c l e i and e l ec t rons c o n t r i b u t e ) . P o l a r i z a t i o n processes i n t h i n amorphous f i l m s are com-p l i c a t e d by the. amorphous nature of the f i l m . However, i t i s reasonable to assume that e l e c t r o n i c and e l a s t i c i o n i c displacements are not e s s e n t i a l l y changed from those i n r egu l a r c r y s t a l s . The p o l a r i z a t i o n processes , of p a r t i c u l a r i n t e r e s t i n the below i n f r a red range, i n v o l v e the movement of ions and. e l ec t rons between energy w e l l s i . e . these are r e l a x a t i o n processes having cha rac t e r -i s t i c f requencies below the i n f r a red. range. A s i n g l e r e l a x a t i o n d i e l e c t r i c mechanism (Debye process) g ives the f o l l o w i n g equations 9 £ ~ E . e ' = e + — — ~ V (2.6) 1 + co - X (2.7) 2 ^ 2 1 + co t where e* i s the r e a l par t of d i e l e c t r i c p e r m i t t i v i t y . e" i s the imaginary par t of d i e l e c t r i c p e r m i t t i v i t y . £ S i s the s t a t i c d i e l e c t r i c p e r m i t t i v i t y . £ . i s the d i e l e c t r i c p e r m i t t i v i t y at h igh f requencies i s the d i e l e c t r i c r e l a x a t i o n t ime. The d i s s i p a t i o n f a c t o r ; tan S , i s g iven by tan & - = (e -e ) ~~T~b~ (2.8) £' - v c s W 2 ,2 £ + £ CO- ' - ^ and goes through a maximum at the frequency Im - 2ic V £ (2.9) when (e -e ) tan's =—^=^- (2.10) m 2»£ £ ! s ^° For amorphous f i l m s one expects a spread i n the cha rac t e r -i s t i c f requencies so that a broader absorp t ion band w i l l be observed ; ( e .g . ) below i n f r a red f requencies Ta o 0 r - d i s p l a y s l o s se s which (12) are almost independent of frequency over a wide range of f requencies The i o n i c r e l a x a t i o n model w i t h r e l a t i v e l y f l a t d i s t r i b u t i o n of •' i ^ 3 ) ' a c t i v a t i o n energies has been claimed v to adequately descr ibe the frequency and temperature dependence of many amorphous f i l m s . Using t h i s model, one gets fo r i o n i c p o l a r i s a t i o n current (Appendix) , 2 _„ °P where E : i s the average e l e c t r i c f i e l d JP - ^ £ " f . ( 2 - 1 1 ) 10 t ': i s the t ime. 2 . 3 D i e l e c t r i c Br eakd own 2-3-1 Types of D i e l e c t r i c Breakdown With o rd ina ry hulk m a t e r i a l s , s e v e r a l forms of breakdown may be d i s t i n g u i s h e d which have the common feature of o c c u r r i n g when h igh enough f i e l d s are a p p l i e d . a. Thermal Breakdown In t h i s type of breakdown heat i s produced by jou le hea t ing or , w i t h A . C . Currents f l o w i n g , by d i e l e c t r i c hea t ing f a s t e r than i t can be conducted away. This i n t u rn may cause an increase i n the c o n d u c t i v i t y which causes a runaway process . I f a t h i n homogeneous f i l m cou ld be prepared ( i . e . wi thout any weak spots) the breakdown should occur at a d e f i n i t e vo l t age and occur s imul taneous ly over most of the i n s u l a t o r ^ 4 ) ^ Gene ra l l y thermal breakdown s t r eng th , when the app l i ed f i e l d i s increased s l o w l y , depends on the shape and s i z e of the sample, on the geometry and (13 thermal p r o p e r t i e s of the e lec t rodes and the ambient c o n d i t i o n s b . E l e c t r i c Breakdown the l i t e r a t u r e under va r ious names, such as i n t r i n s i c , avalanche and f i e l d emiss ion breakdown. The inc idence of breakdown does not occur at a sha rp ly def ined f i e l d and i t des t roys a very sma l l par t of the i n s u l a t o r and t h i s cannot be avoided owing to the r a p i d nature of the process . In p r a c t i c e e l e c t r i c breakdown c o n s i s t s of 2 stages 1. Formation of a conduct ing channel i n the i n s u l a t o r , fo l lowed E l e c t r i c breakdown r e l a t e s to events which are known i n (16) by: 2. d ischarge of the specimen's s tored energy through t h i s channel . 11 The second process can be expla ined i n terms of hea t ing and evapora t ion of the d i e l e c t r i c . The evidence so f a r presented i n d i c a t e s that some form of e l e c t r o n i c avalanche process (17) a s soc ia t ed w i t h f i e l d emiss ion at the cathode i s r e spons ib l e f o r the f i r s t process and that the breakdown s t r eng th i s g r e a t l y a f fec ted by the presence of f laws and i m p u r i t i e s . c . 'Heterogeneous Breakdown Breakdown may occur due to the presence of f laws ( c r acks , f i s s u r e s , p i t s , pores , weak spots) and inhomogenei t ies i n . t h e d i e l e c t r i c m a t e r i a l r a the r than the p rope r t i e s of the homogeneous m a t e r i a l . I t i s c l e a r that breakdown occurs at such inhomogenei t ies at f i e l d s t rengths much lower than would cause breakdown f o r homogeneous f i l m s . 2 .3-2 S e l f - H e a l i n g Breakdown^ 1 4 ^ . In p r a c t i c e , problems have been encountered w i t h t h i n f i l m s which are known to a r i s e a l so i n t h i c k e r i n s u l a t o r s . The samples shor ted and the breakdown vo l tage measured was not that of the bulk but that of the weakest spot . . 'Condi t ions are d i f f e r e n t when samples w i t h s e l f - h e a l i n g breakdown are used. In such samples, o one e lec t rode at l e a s t i s t h inne r than. 1000 - 2000 A. On break-down the t h i n e l ec t rode r a p i d l y evaporated at. the breakdown s i t e w i t h no genera l d e s t r u c t i o n of the sample. Samples w i t h s e l f - h e a l i n g breakdown have the fo l lowing-advantages i n s tudy ing breakdown p r o p e r t i e s : -1. . D i s t i n c t i o n can be made between p rope r t i e s of weak spots and. those of the b u l k . 12 2. Thermal breakdown s t r eng th can be measured without d e s t r u c t i o n . 3. E l e c t r i c breakdown can be observed w i t h minimal d e s t r u c t i o n . 4. Breakdown p rope r t i e s can be asce r t a ined from success ive measurements on one sample. 1 3 3- ELECTRONIC CONDUCTION CURRENT CIIARACTERISTICS OF THIN AMORPHOUS FILMS 3 • 1 Pr op os G cl t M o cl e l A m e t a l / i n s u l a t o r / m e t a l s t r uc tu r e may be represented by a one d imens iona l model i n which c a r r i e r s of one type only (assumed METAL. INSULATOR i / r i v , 6c ML hi V V / r i  • V F i g . 3.1 Band Diagram of Proposed Model here to be e l e c t r o n s ) are cons ide red . In view of the amorphous nature of the f i l m , the t rap concen t r a t ion i s expected to be h igh and . i s assumed to cover an energy range V/ below the conduct ion band edge. The t rap dens i t y i s assumed to decrease e x p o n e n t i a l l y w i t h t rap energy measured downwards from the conduct ion band. This d i s t r i b u t i o n q u a l i t a t i v e l y desc r ibes the s i t u a t i o n shown i n (18) f i g . ( 2 . 1 . b ) v x o \ To experience a Poo le -Frenke l ' mechanism i n the bulk t raps which e x h i b i t a coulombic a t t r a c t i o n w i t h respect to cur ren t c a r r i e r s (here, e l ec t rons ) are necessary . Thus t raps are assumed to be n e u t r a l when occupied and p o s i t i v e l y charged when empty ( i . e . donor s t a t e s ) . Donor s ta tes may a r i s e ( e .g . ) from the contaminat ion of the f i l m dur ing p repa ra t ion or from d i s a s s o c i a -t i o n of molecules and non s t o i c h i o m e t r i c composi t ion of the f i l m . Since some t raps may not be i n i t i a l l y (at zero f i e l d ) f i l l e d , an a d d i t i o n a l f u l l y i o n i z e d acceptor l e v e l i s i n t roduced , whose charge i s i n v a r i a n t w i t h the a p p l i e d f i e l d , to ensure e l e c t r i c a l n e u t r a l i t y . The space charge l i m i t i n g case, u s u a l l y def ined when 14 a l l trapped charges are i n excess of e l e c t r i c a l n e u t r a l i t y , can be obtained as a s p e c i a l case -when the trapped states are i n i t i a l l y f u l l y i o n i z e d . E l e c t r o n s are assumed to flow from the cathode i n t o the conduction l e v e l of the f i l m by Schottky thermionic emission. Since i n d i e l e c t r i c amorphous f i l m s , the mean free path between s c a t t e r i n g ^ o events i s short (a few A), and thus the energy gained by an e l e c -t r o n between s c a t t e r i n g events i s l e s s than the thermal energy, the m o b i l i t y i s assumed to be independent of f i e l d s trength. 3 . 2 Basic Equations 1 . Schottky thermionic emission i s 1 J.= J Q exp p s E 2 ( 0 ) ( 3 . 1 ) where J o = AT 2 exp ~&=& ( 3 . 2 ) 2 . C o n t i n u i t y of steady-state current i n s i d e the d i e l e c t r i c f i l m r e q u i r e s that ( n e g l e c t i n g d i f f u s i o n since the e l e c t r i c f i e l d i s h i g h ) : J = en c(x) uE(x) • ' ( 3 . 4 ) where n c ( x ) i s the free e l e c t r o n d e n s i t y u i s the e l e c t r o n m o b i l i t y i s constant and independent of the p o s i t i o n "x". 3 . Poisson's equation i s where E i s the e l e c t r i c f i e l d , whose d i r e c t i o n i s the negative x a x i s . 15 f : i s the t o t a l charge density i n s i d e the f i l m fix) = e Q l t ( l - f ( x ) ) - N A - n c ( x ) where N_j_ i s the trap d e n s i t y f designates the p r o b a b i l i t y of occupation of a trap i s 'the acceptor d e n s i t y I f *!.{.(x) i s the f i l l e d trap d e n s i t y at p o s i t i o n x then, f(x) = e|~(Nt-NA) - (nt(x)+nc(x))l (3,7) Thus equation (3-5) becomes dE _ e dx ~ e (n t(x)+n c(x) - (N t-N A) (3-8) 4 . n^( x') (steady-state value) i s . obtained by equating the rate of e l e c t r o n capture from the conduction band to the rate of e l e c t r o n release from t r a p s , thus i f i s the trap density per u n i t energy, one may w r i t e r ^ ( r a t e of e l e c t r o n capture) = ncN^ -_j_ ( l - f )o" "V^^ds^. where N^de^ i s the trap d e n s i t y i n the energy range between and e^+ds.^ rj— i s the capture cross s e c t i o n v_j_k i s the e l e c t r o n thermal v e l o c i t y . The rate of e l e c t r o n release from traps i s governed by the e l e c t r o n r e l a x a t i o n time "o given by ' 1 ( e t - A e J / K I where Ae_j. i s the f i e l d lowering of trap depth given by r Ae^ = Bp p KT E 2 (assuming Poole-Frenkel e f f e c t ; see F i g . 2 . 3 ) . 9 i s the atterirpt to escape frequency Thus one may w r i t e , 1 ~ — fl E 2 , . jlT p p F r-girate of e l e c t r o n r e l ease ) = Ny^f ve e de^ For the s teady-s ta te n o n - e q u i l i b r i u m c o n d i t i o n we have, r l = r 2 then the occupancy f a c t o r f i s f ;= n cr v , , c u th n c ° ~ t h + e — fi E^ KT P P . F 1 ( 3 . 9 ) using, equations 3.1 and 3-4, f ( x ) = i - ~ - — ( 3 - 1 0 ) 1 + ) . B ( I , e^-^) ;V^ ( 0 ) e"^ r o c r V t h The trapped e l e c t r o n s / u n i t volume i n the energy range between and + ds^ _ are g iven by N i r , de. / \ Wt t dn , (x) t V A / - 1 1 £ t 1 + ( - 3 U f i ~ ) E(x)e P ' F ( x ) e 3 e K i Jo^ v t h For amorphous f i lms t raps are assumed to cover an energy range W below the conduct ion band. The t rap d e n s i t y / u n i t energy i s assumed i i. (18) to be N w t = A t e MKT ( 3 . 1 l ) where A.j. and N : are constants depending on. the f i l m s t r u c t u r e . Thus the t o t a l number of trapped e l e c t r o n s i s W A t d £ t n ' - ' ....... t v ' ~ J - i i 0 , P P.pB 2(x) - M 2 ( o ) - i | ( ^ > e + - 7 — 1 Mx) e e e J o ^ v t h . The voltage drop across the f i l m i s given by d v = J E(x) d x ( 3 -13 ) where 0 d i s the f i l m t h i c k n e s s . 3 . 3 Computational Procedure I f normalised v a r i a b l e s J ' = E = 3^E: x = ' n J ' n K s n d o ar e • s u b s t i t u t e d i n equations 3-l> 3 - 4 , 3-8 and 3 . 1 2 , 3 .13 one gets, 1 J (x ) = exp E 2 ( x ) ( 3 -14 ) n n y n n w dE d x n 1 t ( n t ( x N ) 7 V n c ( x n ) A T c . ) . - l ] (3.15) rw A t d e t ( ) = 1 — t ( 3 - 1 6 ) • J _ i i - £ t / N - i x o K T N „ Q '•' / \ K T 1 N J e + C 2 • S (xn)<>e E"^(x 0 n n n Q ( x ) ~ C (3.17) c n ^ E x . n n . 1 E f. E (x,-) dx (3.18) a v n / n \ rr n w ; where 2 E ^ ( x J -E^(0) S(x ) = E ( x J e ^ n ^ V e ^ n 1 11 I.l x l 18 C, = (N • d • 6 2 1 o s 2 = j " c - v + J 2 . ( 5 ' 1 9 ) o t l r s e u By s e l e c t i n g a value f o r J , the e l e c t r i c f i e l d at the metal i n s u l a t o r i n t e r f a c e can be computed from equation 3 .14. Poisson's equation 3.15 can now be solved n u m e r i c a l l y f o r a given set of parameters to obtain the f i e l d d i s t r i b u t i o n i n s i d e the f i l m (the method used was a Runge-Kutta method of order 4 ^ ^ ) . Then, the corresponding average field i s computed numerically using equation 3 '18 (che numerical i n t e g r a t i o n method used was Simpson's ( 1 Q N r u l e 1 ). Thus a. p l o t f o r the d.c. conduction c h a r a c t e r i s t i c can be obtained. 3.4 Numerical Results and D i s c u s s i o n The absolute values of the s o l u t i o n s of the previous equations depend d i r e c t l y on the values chosen f o r the disposable parameters. The d i f f e r e n t p o s s i b i l i t i e s of the model can be shown i n the f o l l o w i n g cases: a. High Values of the N d Product ° p_ • Whenever the N d product i s high enough, the e l e c t r i c f i e l d changes from the Schottky value at the M e t a l / I n s u l a t o r i n t e r f a c e towards a constant value i n the bulk which can be evaluated ( n e g l e c t i n g free e l e c t r o n s ) by p u t t i n g — N = n A ( x ) : provided x <<- d\ e o t ' 1 Thus from equation (3-16) ~ SORT (FIELD)] x W (V/cm) Q F i g . 3.2 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Changing N d H 20 V e f z A, de, WQ = / — e + C 9 (3 E(,xj e e At high e l e c t r i c f i e l d s , the f i r s t term i n the denominator may "be neglected thus, x (3 E 2 ( 0 ) g- v., A. KTN J e s  N - ^ ^ t . _o _ 1  or •where ° y \ie (N-1) 0 pE 2(x) E(x) e J = ^ E(x) e ^ ( 3 . 2 0 ) ^ ~ cr v.. A, N KT O.^ -W th t Thus the d.c. conduction c h a r a c t e r i s t i c tends towards a pure Poole-Frenkel mechanism. P i g . 3-2 shows c l e a r l y such behaviour f o r r e l a t i v e l y high N d products. b. Low Values of the N d Product o I f the N Qd product i s s u f f i c i e n t l y low, there w i l l be no s i g n i f i c a n t d i f f e r e n c e between the e l e c t r i c f i e l d i n the bulk and that at the m e t a l / i n s u l a t o r i n t e r f a c e , and hence the average e l e c t r i c f i e l d =a» = I = E<°> (3-22) and J = J q exp BgE'2" ( 3 . 2 3 ) i . e . the d.c. conduction mechanism w i l l tend to be a. pure Schottky process. The d e v i a t i o n of the Schottky slope at low f i e l d s ( f i g u r e s 3-2 and 3.3) i s due to a high trapped e l e c t r o n density which may need a higher average f i e l d than the Schottky value to passthe same F i g . 3-3 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Negative Space Charge 22 cu r r en t . This d e v i a t i o n i s more pronounced fo r a h igh N Q d p ro -duc t . To show c l e a r l y the e f f ec t of negat ive space charge another set of parameters was chosen.- F igure 3-3> the computa-t i o n a l r e s u l t s , ' s h o w s that we have two negat ive space charge r e g i o n s . F i r s t r e g i o n , at low current ( f i e l d ) va lue s , i s due to the h igh trapped e l e c t r o n d e n s i t y . Second r e g i o n , at h igh current ( f i e l d ) r e s u l t s from the increase of free c a r r i e r d e n s i t y . Both regions become more pronounced as the f i l m th ickness i n c r e a s e s . c ' For Intermediate Values of the N d Product o G e n e r a l l y , the d . c . conduct ion c h a r a c t e r i s t i c w i l l c o n s i s t of d i f f e r e n t r e g i o n s . In each r e g i o n , one mechanism (space charge, Schot tky emiss ion or the P o o l e - F r e n k e l mechanism) predominates i n a way s i m i l a r to tha t d i scussed i n par t s ' a ' and ' b ' . In the t r a n s i t i o n r e g i o n between the. P o o l e - F r e n k e l and Schot tky reg ions we may encounter a value fo r the s lope d l o g J / d E 2 l e s s than the t h e o r e t i c a l Schot tky s l o p e . This i s to be expected s ince the • r e l a t i v e change .of the e l e c t r i c f i e l d from i t s value at the m e t a l / i n s u l a t o r i n t e r f a c e decreases as the current i n c r e a s e s . F i g . 3-2 shows t h i s p o s s i b i l i t y . From the previous d i s c u s s i o n , and i n p a r t i c u l a r w i t h reference to f i g u r e s 3-2 and 3-3 the conduct ion current mechanism as a f u n c t i o n of the N d product th ickness may be ou t l i ned schemat i -c a l l y and shown i n f i g . 3-4: where I : Steep r i s e , space charge region I I : P o o l e - F r e n k e l r eg ion I I I : T r a n s i t i o n r e g i o n ; the s lope d l o g J / d E 2 changes "from the •9 CURRENT DENSITY F i g . 3.4 'D.C. Conduct ion Mechanisms P.P. Value to Schot tky v a l u e ; s lope l e s s than Schottky value may be encountered. IV: Schot tky r e g i o n V: Free e l e c t r o n and breakdown r e g i o n 1. For a g iven set of parameters, as the N d product tends to sma l l v a l u e s , the P o o l e - F r e n k e l r eg ion disappears and we have only s'pace charge and Schot tky r e g i o n s . On the other hand as the N d product tends to l a r g e v a l u e s , the Schot tky r e g i o n does not appear and thus the d . c . conduct ion c h a r a c t e r i s t i c .contains .only space charge and P o o l e - F r e n k e l reg ions ( f i g . 3-2). 2. For h igher va lues of A or sma l l e r va lues of C^, f o r a g iven set of parameters, more e l e c t r o n s w i l l be trapped i n c r e a s i n g the space charge- reg ion and d e l a y i n g other r e g i o n s . The increase of trapped, e l e c t r o n d e n s i t y r e q u i r e s a h ighe r f i e l d to pass the same c u r r e n t . At h igh f i e l d s the trapped e l e c t r o n s have a l i t t l e e f f e c t and a l l curves tend to c o i n c i d e . The s t a r t i n g of c o i n c i -dence depends on the N d product . As C , inc reases the free e l e c t r o n o 3 r e g i o n increases ( f i g . 3-5). 3. For lower va lues of N and the same t r ap dens i ty fo r a g iven set of parameters, the same e f f e c t d iscussed i n (?.) occurs . F i g . 3.5 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Changing C 9 and C - , 4 ! o o 2\ 5.0 10-0 7 '15.0 ~SQRT(FIELD)! xl02(V/cm) 4 p i g . j.6 Computed Conduction C h a r a c t e r i s t i c s : E f f e c t of Changing N 20.0 ro 26 The point of coincidence occurs at a f i e l d i n t e n s i t y which depends on the R d product ( f i g . 3.6). 3.5 Transient E l e c t r o n i c Discharge Current •On short c i r c u i t i n g a sample, the space charge e x i s t i n g i n s i d e the f i l m i s released and a part of i t c o n t r i b u t e s to the ( 2 0 2 1 ) e x t e r n a l discharge current ' . According to the proposed model, the ne~ p o s i t i v e charge density i n s i d e the f i l m , n e g l e c t i n g free, e l e c t r o n s , i s given by f (x,t) = e[§H o - n t ( x , t j ] ( 5 . 2 1 ) a. Space charge s u f f i c i e n t f o r x* to e x i s t . b. Space charge s u f f i c i e n t l y small no x* to e x i s t F i g . 3.7 Band Diagram f o r M e t a l / I n s u l a t o r / M e t a l during Discharge The e l e c t r i c f i e l d may become zero at' a p o s i t i o n "x*" i n s i d e the f i l m , i n t h i s case, by Gauss' theorem, i f the charge on counter electrode 1 i s Q^(x,t) where 1 = A 0 { n t ( x , t ) - f N 0] dx (3-22) and A i s the counter electrode area 1 i ° i A dt x" n t ( x * , t ) - f N o dx* dt dn^ dt dx (3-23) 27 But the f i n a l term i s the t o t a l i n t e r n a l current ( neg l ec t i ng t r a n s i t time) hence 'the e x t e r n a l current I - A e e n . f c ( x * , t ) - f N o dx* dt . (3-24) In the f o l l o w i n g , I w i l l cons ider only the case when trapped e l ec t rons are in -excess of e l e c t r i c a l n e u t r a l i t y . To f i n d an express ion fo r n^_(x,t) , cons ider an i n f i n i t e s i m a l energy range between e^ . and + ds^; the trapped e l e c t r o n d e n s i t y i n t h i s energy range Is g iven by (equat ion 3-12) The ra te of re lease of e l ec t rons w i t h b i n d i n g energy i s assumed to be p r o p o r t i o n a l to l / r ^ - 1/rCQ expC -e^/KT) . Hence one may w r i t e _, ij • d n t ( x , t ) = d n t ( x , 0 ) e A ^ (3-26) I n t e g r a t i n g equat ion 3-26 over the energy between = 0 and ^ W ( x , t ) = / dn ( x , t ) (3-27) n t 0 Thus, the volume ra te of r e l ease of e l ec t rons from t raps i s W dn^ ' dn^(x,0) _ t dt x • e - t A t j o to s i m p l i f y equat ion 3-28, l e t u = exp (-e (3-28) t M i , = exp ( - t / ' i £ ) (-3-29) e-W /ET dllj. rrm "' A r ' ' b a U du (3-30) dt jy - l , N - 1 N I U " + C 2 S u To ob ta in a c losed form fo r the above i n t e g r a l the f o l l o w i n g assump-t i o n s w i l l be made: W/KT » 1 , t /•t? 1, SC,,u<£l which may be a reasonable assumptioia fo r deep t r aps , which presumably cause the l o n g 28 discharge cu r ren t , e s p e c i a l l y at. low- p reapp l i ed vo l t ages and N i s s u f f i -c i e n t l y l a r g e to j u s t i f y tha t Jr^ i s almost constant over the e f f e c t ! energy range (deep traps range) , l e t i t s average va lue over t h i s range he a, then dn, • KT a A, _ — I _ 1 . , , 3 1 y dt - -t G e n e r a l l y , the e x t e r n a l cur rent depends on the f i e l d d i s t r i b u t i o n I n s i d e the f i l m and the electrodes ' 1 work f u n c t i o n s . In the case of e l ec t rodes w i t h d i f f e r e n t work func t ions and of (p i ) space charge s u f f i c i e n t l y sma l l " fo r no x* to e x i s t , we have •from P o i s s o n ' s equat ion f^f- = " fye = e n t ( x , t ) / e ( 3 - 3 2 ) I n t e g r a t i n g ( 3 - 3 2 ) t w i c e , one obta ins Y(a)-y(o) = d + r f | ^  ,^ i t ) & „ L ^ = 0 0 ( 3 . 3 3 ) where 0-^  and 0^  are the work funct ions of the two e l e c t r o d e s . I f 0p^ >02> the f i e l d i s everywhere such as to ca r ry e l ec t rons e lec t rode 2 and hence the decrease i n i s due e n t i r e l y to the e x t e r n a l cur ren t e • dt • dt dx t o 0 0 eKTaA Ad ( 3 - 3 4 ) 2 t • , For the above "cases, the e l e c t r o n i c d ischarge current f o l l o w s a l / t law s i m i l a r to the i o n i c p o l a r i z a t i o n cu r r en t . But , i n contrast to the i o n i c p o l a r i z a t i o n cu r r en t , i t i s approximately-independent of the p reapp l i ed v o l t a g e ! 29 4 . EXPERIMENTAL PROCEDURES AND RESULTS 4 • 1 S am vi 1 e P r e p a. r a t i o n 4 . 1 . 1 Tantalum Surface P repa ra t i on Both s i n g l e c r y s t a l and sput tered B-tantalum on Corning 7059 g l a s s subs t ra tes were used. The s i n g l e c r y s t a l surfaces were e l e c t r o - c h e m i c a l l y p o l i s h e d . The tantalum c r y s t a l was made the anode i n a bath of 90% (by volume) H^SO^ (96% reagent) and 10% HP ( 4 9 % reagent) and a cur rent dens i t y of about 100 ma/cm was used fo r about 10 minutes . E l e c t r o - c h e m i c a l p o l i s h i n g was p re -ceded by abras ion down to 2/0 paper. Sput tered tantalum samples were degreased.. by Immersion i n s u l p h u r i c a c i d (96%) sa tura ted w i t h potassium dichromate. The samples were then r i n s e d i n d i s t i l l e d water . 4-1.2 F i l m Growth a. E l e c t r o l y t e S o l u t i o n Grown F i lms The prepared tantalum samples were anodized i n 0.5% by volume s u l p h u r i c a c i d at room .temperature f i r s t at constant cur rent (.5 ma/cm ) and then under constant vo l tage f o r per iods up to 12 hours . _ v The f i n a l leakage current d e n s i t i e s were l e s s than 10 'x-anodizing v o l -tage (amp/cn/") . b. Plasma-Grown F i lms The f i l m was prepared on a tantalum s i n g l e c r y s t a l by a n o d i z a t i o n i n a d . c . glow discharge i n the apparatus descr ibed i n r e fe rence v , at a sample temperature of 35- C and i n the p o s i t i v e column of the d i scha rge . The oxygen pressure was 80 mtorr . and a n o d i z a t i o n was c a r r i e d out at constant cur ren t d e n s i t i e s of 0 .2 , 1.0 and 2.0 ma/cm 2 . 30 F i l m ' t h i c k n e s s e s were est imated u s ing a model 14-Cary (''? \ spectrophotometer and the curves on page 80 of reference , 4.1-3 C o un t e r e1e c t r od e s Gold counter e l ec t rodes were evaporated u s ing photo-etched b e r y l l i u m copper masks and a shutter- i n a conven t iona l b e l l - j a r system (Veeco 400) which employed a d i f f u s i o n pump and l i q u i d n i t r o g e n trap- The evapora t ion was c a r r i e d out at 10 t o r r or l e s s , and in- a l l cases the evapora t ion was s t a r t e d before opening a shu t t e r and exposing the oxide f i l m to the source . The d i s tance between source and f i l m was approximately s i x i nches . Coun te re lec -ir-ode areas were est imated by us ing br idge measurements at 1 0 0 0 Hz and assuming . = 27-6 fo r so lu t ion-grown f i l m s and e^ = 18 f o r plasma-grown f i l m s . Unless otherwise s t a t ed , the coun te re l ec t rode -3 2 -3 2 area was 6-+.1 x 1 0 cm' fo r so lu t ion-grown and 3-6 +.1 x 1 0 cm fo r plasma-grown f i l m s . Gold as the coun te re lec t rode m a t e r i a l was chosen s ince i t i s known that i t g ives s a t i s f a c t o r y devices, w i t h h igh breakdown s t r e n g t h . S i l c o x and M a i s s e l ^ 2 ^ a t t r i b u t e d t h i s to a tendency of gold to br idge f i ne cracks i n the oxide surface which other m a t e r i a l s , e . g . A l , tended, to penetrate l e a d i n g to .poor i n s u l a t i o n p r o p e r t i e s . 4•2 E l e c t r i c a l Measurements Conduction and step response cur ren ts were measured us ing a K e i t h l e y type - 4 1 7 h i g h speed picoamrneter and an x-y recorder fo r mapping the current t r a n s i e n t . Capaci tance and d i e l e c t r i c l o s se s were measured i n 3 t e r -minal connect ion us ing a G..R. 16.15 A br idge i n the frequency range .100 Hz - 1 0 0 kHz. -Non d e s t r u c t i v e breakdown t e s t s were used to compare f i l m s grown by d i f f e r e n t p r epa ra t i on procedures and a l s o diodes of d i f f e r e n t coun te re lec t rode areas prepared on the same f i l m . A Hewlett Packard low frequency f u n c t i o n generator type 202A was used to generate t r i a n g u l a r wave forms of frequency such that the ra te of r i s e of vo l t age was 2V/sec . A l l measurements were c a r r i e d out i n a i r at room temperature w i t h the sample i n an e l e c t r i c a l l y sh i e lded dark chamber. The s p e c i f i e d accuracy of the picoammeter was +3/£ f o r —12 a l l ranges except the lowest (lO amp f . s . ) where i t was +5%. The G . R . br idge accuracy was 0.01$ fo r f < 10 kHz and to 0.2^ at 100 'kHz. The l o s s tangent measurement was accurate to 0.1%. 4•3 D .C . Conduction C h a r a c t e r i s t i c s Samples were s h o r t - c i r c u i t e d fo r per iods up to 12 hours before any measurement was made. On a p p l y i n g a d . c . vo l t age to a sample, a p o l a r i z i n g current s t a r t s to flow which decays towards a steady s ta te v a l u e . • The d . c . vo l t age was appl ied ' fo r more than 1000 s e c . , and the d . c . conduct ion current was obtained by e x t r a -p o l a t i o n of the obtained r e s u l t s . F i g . 4.1 shows l o g J — ^ charac-t e r i s t i c s fo r f i v e d i f f e r e n t th icknesses of oxide prepared on Ta s i n g l e c r y s t a l by s o l u t i o n a n o d i z a t i o n . F i g . 4 . 4 ( l e f t par t ) shows l o g J ~{v c h a r a c t e r i s t i c s f o r f i l m s prepared by d i f f e r e n t procedures . The r e s u l t s were f i t t e d approximately by J = C exp (3 E"2" ( 4 . 1 ) where C and (3 are constants depending on f i l m t h i c k n e s s , p r epa ra t i on procedure and on temperature. 52 -8 V£ , 10 x VOLT 2 cm 2  d P i g . 4.1 D.C. Conduction C h a r a c t e r i s t i c s f o r l a - P o s i t i v e Voltages 33 A J = 10 amp /cm l00 800 1200 1600 2000 2400 ~2800 3200 FILM THICKNESS, A F i g . 4.2 Voltage-Thickness R e l a t i o n For Constant Current F i g . 4-3 D.C. Conduction C h a r a c t e r i s t i c s f o r Ta-Negative Voltages - / r -8 5 C3 x PLASMA GROWN (950 A) SOLUTION GROWN ON O a , SINGLE CRYSTAL (1080A) & b. SPUTTERED Ta (1050 A) 7 1 'OLTAGE . VOLT (Ta 2 ve) 3 .-. D i f f e r e n t va lues fo r "(3".for f i l m s prepared by plasma a n o d i z a t i o n may be expected s ince a d i f f e r e n t r e l a t i v e p e r m i t t i v i t y (21) at h i g h f requencies was repor ted . . F i g . 4 . 2 i s . deduced from f i g . 4 - 1 , i t shows the r equ i r ed vo l tage to pass .a s p e c i f i e d current , as a f u n c t i o n of f . i lm t h i c k n e s s . The decrease of the r equ i r ed o /~\ . f i e l d fo r t h i c k f i l m s (>1000 A) may. be due to f laws D or due to s m a l l space charge i n s i d e the bulk (see chapter 3) or bo th . F igu res 4-3 and 4-4 ( r i g h t par t ) show a J - ^ c h a r a c t e r -i s t i c - fo r Ta n e g a t i v e . • Fig-. 4 .4 shows a c l e a r r e c t i f i c a t i o n p ro -cess . R e c t i f i c a t i o n . p r o c e s s e s i n m e t a l / i n s u l a t o r / m e t a l diodes have been a t t r i b u t e d by-some inves t iga . to r s to the presence of s t r u c t u r a l ' ' ""' ( •< p. h 2 6) defec ts i n the f i l m s such as flaws- and m i c r o f i s s u r e s ..' ' ( 2'7 28 29) A'number of i n v e s t i g a t o r s ' b e l i e v e that the r e c t i f i c a t i o n i s . b a s i c a l l y due to the .exis tence of -p-n or • p- ' i- 'n j u n c t i o n s . T h i s , does not exclude the s u p e r p o s i t i o n .of other mechanisms. ' . 4.4 Step Response Measurements • P o l a r i z a t i o n mechanisms were' inves t iga ted- by : 'measuring the current response to step vol tages , . To avoid i n t e r a c t i o n between, the leakage current and . p o l a r i z a t i o n cu r r en t , the method used was "b measure the current upon remova l .o f the d . c . a p p l i e d v o l t a g e . The charging vo l t ages were app l i ed i n s u i t a b l e steps and were, l e f t on fo r at l e a s t 1000' sec . On the removal of the d . c . voltage. , the d ischarge cur ren t -for a l l prepared f i l m s fo l lowed a r e l a t i o n s h i p J , ( t ) y\ ( E , d ) — • ' •' ' (4 . 2 ) ex u l- ' -, m . 1 1 "G whe.i'e m i s a constant ranging-b etween 0.9 and 1.1 • (r i>l was. obtained only fo r the plasma grown f i l m ) • 37. F i g . 4.5 Discharge Current as Function of F i e l d f o r Films of D i f f e r e n t Thickness (Ta-positive) 38 .c; 8 k l 1.3 12 1.1 1.0 • 9 .8 • 7 .6 - X 0 4 x PLASMA GROWN (950 A) SOLUTION GROWN ON • a. SINGLE CRYSTAL (1080A) 0 b. SPUTTERED Ta (1050 A) 8 12 5 15 20 ' ^ Z J 10 v/cm — F i g . 4 . 6 Discharge Current as Function of F i e l d f o r Films Prepared by D i f f e r e n t Procedures (Ta-positive) .\(B,d) i s a c o n s t a n t w h i c h d e p e n d s on b o t h e . p p i i e d f i e l d a n d f i l m t h i c k n e s s . .. F i g . . 4 . 5 s h o w s t h e q u a n t i t y ' ' • . • > l ( E , d ) = (4.3) f o r f i l m s • o f d i f f e r e n t t h i c k n e s s g r o w n ' o n t a n t a l u m s i n g l e c r y s t a l b y s o l u t i o n a n o d i z a t i o n . F i g . 4.6 s h o w s r ( j E , d ) f o r f i l m s p r e p a r e d b y d i f f e r e n t p r o c e d u r e s . -' r l j E j d ) i s i n d e p e n d e n t o f t i m e a n d s h o u l d n o t v a r y w i t h a p p l i e d f i e l d i f - s p a c e c h a r g e e f f e c t s a r e a b s e n t i . e . i f t h e d i s c h a r g e c u r r e n t i s l i n e a r w i t h a p p l i e d v o l t a g e . The n o n - l i n e a r i t y o f f i g s . 4.5 a n d 4.6 c o n f i r m s t h e p r e s e n c e o f s p a c e c h a r g e i n s i d e , t h e f i l m . T h u s i t i s r e a s o n a b l e t o a s s u m e t h a t a p a r t o f t h e o b s e r v e d e x t e r n a l d i s c h a r g e c u r r e n t i s d u e t o t h e s p a c e c h a r g e i n s i d e t h e a m o r p h o u s f i l m i . e . ' J + = J + J . (4.4) e x t e P w h e r e J . . i s t h e t o t a l e x t e r n a l d i s c h a r g e c u r r e n t d e n s i t y e x t '•' J J g i s t h e e l e c t r o n i c d i s c h a r g e c u r r e n t d e n s i t y . J - i s t h e i o n i c p o l a r i z a t i o n c u r r e n t d e n s i t y . " The d e p e n d e n c e o f ' ' ^ ( E , d ) o n f i l m t h i c k n e s s may be d u e ' t o f i l m p a r a m e t e r s , e . g . . t h e c o n t a m i n a t i o n o f t h e f i l m a n d s u r f a c e e f f e c t . One may e x p e c t h i g h e r l o s s e s i n t h e c a s e o f t h i n n e r f i l m s r e s u l t i n g i n h i g h e r v a l u e s o f " ' ^ ( E , d ) . 4 . 5' A . C . B r i d g e ' M e a s u r e m e n t s F i g . 4-7, 4-9 show t h e v a r i a t i o n o f c a p a c i t a n c e a n d e q u i v a l e n t s e r i e s r e s i s t a n c e (R ) ( t h e m e a s u r e d v a l u e s a r e c o r r e c t e d © d = 280 A L • 1 : : : i : . : I 2xW2 W3 JO4 W5 FREQUENCY Hz F i g . "4.7 Frequency Dependence of Capacitance f o r Films of D i f f e r e n t 'Thickness -F^ F i g . 4 . 8 Frequency Dependence of Equivalent S e r i e s Resistance- f o r Films of D i f f e r e n t Thickness 7.02 § -38 O t /T .96 • 94 .92 2x10 10 3 10 4 FREQUENCY Hz 10 5 A : Q TT requency Dependence of Capacitance f o r Films prepared by D i f f e r e n t Procedures 4 ^ 43 10 10 0 CD 10 •3 70" x PLASMA GROWN (950 A) SOLUTION GROWN ON o a. SINGLE CRYSTAL (1080 A) O b. SPUTTERED Ta (1050 A) 3 W 10 FREQUENCY Hz 10 2 F i g . 4-10 Frequency Dependence of Equivalent Series Resistance f o r Films Prepared by D i f f e r e n t Procedures . ( 2 2 \ f o r s e r i e s r e s i s t a n c e due to contacts, lead r e s i s t a n c e , e t c . 'j w i t h frequency f o r solution-grown f i l m s of d i f f e r e n t t h i c k n e s s . F i g s . 4 . 8 and 4 . 1 0 show such v a r i a t i o n f o r f i l m s grown wi t h d i f f e r e n t p reparation procedures. S l i g h t but d e f i n i t e decrease of capacitance was observed as the app l i e d s i g n a l frequency increased i n d i c a t i n g that the d i e l e c -t r i c constant decreases s l i g h t l y w ith i n c r e a s i n g frequency i n the" frequency 1 0 0 Hz - 1 0 0 kHz. This decrease maybe expected-since -at high frequencies,' some p o l a r i z a t i o n processes of r e l a t i v e l y long r e l a x a t i o n time w i l l not c o n t r i b u t e . The same behaviour was observed f o r l o s s tangent. . Equivalent s e r i e s r e s i s t a n c e R i s given by R = . ( 4 . 5 ) s coC . . s Since no great changes occur i n the p o l a r i z a t i o n processes in'the -frequency range 1 0 0 Hz - 1 0 0 k H z , —2-^- i s almost constant and one. s 1 expects that R i s n e a r l y jxroportional to (—). F i g . 4 . 8 gives R a -, ^ oc- . At higher frequency ranges, where p o l a r i z a t i o n processes change, the frequency dependence of R q may deviate considerably from the ~ r e l a t i o n . At very low frequencies, s c a t t e r i n g p r o c e s s e s ^ ^ may also change such dependence. ' F i g s . 4 . 1 1 and 4 . 1 2 show the v a r i a t i o n of capacitance and l o s s tangent w i t h d.c. bias (Ta electrode negative) f o r solution-grown f i l m s on s i n g l e tantalum c r y s t a l . F i g . 4 .13 shows the e l e c t r i c a l c i r c u i t used f o r measurement where the -transformer bridge i s the . G-.R. 1 6 1 5 A bridge. I t could be assumed that e l e c t r o n i c phenomena are r e s -ponsible f o r such an increase because of space charge formation 45/ VOLTAGE, VOLT F i g . 4.11 Capacitance Dependence on d.c. Bias (Ta~.nega.tive) 46 A 10 8 c: 2 o A o A = 3700 A G d= 1080 • =2050 4 12 16 20 VOLTAGE , VOLT Mg. 4.12 Loss Tangent Dependence on d.c. Bias (Ta-negative) 47 DETECTOR 'd. c, SUPPLY GENERATOR F i g . 4.13 Apparatus fo r Measuring Capacitance and Less Tangent Dependence on d . c . B ias which i s presumably h igh fo r Ta n e g a t i v e . Another exp l ana t i on may be the -reduction of the e f f e c t i v e d i e l e c t r i c th i ckness which may be due to oxygen t r a n s f e r through the ox ide . (34) 4•6 Breakdown Tests As a q u a l i t a t i v e measure of breakdown, a s imple dynamical method >2,33) shown s c h e m a t i c a l l y i n f i g . 4-14 was used. A Hewlet t Packard low frequency f u n c t i o n generator type 202A was used. T r i -angular s i g n a l s a p p l i e d to the samples were adjusted to increase at about + 2 V / s e c . - The output of a K e i t h l e y 417 h igh speed picoammeter and the. vo l t age from the potent iometer were d i s p l a y e d on an x - y r e c o r d e r . . . . . . -The cur ren t measured by the picoammeter i s -given by: I = c g - + I L . (4.6) SAMPLE L. F. GENERATOR \ . PICO-AMMETER RECORDER F i g . 4 .J-4 Breakdown Measurement Apparatus 49 where . • C ; i s the diode capaci tance • V i s the app l i ed vo l tage . •• • I-j^  i s the leakage .current . The q u a l i t a t i v e measure of breakdown vol tage has been de f ined , • f o l l o w i n g Schwartz et a l . > .. as the vo l t age at which the leakage cur ren t becomes'• equal to the charging cur ren t - i . e . . - . I = 2 C | (4.7) The peak amplitude, of the t r i a n g u l a r waveform was increased i n s u i t a b l e steps u n t i l the V - I r e l a t i o n recorded by the x -y recorder s a t i s f i e d equat ion 4.7 and became r e p e t i t i v e over s e v e r a l c y c l e s . Then the f i n a l a p p l i e d vo l tage was recorded and the breakdown s t r eng th c a l c u l a t e d accord ing to E b = i , (4.3) where V-p i s the f i n a l a p p l i e d • vo l t age ' • • d i s f i l m th ickness ' ' F i g . 4-15 shows c l e a r l y the evapora t ion of the counter e lec t rode at the weak spots as the a p p l i e d vo l tage i n c r e a s e s . F i g . 4.16 shows a t y p i c a l f i n a l V - I c h a r a c t e r i s t i c . The v a l i d i t y of the above measure was examined s e v e r a l times by i n c r e a s i n g the a p p l i e d vo l tage by a few percent above that obtained by the above procedure; a d e s t r u c t i v e breakdown was observed •in - a l l cases F i g s . 4-17.and 4-13 show the measured breakdown s t r eng th , as a f u n c t i o n of counter e l ec t rode area and f i l m th ickness fo r f i l m s prepared on tantalum s i n g l e c r y s t a l and sput tered tantalum r e s p e c t i v e l y VOLTAGE F i g . 4 . 1 5 ' E f f e c t of Weak Spots and' '• Counterelectrode Evaporation on I-V C h a r a c t e r i s t i c Q: VOLTAGE F i g . 4.16. A T y p i c a l F i n a l 1-V C h a r a c t e r i s t i c <fV/cm 51 5 8 o | 4.5| N Q •—i O Q Uj 4 3.5 3 0 F i g . .4 .8 7.2 16 2.0 AREA, mm2 2.4 2.8 4.17 Breakdown St rength as a f u n c t i o n of Countere lec t rode F i lms of D i f f e r e n t Thickness Grown on S ing l e C r y s t a l p o s i t i v e ) . 8 o Q --4 u: o 03 4.5 4 3.'51 0 32 area f o r Ta ( T a -o d= 1540 A = 1030 A = 430 A .4 .8 1.2 1.6 2.0 2.4 AREA , mm 2 28 3.2 F i g . 4 .18 Breakdown St rength as a F u n c t i o n - o f Counterelectrode area " f o r F i l m s of D i f f e r e n t Thickness Prepared on Sput tered Ta ' ;.. • 52 o Counterelectrodes were gold of thickness l e s s than 500A thus, on l o c a l i z e d breakdown, the t h i n electrode evaporated at break-down s i t e s before sample d e s t r u c t i o n occurred ( f i g . 4.15). / 53 . 5 • Discussion 5-1 D.C. E l e c t r o n i c Conduction Ore of the troublesome features of e l e c t r o n i c conduction i n t h i n amorphous f i l m s i s i t s time dependence. A- time dependence would be expected due to the.step-response when the voltage i s changed ( p o l a r i z a t i o n e f f e c t s ) but, i n a d d i t i o n , there seems to be an e x t r a time dependence which may be associated w i t h the development of space charge. The steady s t a t e conduction current was a t t a i n e d i n a period which depends on the a p p l i e d voltage. At higher f i e l d s , the steady s t a t e was reached f a s t e r than that .at low f i e l d s suggesting that space charge develops more slowly at. low f i e l d s . Since, the r a t e of i n j e c t i o n of e l e c t r o n s (taken to be Schottky thermionic emission) and the bulk conduction (taken to b e ' f i e l d enhanced thermal emission of e l e c t r o n s from coulombic-traps ( i . e . Poole-Frenkel emission) w i l l net be balanced- • •. at equal f i e l d s , the space charge d i s t r i b u t i o n i n s i d e the f i l m w i l l tend to change and hence the f i e l d d i s t r i b u t i o n a l s o u n t i l a steady s t a t e i s reached. This adjustment of f i e l d d i s t r i b u t i o n , i n a d d i t i o n to causing a time dependence, gives r i s e to a. change i n dlogJ/dE 2 from the t h e o r e t i c a l value f o r a pure Schottky mechanism or pure Poole-Frenkel mechanism. I t i s expected,and the numerical r e s u l t s show, that f o r t h i n enough f i l m s , the i n j e c t i o n mechanism predominates and thus the value of dlogJ/dE 2 tends to the Schottky \ slope.. On the other hand f o r r e l a t i v e l y t h i c k f i l m s the bulk e f f e c t A. swamps the i n j e c t i o n e f f e c t , thus dlogJ/d.E 2 tends to the Poole- . Frenkel slope. Generally, the d.c. conduction.'characteristic may be-divided i n t o d i f f e r e n t regions, each region being charac-t e r i s e d by one conduction mechanism (space charge, Schottky, 54 Poole-Frenkel and t r a n s i t i o n r e g i o n s ) . According to the f i l m parameters, some regions may.disappear and the others predominate. The p r e v i o u s l y . d i s c u s s e d f a c t o r s and the e f f e c t of flaws i n t h i n i n s u l a t i n g amorphous f i l m s are presumably the reasons why i t i s d i f f i c u l t to obtain good r e p r o d u c i b i l i t y and why l i t t l e suc-cess has been obtained i n numerical comparison of the data obtained by d i f f e r e n t experimentors f o r supposedly i d e n t i c a l c o n d i t i o n s . An asymmetric r e c t i f i c a t i o n conduction mechanism was observed experimentally f o r the prepared Ta/Ta20^/A.u diodes. This may be explained i n view of the f a c t that anodic oxide f i l m s are (3 24 25) ge n e r a l l y heterogeneous c o n t a i n i n g flaws and weak spots ' ' . (26 2T 28) Another p o s s i b i l i t y ' ' i s the existence of p-n or p - i - n junc t i o n s w i t h i n the f i l m . 5 .2 Step Response .'Measurements I f l i n e a r response theory a p p l i e s at some values of preapplied v o l t a g e , then the r e a l and imaginary parts of p e r m i t t i v i t y are l i n k e d to the step response-, " ^ ( t ) , -f (t) = - 2 U | ^ i t-m (5.D as below: s ' (w) ='—i c^ + / ^ ( t ) cos cot dt J (5.2) °o L " o » (co) = i - j ^ + . f \ ( t ) s i n cot d t ] (5.3) where c^ i s the capacitance of the sample at very high frequencies C q - i s the capacitance•of the electrodes when sample i s replaced by a i r . G- i s 'the steady state,d.c. conduction J 2. o Q. G •o- •o-O BRIDGE MEASUREMENT A STEP RESPONSE MEASUREMENT V= 10 VOLT • STEP RESPONSE MEASUREMENT V= 7 VOLT 0 •3 - 7 + 1 + 3 + 5 • -f, logW(/lHZ} F i g . -5.1 V a r i a t i o n of D i e l e c t r i c Loss w i t h Frequency: Comparison' of Bridge and Step • • Response Results' . - . - • . • ' ' VJ1 (^4) S u b s t i t u t i n g from (5.1) i n t o ( 5 o ) , one obtains ^ Ham on has shown that f o r the range Q.3< n <1.2, equation (5.4) can be reduced to -.'•£-£"•= ° n \ 1 0 (5.5) c co o For amorphous i n s u l a t i n g f i l m s , G Q may be neglected, then equation (5.5) becomes I (t) t e {to) _ 2 c y 0 (5.6) 65 £ „ = i _ °o T (E, d) n-1 v- N nit + — W 1 U-nJ cos — G o (5.4) w i t h co = Equation (5.6).was used 'by' Hamon f o r the r a p i d evaluation,of l o s s f a c t o r at frequencies below ".01" Hz where A.C. measurement would be tedious or i m p r a c t i c a l . E i g . 5-1 shows, the frequency, dependence of --C e" ..for a o 1050 A T^O^. f i l m grown on sputtered tantalum by s o l u t i o n anodiza-t i o n . The high-frequency points are those obtained by d i r e c t A.C. bridge measurements and the low frequency points were c a l c u l a t e d from step - response measurements and equation 5-6. Due to the i n i t i a l n o n l i n e a r i t y between discharge current and the preapplied.voltages, the c a l c u l a t e d values depend on the preapplied voltage f o r low f i e l d s trength and become independent of the preapplied voltage f o r r e l a t i v e l y high f i e l d strengths (^>lMV/cm). In view of t h i s r e s u l t and" the. r e s u l t obtained, in. Section (3-5) which shows ' that . J/'V decreases as V increases,, one may deduce that'the measurements of low frequency d i e l e c t r i c losses' using a step response technique are complicated' by space charge e f f e c t s only when the preapplied voltage i s r e l a t i v e l y low.' On the' other hand' at r e l a t i v e l y high .preapplied 57 vo l t age , • equat ion 4-4 becomes J , 2---J (5-7) ext p v ' y and thus the above l i n e a r theory a p p l i e s . 5-3' "Breakdown St rength The t rue breakdown s t r eng th does depend on the a p p l i e d vo l tage waveform ( d . c , sawtooth, p u l s e , . - . . ) and on the waveform parameters such as r i s e t ime, pulse wid th and r e p e t i t i o n r a t e . The exper imenta l c o n d i t i o n s and environment have a pronounced e f f e c t on the measured breakdown s t r eng th . ' - . .•' In t h i s work, a simple• dynamical method was used to ob ta in a q u a l i t a t i v e measure of breakdown s t reng th ;and to i n v e s t i g a t e -the e f f e c t ' o f coun te re l ec t rode area on the breakdown. -F i g s . 4-17 and 4•18 i n d i c a t e that breakdown.strength '. decreases s l i g h t l y .with i n c r e a s i n g coun te re l ec t rode area . Schwartz (32 33) ; and co-workers ' ,• u s ing a very s i m i l a r exper imenta l procedure . obtained low breakdown s t r eng th (about 1.2 MV/cm) f o r counter e l ec t rode 2 - • areas l a r g e r than 0.1 mm and t h i s was a t t r i b u t e d to the presence of - f l a w s . A low breakdown s t r eng th may be expected i f th ick- , counter e l ec t rodes were used and so the measured value i s not a bu lk p roper ty .bu t i s due to weak spots and f l a w s . By making, at l e a s t , one counter e l ec t rode l e s s than 1000 A such that i t evaporates at the breakdown s i t e s before sample d e s t r u c t i o n , - i t was p o s s i b l e to. prepare samples which could w i t h -stand under s l o w l y a p p l i e d v o l t a g e , a. f i e l d s t r eng th approaching - -the format ion f i e l d 5 MV/cm). The r e l a t i v e l y s l i g h t decrease of breakdown s t r eng th w i t h i n c r e a s i n g coun te re lec t rode area may be a t t r i b u t e d to the presence 58 of a higher number of flaws assuming that the number of flaws per u n i t area i s s t a t i s t i c a l l y constant. The d e s t r u c t i o n of these s i t e s may a f f e c t the counterelectrode- : c o n d i t i o n s ' and probably the bulk p r o p e r t i e s also, r e s u l t i n g i n a reduction of the. breakdown strength. ; '. E l e c t r i c breakdown of ^a^O^ f i l m s may occur i n two steps. F i r s t i s the formation of a conducting channel i n the i n s u l a t o r which may be due to some form of e l e c t r o n i c avalanche process, associated with the emission at the cathode and that process i s g r e a t l y a f f e c t e d by the presence of flaws and i m p u r i t i e s . Second i s the discharge of the specimen stored energy through the formed, channel by heating and evaporation of the d i e l e c t r i c s . 59 6. CONCLUSIONS • "A model f o r t h i n i n s u l a t i n g amorphous f i l m s has been discussed and the d.c. conduction c h a r a c t e r i s t i c computed. The numerical r e s u l t s show that space charge tends to develop, thus i n f l u e n c i n g the e l e c t r i c f i e l d d i s t r i b u t i o n u n t i l a balance between the rate of i n j e c t i o n of el e c t r o n s (taken to be Schottky thermionic emission) and the bulk conduction (taken to be Poole-Frenkel e f f e c t ) i s achieved. The computed c h a r a c t e r i s t i c shows the' p o s s i b i l i t y of obtaining a Schottky slope f o r t h i n enough films-and a Poole-Frenkel slope f o r t h i c k enough f i l m s . Generally the computed d.c. conduc-t i o n c h a r a c t e r i s t i c may be d i v i d e d i n t o d i f f e r e n t regions; each one being c h a r a c t e r i s e d by one conduction mechanism (space charge -Schottky emission - Poole-Frenkel emission and t r a n s i t i o n r e g i o n s ) . Transient currents were discussed oh the ba s i s of the proposed model and i t was p o s s i b l e to obtain a " l / t " law. In contrast, to the i o n i c discharge current which increases l i n e a r l y w i t h preapplied volt a g e , the e l e c t r o n i c discharge current was found to be approximately independent of preapplied v o l t a g e . " Ta/Ta20^/Au diodes were f a b r i c a t e d , the parent metal was e i t h e r tantalum s i n g l e c r y s t a l or sputtered [3-tantalum. Ta20^ f i l m s were e i t h e r solution-grown or plasma-grown f i l m s . Space charge inside- the i n s u l a t i n g f i l m was found to have a pronounced e f f e c t on the. e l e c t r i c a l c h a r a c t e r i s t i c s of the prepared diodes. " The release of the charge i n s i d e the f i l m , on short c i r c u i t i n g the sample, gives r i s e to an e x t e r n a l e l e c t r o n i c current which i s added to the i o n i c p o l a r i z a t i o n current. The t o t a l e x t e r n a l current was• found to be ' 60 where (E,d) i s a non l inea r f u n c t i o n of ' E . Such n o n l i n e a r i t y of the e x t e r n a l current w i t h the p r e - a p p l i e d vo l tage may.lead to serious e r r o r s i f low frequency d i e l e c t r i c l o s se s are determined •by s tep response measurement. Over the frequency range 100Hz - 100kHz capaci tance and l o s s tangent were found to decrease s l i g h t l y w i t h i n c r e a s i n g f r e -quency wh i l e the equ iva len t s e r i e s r e s i s t a n c e was found to be approx--1 05 ima te ly p r o p o r t i o n a l to to ' , sugges t ing that p o l a r i z a t i o n processes do not change cons ide rab ly over t h i s frequency range. A d e f i n i t e • increase i n capaci tance w i t h d . c . a p p l i e d vo l tage (Ta -ve) was observed i n so lu t ion -g rown f i l m s prepared on tantalum s i n g l e c r y s t a l . • A l l prepared.diodes w i t h s e l f - h e a l i n g breakdown were found to withstand,under s l o w l y a p p l i e d f i e l d s , f i e l d s t rengths approaching the formation f i e l d (—5 MV/cm). A s l i g h t decrease- of breakdown s t r eng th w i t h i n c r e a s i n g counter e l ec t rode area was observed and • may be a t t r i b u t e d to the presence of h igher number of f laws assuming that the number of flaws per u n i t area i s s t a t i s t i c a l l y cons tant . 61 APPENDIX Ioni c . P o l a r i z a t i o n Current, for' 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 In t h i s appendix, i t i s intended to derive the " l / t " law f o r i o n i c p o l a r i z a t i o n current using the i o n i c r e l a x a t i o n model wi t h r e l a t i v e l y 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 . A model i n which ions make f i e l d a s s i s t e d ' t h e r m a l l y ac-t i v a t e d hops between adjacent s i t e s with r e l a x a t i o n time . gi v e s , = (A-l) where X = exp w/KT (A-2) /"£ o = constant (inverse of jump frequency) w := a c t i v a t i o n :-energy p .= p o l a r i z a t i o n ^ s = s t a t i c s u s c e p t i b i l i t y due to t h i s process. A s i n g l e value of a c t i v a t i o n , energy w gives Debye l o s e s £< = £ + — ( A - 3 ) 0 0 l + o o 2 ^ 2 ( c ! V - £ ^ )w X 1 + CO c where .£',£" are the r e a l and imaginary parts of d i e l e c t r i c p e r m i t t i v i t y £ .£ are the values of p e r m i t t i v i t y at zero and i n f i n i t e S 03 . ^ frequencies r e s p e c t i v e l y . For amorphous f i l m s , the above s i t u a t i o n , of one s i t e w ith s i n g l e a c t i v a t i o n energy w i l l not be v a l i d and we have to 6 2 consider many s i t e s with some d i s t r i b u t i o n of a c t i v a t i o n energy f(w), thus equations A-3 and A-4 become 0 1 + oa X e" = U^-E ) J " ^ w ) " V ( A-6) : o 1 + oj C D e f i n i n g an a c t i v a t i o n energy W q corresponding to r e l a x a t i o n time equal l / f ; . where f i s the frequency of the ap p l i e d f i e l d ' i . e . | = ,e o / .' • (A-7) Thus A-5 becomes C' - % + ( e s ^ } J r (A-8). 0 l + ( 2 j r ) 2 exp 2 _ ^ V KT The denomenator i n the integrand can be approximated to one f o r w,<.wQ and to i n f i n i t y f o r w>w , thus w £ ' = E + (e„ -e,. ) I f(w) dw (A-9) O S i m i l a r l y w-wQ e" = ( e s - e ) J £ l ! i J ! ! ^ f L p Z _ *• (A-10) S « / N 2 2 (W-W J o 1 + ( 2 J C J exp o KT Assuming that f(w) i s f l a t enough D so that i t does not change much i n the region, centered at w , then f(w) may be assumed con-stant equal to f (w ) , thus we get e" = ( e - - e . ) KT f ( w ) § . ( A - l l ) On the removal of step voltage t i o n A - l gives P - P,, o-^-r ( A - 1 2 ) where p o = e yC E = s t a t i c p o l a r i s a t i o n due to t h i s process 63 = (r. -c )E . (A-l3) The displacement current d e n s i t y i s given by J = £f = D ^ F + £ 0 E ) . .(A-14) P dt.. dt . which becomes f o r constant f i e l d s JP = SI <*-*5) Equations A-12 and A-15 give r p J = P s e~ t /'^ (A-16) I n t e g r a t i n g equation.A-16 over the d i s t r i b u t i o n of a c t i v a t i o n energies f(w) J P = P S f f ( " ) e " . t A ! a» ( A . 1 7 ) Using equation A-2 r - t / t J = P Q K T J f(w) d ^ ( A" 1 8) p s Q C <± Assuming that f(w) i s n e a r l y f l a t over the range where the value — t 2 of e / i s more e f f e c t i v e , then f(w')may be evaluated at the peak ' f = t/2 = ~^ e^ w a n (} taken outside the i n t e g r a l , thus . J = VL T f(w) (A-19) p t . • Now the angular frequency may be determined by s e t t i n g w'=wo which gives w =-4tt/t. Combining equations A - l l , A-13, and A-19, one gets, 64 REFERENCES D.R. Lamb, E l e c t . C oriel. Mech. i n Thin I n s u l a t i n g F i l m s , -Methuen Co. L t d . , 1967.' 2. A . K . Jonscher , Thin S o l i d F i l m s , 1, 2 1 3 , 1 9 6 7 - '. 3. D'. Vermllyea, J . A p p l . Phys.,"26, 3 6 6 3 , 1 9 6 5 . 4. D. V e r m i l y e a , T e c h n i c a l Informat ion S e r i e s , G . E . , Report No. 67 - C - 0 7 4 , 1 9 6 7 . 5. C . A . Mead, Phys.' Rev., 128, 2088, 1 9 6 2 . 6. J . F r e n k e l , Tech. 'Phys. U.S.S.R., 5., 6 8 5 , 1 9 3 8 ; Phys. R e v . , 5_4, 617, 1 9 3 8 . .. 7. J . J . O'Dwyer, J . A p p l . P h y s . , 17, 5 9 9 , 1 9 6 6 . 8. H . F r o h l i c h , P r o c . Roy. S o c , A188, 521, 1947. 9. R. Frank and J . Simmons, J . A p p l . P h y s . , 28, 832, 1 9 6 7 . 1 0 . H . F r o h l i c h , Theory of D i e l e c t r i c s , Oxford U n i v e r s i t y P re s s , 1 9 4 9 . 11 . B. S z i g e t i , Trans . Faraday S o c , 45_, 1 5 5 , 1 9 4 9 . 1 2 . C. C h e r k i , R. Coelho and L . M a r i a n l , S o l i d State Comm., 1, 411, 1 9 6 6 . 1 3 . M. Gevers and F . Du Pre', P h i l i p s Research Repor t s , 1, 298, • 4 7 7 , 1 9 4 6 . '14. N . K l e i n , J . E lec t rochem. S o c , 1 1 6 , 9 6 3 , 1 9 6 9 -1 5 . J . J . 0'Dwyer, Theory of Breakdown of S o l i d s , ' O x f o r d U n i v e r s i t y P re s s , 1 9 6 4 . • 1 6 . C . J . D e l l ' O c a , D.L. P u l f r e y and L . Young, Phys ic s of Th in F i l m s , . i n the p r e s s . 1 7 . J . J . 0'Dwyer, J . Phys. Chem. S o l i d s , 28, 1 1 3 7 , 1 9 6 7 -18. A . Rose, Phys. R e v . , 21, 1 5 3 8 , 1 9 5 5 -1 9 . S .D. Conte, Elementary Numerical A n a l y s i s , M c G r a w - H i l l , 1 9 6 5 . 20. J . Lindmayer, J . A p p l . P h y s . , '36, .196, 1965.-'' 21. D . L . P u l f r e y , P . S . Wi lcox and L . Young, J . A p p l . P h y s . , 10, 3891,' 1 9 6 9 - ' . • • 6'.5 22. W.L. Lee, D.L. P u l f r e y and L. Young, Ext. Abstr., New York Electrochem. Soc. Meeting, 1969. 23. L. Young, Anodic Oxide Eilms, Academic Press, London, 1961. 24- N. Sil.cox and L. M a i s s e l , I.B.M. Technical Report 22.061, Components Div., I.B.M., Poughkeepsie, New Yrok. 25- D. Vermilyea, J . Appl. Phys., 27_, 963, 1956. 26. L. Young, Trans. Faraday' S o c , 5J5_, p . . 437 and p. 842, 1959-27. Y. Sasaki, J . P h y s . Chem. S o l i d s , ±1, 177, I960. 28. F. Huber, S o l i d State E l e c t r o n i c s , 5_, 410, 1962. 29. F. Huber, J . Electrochem. S o c , 110, 846, 1963. 30. F. A r g a l l and A.K. Jonscher, Thin S o l i d Films, 2, 185, 1968. 31. D.A. Smith and G-.A. Sh i r n , D a l l a s Electrochem. Soc. Meeting, 1967. ' 32. N. Schwartz and M. Gresh, J . Electrochem. S o c i e t y , 112, 295,. 1965. 33. N.N. Axelrod and N.'. Schwartz, i b i d , 116, 4 6 0 , ' l 9 6 9 . " 34. M.E. B a i r d , Rev. Mod. Phys., 40 , . 219 , 1968. 35. B, Hamon, Proc. I.E.E., 22., P- 151, 1952. • 

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