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Space charge and high field effects in thin amorphous films Shousha, Abdel Halim Mahmoud 1971

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SPACE CHARGE AND HIGH FIELD EFFECTS IN THIN AMORPHOUS FILMS by ABDEL HALIM MAHMOUD SHOUSHA B . S c , Cairo U n i v e r s i t y , 1965 M.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard Research Supervisor f  Members of the Committee Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA ..August, 1971 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 o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 reference and study. I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or 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 ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of ^ / e C ^ £y> The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The present t h e s i s i s concerned mainly w i t h space charge and h i g h f i e l d e f f e c t s on the e l e c t r i c a l p r o p e r t i e s of t h i n amorphous f i l m s . A theory of space charge c o n t r i b u t i o n to the p o l a r i z a t i o n current i n t h i n d i e l e c t r i c f i l m s i s proposed. The t r a n s i e n t current on s h o r t - c i r c u i t i n g a t h i n d i e l e c t r i c f i l m i s b e l i e v e d to c o n s i s t of two components, one due to the d i e l e c t r i c p o l a r i z a t i o n and the other due to trapped space charge. The space charge c o n t r i b u t i o n i s i n v e s t i g a t e d u s i n g a model f o r a f i l m c o n t a i n i n g d i s -t r i b u t e d t r a p s . Computed r e s u l t s seem to be c o n s i s t e n t w i t h experimental r e s u l t s on Ta/Ta^O^/Au diodes, so that space charge e f f e c t s are more important at low p r e a p p l i e d f i e l d s . The a p p l i c a b i l i t y of step response techniques to determine low frequency d i e l e c t r i c l o s s e s i s discussed and the e f f e c t of space charge on the d i e l e c t r i c l o s s e s i s analysed. The theory of thermoluminescence and t h e r m a l l y s t i m u l a t e d c u r r e n t s i s extended to the case of traps w i t h d i s t r i b u t e d b i n d i n g energies to i n v e s t i g a t e the p o s s i b i l i t y of d i s t i n g u i s h i n g between d i s t r i b u t e d and d i s c r e t e trap l e v e l s . I t seems p o s s i b l e to d i s t i n g u i s h e x p e r i m e n t a l l y between d i s t r i b u t e d and d i s c r e t e traps by using d i f f e r e n t doses of o p t i c a l r a d i a t i o n to o b t a i n i n i t i a l l y d i f f e r e n t amounts of trapped charges, and by v a r y i n g the frequency of o p t i c a l e x c i t a t i o n over a s u i t a b l e frequency range to a l l o w only c e r t a i n energy l e v e l s to be occupied by e x c i t e d e l e c t r o n s . High f i e l d e l e c t r o n i c conduction through very t h i n f i l m s sandwiched between two metal e l e c t r o d e s i s analysed. I n view of the f a s t t u n n e l i n g time of e l e c t r o n s through very t h i n f i l m s , MIM s t r u c t u r e s can be used f o r microwave d e t e c t i o n . I t i s shown that the maximum responsivity-bandwidth product of such detectors i s obtained when they are b i a s e d at a v o l t a g e equal to the anode work f u n c t i o n ( i n v o l t s ) , and that the presence of i n v a r i a n t p o s i t i v e space charge i i i n c r e a s e s the magnitude of t h i s maximum. In c o n s i d e r i n g h i g h f i e l d s w i t c h i n g i n t h i n f i l m s of semiconducting g l a s s e s , i t i s suggested that J o u l e h e a t i n g , which could account f o r the delay times observed e x p e r i m e n t a l l y , serves only to i n i t i a t e an e l e c t r o n i c s w i t c h i n g mechanism. A model f o r c u r r e n t - c o n t r o l l e d negative r e s i s t a n c e due to space charge formation i s proposed and i t s dc c h a r a c t e r i s t i c s are computed. C a r r i e r i n j e c t i o n from the e l e c t r o d e s i s taken to occur e i t h e r by Schottky thermionic emission or a Fowler-Nordheim t u n n e l i n g mechanism. The i n j e c t e d c a r r i e r s develop space charge regions near the e l e c t r o d e s by impact i o n i z a t i o n . The p o s i t i o n dependent generation-recombination r a t e i s disc u s s e d . The s m a l l ac s i g n a l e q u i v a l e n t c i r c u i t of the model i s given. The formation of current f i l a m e n t s i s analysed. Memory devices are. discussed i n terms of f i l a m e n t formation and phase change mechanisms due to excessive h e a t i n g . Filamentary breakdown has been observed i n anodic f i l m s grown on Ta, A l , Nb and T i . A d e t a i l e d experimental study of f i l m growth and the e f f e c t s of growth c o n d i t i o n s , f i l m t h i c k n e s s , counterelectrodes and temperature on breakdown s t r e n g t h has been c a r r i e d out. A p o s s i b l e mode of breakdown, i n which breakdown can r e s u l t from thermal e f f e c t s f o l l o w i n g a n o n - d e s t r u c t i v e e l e c t r o n avalanche, i s proposed and i t s l i m i t a t i o n s are pointed out. I t i s concluded t h a t breakdown i n t h i n anodic f i l m s would occur due to d i s r u p t i o n of the chemical bonds as the a p p l i e d f i e l d approaches the formation f i e l d . The product of the molecular d i s s o c i a t i o n and the presence of e n e r g e t i c e l e c t r o n s could s t a r t an accumulative process which might end w i t h the formation of a h i g h l y conducting channel. The i n j e c t e d e l e c t r o n s , f i e l d d i s t o r t i o n and thermal runaway could a s s i s t i n the channel development. Once the channel i s developed, the sample's s t o r e d energy s t a r t s to d i s s i p a t e through the channel. The v o l t a g e c o l l a p s e has been found e x p e r i m e n t a l l y to occur i n a time of l e s s than 200 nanoseconds. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v i i i ACKNOWLEDGEMENT x i i i 1. INTRODUCTION 1 1.1 General 1 1.2 Scope of the Present Thesis 3 2. SPACE CHARGE CONTRIBUTION TO THE POLARIZATION CURRENTS 4 2.1 Introduction 4 2.2 Analysis of External Discharge Current 7 2.3 P o l a r i z a t i o n Current of a D i e l e c t r i c with 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..... 9 2.4 Space Charge P o l a r i z a t i o n Current 12 2.4.1 Model and Basic Equations 12 2.4.2 Normalized Equations I 6 2.4.3 Method of Solution 16 2.4.4 Results and Discussion....... 18 2.5 Experimental Procedures.and Results 25 2.5.1 Tantalum Preparation 25 2.5.2 Anodization of Tantalum 25 2.5.3 Deposition of Counterelectrodes 26 2.5.4 E l e c t r i c a l Measurements 26 2.5.5 Experimental Results.... 27 2.6 The A p p l i c a b i l i t y of Step Response Method 29 2.6.1 Step Response Method , 29 2.6.2 Space Charge E f f e c t s 30 •iv 2.7 Space Charge E f f e c t s on the Small Ac Signal D i e l e c t r i c Losses 32 2.8 Discussion 35 3. TRAP DETECTION: THERMALLY STIMULATED CONDUCTIVITY AND LUMINESENCE..... 3 7 3.1 Introduction 37 3.2 Bas i c Equations 3 8 3.3 A n a l y t i c a l Solutions 40 3.3.1 Monoenergetic Traps 40 3.3.2 Exponential Trap D i s t r i b u t i o n 42 3.3.3 Uniform Trap D i s t r i b u t i o n 42 3.4 Numerical Results and Discussion 43 4. HIGH FIELD EMISSION: TUNNELING AND SCHOTTKY CURRENTS IN VERY THIN . FILMS 4 9 4.1 Introduction 49 A. 2 Formulation 50 4.3 T r a n s i t i o n from Tunneling Mechanism to Schottky Thermionic Emission 53 4.4 Space Charge E f f e c t s on Tunneling Currents 55 4.5 Optimum Bias f o r MIM Detectors 57 4.6 Summary.... 60 5 . HIGH FIELD SWITCHING IN THIN AMORPHOUS FILMS 6 1 5.1 Introduction •••• ^ 5.2 Electrothermal I n i t i a t i o n of the Switching Mechanism 64 5.2.1 General Formulation 6 5 5.2.2 S t a t i c C h a r a c t e r i s t i c s . . . . 6 8 5.2.3 Delay Time 71 IT Page 5.3 S w i t c h i n g Mechanism ... 76 5.4 Current C o n t r o l l e d Negative D i f f e r e n t i a l C o n d u c t i v i t y due to Space Charge B a r r i e r s 78 5.4.1 Model and B a s i c Equations 7 3 5.4.2 Normalized Equations 81 5.4.3 Method of S o l u t i o n 82 5.4.4! Symmetrical Case 83 5.4.5 Asymmetrical Case 87 5.4.6 D i s c u s s i o n 88 5.4.7 Small S i g n a l E q u i v a l e n t C i r c u i t v 89 5.5 Formation of Current Filaments due to R a d i a l D i f f u s i o n . . . 92 5.6 Memory Sta t e Formation , . i 96 5.7 Summary. 97 FILAMENTARY BREAKDOWN IN THIN ANODIC FILMS 99 6.1 I n t r o d u c t i o n •••• 9^ 6.2 E l e c t r i c Breakdown Theories 100 6.3 Breakdown Tests: Experimental Procedure 104 6.3.1 Sample P r e p a r a t i o n 104 6.3.2 E l e c t r i c a l Measurements 106 6.4 Breakdown Tests: Experimental R e s u l t s and D i s c u s s i o n 106 6.4.1 Prebreakdown Conduction 106 6.4.2 Filamentary Breakdown 11° 6.4.2.1 Optical Microscope Observations HO 6.4.2.2 Observation of Voltage C o l l a p s e 112 6.4.2.3 P o s s i b l e P h y s i c a l Mechanisms f o r Channel Formation. 114 vi Page 6.4.3 Breakdown S t r e n g t h o f T a 2 ° 5 F i l m s v 1 1 6 6.4.3.1 T h i c k n e s s Dependence o f Breakdown S t r e n g t h H 6 6.4.3.2 Breakdown Dependence on P u l s e W i d t h H 8 6.4.3.3 Temperature Dependence o f Breakdown S t r e n g t h H 9 6.4.3.4 Breakdown Dependence on t h e F o r m a t i o n C u r r e n t D e n s i t y 1 2 1 6.4.4 Breakdown Dependence on D i e l e c t r i c C o n s t a n t 122 6.4.5 E l e c t r o d e E f f e c t s . . 1 2 2 6.5 On t h e P o s s i b i l i t y o f Th e r m a l Breakdown. 1 2 4 6.5.1 B a s i c E q u a t i o n s 125 6.5.2 R e s u l t s and D i s c u s s i o n 127 6.6 Breakdown Mechanism 132 6.6.1 S t a g e I : F o r m a t i o n o f a C o n d u c t i n g C h a n n e l . . . . . . 133 6.6.2 S t a g e I I : D i s c h a r g e o f Sample's S t o r e d E n e r g y I 3 4 7. CONCLUSIONS 1 3 7 APPENDIX 1. CALCULATION OF RELAXATION TIME SPECTRA IN AMORPHOUS FILMS 1 4 1 APPENDIX 2. GROWTH AND IONIC CONDUCTIVITY OF T a ^ FILMS i 4 3 A2.1 T h e o r i e s o f I o n i c C o n d u c t i o n i 4 3 A2.2 Sample P r e p a r a t i o n I 4 4 A2.3 Measurements I 4 4 A2.4 S m a l l S i g n a l F i l m Impedance ;. 1 4 6 A2.5 E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n I 4 7 BIBLIOGRAPHY 1 5 3 v i i LIST OF ILLUSTRATIONS Figure Page 1.1 Density of states i n a n o n c r y s t a l l i n e semiconductor, the l o c a l i z e d states are shown shaded. In (a) there i s an energy gap; i n (b) the energy gap disappears 2 2.1 Metal/amorphous film/metal structure during discharge. (a) Energy band diagram; (b) 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 . S o l i d l i n e s , negative space charge; broken l i n e s , p o s i t i v e space charge 7 2.2 Space charge p o l a r i z a t i o n currents for exponential s p a t i a l d i s -t r i b u t i o n of trapped charges 19 2.3 The e f f e c t of capture cross s e c t i o n on space charge p o l a r i z a t i o n current 19 2.4 Decay of trapped charges i n s i d e a f i l m having constants of F i g . 2.2. S o l i d l i n e s , C = 0.5; dotted l i n e s , C = 0.2 20 2.5 Time dependence of trapped charges. S o l i d l i n e s , computed f o r a f i l m having constants of F i g . 2.2 with C = 0.2; broken l i n e s computed f o r a f i l m having constants of F i g . 2.3 with a =0.01. 20 n 2.6 V a r i a t i o n of trapped charge density and f i e l d with p o s i t i o n i n a f i l m having constants of F i g . 2.3 with a =0.01 22 n 2.7 Time dependence of x* and n t ( x * , t ) for a f i l m having constants of F i g . 2.2.. 22 2.8 The e f f e c t of the presence of i n v a r i a n t p o s i t i v e space charge, (a) Space charge p o l a r i z a t i o n current; (b) v a r i a t i o n of trapped charge density and f i e l d with distance i n s i d e the f i l m 24 2.9 The e f f e c t of metal work function on space charge p o l a r i z a t i o n current 24 i H ^ -i Figure Page 2.10 . Discharge currents versus time as a function of preapplied voltage. (Ta-positive) 28 2.11 Discharge currents 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) '28 2.12. Low frequency d i e l e c t r i c losses. (a) The e f f e c t of preapplied f i e l d on the calculated d i e l e c t r i c lenses using step response measurements, (b) comparison of bridge and step response r e s u l t s 31 2.13 Loss f a c t o r dependence on dc bias 35 3.1 Glow curves for d i f f e r e n t trap d i s t r i b u t i o n s , (a) monoenergetic traps; (b) exponential trap d i s t r i b u t i o n 44 3.2 E f f e c t of the i n i t i a l amount of trapped charge density on the computed glow curves, (a) monoenergetic traps; (b) exponentially d i s t r i b u t e d traps 46 3.3 E f f e c t of retrapping on the computed glow curves, (a) mono-energetic traps; (b) exponentially d i s t r i b u t e d traps 47 4.1 T r a n s i t i o n from tunneling mechanism to Schottky termionic emission 54 4.2 Voltage-current c h a r a c t e r i s t i c s of a MIM tunneling diode, the presence of p o s i t i v e space charge increases the current density... 56 4.3 P o t e n t i a l b a r r i e r lowering as a function of the applied f i e l d 56 4.4 V a r i a t i o n of the detector responsivity-bandwidth product as a function of the bias voltage 59 5.1 Types of CCNR c h a r a c t e r i s t i c s " 61 5.2 Discrete-space continuous-time analogue representation of e l e c t r i c a l heating of a t h i n f i l m ( e l e c t r i c a l equivalent c i r c u i t for Eq. (5.5)) 6 ? i x Figure Page 5.3 Computed voltage-current c h a r a c t e r i s t i c s f o r a t h i n c y l i n d r i c a l s t r u c t u r e 67 5.4 Current density and temperature d i s t r i b u t i o n s w i t h i n the semi-conducting f i l m . The given d i s t r i b u t i o n s are those associated with points 1, 2, 3 shown i n F i g . 5.3 70 5.5 Exact and approximate solutions f o r threshold f i e l d 70 5.6 Computed delay time, c r i t i c a l temperature r i s e and e l e c t r i c energy needed f o r switching to occur versus applied voltage.... 73 5.7 Exact, , and approximate, — , delay time as a function of applied voltage f o r d i f f e r e n t ambient temperatures. The broken l i n e s show the corresponding c r i t i c a l temperature r i s e . 73 5.8 The e f f e c t of external r e s i s t a n c e on delay time 75 5.9 Ac switching c h a r a c t e r i s t i c s : threshold f i e l d , delay time and e l e c t r i c energy needed f o r switching to occur. Applied voltage = V s i n 2 u f t 75 o 5.10 Computed voltage-current c h a r a c t e r i s t i c s : i n j e c t e d c a r r i e r s are taken to be given by Schottky thermionic emission; C2-O.I, a=5, m=8 and E = 50 85 o 5.11 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 w i t h i n a f i l m having constants of F i g . 5.10 with a = 0.5 85 5.12 Computed voltage-current c h a r a c t e r i s t i c s : i n j e c t e d c a r r i e r s are taken to be given by the Fowler-Nordheim tunneling mechanism; H= 400, a = 0.2, m = 8 and E =50 86 o 5.13 Computed voltage-current c h a c t e r i s t i c s : i n j e c t e d electrons are taken to be given by Schottky thernuaiic- emission; C2=0.1, k - 0.4, x Figure Page 3 = 0.125 and E = 50 86 o •5.14 Small ac s i g n a l equivalent c i r c u i t 91 5.15 S p a t i a l v a r i a t i o n of current density through the current filament 96 6.1 Schottky plots f o r T a ^ fil m s . . . . . . . . 108 6.2 Temperature dependence of conduction current: applied voltage = 25V; f i l m thickness = 1820 & 108 6.3 Energy band diagram for a m e t a l / d i e l e c t r i c / m e t a l showing the e f f e c t of a p o s i t i v e i on at a depth of a. The d i f f e r e n t b a r r i e r shapes are for d i f f e r e n t impact radius r (Ref. 81) 109 6.4 Current-voltage c h a r a c t e r i s t i c s of a Ta^O^ f i l m using an applied ramp voltage ( f i l m thickness = 2050 X) 109 6.5 T y p i c a l s i n g l e channel breakdowns (2.5um/div) H I 6.6 Extended breakdown patterns (10 um/div) • .. H i 6.7 Voltage waveform across a sample. The number of breakdown events increase as the magnitude of the applied pulse increases. H o r i -—5 zontal scale 10 sec/div and v e r t i c a l Scale 15 V/div.. H 3 6.8 . Voltage collapse on s i n g l e breakdown event. Hor i z o n t a l scale -7 2 x 10~ sec/div and v e r t i c a l s c a le 15 V/div H 3 6.9 Channel development due to a l o c a l i z e d defect . H 5 6.10 Thickness dependence of breakdown strength 117 6.11 Breakdown dependence on pulse width 117 6.12 Temperature dependence of breakdown strength ( f i l m thickness = 2050 A) 120 6.13 Breakdown dependence on the formation current density 120 x i Figure Page 6.14 Breakdown dependence on d i e l e c t r i c constant and melting point ( d i e l e c t r i c s are A l ^ , T a ^ , N b ^ and T i O p 123 6.15 Electrode e f f e c t on breakdown strength. (Metals used are Au, A l and In) 123 6.16 Thermal runaway: channel current and maximum temperature v a r i a -7 —2 t i o n versus times. (Constants used are x/d= 10 , a = 10 , pc = 4 x 10" 6, ty - 0.6 eV, B/k = 0.2 eV, K=0.5, r £ = 20A, 6 =5A. Unless otherwise stated, M.K.S. units are used) 128 6.17 V a r i a t i o n of delay time to breakdown with the i n i t i a l temperature r i s e for d i f f e r e n t power inputs 131 6.18 V a r i a t i o n of c r i t i c a l f i e l d with the i n i t i a l channel temperature r i s e f o r d i f f e r e n t power losses 131 6.19 Computed voltage collapse' and channel res i s t a n c e during the discharge. 136 A2.1 C i r c u i t employing lock i n am p l i f i e r s to measure the f i l m impedance during f i l m growth 145 A2.2 P a r a l l e l and s e r i e s equivalent c i r c u i t s 146 A2.3 Dependence of f i l m resistance and capacitance on the charge passed during anodization 148 A2.4 kog^Q J-E c h a r a c t e r i s t i c s of grown films ( t a f e l p l o t ) 148 A2.5 Film resistance dependence on the forming current density 150 A2.6 Frequency dependence of i o n i c conductivity 150 x i i ACKNOWLEDGEMENT I am most g r a t e f u l to Dr. L. Young f o r h i s i n s p i r i n g supervision and invaluable guidance during the course of t h i s work. Grate f u l acknowledgement f o r f i n a n c i a l support i s given to the National Riie&rch Council (operating grant # A3392), the B r i t i s h Columbia Telephone Company (fellowship awarded 1969-1970) and the U n i v e r s i t y of B r i t i s h Columbia (fellowship awarded 1970-1971). I am indebted to Dr. D. L. P u l f r e y for many useful discussions. H e l p f u l eeeptration and assistance from my fellow graduate students i n the S o l i d S t a t t E l e c t r o n i c s Group i s g r a t e f u l l y acknowledged. I thank Messrs. H. Black and A. MacKenzie f o r t h e i r valuable t e c h n i c a l assistance and Miss L. Morris f o r typing the t h e s i s . 1 1. . INTRODUCTION 1.1 General Due to t h e i r importance i n monolithic, hybrid and t h i n f i l m integrated c i r c u i t s , t h i n amorphous films have been a subject of extensive experimental 1-3 and t h e o r e t i c a l studies which have l e d to an understanding of t h e i r properties and much confidence i n t h e i r use. Their applications i n MOS f i e l d e f f e c t t r a n s i s t o r s , t h i n f i l m t r a n s i s t o r s , switching d e v i c e s , e l e c t r o l y t i c and t h i n f i l m capacitors and R-C d i s t r i b u t e d c i r c u i t s are of i n c r e a s i n g importance. A p a r t i c u l a r a p p l i c a t i o n may pose stringent requirements on the f i l m p r o p e r t i e s , and consequently i t may be required to optimize many of the f i l m p r o p e r t i e s . The study of the properties and a b e t t e r understanding of the p h y s i c a l processes involved are v i t a l to the improvement of device c h a r a c t e r i s t i c s and performance. The performance of a f i l m i s greatly affected by i t s s t r u c t u r a l p r o p e r t i e s which i n turn, depend on the material and the methods used f o r i t s preparation. Experimental analyses of n o n c r y s t a l l i n e films show c l e a r l y the existence of a short range order and equally c l e a r the absence of a long range order i n the atom d i s t r i b u t i o n . However, these analyses cannot d i s t i n g u i s h between the two e s s e n t i a l s t r u c t u r a l models for amorphous 4 5 materials ' . The continuous model assumes that the i n d i v i d u a l bonds are s l i g h t l y disordered both i n d i r e c t i o n and i n extension, but that the cumulative e f f e c t of t h i s disorder i s to produce a complete smearing out of the long range order. This model represents an i d e a l i z e d amorphous continuum which i s s t a t i s t i c a l l y homogeneojus.everywhere. The other model considers the amorphous s o l i d s as a discontinuous arrangement of c r y s t a l l i t e s separated by intermediate regions which accommodate the m i s f i t s between the c r y s t a l l i t e s . 2 These two s t r u c t u r a l models are r e a l l y only l i m i t i n g cases among 4 many models which may represent the actual structure. A u n i f i e d model can explain a l l the experimental observations. In t h i s model, the structure i s not assumed to be quite uniform but has regions with more or l e s s order on a scale much greater than that of interatomic spacing. In some cases c r y s t a l l i t e s may be more d e f i n i t e while i n others the structure may be c l o s e r to a homogenous but i r r e g u l a r network. The t r a n s i t i o n from one extreme case to the other i s gradual and i t depends on the material used and preparation techniques. T h e o r e t i c a l p r e d i c t i o n s of the energy band diagram and c h a r a c t e r i s t i c s of amorphous materials d i f f e r according to the p a r t i c u l a r model used. However, since the basic features of the energy band diagram are determined p r i m a r i l y by the short range order, or more p r e c i s e l y by the actual bonds between atoms, the general features of the energy band diagram are preserved i n the t r a n s i t i o n from the c r y s t a l l i t e to amorphous stat e s . The disappearance of long range order leads to a new concept, that of l o c a l i z e d s t a t e s . An e l e c t r o n i n a l o c a l i z e d state can be described as trapped. This means that the absence of long range order produces a high density of traps. N(6) N(8) F i g . 1.1 Density of states i n a n o n c r y s t a l l i n e . - semiconductor, the l o c a l i z e d states are shown shaded. In (a) there i s an energy gap; i n (b) the engrgy gap disappears The energy band gap of amorphous materials e i t h e r p e r s i s t s or i t may be replaced by a minimum i n the allowable density of states (pseudogap) . Fo energies near the extremities of the free bands or i n the pseuodgap, the wave functions become l o c a l i z e d and the m o b i l i t y i s much l e s s than that i n the free bands of a c r y s t a l , t y p i c a l l y by. some order of magnitude. It should be stated that there are s t i l l many unsolved problems i n the quantum theory of amorphous materials (some have recently been pointed out by Mott'7) and that there are many t h e o r e t i c a l disagreements between d i f f e r e n t authors at the present time^. 1.2 Scope of the Present Thesis This thesis i s concerned mainly with space charge and high f i e l d e f f e c t s i n t h i n amorphous f i l m s . In Chapter 2, a theory of space charge co n t r i b u t i o n to p o l a r i z a t i o n currents i s given, and the e f f e c t of the presence of space charge on d i e l e c t r i c losses i s discussed. To i n v e s t i g a t e the p o s s i b i l i t y of d i s t i n g u i s h i n g between d i s t r i b u t e d and d i s c r e t e trap l e v e l s , the theory of thermoluminescence and thermally stimulated currents i s extended to , the case of traps with d i s t r i b u t e d binding energies (Chapter 3). In Chapter A, high f i e l d emission and MIM Tunneling diodes as wave detectors are analysed. Chapter 5 deals with high f i e l d switching i n t h i n amorphous f i l m s . The delay times are analysed i n terms of sample s e l f - h e a t i n g while the switching mechanism i s considered e l e c t r o n i c i n nature. A model f or cu r r e n t - c o n t r o l l e d negative resistance due to space charge formation i s developed. The formation of current filaments and memory devices are discussed. In Chapter 6, d e t a i l e d experimental and t h e o r e t i c a l studies on breakdown i n anodic oxide f i l m s are presented and a pos s i b l e mechanism f o r breakdown i s proposed. The main r e s u l t s and conclusions to be drawn from the present thesis are given i n Chapter 7. 2. SPACE CHARGE CONTRIBUTION TO THE POLARIZATION CURRENTS 2.1 Introduction 8 9 In considering the 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 , 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 d i s p l a c e -ments* are not e s s e n t i a l l y changed from those i n regular c r y s t a l s , except that i n amorphous films there w i l l be a spread i n the c h a r a c t e r i s t i c frequencies so that a broader absorption band w i l l be observable i n the u l t r a v i o l e t and i n f r a r e d range. Below the i n f r a r e d range, the p o l a r i z a t i o n processes are r e l a x a t i o n type processes which involve the movement of ions and electrons between energy w e l l s * * . In accordance with the nature of amorphous f i l m s , the associated a c t i v a t i o n energies w i l l d i f f e r f o r d i f f e r e n t ions. Even f o r a p a r t i c u l a r i on or electron the force f i e l d i n which i t t r a v e l s w i l l be rather i r r e g u l a r and accordingly the p o t e n t i a l b a r r i e r height w i l l vary from one point to another. Thus, one would expect a spread i n the c h a r a c t e r i s t i c Since e l e c t r o n i c and,ionic p o l a r i z a t i o n cannot be completely separated because i o n i c displacement induces e l e c t r o n i c displacement, i t seems b e t t e r to characterize 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 d i s -placement) or i n f r a r e d (both e l e c t r o n i c and i o n i c displacement contribute) ** The further mechanism of o r i e n t a t i o n of molecules with permanent dipoles i s u n l i k e l y to a r i s e i n amorphous films since i t i s improbable that molecular groups having permanent dipole moments could be incorporated i n the f i l m i n such a way as to rotate with the f i e l d , though the motion of molecular chains may occur as postulated f o r glasses^-. 12 frequencies. Another complication could a r i s e from space charge formation (e.g., trap charging and discharging) which i s a time dependent process and contributes to the p o l a r i z a t i o n currents. The magnitude and character of the space charge p o l a r i z a t i o n current i s determined by the presence of ele c t r o n traps. In amorphous fi l m s , the trap density i s expected to be high and to strongly a f f e c t the measured p o l a r i z a t i o n current. 13 14 D i e l e c t r i c measurements on several amorphous d i e l e c t r i c s ' (e.g., SiC^, Ta20^, Al^O^) have shown that the d i e l e c t r i c constant, e', and the loss factor, tan 6, at. room temperature are r e l a t i v e l y independent of frequency i n the audio range. Several models have been proposed to explain the near constancy of e' and tan 6. Young1"' has used the Maxwell l a y e r model i n which there was an exponential dependence of the conductivity on distance i n t o the d i e l e c t r i c f i l m , but i t i s d i f f i c u l t to account f o r the required dependence of parameters on f i l m thickness. It seems more l i k e l y that f i l m d i e l e c t r i c properties are re l a t e d to ion or ele c t r o n hopping processes between energy wells. A r g a l l and Jonscher 1^ have discussed e l e c t r o n hopping i n r e l a t i o n to SiO f i l m s . M aserjian 1^ has analysed e l e c t r o n trap to conduction 18 band t r a n s i t i o n s i n connection with Itt^O,. anodic f i l m s . Gevers and DuPre have proposed a model, for a d i e l e c t r i c with 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 i n which ions make f i e l d - a s s i s t e d thermally-activated jumps between contiguous equilibrium s i t e s separated by p o t e n t i a l b a r r i e r s , which has been shown to adequately describe the frequency and temperature dependences of e' and tan 6 of many amorphous d i e l e c t r i c s . The step response can be r e l a t e d to the ac response by the Fouri e r transform, i f i i n e a r d i e l e c t r i c theory applies. Transient measurements have 6 19 been used by Hamon to determine the d i e l e c t r i c losses at very low frequencies. Transient discharge currents observed experimentally t y p i c a l l y follow a 1/ t U law (e.g., n=l f o r la^O^^ and S i O ^ f i l m s ) . They have been i n t e r p r e t e d as 21 due e n t i r e l y to the step response of a l i n e a r d i e l e c t r i c or to e l e c t r o n i c 20 22 23 space charge ' ' . The p o l a r i z a t i o n current for a l i n e a r d i e l e c t r i c i s l i n e a r l y dependent on the preapplied f i e l d , while space charge e f f e c t s are expected to be nonlinear with the preapplied f i e l d . Thus, i f space charge e f f e c t s are present, the r e l a x a t i o n function <)>(t) ( i . e . , the current I ( t ) as a function of time on removing a unit step voltage) w i l l be a non-linear 14 function of the preapplied f i e l d . Thus as P u l f r e y , Wilcox and Young noted, i t should be p o s s i b l e to experimentally d i s t i n g u i s h between the two p o s s i b l e models ( l i n e a r d i e l e c t r i c model and e l e c t r o n i c space charge model) by using the expected n o n - l i n e a r i t y i n the r e l a x a t i o n function. 24 25 In a previous work ' , the high f i e l d e l e c t r o n i c conduction f o r a model of an amorphous d i e l e c t r i c f i l m containing traps with d i s t r i b u t e d binding energies, has been inv e s t i g a t e d . Such a model seems relevant to t y p i c a l t h i n f i l m d i e l e c t r i c s such as Si02 and Ha.^^ which are used i n microelectronic devices. The i n j e c t i o n of electrons at metal/film i n t e r f a c e was taken to be governed by Schottky thermionic emission. The space charge density was taken as determined by the balance between detrapping due to the Poole-Frenkel e f f e c t and f i e l d independent trapping. I t was noted that the steady state charge d i s t r i b u t i o n of trapped electrons i n t h i s system was the appropriate s t a r t i n g point for an analysis of the e f f e c t s of e l e c t r o n i c space charge on the discharge current obtained by subsequently shorting a f i l m pre-v i o u s l y held f o r sometime at constant applied f i e l d . In t h i s chapter, the contribution of both space charge and d i e l e c t r i c p o l a r i z a t i o n to the external discharge current and t h e i r consequence i n determining the low frequency d i e l e c t r i c losses using step response 13 2 6 ' measurements ' are inv e s t i g a t e d . Space charge e f f e c t s on the small ac s i g n a l d i e l e c t r i c losses are also analysed. 2.2 Analysis of External Discharge Current On shorting a me t a l / d i e l e c t r i c / m e t a l structure previously held f o r some time at constant e l e c t r i c f i e l d , the e l e c t r i c f i e l d E(x,t) i s taken as changing from E(x,-0) to E(x,+0) instantaneously ( r e l a t i v e to the long d i s -charge time) r e s u l t i n g i n a sudden change i n surface charge from Q(0,-0) to Q(0,+0). This sudden change w i l l cause a current impulse A<5(t) i n the external c i r c u i t , i . e . , (2.1) where J e x t ( t ) = A6(t) + J ( t ) J ( t ) i s the non impulsive part of J e x t ( t ) a n ( ^ <5(t)' i s the dirac d e l t a function The continuous discharge current density, J ( t ) , i s due to some long r e l a x a t i o n processes which are d i e l e c t r i c , p o l a r i z a t i o n processes and processes associated with the release of trapped space charge. I (a) Fig. 2.1 Metal/amorphous film/metal structure during discharge, (a) Energy band diagram; (b) 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 . S o l i d l i n e s , negative space charge; broken l i n e s , p o s i t i v e space charge. 8 Fi g s . 2.1-a and 2.1-b show an energy band diagram and 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 f o r a f i l m containing space charge with shorted d i s s i m i l a r electrodes. Due to the presence of space charge we may f i n d a point, x*, where the f i e l d i n s i d e the f i l m i s zero. In t h i s case, using Gauss' theorem, the time rate of change of the free charge per unit area on the cathode, assuming that the trapped charge density i s much greater than the free charge density, i s given by dQ = d D(0,t) dt d t = •§£-[ /* e[N Q(x) - n t ( x , t ) ] d x + P(x*,t)] x And by applying L e i b n i t z ' s rule f o r the d i f f e r e n t i a t i o n of an i n t e g r a l , we obtain f = e [ n t ( x * , t ) - V x * ) ] ^ _ + e / 0 dx (2.2) , d P(x*,t) dt where D(0,t) i s the e l e c t r i c displacement at the cathode, n^(x,t) i s the trapped space charge density, N^(x) i s the i n v a r i a n t p o s i t i v e space charge i n s i d e the f i l m , P(x,t) i s the d i e l e c t r i c p o l a r i z a t i o n and e i s the magnitude of the e l e c t r o n i c charge. The conduction current density i n s i d e the f i l m at the m e t a l / d i e l e c t r i c i n t e r f a c e can be determined by using the current c o n t i n u i t y equation x * 3 J i n ( x , t ) x* a(n t(x,t)-N o(x))_ X ^ dx = e — dx 0 9x 0 3t Assuming that J \ n ( x * , t ) i s n e g l i g i b l y small (since the e l e c t r i c f i e l d at x* equals zero), the above r e l a t i o n reduces to 9 x* 8 n t ( x , t ) J L N ( 0 . t ) = - e / 0 dx (2.3) u s i n g Eqs. (2.2) and (2.3) and the c o n t i n u i t y equation « « > - f + J l n « ' t > the e x t e r n a l discharge current J ( t ) can be given by J ( t ) = e [ n t ( x * , t ) - N <x*>] ^  + ( 2. 4) Thus one can w r i t e J ( t ) = J g c ( t ) + J p ( t ) (2.5) where J c i s the space charge c o n t r i b u t i o n to the e x t e r n a l current d e n s i t y and Jp i s the c o n t r i b u t i o n of d i e l e c t r i c p o l a r i z a t i o n processes. Eq. (2.4) has two l i m i t i n g cases: f i r s t , when space charge d e n s i t y i s n e g l i g i b l y s m a l l , Eq. (2.4) reduces to J ( t ) = J p ( t ) = (2.6) And f o r amorphous f i l m s , one can de r i v e an expression f o r Jp usin g a d i e l e c t r i c p o l a r i z a t i o n model w i t h 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 (Sec. 2.3) Secondly, when space charge predominates, Eq. (2.4) becomes J ( t ) = J ( t ) = e[n ( x * , t ) - N (x*)]. ~ ( 2.7) sc t o dt This case w i l l be considered i n some d e t a i l i n Sec. 2.4. 2.3 P o l a r i z a t i o n Current of a D i e l e c t r i c w i t h 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 The d i e l e c t r i c p o l a r i z a t i o n response to a step f i e l d f o r a model In 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 - a c t i v a t e d hops between adjacent g e q u i l i b r i u m s i t e s separated by a p o t e n t i a l b a r r i e r of he i g h t q i s given by £ = i (P - P) (2.8) dt T 5 where T x = -f exp(q/kT) ' (2.9) i s a constant equal to the inverse of the jump frequency P i s the s t a t i c p o l a r i z a t i o n s . k i s Boltzmann's constant T i s the temperature When only one process with a c t i v a t i o n energy q i s operative, and a s i n u s o i d a l f i e l d represented by E = e exp (jut) i s applied, the d i e l e c t r i c p e r m i t t i v i t y , e(w), can be obtained by s u b s t i t u t i n g P(u) = (E(U) - e j e exp (jut) T, / * ~ ^, % (2.10) P s = ( e s ~ e r o)e exp ( ju)t) i n Eq. (2.8), where £ g and are the values of the d i e l e c t r i c p e r m i t t i v i t y at zero and i n f i n i t e frequencies r e s p e c t i v e l y . Thus, we get the Debye equation f o r the r e a l , e ' ( c o ) , and the imaginary, E " ( U ) , components of the d i e l e c t r i c p e r m i t t i v i t y : E £ e'(w) = £„ + 5 I / (2.11) 1+U T e e *e"(a>) = 5 ~ (2.12) 1+0) T For amorphous f i l m s , the above s i t u a t i o n of one s i t e with 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 consider many s i t e s with s u i t a b l e d i s t r i b u t i o n of a c t i v a t i o n energies G(q) where 1^ G(q)dq = 1 (2.13) Introducing a c e r t a i n a c t i v a t i o n energy q^, being the value of q corresponding to the case OJT = 1, i . e . , 2 w = ~ exp(-q o/kT) Eqs. (2.11) and (2.12) become 00 e'((o) = e + (e - e )/„ G ( ^ , c^prr dq (2.14) » s » 0 l+exp(2(q-q o)/kT) n ^ G(q)exp(2(q-q o)/kT) e"(a)) = (e - e )/„ -rv 7-^ 7 7 7 — — dq (2.15) s °°' 0 l+exp(2(q-q )/kT) The denominator i n the integrand of Eq. (2.14) can be approximated to one f o r q£q o and to i n f i n i t y f o r q?<l0- Thus, Eq. (2.14) reduces to q e'Oo) = e + (e - e )/° G(q)dq (2.16) 00 s U 18 Gevers and Dupre have showed that i f a n e a r 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 G(q) i s present i n the d i e l e c t r i c then Eq. (2.15) reduces to e"(u) = ( e s - e j f kT G(q Q) (2.17) The p o l a r i z a t i o n current i s given by J p = f (2.18) Eqs. (2.8) and (2.18) f o r a s i n g l e p o l a r i z a t i o n process, give P . J p = -f- e " t / T (2.19) For a d i s t r i b u t i o n of a c t i v a t i o n e nergies, one could w r i t e t / t — ^ *i G(q)P e Using Eq. (2.9) we get j , P k T r G(q)e~ f c /1 d T ( 2 < 2 0 ) P • s 0 2 T Since G(q) i s assumed n e a r l y f l a t , i t may be evaluated at the peak of -tlx 2 e IT , and taken o u t s i d e the i n t e g r a l . Eq. (2.20) can be e a s i l y i n t e g r a t e d to give 12 J p = k T ) P (2.21) *r t S where q' = kT l n ( t / t ) (2.22) o Thus, on removing a p o l a r i z a t i o n f i e l d , i . e . , E = E Q (1 - u ( t ) ) , the p o l a r i z a t i o n J p = 1 . 8 * E Q u ( t ) ( 2 ' 2 3 ) current i s given by k T . G ( q ' . ) ( £ s - e J t Combining Eqs. (2.17) and (2.23) gives T - 1 e"(o>) G(q') , . ' J P " TT ~ G(^) E 0 U ( t ) ( 2 ' 2 4 ) Again, making use of the assumed 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, Eq. (2.24) reduces to j =1 £ u ( t ) ( 2 . 2 5 ) r TT t o Thus, the p o l a r i z a t i o n current of the above model follows a law with a l i n e a r dependence on the preapplied f i e l d . 2.4 Space Charge P o l a r i z a t i o n Current 2.4.1 Model and Basic Equations A. m e t a l / d i e l e c t r i c film/metal sandwich may be represented by a one-dimensional model i n which c a r r i e r s of one type only, assumed here to be e l e c t r o n s , are considered. In view of the amorphous nature of the f i l m , the trap concentration might be expected to be high i n an energy range W below the conduction band edge. The trap density/unit energy i s assumed to be N W t = A texp(-e t/NkT) (2.26) where E i s the trap energy measured downwards from the conduction band, and AFC and N are constants which define the shape of the d i s t r i b u t i o n of traps with respect to energy. In the ordinary d e r i v a t i o n of the Poole-Frenkel equation, traps which e x h i b i t a coulombic a t t r a c t i o n with respect to current c a r r i e r s 27 (electrons i n t h i s case) are necessary . Thus traps 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 . , they are donors). I t seems cl e a r that even with the use of the high frequency per-m i t t i v i t y the Poole-Frenkel law (and the Schottky law) are based on no more than s e m i c l a s s i c a l treatments and are not to be expected to be accurate. The idea of a simple coulombic force between electrons and e i t h e r donor traps or metal i s not more than a simple convenient approximation. F i e l d enhanced release from acceptor l e v e l s (defined as l e v e l s which are n e u t r a l when emptied of an electron) may conveniently and probably j u s t as accurately be described by the same Poole-Frenkel equation. The actual mixture of donor and acceptor traps i s represented by introducing a v a r i a b l e parameter N^, representing the space charge which would e x i s t i f a l l traps were donors and were f u l l y i o n i z e d . By varying the range from a l l donors to a l l acceptors may be considered, with a corresponding v a r i a t i o n i n the concentration of electrons i n the conduction band, and hence the conductivity, i n e l e c t r i c a l l y n e u t r a l d i e l e c t r i c . 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 i n the f i l m i s governed by Poisson's equation ||= - f [ N t ( l - F ( x , t ) ) - N A - n c ( x , t ) ] (2.27) 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 axis. F designates the occupation of the whole set of traps n i s the free c a r r i e r density, c J Let n (x) be the f i l l e d trap density at p o s i t i o n x and N = N -N , 14 thus Eq. (2.27) can be s i m p l i f i e d to | | - f [ n t ( x , t ) + n c ( x , t ) - N Q ( X ) ] (2.28) During the t r a n s i e n t , the rate of change of charge density i s r e l a t e d by the c o n t i n u i t y equation 9(n +n ) 8n a_ 8 2n 3t ~8x y n c 9x ~~2 ( 2 ' 2 9 ) OX where u i s the c a r r i e r m o b i l i t y D i s the d i f f u s i o n constant. The rate of change of trapped charge can be obtained from the dif f e r e n c e between the rate of electron capture from the conduction band (r^) and the rate of e l e c t r o n release from traps ( r ^ ) • Considering an i n f i n i t e s i m a l range of trap energies between e t and e + de^, the expression f o r r ^ and can be written as r l = n c \ t ( 1 - f D > ° V t h d £ t ( 2 ' 3 0 ) where a i s the capture cross s e c t i o n . V\, i s the e l e c t r o n thermal v e l o c i t y th fp i s the occupancy factor of a trap such that P(x) -/ Wfr.N T^de 0 D_Wt t W X N„. de 0 Wt t The rate depends upon the concentration of centres which are occupied by electrons and on the e l e c t r o n r e l a x a t i o n time T. In the stated model x i s given by an expression of the form T = ^ exp((e t - Ae t)/kT) ( 2 . 3 1 ) where v i s the attempt to escape frequency Ac f c represents the lowering of the trap b a r r i e r height assuming a Poole-Frenkel mechanism. 15 1/2 ' Ae t = 6 p FkTE and . 8 = — f^-PPF kT H E I t i s then possible to write r 2 = \ t fD V e X p ( " k T ^ ^ ^ P ^ 1 7 2 ^ ^ (2.32)-combining Eqs. (2. 30)and (2. 32), we get = n c N W t ( 1 - f D ) G V t h d £ t - W e X P ( - fe> e x p ( 6 p F E 1 / 2 ) d E t (2.33) Assuming that most of the trapping takes place between the thermal equilibrium 20 l e v e l and quasi fermi l e v e l such that f^ equals one over the energy range between e and ej and equals zero elsewhere., and since >> NkT, one may take equal to i n f i n i t y , and the integration of Eq. (2.33) yie l d s 9n .. e a t " = n c GvthtVwt d £ t " L Nwtdet] " v e xP< ePF E >C NwtexP(- kr ) d et fn fn Substituting for N form Eq. (2.26) the above equation reduces to Wt an vN n 1+N •aT = V ^ V V " 1+1 ^ > - p ( B p F E 1 / 2 ) (2.34) We are now i n position to solve Poisson's equation (Eq. (2.28)), thus giving an expression for the voltage drop across the f i l m , namely /* E(x,t)dx = (<r)2 - •p/e- (2.35) where <j>^  and fy^ are the work functions of the two electrodes. F i n a l l y the external space charge p o l a r i z a t i o n current i s given by J ( t ) = e(n. (x*,t) - N ( x * ) ) ^ (2.36) sc t o dt 16 2.4.2 . Normalized Equations I f the normalized v a r i a b l e s x n = x/d, E n = E / ( f N t d ) , * n - E ( * 2 - * 1 ) / e 2 d 2 N t  n t n = V Nt' n c n = n c / N t ' N o n = V Nt a n = a v t h N t ( l + N ) / v , = e y N j l + N ) / e v , J g c n = J ( i + N ) / v e l T t f and t = tv/(l+N) n are s u b s t i t u t e d i n Eqs. (2.28) (2.29), (2.34), (2.35) and (2.36), we get . 3E ' n = n + n - N (2.37) where 9x tn cn on n ^ = a n (1-n. ) - (n. ) 1 + N exp(C.E 1 / 2 ) (2.38) 9t n cn t n tn I n n 3(n +n ) 9n 3 2n t n cn „ cn , , • „ , cn r r = y E + y n (n +n -N ) - c~ y — ^ — ,„ 3t n n 3x. n cn t n cn on 2 n . 2 (2.39) n n 3x n E dx = <j> (2.40) 0 n n n dx* J = (n. ( x * , t ) - N (x*)) -r-S- (2.41) sen tn n n o n dt n C x = B p F ( e N t d / e ) 1 / 2 and C 2 = ED/(eyN td 2) 2.4.3 Method of S o l u t i o n 28 29 F i n i t e d i f f e r e n c e methods ' were used to s o l v e the above equations. The p a r t i a l d i f f e r e n t i a l equations are approximated by f i n i t e d i f f e r e n c e equations through the t r a n s f o r m a t i o n scheme below. The values of a v a r i a b l e y ( x , t ) are then determined at a d i s c r e t e mesh of p o i n t s i n ( x , t ) space. ,The d e r i v a t i v e s of the quantity y(x,t) are approximated by iZ /„ T N ~ y(x+h,t) - y(x-h.t) 8x ^ X , t ; 2h 8 2Y / v j _ \ - y(x+h,t) - 2 y(x,t) + y(x-h.t) ' (2.42) 2 U » t ; " .2 3x . h lZ/Y .*„ y(x,t+k) - y(x,t) . a t u , t ; " K Thus, Eq. (2.38) becomes • n (x,t + k) = n (x,t) + k a n (x ,t )[1-n „(x ,t )] . tii tn n cn n n t n n n - k[n. ( x . t ) ] 1 + N e x p ( c . / I ) (2.43) tn n n i n Combining Eqs. (2.38) and (2.39), we get IJ-M ^_ u E^(x ,t ) c_y r / M I + N , /jr .. / • i . ^ \ r n n n n 2 n-, n (x ,t )] exp(c /E )= n (x -h,t )[ - rr x~\ t n n n i n cn n n 2n , z ' . ' n + n (x ,t ) [- \ + a (1-n (x , t ) cn n n k, n tn n n 2u c + y-(n (x ,t ) 4 u ( x ,t )-N ) + — ~ ] n tn n n cn n n on , /. i h y E ( x , t ) c„u , / . w n n n n I n-i + n (x +h,t ) [ — — ] cn n n 2h ,2 n + n (x ,t + k)/k (2.44) cn n n For high r e s i s t i v i t y materials with high trap density, free c a r r i e r s can be neglected with respect to trapped charges. Thus, Eq. (2.44) can be reduced to a system of l i n e a r equations whose matrix has a l l i t s elements equal to zero except those i n the main diagonal and on the two adjacent diagonals. Such a system of equations can be solved for a known d i s t r i b u t i o n of trapped charge to determine the free c a r r i e r concentration n c n ( x n » t n ^ without p i v o t i n g . Once n (x ,t ) and n (x ,t ) were determined Eq. (2.43) * tn n' n cn n n 18 was used to determine n (x , t + k). The f i e l d d i s t r i b u t i o n tn n n E^Cx^,^) was obtained by i n t e g r a t i n g Poisson's equation (Eq.(2.37)). E (x ,t ) = E (0,t ) + A(n -N )dx (2.45) n n n n n o tn on n The i n t e g r a t i o n was evaluated using Simpson's r u l e , and E n ( 0 , t n ) was determined from 1 x E (0,t )• = <fc - X n / n n ( n t -N )dx'dx (2.46) n ' n rn 0 0 tn on n n The zero f i e l d point was determined by i n t e r p o l a t i o n and the external current i s r e a d i l y obtained using Eq. (2.41). 2.4.4 Results and Discussion The d i s t r i b u t i o n of trapped charge which b u i l d s up during the a p p l i c a t i o n of a dc e l e c t r i c f i e l d depends on the p h y s i c a l mechanisms involved. A p o s s i b l e d i s t r i b u t i o n may be estimated using one of the various models f o r space charge l i m i t e d e l e c t r o n i c conduction. The d i s t r i b u t i o n assumed i n the-present study i s based on the model discussed i n references 24 and 25. F i g s . 2.2 and 2.3 were computed f o r the case where a l l trapped charges were i n excess of e l e c t r i c a l n e u t r a l i t y . The computed r e s u l t s show that the external e l e c t r o n i c currents obey, a f t e r some i n i t i a l time, a l / t n law, i . e . , J = K / t n (2.47) sc sc where n i s a constant very close to one. The magnitude of K depends on the i n i t i a l s p a t i a l trapped charge d i s t r i b u t i o n . However, f o r a given d i s t r i b u t i o n but with d i f f e r e n t i n i t i a l amount of trapped charge, K g tends to the same value ( F i g . 2.2) since the s p a t i a l d i s t r i b u t i o n s tend to be the same (Fig. 2.4). -0, -WL 2; + o -12 -14 -16 5.0 C = 0.8 C, -N = 10 7-5 10-0 iog]0 (t/(1+N)i/-') 12.5 F i g . 2.2 Space charge p o l a r i z a t i o n currents f o r exponential s p a t i a l d i s t r i b u t i o n of trapped charges. F i g . 2.3 The e f f e c t of capture cross s e c t i o n on space charge p o l a r i z a t i o n current. 20 c 0.5 •INITIAL log tn = 10.0 DISTANCE, Xn F i g . 2.4 Decay of trapped charges i n s i d e a f i l m having constants of Fig. 2.2 S o l i d l i n e s , C = 0.5; dotted l i n e s , C = 0.2. 12.5 iogw tt/P+N)ir') F i g . 2.5 Time dependence of trapped charges. S o l i d l i n e s , computed fo r a f i l m having constants of F i g . 2.2 with C = 0.2; broken l i n e s computed f o r a f i l m having constants of F i g . 2.3 with o = 0.01. n 21 The l o c a l s p a t i a l r a t e of r e l e a s e from traps may be approximated, i n some stage of the disch a r g e , by a 1/t law ( F i g . 2.5). Such time dependence can be d e r i v e d , under s i m p l i f i e d c o n d i t i o n s , using the present model. Eq. (2.38) n e g l e c t i n g o^ and c^, can be reduced to d n t n , ,1+N ~dt~ = ^ t n ) n and thus tn n _ ( t ) . H (2.48) d i f f e r e n t i a t i n g Eq. (2.48), and assuming N >> 1, we may get d t n (1+Nt (0) ) n t n (2.49) N now i f N t n n t n ( 0 ) >> 1, Eq. (2.49) reduces to dn t -n t(0) dt Nt However, a l o c a l s p a t i a l r a t e of r e l e a s e from traps obeying a 1/t law does not n e c e s s a r i l y r e s u l t i n an e x t e r n a l c u r r e n t obeying the same law and the current may even be i d e n t i c a l l y zero. To observe any e x t e r n a l current the s p a t i a l d i s t r i b u t i o n must vary w i t h time ( i . e . , n t ( x , t ) ^ n ^ ( x ) n 2 ( t ) ) i n such a way that the r a t e of change of the zero f i e l d p o i n t e x i s t s ( F i g . 2.6). F i g . 2.7 shows the time dependence of x* and n (x* t ) f o r an e r n t n n' n ex p o n e n t i a l i n i t i a l trapped charge d i s t r i b u t i o n . The v a r i a t i o n of n t n ( X n » t n ^ i s s m a l l over a long time and may be considered constant i n c a l c u l a t i n g the e x t e r n a l c u r r e n t . The zero f i e l d p o i n t may be given by x* = A l n ( t / t ) (2.50) o and thus dx* dt = A/t (2.51) DISTANCE, Xn F i g . 2.6 V a r i a t i o n of trapped charge density and f i e l d with p o s i t i o n i n a f i l m having constants of F i g . 2.3 with a = 0.01. 5-0 7.5 10-0 12.5 l°9l0(t/(1+N)v~1) F i g . 2.7 Time dependence of x* a n d ( x * , t ) f or a f i l m having constants of F i g . 2.2. 23 where A and t are constants. However, since there i s no simple way to derive an expression f o r x * ( t ) , which depends d i r e c t l y on the shape of the s p a t i a l d i s t r i b u t i o n of trapped charges, i t w i l l be postulated that to have an external current obeying a 1/t law the zero f i e l d point must change i t s p o s i t i o n l o g a r i t h m i c a l l y with time (Eq. (2.50)). For the case where i n v a r i a n t p o s i t i v e space charge i s present and the i n i t i a l amount of trapped electrons i s such that the net charge i n s i d e the f i l m i s p o s i t i v e , more electrons w i l l be trapped to achieve n e u t r a l i t y and thus the zero f i e l d point w i l l move towards the cathode (Fig. 2.8b). tn t h i s case, an almost time-independent external current w i l l r e s u l t (Fig.' 2.8a). The e f f e c t of some model parameters was studied. Computed r e s u l t s show that the external space charge currents increase with increase of capture cross s e c t i o n ( F i g . 2.3), d i f f u s i o n constant or inverse of c a r r i e r m o b i l i t y (not shown i n F i g . 2.3). As expected, the magnitude of external current depends on the d i f f e r e n c e between the two metal work functions ( F i g . 2.9). These r e s u l t s can be explained i n terms of the r e l a t i v e changes i n the s p a t i a l trapped charge d i s t r i b u t i o n . A slowly varying s p a t i a l d i s t r i b u t i o n r e s u l t s i n a low value f o r the rate of change of the zero f i e l d point and thus small e x t e r n a l current. D i f f e r e n t rates can r e s u l t from d i f f e r e n c e s i n free c a r r i e r concentration. For higher c a r r i e r concentration, one would expect f a s t e r changes i n the s p a t i a l d i s t r i b u t i o n and thus higher currents. In concluding t h i s s e c t i o n , we postulate that space charge p o l a r i z a t i o n currents can follow a 1/t law, i . e . . J - K / t and that K divided by » sc sc sc preapplied f i e l d i s a decreasing function of preapplied f i e l d . (e) N , -2 -4 to. c2.n n. NM.CO.I 40 5.0 6.0 \0.2S, OA "55? 51 DISTANCE. Xn F i g . 2.8 The e f f e c t of the presence of i n v a r i a n t p o s i t i v e space charge, (a) Space charge p o l a r i z a t i o n current; (b) v a r i a t i o n of trapped charge density and f i e l d with distance i n s i d e f i l m . F i g . 2.9 The e f f e c t of metal work function on space charge p o l a r i z a t i o n current. 25 2.5 Experimental Procedures and Results 2.5.1 . Tantalum Preparation Both s i n g l e c r y s t a l s and sputtered 6 tantalum on Corning 7059 glass substrates were used. The s i n g l e c r y s t a l s were e l e c t r o p o l i s h e d i n a bath of 90% (by volume) K^SO^ (98% reagent) and 10% HF(48% reagent) using 2 a current density of about 100 ma/cm for about 10 minutes. Sputtered tantalum samples were degreased by immersion i n sulphuric acid saturated with potassium dichromate. 2.5:2, Anodization of Tantalum a. Solution Anodization Tantalum samples were anodized i n 0.5% by volume sulphuric a c i d 2 at room temperature f i r s t by applying a constant current density (0.5 ma/cm ) u n t i l a predetermined voltage was reached. The terminal voltage was then l e f t on the sample for periods up to 12 hours. b. Plasma Anodization Plasma anodization of a tantalum s i n g l e c r y s t a l was c a r r i e d out i n a dc glow discharge i n the apparatus described i n Reference 30 at a sample temperature of 35°C and i n the negative glow of the discharge. The oxygen pressure was 80 mtorr and anodization was c a r r i e d out at constant current density. The thickness of the grown f i l m was estimated using the spectro-photometric method described i n Reference 3 making no allowance f o r the di f f e r e n c e i n substrate. 26 2.5-3 D e p o s i t i o n of Counterelectrodes Counterelectrodes of gold dots were evaporated u s i n g photo-etched b e r y l l i u m copper masks and a s h u t t e r i n a conventional 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 t r a p . T y p i c a l l y , the background pressure i n s i d e the vacuum b e l l j a r was about 10 t o r r or b e t t e r . Evaporation was s t a r t e d before opening a s h u t t e r and exposing the oxide f i l m to the source. The di s t a n c e between source and f i l m was approximately s i x inches. Diameters of evaporated dots were measured by a t r a v e l l i n g microscope and t h e i r areas were estimated by usi n g b r i d g e measure-ments at 1000 Hz and assuming = 27.6 f o r solution-grown f i l m s and = 18 f o r plasma grown f i l m s . 2.5.4 E l e c t r i c a l Measurements Step response currents were measured by using a K e i t h l e y type 417 high speed picoammeter (accuracy: +3% f u l l s c a l e f o r a l l ranges except the lowest, which i s +5%). AM o s e l e y 7100 BM s t r i p chart r e c o r d e r , connected to the output of the Picoammeter, was used f o r mapping the current t r a n s i e n t . Capacitance and d i s s i p a t i o n f a c t o r s were measured i n a 3 t e r m i n a l connection u s i n g a GR 1615A capacitance b r i d g e i n the frequency range 100Hz~ 100 kHz. The GR 1615A i s a transformer arm r a t i o b r i d g e w i t h the f o l l o w i n g r a t e d a c c u r a c i e s C = C + 2 x l 0 ~ 5 f/kHz% + 2 x l 0 ~ 3 ( C / p f ) ( f / k H z ) 2 % ; me as -D = D + .1 x(D )% + .001 meas - meas 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 s i d e a grounded, l i g h t t i g h t metal box. E l e c t r i c a l contacts 27 to the electrodes were made by using gold plated beryllium-copper springs mounted on Kulicke and Soffa micromanipulators. 2.5.5 Experimental Results To avoid the superposition of the transient current and the dc conduction current, the technique used was to measure the current upon removal of the applied e l e c t r i c f i e l d . The i n i t i a l t r a n s i e n t current, A6(t), can be neglected f o r t > 1 sec. since the time constant of the measuring c i r c u i t i s much less than 1 sec. (the sample capacitance was of the order of 3 2 10 pf and the input resistance of the picoammeter used increases from 10 ft i n the 10 "*A range to l O ^ f t i n the 10 "*"^A range i n the decade st e p s ) . F i g . 2.10 shows the discharge currents as a function of time with the preapplied voltage as a parameter for a plasma grown f i l m . The discharge currents can be approximated by J( t ) = n(E) E / t n . (2.52) where n(E) i s a p r o p o r t i o n a l i t y constant and i s a function of the preapplied f i e l d , E i s the average preapplied f i e l d (= ~) and n i s a constant close to one. The quantity n(E) i s independent of time and should not vary with the preapplied f i e l d i f space charge i s absent. Fig. 2.U shows n ( E ) f o r several films prepared by d i f f e r e n t procedures. The observed n o n l i n e a r i t y can be explained i n terms of the presence of trapped space charge. The dotted curve i n F i g . 2.11 was drawn f or n(E) given by n(E) =0.65 +0.8/E (2.53) Now, i f space charge p o l a r i z a t i o n current i s almost independent of preapplied f i e l d or, at l e a s t , increases l e s s than l i n e a r l y with preapplied f i e l d (Sec. 2.4), the second term i n Eq. (2.53) may be a t t r i b u t e d to the release of trapped charges while the f i r s t term could be due to the d i e l e c t r i c p o l a r i z a t i o n processes. -10 -11 o N CJ -13 -14 PLASMA-GROWN FILM FILM THICKNESS =950 X FILM AREA 3.7xW~3cm2 log { Q ft/sec) F i g . 2.10 Discharge currents versus time as a f u n c t i o n of p r e a p p l i e d v o l t a g e . ( T a - p o s i t i v e ) . E,10° VOLT/cm F i g . 2.11 Discharge c u r r e n t s as f u n c t i o n of f i e l d f o r f i l m s prepared by d i f f e r e n t procedures ( T a - p o s i t i v e ) . oo 29 2.6 The A p p l i c a b i l i t y o f Step Response Method 2.6.1 S t e p Response Method S t e p r e s p o n s e measurements a r e o f t e n r e g a r d e d as a s u i t a b l e means -2 f o r o b t a i n i n g d i e l e c t r i c l o s s e s a t v e r y low f r e q u e n c i e s ( l e s s t h a n 10 Hz) at w h i c h ac b r i d g e measurements a r e t e d i o u s o r i m p r a c t i c a l . I f l i n e a r r e s p o n s e t h e o r y h o l d s , t h e n E'(OJ) and e"(o)) a r e l i n k e d t o t h e r e l a x a t i o n f u n c t i o n <j>(t) as f o l l o w s 1 " * e'(u) = i [ C m + <t>(t) cos wt d t ] (2.54) o e"(u) - hi- + C <Kt) s i n cot d t ] (2.55) C to u o where is t h e c a p a c i t a n c e o f t h e sample a t v e r y h i g h f r e q u e n c i e s . C Q is t h e c a p a c i t a n c e o f t h e e l e c t r o d e s when t h e sample i s r e p l a c e d by a i r G is t h e s t e a d y s t a t e dc c o n d u c t a n c e . The r e l a x a t i o n f u n c t i o n <j>(t) f o r many m a t e r i a l s a t f i x e d t e m p e r a t u r e obeys t h e r e l a t i o n <|>(t) = A t ~ n (2.56) where A and n a r e c o n s t a n t s . U s i n g t h i s e x p r e s s i o n f o r $(t) i n Eqs. (2.54) and ( 2 . 5 5 ) , we may get e*(w) = 4 [C + Aw 1 1 - 1 T ( l - n ) c o s ( i - n ) o<n<l (2.57) o e"(a)) = 4 t - + A o ) 1 1 - 1 ( l - n ^ o s ? 1 ] 0<n<2 (2.58) O The d i e l e c t r i c l o s s can be e x p r e s s e d i n terms o f t h e r e l a x a t i o n f u n c t i o n at a p a r t i c u l a r t i m e t . as l e"((o) = 4>(t )/wC + G/wC o (2.59) 30 provided that (o and t ^ are r e l a t e d by 1/n tot. = [r(l-n)co3(mr/2)] (2.60) 13 Hamon has shown that the r i g h t hand s i d e of Eq. (2.60) i s almost independent of n i n the range 0.3<n<1.2, having the mean value 0.63 w i t h an accuracy of +3%, i . e . , oo. = 0.63/t 1 i I f discharge currents are considered, then the conduction current i s zero and thus, Eq. (2.59) can be w r i t t e n as e i jo^i/.) WC V o or „ ( . J(0.63/co) (2.61) e K w } a) e E o Eq. (2.6l)was used by Hamon f o r the r a p i d e v a l u a t i o n of l o s s f a c t o r at very low frequ e n c i e s . 2.6.2 Space Charge E f f e c t s In d e r i v i n g the above equations, i t has been assumed that the s u p e r p o s i t i o n p r i n c i p l e h o l d s . Thus, i f space charges are present, the a p p l i c a b i l i t y of Hamon's method may be v i o l a t e d . F i g . 2.12.b shows the frequency dependence of C e" f o r 1050 X s o l u t i o n grown f i l m on spu t t e r e d tantalum. The h i g h frequency p o i n t s are those obtained by d i r e c t ac b r i d g e measurements w h i l e the low frequency p o i n t s were c a l c u l a t e d from step response measurements and Eq. (2.61). The equation of the s o l i d l i n e i s given by Lo g 1 0 ( C o e " y p f ) = 1.42 - 0.09 l o g 1 0 ( f / H z ) The observed n o n l i n e a r i t y between the discharge current and pre-a p p l i e d f i e l d r e s u l t s i n field-dependent values f o r c a l c u l a t e d d i e l e c t r i c E,10° VOLT cm' 2 (b) 3 o ° o .1 o BRIDGE MEASUREMENTS • STEP RESPONSE MEASUREMENTS USING E>1MV cm'1 I ; I I I » » - J 0 j 3 5 logm(f/Hz) F i g . 2.12 Low frequency d i e l e c t r i c losses. (a) The e f f e c t of pre-applied f i e l d on the calculated d i e l e c t r i c losses using step response measurements, (b) comparison of bridge and -step response r e s u l t s . losses ( F i . g 2.12a). However, the d i e l e c t r i c losses become field-independent for r e l a t i v e l y high f i e l d s and may be extrapolated from t h e i r high frequency values. The apparent a p p l i c a b i l i t y of step response procedures at high pre-applied f i e l d s gives us more evidence that space charge p o l a r i z a t i o n currents are only appreciable at low preapplied f i e l d s . I t i s to be noted, however, that the d i r e c t l y measured d i e l e c t r i c losses may be a f f e c t e d by the presence of a dc b i a s i n g voltage and t h e i r values may be s i g n i f i c a n t l y d i f f e r e n t from the small ac s i g n a l values. 2.7. Space Charge E f f e c t s on the Small Ac Signal D i e l e c t r i c Losses The c a r r i e r transport equation and Poisson's equation can be written, neglecting d i f f u s i o n , as J £ ( x , t ) = eyn c(x,t)E(x,t) (2.62) J ( t ) - J c(x,t) + e d M ^ ( 2 > 6 3 ) 3E(x,t) = £ t ( x > t ) + n ( X > t ) - N (x)] (2.64) d X £ L. C ' O d / d E(x,t)dx = V(t) (2.65) where J and J are the conduction and the t o t a l current whose d i r e c t i o n c is the negative.x axis. E is 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 axis. V is the t o t a l applied voltage i n c l u d i n g the d i f f e r e n c e between the two metal work functions. e, is the d i e l e c t r i c constant, a ed J £ d For an applied ac small s i g n a l superimposed on a dc b i a s , i . e . , V ( t ) = V q + v exp (jut) V « V Q ( 2 . 6 6 ) and making the conventional small ac s i g n a l approximations: y(x,t) = Y q ( X ) + y(x) exp(jwt) ( 2 . 6 7 ) where y represents anyone of the above p h y s i c a l q u a n t i t i e s , Eqs. ( 2 . 6 3 ) , ( 2 . 6 4 ) and ( 2 . 6 5 ) reduce to J = eu[fi E + n e + joie.e] ( 2 . 6 8 ) c o co d = - [fi + fi ] ( 2 . 6 9 ) dx e d d £ 6(x)dx = v ( 2 . 7 0 ) Eq. ( 2 . 6 8 ) can be integrated to y i e l d d d J d = ep [ / A n E dx + /. n e dx + j c o e , v] ( 2 . 7 1 ) O c o 0 co J d The v a r i a t i o n i n trapped charge density, fit(x), can be determined from the equation governing the c a r r i e r release from traps. Eq. ( 2 . 3 4 ) can be w r i t t e n , neglecting c^, as Sn (x,t) v N n 1+N -V- = v v t h ( N t _ n t ) ~ T+I ^ ( 2 - 7 2 ) which, with the help of Eq. ( 2 . 6 7 ) , gives fi aV (N -n ) c tn t to n. = ' aV . n + v(n /N ) +jc tn co to t A a_ ~ 6+jw nc where a = aV.. (N -n ) th t to and 8 = a V , n + v(n /N ) N tn co t 0 t ( 2 . 7 3 ) Thus, Poisson's equation becomes ,~ n a de _ e,~ , c , dx e^ nc 8+jV V B2+a,2 and thus, f a t u s g ^ s ) , * ( 2 . 7 5 , d 8 + 0) Eq. (2.71) can now be written as Jd - ey/!j fi E dx + J / fi (x) /* fi (x')[l+ dx'dx O c o E, U co ( J c .1,1 d 8 + 0 ) + ju(ej - jepv = ju)(s' - j e " ) v (2.76) For the l i m i t i n g cases where there i s no phase d i f f e r e n c e between the d i f f e r e n t p h y s i c a l q u a n t i t i e s , i . e . , when the applied s i g n a l frequency tends to zero or i n f i n i t y , the d i e l e c t r i c losses may be wr i t t e n as - d es' d x e"(o>) = e"(u) + ^ r [ e y / n n E dx +~^fn n (x) fn n f x ' ) [ l + ] dx'dx d OJV ( J c o „ z u co O c „z . I B+o) (2.77) The above equation shows c l e a r l y that the d i e l e c t r i c losses increase as the dc f i e l d and/or trapped charge density increase ( F i g . 2.13). Further, the e f f e c t of trapped charge i s frequency dependent and tends to zero as the frequency tends to i n f i n i t y . o c X >^ c CD d= 1080 A = 2050 A 3400 A F i g . 2.13 L o s s - f a c t o r dependence on dc bias. 2.8 Discussion Space charge c o n t r i b u t i o n to p o l a r i z a t i o n currents i s mainly determined by the nature and magnitude of l o c a l i z e d energy states i n s i d e the f i l m which act as el e c t r o n traps. E l e c t r o n traps could be n e u t r a l , p o s i t i v e l y or negatively charged when unoccupied. Experimental studies on e l e c t r o n i c conduction through t h i n amorphous films i n d i c a t e that conduction currents may be governed by Poole-Frenkel mechanism. Thus, one would expect that most traps are charged type traps so that they would be able to e x h i b i t the Poole-Frenkel mechanism. I f t h i s were true then current c a r r i e r s would not get trapped when the applied f i e l d i s s u f f i c i e n t l y high, i . e . , trapped charge density may decrease with in c r e a s i n g preapplied f i e l d . Even i f the present traps are n e u t r a l , there i s an upper l i m i t f o r trapped charge density determined by the a v a i l a b l e traps i n s i d e the f i l m . The present computed space charge p o l a r i z a t i o n currents show that these currents, a f t e r some i n i t i a l time, do not depend on the i n i t i a l amount of trapped charges since they relax to almost the same s p a t i a l d i s t r i b u t i o n . I f one or the other of the above conclusions were true, i t would be reasonable to expect that the magnitude of space charge p o l a r i z a t i o n current divided by preapplied f i e l d should diminish as the preapplied f i e l d i s increased, i . e . , Lim ^sc ^  0 E -> E E On the other hand, l i n e a r d i e l e c t r i c theory p r e d i c t s a l i n e a r dependence of d i e l e c t r i c p o l a r i z a t i o n current on the preapplied f i e l d , i . e . , The above discussion i n d i c a t e s that space charge currents can cons t i t u t e an appreciable part of the external discharge current when the preapplied f i e l d i s s u f f i c i e n t l y low; and there i s a c r i t i c a l f i e l d above which the external current i s mainly due to d i e l e c t r i c p o l a r i z a t i o n processes, i . e . , J ( t ) = J,, + J E < E P S C c while J ( t ) = J D E > E r C The observed nonlinear dependence on preapplied f i e l d of the external discharg currents and the v a l i d i t y of step response procedures to determine low frequen losses only at high preapplied f i e l d s gives us some evidence which confirms the above conjecture. 3. TRAP DETECTION THERMALLY STIMULATED CONDUCTIVITY AND LUMINESCENCE 3.1 I n t r o d u c t i o n As p o i n t e d out i n the previous chapter, the presence of traps i s the main cause f o r the b u i l d up of space charge i n d i e l e c t r i c f i l m s . The nature and d e n s i t y of traps depend on the f i l m s t r u c t u r e . Traps could be an i n t r i n s i c property of amorphous s o l i d s as discussed i n Sec. 1.1. In a d d i t i o n , the presence of i m p u r i t i e s and the n o n s t o i c h i o m e t r i c composition of some t h i n f i l m s ( e s p e c i a l l y compound m a t e r i a l s ) could i n t r o d u c e more t r a p p i n g s t a t e s . 18 -3 A trap d e n s i t y of the order of 10 cm i s not unexpected i n t h i n amorphous f i l m s . Traps i n c r y s t a l l i n e m a t e r i a l s , due to i m p u r i t i e s or d e f e c t s , occupy one or more d i s c r e t e energy l e v e l s i n c o n t r a s t to amorphous m a t e r i a l s where one can have a continuum of energy l e v e l s . , In each of these l e v e l s , the e l e c t r o n i s trapped and unable to move unless i t r e c e i v e s s u f f i c i e n t thermal energy to r e l e a s e i t . Trapped e l e c t r o n s can not tunnel from one trap to another, otherwise the trapped e l e c t r o n s would have f i n i t e m o b i l i t y . Mott^ has shown t h a t , i f two s t a t e s are c l o s e enough together f o r t u n n e l i n g , they s p l i t i n t o two s t a t e s which do not have the same energy. I t would be i n t e r e s t i n g to be able to d i s t i n g u i s h e x p e r i m e n t a l l y between d i s t r i b u t e d and l o c a l i z e d t r a p l e v e l s . One of the most common methods f o r determining e l e c t r o n trap parameters and t r a p p i n g k i n e t i c s i s the thermal glow method, i n c l u d i n g thermo-31 32 33-35 luminescence ' and t h e r m a l l y s t i m u l a t e d current . In the glow method, the sample i s e x c i t e d by u l t r a v i o l e t r a d i a t i o n at low temperature such that e l e c t r o n s i n the valence band and bound s t a t e s are e x c i t e d to the conduction band. The excited electrons may recombine with holes e i t h e r d i r e c t l y or at recombination centers, or may become trapped. Then, i f the sample i s warmed up at constant heating rates, electrons w i l l be expelled from traps i n t o the conduction band. Once i n the conduction band, the excess free electrons could r e s u l t i n an increase of sample e l e c t r i c a l c onductivity or, i n the case of luminescent materials, the free electrons could reach luminescence centers at which they recombine r a d i a t i v e l y . By p o s t u l a t i n g an appropriate model, expressions can be derived to determine trap parameters from the experimentally observed glow curves. Most models, i f not a l l , have assumed one or more of the following assumptions:"^ (1) Traps occupy a s i n g l e energy l e v e l ( i . e . , monoenergetic traps) (2) Retrapping processes are neglected or, only the l i m i t i n g cases of slow and f a s t retrapping are considered. (3) The rate of disappearance of electrons due to recombination may be described by a constant recombination time T. (4) The n e u t r a l i t y condition i s preserved. The purpose of the present chapter i s to study the glow curves f o r a material containing e l e c t r o n traps with d i s t r i b u t e d binding energies, and thus to determine t h e i r b a s i c features which can be u t i l i z e d experimentally to i d e n t i f y trap d i s t r i b u t i o n . To t h i s end, the e f f e c t s of d i f f e r e n t model parameters on the glow curves are inve s t i g a t e d f o r both monoenergetic traps and exponentially d i s t r i b u t e d traps. 3.2 Basic Equations Analysis s i m i l a r to that given i n Sec. 2.3, gives for the rate of change of trapped charge density the following r e l a t i o n : dn -v/°° N y t exp(-e t/kT)de , (3.1) f n > where n i s the free electron density. c • n^ i s the trap density unit energy. e i s the trap energy measured downwards from the conduction band, s i s the capture cross section. V ^ i s the e l e c t r o n thermal v e l o c i t y , v i s the attempt to escape frequncy. The rate of change of free electrons, assuming a constant recombination time T, may be w r i t t e n as dn n dn —9- = —£. £ C3 o) dt T dt 37 The s o l u t i o n of Eq. (3.2) i s given by fc d n t t ' - t d n t \ m - f o A t r d T e x p ( V ^ " T ~ d T ( 3 ' 3 ) The approximation adopted i n Eq. (3.3) i s a good approximation whenever the recombination time i s s u f f i c i e n t l y short, such that dn^/dt<< n / T . The resultant conductivity, a, i s given by a = n eu (3.4) c Where e i s the e l e c t r o n i c charge and y i s the e l e c t r o n m o b i l i t y . Combining Eqs. (3.3) and (3.4), we get dn t a = -eyx (3.5) 39 i n t e n s i t y , A s i m i l a r equation has frequently been used for thermoluminescence dn I = -c —j-£ (3.6) 40 where I i s the thermoluminescence i n t e n s i t y and c i s a p r o p o r t i o n a l i t y constant. Define a new v a r i a b l e , G, given by G r °/eUT ( 3 ' 7 a ) or G - I / c < 3 ' 7 b ) In the glow method, the temperature i s usually increased l i n e a r l y with time, i . e . , T = T + Bt (3.8a) o or dT = Bdt (3.8b) Eqs. (3.1), (3.3), (3.5 or 3.6) and (3.8) can be solved simultaneously to determine the shape of glow curves ( i . e . , 0 vs. I or I vs. T ) . 3.3 A n a l y t i c a l Solutions ' 3.3.1 Monoenergetic Traps In t h i s case, a l l present traps are located at a s i n g l e energy l e v e l £ eV below the conduction band, i . e . , V - N t 6 ( e t - V ( 3 - 9 ) where Nfc i s the trap density 6 i s the Dirac d e l t a function Combining Eqs. (3.1), (3.3), (3.8) and (3.9), we get dn» v nj. exp(-C,t/kT) dt ~ " B 1 + a(l-np ( 3 , 1 0 ) where i s the normalized trapped charge density = n t / N t a . - x s V t h N t Eq. (3.10) can be solved a n a l y t i c a l l y f o r some l i m i t i n g cases. a. Slow Retrapping (q<<!) Eq. (3.10) can thus be reduced to dn . = - | n f c exp(-5 t/kT) (3.11) The s o l u t i o n of the above equation i s given by nfc(T) = n t(T o) exp[- ~ exp(- ~ | ) d t ] (3.12) o and thus, a = n t ( T o ) T e y v e x p [ - ^ -/J | exp(- ^ ) d T ] (3.13) o The maximum conductivity occurs where „ C. vkT^ m t where T^ i s the temperature at which conductivity i s a maximum. I t i s assumed that v i s temperature independent and that over the temperature span of the glow curve, the v a r i a t i o n of y and T with temperature can be ignored. n t b. P a r t i a l l y F i l l e d Traps and Fast Retrapping (^_n, =n^;a>>l) t The reduced form of Eq. (3.10) under the above conditions i s dn. - ^ = - ^ n t e x p ( - y k T ) (3.15) Eq. (3.15) i s the same as Eq. (A3.11) i f v i n Eq. (A3.11) i s replaced by v/a. Thus the maximum conductivity occurs when {, vkT 2 • " • ^ " i s r ( 3 - 1 6 ) tn t The temperature at which conductivity i s a maximum decreases as the r a t i o v/a increases. This r e s u l t i s in agreement with computed r e s u l t s ( F i g . 3.3). 3.3.2 Exponential Trap D i s t r i b u t i o n In t h i s case the trap density per unit energy i s taken to be given by N w t - Afc exp(- ^ / k T ^ (3.17) where A and T are constants which define the shape of the d i s t r i b u t i o n of t c . traps i n energy. S u b s t i t u t i n g Eq. (3.17) i n Eq. (3.1) and i n t e g r a t i n g , we get T c . dn vN n 1+ — I + — T Combining Eqs. (3.3), (3.8) and (3.18), we have T dT B- (3.19) (1+ - ^ ) ( l + a ( l - n p ) This equation i s best solved numerically. 3.3.3 Uniform Trap D i s t r i b u t i o n In t h i s case, the trap density per unit energy i s taken to be given by N t N W f c = 1 f [ u ( £ t ) - u(£ t - W)J (3.20) where W i s the energy range below the conduction band where traps are uniformly d i s t r i b u t e d (W>>kT) u i s the unit step function Using Eq. (3.20), Eq. (3.1) becomes dn. vkTN „ TF " V V t " V " — e x p ( - EF ( 1 " V V > . C 3 - 2 1 ) combining Eqs. (3.3), (3-8) and (3.21), we get dn' vkT e x p ( - | T ( l - n ; ) t - (3.22) dT 6W(1 + a(l-nj.)) Once n^ becomes much l e s s than unity, the rate of change of trapped charge (and hence conductivity) becomes independent of the magnitude of trapped charges. Eqs. (3.10), (3.19) and (3.22) arp nonlinear d i f f e r e n t i a l equations t h e i r general s o l u t i o n s have been computed on an IBM360/67 machine using fourth order Runge Kutta numerical method. 3.4 Numerical Results and Discussion Glow curves f o r a range of values of model parameters have been computed and they are displayed g r a p h i c a l l y i n Figs. 3.1 through 3.3. However the absolute values of the solutions of the model equations depend d i r e c t l y on the absolute values chosen for the disposable parameters. The following features are common to a l l computed glow curves: (1) Glow curves computed f o r both monoenergetic traps and exponentially d i s t r i b u t e d traps a t t a i n a maximum at a temperature, T^, defined by the model parameters. The computed curves f o r monoenergetic traps are more symmetrical around T^ than those computed for exponentially d i s t r i b u t e d traps, also the l a t t e r curves are wider than those computed for monoenergetic traps ( F i g . 3.1) This i s to be expected. When a d i s t r i b u t i o n of trap l e v e l s e x i s t s , rather than a s i n g l e l e v e l , the trapped electrons are permitted to vary with temperature over a wider range of temperature. (2) The temperature at which the maximum glow, G^, occurs as w e l l as i t s value depend on the energy d i s t r i b u t i o n of traps ( F i g . 3.1). As the 44 I o 8 in. I UJ o 4 0 kt=oJ v= 109 sec~f = / s e c ~ ; nt (TQ) = 0.4 (a) 200 400 600 TEMPERATURE, °K u 0) i ? 2 o —i CD 0 0 Tc - WOO •v = 109 sec~^ (3 = 1 sec n\(TQ) ,0.4 (b) 200 400 TEMPERATURE, ~K 600 800 F i g . 3.1 Glow curves f o r d i f f e r e n t trap d i s t r i b u t i o n s , (a) monoenergetic traps; (b) exponential trap d i s t r i b u t i o n . trap energy £ or the c h a r a c t e r i s t i c temperature increases both as we l l as 1/G increase, and consequently the glow curves become wider, m (3) For a given set of mbdel parameters, the area under the glow curve i s prop o r t i o n a l to the i n i t i a l trapped charge, n^CT^). However, the de t a i l e d shape of the glow curves i s dependent on n^CT^) except i n a few cases, e.g., the case of monoenergetic traps with slow retrapping. (4) I t may be u s e f u l to note that, even though i s almost independent of the i n i t i a l magnitude of trapped charges i n the case of monoenergetic traps, i t changes considerably as the i n i t i a l trapped charges i n exponentially d i s t r i b u t e d traps change ( F i g . 3.2). (5) As retrapping processes become more s i g n i f i c a n t , the maximum of the glow w i l l s h i f t to a higher temperature while the magnitude of the maximum w i l l decrease, r e s u l t i n g i n a wider curve. On the other hand, as retrapping becomes l e s s important the glow curve w i l l be completely defined by the recombination process (Fig. 3.3). (6) The increase of the heating r a t e , 8 , has the e f f e c t of incr e a s i n g the magnitude of the maximum glow and s h i f t i n g the temperature at which the maximum occurs to a higher value. The above features could enable us to d i f f e r e n t i a t e experimentally between monoenergetic traps and traps d i s t r i b u t e d over a range of binding energies. Apart from the c l e a r differences i n glow curves f o r both cases, the following experiments may be u s e f u l to confirm the nature of the trap d i s t r i b u t i o n : (1) D i f f e r e n t doses of o p t i c a l r a d i a t i o n ( i . e . , d i f f e r e n t e x c i t a t i o n times) can be used to obtain d i f f e r e n t i n i t i a l trapped charges. Then the dependence of T^ on the e x c i t a t i o n time could be u t i l i z e d to i n v e s t i g a t e the nature of the trap d i s t r i b u t i o n . WO 200 300 . 4 0 0 TEMPERATURE, °K F i g . 3.2 E f f e c t of the i n i t i a l amount of trapped charge density on the computed glow curves, (a) monoenergetic traps; (b) exponentially d i s t r i b u t e d traps. 47 0\ 1 i L i i i i I 200 400 600 800 TEMPERATURE, °K F i g . 3.3 E f f e c t of retrapping on the computed glow curves, (a) monoenergetic (2) The frequency of o p t i c a l e x c i t a t i o n may be varied over a s u i t a b l e range of frequencies to allow only c e r t a i n energy l e v e l s , i f present, to be occupied by excited electrons. The dependence of the magnitude of the maximum glow on the o p t i c a l wavelength should decrease monotonically with increasing o p t i c a l wavelength. F i n a l l y , i t should be pointed out that the presence of several glow peaks does not exclude the p o s s i b l i t y of the presence of d i s t r i b u t e d traps. I t may only i n d i c a t e that the trap d i s t r i b u t i o n does not follow a simple a n a l y t i c r e l a t i o n but that i t could contain several peaks at c e r t a i n energy l e v e l s depending on the preparation technique. 49 4. HIGH FIELD EMISSION: TUNNELING AND SCHOTTKY CURRENTS  IN VERY THIN FILMS 4.1 Introduction High f i e l d emission from the cathode can modify the conduction properties of a t h i n f i l m sandwiched between two metal electrodes. In 25 considering the e l e c t r o n i c conduction , the bulk properties as w e l l as the electrode emission properties should be considered. However, for very t h i n f i l m s ( f i l m thickness l e s s than, say, I O OX) the e l e c t r o n i c conduction i s mainly governed by the metal/film i n t e r f a c e . Current c a r r i e r s are i n j e c t e d from the metal electrode by e l e c t r o n tunneling and/or thermionic emission. The primary c r i t e r i o n for d i s t i n g u i s h i n g between tunneling and thermionic emission currents has been the temperature dependence of the measured conduction current. The temperature dependence of tunneling current i s very Weak compared to that of thermionic current, and i t a r i s e s mainly from the v a r i a t i o n of the a v a i l a b l e electrons for tunneling the v a r i a t i o n i n the b a r r i e r shape and the temperature v a r i a t i o n of the f i l m energy-momentum r e l a t i o n s h i p . In c a l c u l a t i n g Schottky thermionic currents, the o p t i c a l d i e l e c t r i c constant should be used (an e l e c t r o n emitted with v e l o c i t y of 10"* m/sec. 2 ° -13 w i l l cover 10 A i n 10 s e c ) . However, the appropriate d i e l e c t r i c constant for c a l c u l a t i n g tunneling currents seems to be d i f f e r e n t from the o p t i c a l 40 value . This i s because the time of i n t e r a c t i o n between the e l e c t r o n and the p o t e n t i a l b a r r i e r i s s u f f i c i e n t l y long such that the ions i n the b a r r i e r can follow the passing e l e c t r o n through the t r a n s i t i o n . The i n t e r a c t i o n time decreases as the b a r r i e r width decreases. Thus, the appropriate d i e l e c t r i c constant f o r tunneling might be assumed to decrease as the e l e c t r o n energy 50 approaches the top of the b a r r i e r where i t s width i s smaller. In t h i s study, we w i l l uSe the s t a t i c d i e l e c t r i c constant for c a l c u l a t i n g the tunneling current ( r e c a l l i n g that the major part of the tunneling current takes place around the Fermi l e v e l of the cathode) and the o p t i c a l d i e l e c t r i c constant f o r Schottky thermionic emission. The object of t h i s chapter i s to i n v e s t i g a t e the t r a n s i t i o n from tunneling currents to thermionic currents, the e f f e c t of the presence of space charge on tunneling currents and to determine the optimum b i a s i n g f o r a metal/very t h i n film/metal detector. 4.2 Formulation The tunneling current density from one metal electrode to the other across a d i e l e c t r i c (or, semiconductor) f i l m i s given by^ X _ _ 4 Time °° E J12 " h 3 J Q d E J > f ( E 1 ) [ l - f ( E 2 ) ] / 0 T ( E x ) d E x (4.1) where m i s the e l e c t r o n mass. e i s the e l e c t r o n charge. h i s Planck's constant. f(E) i s the Fermi-Dirac function; E r e f e r s to the t o t a l e l e c t r o n energy. E 1 , E 2 r e ^ e r t 0 the i n i t i a l and f i n a l e l e c t r o n energies r e s p e c t i v e l y . T(E^) i s the t r a n s i t i o n p r o b a b i l i t y ; E^ r e f e r s to e l e c t r o n energy i n the x d i r e c t i o n . E = E - E - L x Using a s i m i l a r expression f o r the current i n the opposite d i r e c t i o n , the net current density i s the d i f f e r e n c e between the two expressions. Thus, E J - ^ § /JJ dEjf(\) - f ( E 2 ) ] fQ T ( E x ) d E x (4.2) h S u b s t i t u t i n g f o r the Fermi-Dirac function f ( E 1 ) = l / ( l + e x p ( ( E f - E 1 ) / k T ) ) f ( E 2 ) = f(E x+eV) = l/(l+exp((E f-E 1-eV)/kT)) 42 Eq. (4.2) reduces to — ; 0 l n [ l - r e x p ( ( E f - E x - e V ) y k T ) 3 T ( E x ) d E x ( 4 ' 3 ) I t i s convenient to define a new energy parameter e = E - E. x f Thus, Eq. (4.3) becomes AT c- l+exp(-e4cT) J '*» -T- / „ In [-rr m : wx J J T ( e ) d e (4.4) • 2 k - E f ll+exp(-(e+eV)AcT ) 4Timek 1 • Where A = , 3 h 2 2 = 120. amp/cm deg I t should be noted that the above i n t e g r a l i s not s e n s i t i v e to the absolute value of E^ provided i t i s s u f f i c i e n t l y high (few eV) since T(e) decreases r a p i d l y with decreasing energy. The current magnitude depends c r i t i c a l l y on the p o t e n t i a l b a r r i e r i n the f i l m which determines the transmission p r o b a b i l i t y . The transmission p r o b a b i l i t y i s obtained by matching the Bloch waves at the two metal/film i n t e r f a c e s . However, an approximate s o l u t i o n using the WKB method (for slowly varying p o t e n t i a l b a r r i e r s ) gives T(e) = exp [-2 / 2 U (x) - e) dx] (4.5) X l / ti / 52 with ^ .<fr(x) = <t> +(( + „ - <f> ) - eV) *• +' 4 + <j>T (4.6) C 3. C Q S C Id where <|> and <f) are the cathode and anode work function respectively, c a d i s the f i l m thickness. <j> i s the contr i b u t i o n of any present space charge. SO <j>j i s the image force c o r r e c t i o n . 8TTE e 2x i // j\2 2. nd o r £ n=l ((nd) -x ) 2 2 " .05e 2 d 2 ( 4 > ? ) ire e d x(d-x) o r Nonstoichiometric films may contain unneutralized space charge, e.g., i n the case of t h i n oxide films i t seems reasonable to assume that near the substrate metal an incomplete oxidation process gives r i s e to a 43 region where excess p o s i t i v e ions e x i s t . Assuming that the charge decreases exponentially with the distance from the substrate metal, we may write p(x) = N exp(- ^ ) (4.8) where N and m are constants. The s o l u t i o n of Poisson's equation, when the substrate metal i s biased negatively, gives 2 M,2 <|>sc(x) = - ^-^-J t ( l " exp(- | ( i _ exp(-m))] (4.9) e e m o r while f o r p o s i t i v e substrate metals, the s o l u t i o n i s 2 2 * s c ( x ) = - ^ - ^ - J ^ [ ( i " e x p ( m f ) ) _ f ( 1 " e x P ( m ) ) J ( 4 ' 1 0 ) E e m o r 53 Once the shape of the p o t e n t i a l b a r r i e r i s determined, the trans-, mission p r o b a b i l i t y can be c a l c u l a t e d and the voltage-current c h a r a c t e r i s t i c s of the tunnel diode may be obtained. 4.3 T r a n s i t i o n from Tunneling Mechanism to Schottky Thermionic Emission In evaluating tunneling currents, the upper i n t e g r a t i o n l i m i t i n Eq. (4.4) should be the p o t e n t i a l b a r r i e r maximum, e . However, numerical c a l c u l a t i o n s show that the major part of tunneling current takes place wi t h i n a narrow energy range around the Fermi l e v e l of the cathode. On the other hand, Schottky thermionic emission occurs when electrons have s u f f i c i e n t k i n e t i c energy to surmount the p o t e n t i a l b a r r i e r rather than to tunnel through i t . An expression for Schottky thermionic emission can be obtained by s u b s t i t u t i n g T(e) = 1 and assuming e>>kT i n Eq. (4.4). Thus, we get J s = TT f°° e x P < - e / k T ) d e m = AT 2 exp(- <f>c/kT)exp(Ae/kT) (4.11) where Ae i s the b a r r i e r height lowering. Eq. (4.11) shows that Schottky currenc i s very s e n s i t i v e to temperature v a r i a t i o n i n contrast to tunneling current which v a r i e s very s l i g h t l y with temperature. F i g . 4.1 shows that the r a t i o ^ip/^g increases as: 1. the ambient temperature decreases. 2. the applied voltage increases, except f o r very t h i n films and low applied voltages where the anode work functions could have pronounced e f f e c t VOLTS F i g . 4.1 T r a n s i t i o n from tunneling mechanism to Schottky termionic emission 3. the magnitude of the p o s i t i v e space charge increases. 4. the cathode work function decreases (not shown i n F i g . 4.1) Thus the predominant process, whether i t i s tunneling of Schottky emission, depends on f i l m parameters, applied voltage and ambient temperature. I t i s i n t e r e s t i n g to note that both processes could contribute equally to conduction current under c e r t a i n conditions. The rest of t h i s study w i l l be concerned only with tunneling currents. 4.4 Space Charge E f f e c t oh Tunneling Currents F i g . 4.2 shows the voltage-current c h a r a c t e r i s t i c s of a f i l m space charge-free and that of a f i l m containing p o s i t i v e space charge. The main 2 2 feature of the computed c h a r a c t e r i s t i c s i s that d lnl/dV changes sign as the applied voltage passes <j> /e, i . e . , when <j>(d) l i e s below the cathode 44 Fermi l e v e l . I t i s worth noting that Fowler-Nordheim tunneling currents 2 2 have a negative value for . d lnl/dV . Assuming that the tunneling current occurs at z = 0 and that <j>(d) = 0 and neglecting space charge and image force e f f e c t s , we get from Eq. (4.5) I f the a v a i l a b l e electrons for tunneling are almost constant, tunneling current, at high applied f i e l d s , might be proportional to the above quantity. The Fowler--Nordheim expression for tunneling at high f i e l d s i s J. FN 8tihd $ exp(- 8Tid/2m 3h (4.12) F i g . 4.3 P o t e n t i a l b a r r i e r lowering as a function of the applied 57 The presence of the above mentioned i n f l e c t i o n point leads to an i n t e r c e p t i o n between the forward and reversed V-I c h a r a c t e r i s t i c s . The same features are applicable to films containing i n v a r i a n t space charge. However, the presence of p o s i t i v e space charge increases the current density for a given applied voltage since the p o s i t i v e space charge increases the p o t e n t i a l b a r r i e r lowering. The p o t e n t i a l b a r r i e r lowering i s presented i n F i g . 4.3. I t i s r e l a t e d to the applied f i e l d by the r e l a t i o n A<(> = e/v7d where 8 i s a constant depending on the f i l m d i e l e c t r i c constant. The presence of space charge i n s i d e the f i l m and the d i f f e r e n c e between the two metal work functions might change the magnitude of 8 and could cause an appreciable deviation from the above r e l a t i o n e s p e c i a l l y at low applied voltages. 4.5 Optimum Bias for MIM Detectors Because of the nonlinear nature of the voltage-current c h a r a c t e r i s t i c s of very t h i n d i e l e c t r i c (or, semiconductor) films sandwiched between two • 45 46 metal electrodes, MIM diodes can be used as wave detectors ' . The f i g u r e of merit, F, f o r such detectors i s usually taken to be the responsivity-band-width product, where the diode r e s p o n s i v i t y i s defined as the detected voltage d i v i d e d by the input power. In t h i s s e c t i o n i t w i l l be shown that there i s a bias voltage which gives maximum responsivity-bandwidth product. To avoid excessive computing time, i n determining .the small s i g n a l c h a r a c t e r i s t i c s , the following procedure has been used. Around each b i a s i n g p o i n t , the already computed dc c h a r a c t e r i s t i c s have been f i t t e d to a power function using l e a s t 58 squares f i t t i n g : l ( v ) «= Gv + G 2 v 2 + G 3v 3 where v i s the small s i g n a l voltage measured from the bia s i n g point. The computed r e s u l t s show that G^/G i s a maximum at the above mentioned i n f l e c t i o n point (when the b i a s i n g voltage i s <j> /e). This i s true, since i f SL dV then 2 dlh I _ d l , d l _ r , ~W~ ~ ^ 2 1 dV " 2 G 2 / G dlnX But i s a maximum (see F i g . A.2), hence G2/G i s a maximum. The detected dc voltage across the diode, i n response to v = v s i n tot, i s thus given by and the responsivity-bandwidth product i s given by F = (2V D/v 2 G ) x (G/2TTC) * & h + f 2 ( G 3 / G ) v ] ( 4 - 1 4 ) where C i s the detector capacitance. F i g . 4.4 shows the v a r i a t i o n of the diode responsivity-bandwidth product as a function of the bias voltage. The highest value for the responsivity-bandwidth product occurs when the bias voltage i s <j> /e. This i s because G~/G i s maximum at that b i a s . The presence Si of p o s i t i v e space charge increases the magnitude of the maximum, since the p o t e n t i a l b a r r i e r height changes with applied voltage f a s t e r than that f o r space charge-free diodes. I t i s worth noting that the discussed optimum bias i s also applicable LO VOLTS F i g . 4.4 V a r i a t i o n of the detector responsivity-bandwidth product as a function of the b i a s voltage. vo for the case of heterodyning detectors. In t h i s case, we should replace .Eq. (4.13) by VD " V 2 ( G2 / G> ( 4 ' 1 5 ) where and are the magnitudes of the two applied s i g n a l s . Thus, the same r e s u l t follows. 4.6 Summary The predominant conduction mechanism i n very t h i n f i l m s , whether i t i s tunneling or Schottky emission, depends on f i l m parameters, applied voltage and ambient temperature. In view of the fa s t tunneling time of electrons through very t h i n f i l m s , MIM structures can be used f o r microwave detection. I t i s shown that the maximum responsivity-bandwidth product of MIM detectors i s obtained x^hen they are biased at a Voltage equal to the anode work function ( i n v o l t s ) . The presence of in v a r i a n t p o s i t i v e space charge increases the. magnitude of t h i s maximum. 5. HIGH FIELD SWITCHING  IN THIN AMORPHOUS FILMS 5.1 Introduction E l e c t r i c a l i n s t a b i l i t i e s have been observed i n a large v a r i e t y of t h i n amorphous films sandwiched between two metal electrodes. Most of those i n s t a b i l i t i e s are of the current c o n t r o l l e d negative r e s i s t i v i t y (CCNR) type which have been observed i n , e.g., NiO4'', Nb20,.48, Ta^O^^, 49 50 51 52 53-55 H O 2 , ^Oj. ' , Fe^O^ and semiconducting glasses . The observed voltage-current c h a r a c t e r i s t i c s suggest the four f o l d c l a s s i f i c a t i o n shown i n F i g . 5.1"*^. F i g . 5.1 Types of CCNR c h a r a c t e r i s t i c s . 62 (1) The Negative Resistance Device; I t s V-I c h a r a c t e r i s t i c i s retraceable and i t shows an extended negative d i f f e r e n t i a l r e s i s t i v i t y region. (2) The Switching Device: I t has no stable operating point between the. o r i g i n a l high resistance state and the conductive state to which the device switches, at a threshold Voltage V ^ . The conductive state can be maintained only i f the current i s above the sustaining current I g . (3) and (A) The Negative Resistance Device with Memory and The Switching Device with Memory: They are s i m i l a r to (1).and (2) res p e c t i v e l y but with the di f f e r e n c e that once the conducting state i s established, i t p e r s i s t s without noticeable decay. The high resistance state can be re- e s t a b l i s h e d by increasing the current above a c e r t a i n Value and switching i t o f f r a p i d l y . The main experimental features of CCNR phenomena can be summarized: (1) CCNR can occur i n t h i n oxide films only i f a forming process has been done"*^. The forming process consists of passing a c r i t i c a l current 2 (=0.1 Amp/cm ) i n a pulsed or steady-state form. I t i s believed that c o n s i -derable heating of the f i l m and some i r r e v e r s i b l e changes, p r i m a r i l y s o l i d -s t a t e d i f f u s i o n of the metal electrodes i n the f i l m , occur at the i n t e r f a c e s with the metal electrodes. (2) Conduction current through the amorphous f i l m before switching ( i . e . , i n the high resistance state) depends strongly on temperature and voltage and i s prop o r t i o n a l to the f i l m cross s e c t i o n a l area. However, i n the conductive s t a t e , the conduction current has been found to be independent of the f i l m cross s e c t i o n a l area, the current i s concentrated i n narrow f i l a m e n t s ^ . 63 (3) While the t h r e s h o l d v o l t a g e V ^ depends on sample t h i c k n e s s the s u s t a i n i n g v o l t a g e V g i s n e a r l y constant f o r s i m i l a r l y prepared 53 s t r u c t u r e s ; the higher the d i e l e c t r i c constant the lower the value of V , suggesting an i o n i z a t i o n process, s (4) White i l l u m i n a t i o n increases the c u r r e n t , decreases the t h r e s h o l d 57 v o l t a g e , but has no e f f e c t on the s u s t a i n i n g v o l t a g e (5) I n f r a r e d r a d i a t i o n has been observed during s w i t c h i n g , e.g., o 58 v i t r e o u s As^Te^ • A s 2 Se^ at 78 k shows a weak peak at .1.16 eV and a str o n g peak at 0.67 eV, the f i r s t peak is due to interband recombination and the stronger low energy peak may be due to the capture of c a r r i e r s by recombination centers. (6) The s w i t c h i n g time T (the time taken by the sample to undergo s t r a n s i t i o n from the high r e s i s t a n c e s t a t e to the conducting s t a t e ) i s of the order 10 sec. w h i l e the delay time (the time between the a p p l i c a t i o n of the voltage and the s t a r t i n g of the s w i t c h i n g process) may be approximated by = A exp(-BV) —6 where i s approximately 10 sec. at V = and no s w i t c h i n g appears p o s s i b l e below V ^ ( i . e . , = <*>). Sever a l models have been proposed to e x p l a i n CCNR i n t h i n amorphous f i l m s . Some models have considered the phenomena to be e s s e n t i a l l y an e l e c t r o n i c phenomenon,and the others considered i t as a r e s u l t of thermal e f f e c t s . Double i n j e c t i o n " ' 9 , tunnel i n j e c t i o n through f i e l d - d e c r e a s i n g b a r r i e r ^ , space charge f o r m a t i o n ^ ' ^ \ impact i o n i z a t i o n 4 ^ , s e l f - h e a t i n g and f i l a m e n t a r y 62—64 conduction processes have been p o s t u l a t e d as p o s s i b l e mechanisms r e s p o n s i b l e f o r CCNR c h a r a c t e r i s t i c s . In t h i s chapter, e l e c t r o t h e r m a l i n s t a b i l i t i e s w i l l be inve s t i g a t e d and their l i m i t a t i o n s f o r describing CCNR observed i n t h i n films w i l l be pointed out. Sample s e l f - h e a t i n g could account for the observed delay times. However, i t i s believed that the switching process i s an e l e c t r o n i c p r o c e s s ^ . Consequently, an e l e c t r o n i c model i s proposed. The small ac s i g n a l equivalent c i r c u i t of the model i s given. Memory devices w i l l be discussed i n terms of filament formation and phase change mechanisms due to excessive heating. 5.2 Electrothermal I n i t i a t i o n of the Switching Mechanism Pulse measurements^^, on an 0 . 8 um thick layer of amorphous Te^ ,.-A S Q 2 S1Q ^  Ge^ 2> show that the temperature of a switching device r i s e s about 15°C above ambient before switching under s t a t i c conditions. This temperature r i s e can be due to device s e l f - h e a t i n g which w i l l now be discussed i n some d e t a i l . Whatever the true e l e c t r o n i c process, or processes which i n i t i a t e the CCNR c h a r a c t e r i s t i c s are, they w i l l determine the temperature and f i e l d conductivity dependences. The experimentally observed f i l m conductivity i s approximately of the f o r m ^ a(E,T) = a Q exp (-(e - 0E)/2 kT) (5.1) where E i s the average applied f i e l d k i s Boltzmann's constant T i s the temperature aQ, 6 and e are f i l m constants The exponential-dependence of conductivity on temperature could f a c i l i t a t e thermal heating. The sample temperature could eventually reach the c r i t i c a l temperature needed before switching takes place, provided the input power i s s u f f i c i e n t l y higher than the heat losses from the sample surface 5.2.1 General Formulation The sample s e l f - h e a t i n g process i s e s s e n t i a l l y determined by the temperature dependence of the f i l m conductivity and the associated feedback processes, namely: temperature r i s e , conductivity increase, current. increase and higher power d i s s i p a t i o n i n s i d e the f i l m leading to temperature r i s e . Such a closed loop could produce an unstable s i t u a t i o n i n which the current density increases r a p i d l y . Furthermore, whenever the f i l m edges are kept at constant temperature (ambient temperature) the temperature r i s e w i l l tend to be l o c a l i z e d at some point i n s i d e the f i l m . Thus the temperature need only increase i n a l o c a l i z e d area and t h i s i n turn w i l l r e s u l t i n a shorter delay time and l e s s energy w i l l be required before switching takes place. The f i n a l steady state current w i l l be l i m i t e d by the energy los s from the sample, condu c t i v i t y s a t u r a t i o n and the external c i r c u i t . The temperature d i s t r i b u t i o n s i n the f i l m can be expressed by 69 the usual heat conduction equation 9T Pc •—— = V-(KVT) + P (5.2) where P i s the net power input per u n i t Volume i n the f i l m and p, c, K r e f e r to the f i l m density, s p e c i f i c heat and thermal conductivity. I f we assign a constant l/X to the external thermal res i s t a n c e of the sample, then the heat loss per u n i t volume per unit time can be w r i t t e n as P e x t = " V (5.3) where d i s the f i l m thickness and T^ i s the ambient temperature. The e l e c t r i c power input i s given by P. » o(E,T)E 2 (5.4) Combining Eqs. (5.1), (5.A) y i e l d s the relevant equation which could describe CCNR phenomena. A l l the r e s u l t s presented have been obtained for c y l i n d r i c a l geometry with the current flowing along the z-axis and the c y l i n d e r w a l l maintained at the ambient temperature. I t has been assumed that the f i l m has no temperature gradient perpendicular to the e l e c t r i c a l contacts and that the f i l m thermal conductivity i s temperature independent (or, i t s v a r i a t i o n can be neglected compared to e l e c t r i c a l conductivity v a r i a t i o n ) . Thus the heat conduction equation can be w r i t t e n as + I 9 T - X (T r \ ^(E.T) -2 pc 9T ' „ 2 r 3r " dK ( T " V K E + ~K 3t" ( 5 ' 5 ) ar with the boundary conditions f£| = 0; T(a,t) = T (5.6) * r r=o 0 and the i n i t i a l condition T(r,o) = T -asrsja (5.7) o Eq. (5.5) i s a non-linear p a r t i a l d i f f e r e n t i a l equation whose general s o l u t i o n may be obtained using e i t h e r an analogue or a d i g i t a l computer. The discrete-space continuous-time analogue representation of Eq. (5.5) i s shown i n F i g . 5.2 The temperature d i s t r i b u t i o n w i t h i n the f i l m could be obtained using the equivalent c i r c u i t shown, but the necessity of a separate non-linear function generator at each node creates a major problem. The r e s u l t s presented have been obtained an IBM 360/67 machine, using constants given i n Table 5.1 (unless otherwise stated), which are intended to r e f e r to 53 55 chalcogenide glasses ' (o) R*h* C ml. lG ' f E2-(rN_,-TN.,)/2h2(N-1) A IS THE DISTANCE BETWEEN TWO GRID POINTS IN THE RADIAL DIRECTION v^ -I Tfo) A A rM , . TM'TO r - r — i — V S 5.2 Discrete-space continuous-time analogue representation of e l e c t r i c a l heating of a th i n f i l m ( e l e c t r i c a l equivalent c i r c u i t f o r Eq. (5.5). 15 0. W I C5 £ Q: 3 5 o ^o; CONSTANTS OF TABLE I (b) K=l Wm-1^'1 (c) \ = 2000 Wm-2^'1 0 10 2 0 VOLTAGE (VOLT) 5.3 Computed voltage-current c h a r a c t e r i s t i c s f o r a t h i n c y l i n d r i c a l structure 68 TABLE 5.1 CONSTANTS a = 5 x 10 "* i , d = 5 x 10~ 7 m -.3 _ - l -1 a = 10 ft m o e = 1 eV -9 8 = 4.3 x 10 em K = 0.5 Wm"1 ° K _ 1 pc = 10 6 J k i 1 ° K _ 1 X = 5 x 1 0 3 W m~2 ° K - 1 Once the temperature d i s t r i b u t i o n has been obtained, the t o t a l c u r r e n t , I , i s given by I = / a J(r)2Trrdr (5.8) o or 1 = "iF C a ( r ) r d r ( 5 - 9 ) where J i s the current d e n s i t y , V i s the a p p l i e d v o l t a g e and a i s the f i l m r a d i u s . 5.2.2. S t a t i c C h a r a c t e r i s t i c s The s t a t i c c h a r a c t e r i s t i c s are governed by the equation d 2T 1 dT _ \ ( . q(E,T) 2 ( . . 2 + r dr dK ( T V K E ( 5 ' 1 0 ) dr The s o l u t i o n of Eq. (5.10) w i t h the boundary c o n d i t i o n s given by Eq. (5.6) has been obtained n u m e r i c a l l y using the 4th order Runge-Kutta method. The computed r e s u l t s ( F i g . 5.3) e x h i b i t CCNR c h a r a c t e r i s t i c s and a re g i o n where both current and v o l t a g e are decreasing w h i l e the sample i s heat i n g up. The current range where the vol t a g e i s t r i p l e - v a l u e d occurs when the cur r e n t i n c r e a s e i n the c e n t r a l r e g i o n (the p r o s p e c t i v e f i l a m e n t ) i s l e s s 69 than the current decrease i n the low temperature region. This t r i p l e - v a l u e d region could account for the observed h y s t e r e s i s i n a l l switching devices. F i g . 5.4 shows that the temperature d i s t r i b u t i o n w i t h i n the f i l m i s almost uniform u n t i l the onset of the e l e c t r i c a l i n s t a b i l i t y . Thus one could obtain an approximate expression f o r the threshold voltage 2 by assuming V T = 0. The threshold f i e l d can be determined by equating the power input to the power l o s s and by equating t h e i r d e r i v a t i v e s w.r.t. temperature, i . e . , ' 4 (T - T ) * a E 2 exp ((BE., - e)/2kr ) (5.11) d c o o th th c and . (e-BE , ) 9 I * % E t h e x P « g E t h " £> / 2 kV ( 5 - 1 2 ) c Thus by d i v i d i n g Eq. (5.11) by Eq. (5.12), we get 2kT 2 A T c = <VTo> = I T 3 E T > ( 5 ' 1 3 ) tn but since A^c Is always small (of the order of few tens of degrees centigrade), one may write « 2kT Z tn The threshold f i e l d f o r switching can be obtained by s u b s t i t u t i n g Eq. (5.14) i n Eq. (5.11). / XAT E t h - / - d ? ( r 5 V < 5 - 1 5 > where a(T ) = exp ((BE., - c)/2kT ), and 0 0 tn o e = 2.7182818 Eq. (5.15) gives values for the threshold f i e l d very close to that obtained by s o l v i n g the complete thermal conduction equation (Eq. (5.10) whenever X/d i s s u f f i c i e n t l y high such that the energy los s by r a d i a t i o n determines the threshold F i g . 5.4 Current density and temperature d i s t r i b u t i o n s within the semi-conducting f i l m . The given d i s t r i b u t i o n s are those associated with p o i n t s . ! , 2, 3 shown i n F i g . 5.3. log (\ d-^0-1/Ws^ ' n r f 2 V ' ) F i g . 5.5 Exact and approximate solutions f o r threshold f i e l d . 71 f i e l d . On the other hand, i f X/d i s small, the temperature d i s t r i b u t i o n w i t h i n the sample a f f e c t s the threshold f i e l d and i t w i l l be higher than that given by Eq. (5.15) F i g . 5.5. The computed r e s u l t s (Fig. 5.5)show that the threshold f i e l d becomes l e s s dependent on X/d as X/d increases ( i . e . , as the f i l m thickness or the r a d i a t i o n resistance decreases) and such dependence becomes even l e s s as the ambient temperature decreases. As the maximum temperature wi t h i n the sample reaches i t s c r i t i c a l value, the temperature rate of increase w i l l be Very f a s t and the e l e c t r o n i c conduction current w i l l tend to concentrate i n a narrow channel r e s u l t i n g i n the s o - c a l l e d thermal channel. However, i t should be noted that the maximum temperature w i l l not increase i n d e f i n i t e l y (otherwise breakdoxra occurs) but i t w i l l s t a b i l i z e at some value determined by the external c i r c u i t and pos s i b l y by the associated e l e c t r o n i c processes. .. 5.2.3 Delay Time A. Square WaVe Voltage The e l e c t r i c energy d i s s i p a t e d i n the sample during the delay time i s used to heat up the sample which eventually reaches the c r i t i c a l temperature where switching can take place. The c r i t i c a l temperature r i s e i s taken to be given by « . 2kTT where E^ i s the applied e l e c t r i c f i e l d . Delay times were computed from Eq. (5.5) using f i n i t e d i f f e r e n c e methods (c e n t r a l d i f f e r e n c e s f o r s p a t i a l d e r i v a t i v e s and forward di f f e r e n c e s for time d e r i v a t i v e s ) and the c r i t i c a l condition given by Eq. (5.16). F i g . 5.6 shows that the computed delay times could be f i t t e d to 72 L) o O O O with ^(300°) « 4.5 x 10~ 3 sec; and V q(300°) = 4.68 V, but the delay time s t a r t s to deviate considerably from the above r e l a t i o n as the applied f i e l d approaches the c r i t i c a l f i e l d . An approximate expression for the delay time can be derived assuming a uniform temperature d i s t r i b u t i o n (see Sec. 5 . 2 . 2 ) and an input power much greater than energy losses. Pc O E ^ ( 5 . 1 8 ) The s o l u t i o n of the above equation follows as a/T T T = P ca . e „ -a, , c - Ei(£) L D n v2 1 a/T T JT (5.19) u \ o o b oo g — X where E i i s the exponential i n t e g r a l = - f dx, and a = (E-BE )/2k. A comparison between the r e s u l t s obtained from Eq. (5.19) with those obtained from Eq. (5.8) shows that the above approximate s o l u t i o n gives us almost an exact s o l u t i o n when the Voltage pulse i s r e l a t i v e l y high F i g . 5.7. Furthermore, i f T^ i s s u f f i c i e n t l y small, Eq. (5.19) may be approximated as pckT 2 e-BE v o / b T D " 7 ~ i e x p ( - 2 k Y i ) ( 5 ' 2 0 ) " a o \ ° » T exp - ( V / V ) o o where „ pckT T o = — l e xp (ikT->' v o = < 2 kV / B cto E, o o b 73 K5» CD o 25 35 VOLTAGE (VOLT) . 45 4-24 16 ki o US J.08 F i g . 5.6 Computed delay time, c r i t i c a l temperature r i s e and e l e c t r i c energy needed f or switching to occur versus applied voltage. 30 VOLTAGE (VOLT) 40 50 F i g . 5.7 Exact, • — , and approximate, , delay time as a function of applied voltage f o r d i f f e r e n t ambient temperature. The broken l i n e s show the corresponding c r i t i c a l temperature r i s e . 74 The e l e c t r i c energy required before switching can take place i s given by £ = / I V d t nc o = 2TrE2d /. D / a a ( r , t ) rdr dt (5.21) b o o The computed r e s u l t s show that the required e l e c t r i c a l energy i s almost constant except when the applied f i e l d approaches the threshold value f o r switching. The presence of an external resistance would increase the device delay time. The delay time w i l l continue to increase as the external s e r i e s resistance increases ( F i g . 5.8) and eventually reaching the c r i t i c a l c ondition would be impossible ( i . e . , T^-**5). This occurs when the magnitude of the voltage across the device becomes less than the threshold voltage. B. Sinus o i d a l Voltage The b a s i c features of ac switching of a switching device are given i n F i g . 5.9, namely: (1) The threshold f i e l d and the f r a c t i o n a l delay time (T^ f) increase as the applied s i g n a l frequency increases. (2) The e l e c t r i c energy required for switching i s almost constant and i s of the same order as that required i n the case of pulse switching. (3) The above mentioned q u a n t i t i e s and the device c u t - o f f frequency depend on the amplitude of the applied s i g n a l . I t should be noted that the above features are common to a l l voltage wave forms with f i n i t e r i s e time. UJ Q 10 10J Rext (OHMS) 10" 10* F i g . 5.8 The e f f e c t of external resistance on delay time. 65 .32 55 § 45 UJ o S —i VOLTAGE J6^ o c 4 5 logJ0(f/Hz) -1,00 F i g . 5.9 Ac switching c h a r a c t e r i s t i c s : threshold voltage, delay time and e l e c t r i c energy needed f o r switching to occur. Applied voltage= V s i n 2 Tfft. 76 5.3 Switching Mechanism The e x p e r i m e n t a l l y observed s w i t c h i n g time i n chalcogenide f i l m s i s -9 53 as low as 10 sec which can not be accounted f o r by simple thermal runaway. I t would r e q u i r e a much, longer time and temperature i n the current f i l a m e n t which could destroy the sample. In i n v e s t i g a t i n g Chalcogenide g l a s s e s , 56 F r i t z s c h e has shown that i t i s not p o s s i b l e to f i n d a value f o r the current f i l a m e n t cross s e c t i o n a l area which i s small enough to permit a s i z a b l e temperature r i s e but at the same time l a r g e enough to y i e l d the low value of r e s i s t a n c e i n the conductive s t a t e . Furthermore, Eqs. (5.1) and (5.5) cannot represent s t a b l e dynamic 52 o p e r a t i o n which has been observed exp e r i m e n t a l l y . Making the conventional s m a l l s i g n a l approximations: E ( x , t ) = E f c ( x , t ) + e ( x , t ) , a ( x , t ) = a b ( x , t ) + a ( x , t ) , (5.22) and T(x,t) = T b ( x , t ) + T ( x , t ) , Eq. (5.5) can be reduced to v2* = i f - IK + 2%V> + if f <5-23> where 6 i s r e l a t e d to the e x t e r n a l a p p l i e d v o l t a g e V by 3.C e = ( V - 2TTE,R ( fa 5 r d r ) ) / ( d + 2TTR fa o. r d r ) (5.24) ac b s o s o b where R G represents the e x t e r n a l s e r i e s r e s i s t a n c e . Considering the c e n t r a l 2~ p a r t of the formed f i l a m e n t where V. T=o and a, according to Eq. (5.1), i s e l v e n b y e - BE, o * a, ( j)i (5.25) 2K T^ Eq. (5.23) i s thus reduced to 77 (5.26) Neglecting the temperature dependence of e, the s o l u t i o n of Eq. (5.26) i s only s t a b l e i f , X/dVo, (5.27) But since t h i s condition i s not r e a l i z a b l e , one would expect that for stable dynamic c h a r a c t e r i s t i c s , the conduction mechanism should change so that thermal runaway may be avoided. responsible for the observed CCNR i n t h i n chalcogenide semiconductor films are e l e c t r o n i c i n nature. However, i t i s believed that s e l f - h e a t i n g serves to i n i t i a t e the e l e c t r o n i c switching mechanism. Self-heating could i n i t i a t e an e l e c t r o n i c switching mechanism, e.g., by developing high f i e l d regions near the electrodes. Since the voltage, a f t e r the onset of electrothermal i n s t a b i l i t y , i s a decreasing function of the current density, the excess Voltage i s la r g e s t wherever the current density i s largest (current density v a r i a t i o n could be due to v a r i a t i o n s i n the e m i s s i v i t y of the sample el e c t r o d e s ) . A space charge region w i l l develop r e s u l t i n g i n higher f i e l d s at the electrodes. Another p o s s i b i l i t y ^ ' ^ could be a temperature gradient near the electrodes which may produce a e l e c t r i c f i e l d gradient so that higher f i e l d s e x i s t at the electrodes. This, i n turn, can enhance current c a r r i e r i n j e c t i o n through the metal/film b a r r i e r . I f the e l e c t r i c f i e l d i n s i d e the f i l m i s s u f f i c i e n t l y high, c a r r i e r m u l t i p l i c a t i o n by impact i o n i z a t i o n may occur and the f i l m may show a CCNR c h a r a c t e r i s t i c which i s e l e c t r o n i c i n nature. Thus, one can conclude that the switching mechanism and the processes 78 5.4 Current Controlled Negative Differential Conductivity Due to Space Charge  Barriers 5.4.1 Model and Basic Equations A two carrier, one dimensional model may represent a metal/amorphous film/metal diode exhibiting CCNR characteristics. In the present model, entrance of current carriers from the cathode into the conduction band of the film i s taken to be given by either Schottky thermionic emission or the Fowler-Nordheim tunneling mechanism (see Chapter 4), i.e., J n(0) - -An exp(B g Jfffij (5.28) or n n where J (0) = -A'E2(0) exp(-H/E(0)) J^(0) is the electron current density at the cathode. E(0) i s the electric f i e l d at the cathode. A and A' are constants, n n $ s i s the Schottky slope e /"c~ 2kT /TTG" Injected carriers could undergo a multiplication process by impact ionization i f the f i e l d inside the film i s sufficiently high. Near the cathode the electrons, which have just tunneled through the barrier, are moving faster than in the body of the diode and so do not combine so easily with positive holes. That i s why a concentration of positive charge builds up at the cathode before the rate of recombination balances the rate of generation^. The steady state continuity equation can be written as - v.J = -(G - R) e n ' (5.29) - y - j = G - R e p 79 where J and J are the ele c t r o n and the hole current density r e s p e c t i v e l y , n p J c J G and R are the generation and recombination rate function r e s p e c t i v e l y . The electron-hole generation rate due to impact i o n i z a t i o n i s given by (see Reference 71 Chapter 2) G - a n u + a p u (5.30) n n p p where n and p are the ele c t r o n and hole density r e s p e c t i v e l y . y n and are the electron and hole m o b i l i t y r e s p e c t i v e l y . a /E i s the ele c t r o n i o n i z a t i o n rate (= number of electron-hole n p a i r s generated by an e l e c t r o n per unit distance t r a v e l l e d ) . a /E i s the hole i o n i z a t i o n rate. P ct n and cr^ may be approximated by one of the following functions: (1) a = a (E/E ) m o o (2) a = a Q exp (3 E) (3) a = a Q exp(-B/E) m (5.31) where a , E , B and m are constants, o * o For the net generation-recombination rate, one may write G - R - U(E, n, p, x) However, f o r s i m p l i c i t y i t w i l l be assumed that U = G (5.32) or U = 0 depending on the p o s i t i o n i n s i d e the f i l m . •. . . The transport equations f o r J ^ , are 80 T = eu n E + e D Vn (5.33) Jn n n J = ey p E - e D Vp (5.34) p P P For steady state conditions, the total current density is J = J + J = constant n p where D and D are the electron and hole diffusion constant respectively, n p The electric field distribution inside the film can be determined from Poisson's equation V-E.« -(p-n) (5.35) e and the voltage drop across the film is given by V - - / d E dx (5.36) o Eqs. (5.33), (5.34), and (5.35) can be combined to yield a nonlinear differential equation describing the electric field distribution inside the 72 film . To derive such an equation, multiply Eq. (5.34) by 1/6 (<5 =y /y =D /D,) p n p n and subtract the result from Eq. (5.33). Thus, we get J - J /6 - e D (Vn + Vp) - ey E(V-E) (5.37) n . p n n Similary by adding Eqs. (5.33) and (5.34), we get J + J /6 - - E D V(V«E) + ey E(n + p) (5.38) n p n n Now, for a one dimensional model, Eq. (5.37) upon integration gives . sry 7* (J - J /6)dx = eD (n + p) - ~ YT + k' (5.39) o n p n 2 where k' is the integration constant - ^ ~ E2(0) - eDn(n(0) + p(0)) Eliminating (n+p) from Eq. (5.39) by using Eq. (5.38) in one dimensional form, we get •'•••> 2 e ii eD2 4 4 +• U k'E - E 3 + y E / X (J - J /6)dx - D (J +J /5) (5.40) n dx2 n 2 n o n p n n p 81 5.4 .2 Normalized Equations An a l t e r n a t i v e way of expressing the above equations r e s u l t s i f the following normalized v a r i a b l e s : . J J nn A ' pn A ' n d n r n /e 2u d / e y d n v/ eA ' n / eA n v n /ye lu e ' E = -E j~— , V = V ' n n v/dA ' n / , 3 . V n V d A n are s u b s t i t u t e d i n Eqs. (5 .28) , (5.29) and (5.33)-(5.36). The modified equations are then given by J n n ( 0 ) = exp ( /C 2E n (0 ) ) (5.41) or J„n<°> = E ! ( 0 ) exp(-C;/E (0)) nn n 2. n dJ . dJ _ J ™ = U , — E I l = - u (5.42) dx n dx n n n dn nn l dx n n n dP J /6 = C. ~ - + E P (5.44) pn 1 dx n n n dE -r- 2- = n - p (5.45) dx n n n V = f1 E dx (5.46) n o n n 2 3 _ d E E J x ,2 n , , _ n , „ , n CT — r - ^ + kE = -r^ + E / " (J - J / 6 ) d x - C , ( J +J / « ) / c 1 dx^ n 2 n o nn pn 1 nn pn (5 .47; where C, = 1 d y aA n n 82 C =V d 2 y n e and E 2(0) • k = C ^ n (0) + P (0)r 2 I n n For the no d i f f u s i o n case, Eq. (5.47) becomes E 2 ( x ) = E 2(0) + 2 / n (J - J /6)dx (5.48) n n n o n n p n n Eqs. (5.41) and (5.45)-(5.47) represent a general formulation of space charge current flow i n a trap free f i l m (or, when trap e f f e c t s can be neglected) with blocking contacts and f i n i t e generation-recombination rate. The following computations are intended to show that the above model, under c e r t a i n conditions, can e x h i b i t CCNR c h a r a c t e r i s t i c s . 5.4.3 Method of Solution For a given generation-recombination rate function, the s o l u t i o n of the above equation could, i n p r i n c i p l e , be determined such that they s a t i s f y the boundary conditions. To avoid excessive computing time, two r e l a t i o n s may be derived to determine the t o t a l current density and the e l e c t r i c f i e l d at a point, x , i n s i d e the f i l m f or given boundary conditions. Consider the d i f f u s i o n free case: Eq. (5.48) may be w r i t t e n as (subscript n w i l l be omitted) " s » ' f > -sw„-V 4 > ( 5 - w dE * M u l t i p l y i n g Eq. (5.49) by 2 E and i n t e g r a t i n g between x and general point x,we get dE > — = (n -dx  p) = + 4 / / E ( X * } U ^ + I/6)EdE + i ( E - g ) 2 *) (5.50) * *Y E(x) 2 d x x _ x where the sign determines the p o l a r i t y of the net charge. Integrating once more, we have 83 * ^ ,E(x*) EdE x = + J E ( 0 ) fiJu*™. U ( l + l / 6 ) E d E 4 ( E ^ ) 2 *) (5.51) E(x*) 2 d x x t = x The t o t a l current density i s given by J = J (0) + J (0) n p = 2J n(0) + E(0)(P(0) - n(0)) = 2J_(0) + Jl / ( / E ( X A ) U(l+l/6)EdE4(E^|) 2 *) x=x (5.52) E(0) 2 d x x= *For a given i n j e c t e d current from the cathode, ^ n C 0 ) , (or, the it e l e c t r i c f i e l d E(0)) and a boundary condition at the point x , Eqs. (5.51) and (5.52) can be solved to determine the t o t a l current density and the e l e c t r i c f i e l d at x .. The e l e c t r i c f i e l d and charge d i s t r i b u t i o n s may then be determined to give the current-voltage c h a r a c t e r i s t i c s . 5.4.A Symmetrical Case In t h i s case, i t i s assumed that both electrons and holes have s i m i l a r p r o p e r t i e s . The i n j e c t e d hole current at the anode i s taken to be given e i t h e r by Schottky thermionic emission or the Fowler-Nordheim tunneling process i . e . , J p ( l ) - exp( /C 2 E ( l ) (5.53) or, J Cl) = E 2 ( l ) exp ( -c:/E (D) p I Due to symmetry, e l e c t r i c a l n e u t r a l i t y w i l l occur at the middle of * the sample ( i . e . , x=0.5). Thus i f x i s the point at which the n e u t r a l region begins, we have dx'x Assuming u = u and a = a , the generation-recombination rate may n p n p 84 be wr i t t e n as j U _ — f(E) ' 0<:x<a; a<x$l = 0 a*x$l-a (5.54) where f(E) has the same fu n c t i o n a l form of the i o n i z a t i o n r ate. For example, consider the case where . f(E) = a(E/E ) m E z E o o = 0 E < E o Thus, Eqs. (5.51) and (5.52) reduce to * = /E<**> E(x) dE (5.55) E(0) k / f / E m + 1 ( x ) - E m + 1 ( x * ) J = 2J (0) + k*^E m + 1 ( 0 ) - E m + 1 ( x * ) (5.56) n where k = vW(m+l)Ej Eq. (5.55) can be modified to y i e l d f o r the current density 7m o J = B/2 + AB/2) 2 - C where B = 4 J (0) + k 2(E n r i " 1 ( 0 ) - E ^ x * ) ) n C = 4 J 2(0) n Eqs. (5.53), (5.55) and (5.56) have been solved simultaneously to determine J , and E(x ). The voltage drop across the sample can be determined a f t e r s o l v i n g Poisson's equation. The computed V-I c h a r a c t e r i s t i c s , f o r d i f f e r e n t boundary conditions and model parameters, are presented i n Figs. 5.10 and 5.12. The d e t a i l s of computed CCNR c h a r a c t e r i s t i c s c l e a r l y depend on the assumptions made. However, t h e i r b a s ic features are i n general agreement and w i l l probably remain true f o r any reasonable assumptions concerning the ph y s i c a l mechanisms involved. 3 40 60 - 80 90 NORMALIZED VOLTAGE F i g . 5.10 Computed voltage-current c h a r a c t e r i s t i c s : i n j e c t e d c a r r i e r s are taken to be given by Schottky thermionic emission; C„ = 0.1, a = 5, m=8, and E = 50 F i g . 5.11 F i e l d d i s t r i b u t i o n w i t h i n a f i l m having constants of F i g . 5.10 w i t h a =» 0.5 40 60 80 . 90 NORMAL/ZED VOLTAGE F i g . 5.12 Computed voltage-current c h a r a c t e r i s t i c s : i n j e c t e d c a r r i e r s are taken to be given by the Fowler-Nordheim tunneling; H=400, a=*0.2, m=8 and E « 50 NORMALIZED VOLTAGE F i g . 5.13 Computed voltage-current c h a r a c t e r i s t i c s ; i n j e c t e d electrons are taken to be given by Schottky thermionic emission; C =0.1, k=0.4, B«=0.125 and E • 50 2 o 87 5.4.5 Asymmetrical Case In this case, electrons and holes have different properties and the boundary conditions at the cathode and anode are different. Thus, the two electrodes w i l l not be equidistant from the minimum f i e l d point. The position of the minimum f i e l d point depends on the true mechanisms involved. Since i t is believed that thin films exhibiting CCNR contain a neutral region^, i t w i l l be assumed that a charge neutrality region w i l l occur in the space between x and the anode. The generation-recombination rate function may be written as tT A V d J a n ( 1Vp p /Vn n ) ) V A n E Cl+(up /yn )) And i f V n > > u >the above relation may be approximated by U = • \ Thus, we may write y e , Ja n d n A n U = -| f (E) 0 $ x < a = 0 elsewhere For the case where f(E) i s given by f(E) = <xexp(3E) E >, E Q = 0 E < E o Eqs. (5.51) and (5.52) reduce to where x * = . / E ( X } E ( x ) d E (5.57) E ( 0 ) k/f /exp( E(x)) - exp (BE (x*)) J = 2J n(0) + k /J /exp(BE(0) - exp(3E(x*)~) (5.58) k « /(2(1 + l/6)a)/B F i g . 5.13 shows the computed V-I c h a r a c t e r i s t i c s u s i n g Eqs. (5.57) and (5.58). The i n j e c t e d e l e c t r o n s have been assumed to be given by Schottky thermionic emission. The constants used i n computing F i g . 5.13 have been chosen to give c h a r a c t e r i s t i c s s i m i l a r to those obtained f o r the symmetrical case. However, one would expect that a major d i f f e r e n c e between t h i s case and the symmetrical case would he a great r e d u c t i o n i n the absolute value of the current d e n s i t y . 5.4.6 D i s c u s s i o n 6 I t i s b e l i e v e d that some p h y s i c a l processes r e s p o n s i b l e f o r CCNR c h a r a c t e r i s t i c s i n t h i n f i l m s are p o s i t i o n dependent i n a d d i t i o n to being f i e l d dependent. In the present model the generation-recombination r a t e i s taken to be a p o s i t i o n dependent process. However, other processes and f i l m constants could be e q u a l l y p o s i t i o n dependent. A model, f o r a f i l m w i t h b l o c k i n g contacts and c o n t a i n i n g a p o s i t i o n dependent generation-recombination r a t e , has been developed w i t h the a b i l i t y to e x h i b i t CCNR c h a r a c t e r i s t i c s . The computed r e s u l t s have the f o l l o w i n g f e a t u r e s i n agreement w i t h experimental r e s u l t s : (1) 'The t h r e s h o l d v o l t a g e depends g r e a t l y on the model parameters. Such dependence could e x p l a i n the v a r i a t i o n of V ^ from sample to sample, because the generation-recombination r a t e could be s e n s i t i v e to the p r e p a r a t i o n c o n d i t i o n s . (2) The s u s t a i n i n g v o l t a g e i s almost constant and i s r e l a t e d , i n the present model, to the value of the e l e c t r i c f i e l d below which the generation r a t e i s n e g l i g i b l y s m a l l . (3) The re d u c t i o n of the t h r e s h o l d v o l t a g e by white i l l u m i n a t i o n could be ex p l a i n e d i n terms of i n c r e a s i n g the generation r a t e . (4) Recombination r a d i a t i o n i s to be expected s i n c e one of the b a s i c mechanisms i n v o l v e d i n the present model i s the e l e c t r o n - h o l e recombination. Recombination processes could be inter b a n d recombination and/or through recombination centres. 5.4.7 Small S i g n a l E q u i v a l e n t C i r c u i t S p a t i a l charge d i s t r i b u t i o n obtained by using the present model ( F i g . 5.11) suggests that a f i l m could be d i v i d e d i n t o three d i s t i n c t regions. Regions one and three c o n t a i n net p o s i t i v e and negative charge r e s p e c t i v e l y , w h i l e r e g i o n two i s a n e u t r a l one. Space charges d e n s i t y 0^ and are not l i n e a r l y p r o p o r t i o n a l to V- and V. and thus, i t i s more u s e f u l to de f i n e d i f f e r e n t i a l capacitances: C l d = dv7 (5.59) and dQ, :3d = dV. (5.59) In the same way we de f i n e d i f f e r e n t i a l conductances dJ ' i d " dV, i = 1, 2, 3 (5.60) 90 now, i f the applied voltage i s a small ac s i g n a l superimposed on large dc b i a s , i . e . , > V = V + v exp'Cjut) V « V o o the current modulation around i t s dc steady state J q has two components: One i s i n phase and the other out of phase with the applied ac s i g n a l . In region one, the two components are given by J l r = G l d \ dQ dv 1 J l q dt Id dt S i m i l a r l y f o r region three j 3 r = G 2 d *3 dQ 3 dv 2  j 3 q = "dt = C 3 d ~ d t while f o r region two, the capacitive component i s given by dD e d v 2 , c  3 2 q d t ' d j I t ( 5 , 6 1 ) dv = C ——• dt where C a e/d 2 which i s generally small compared with or C ^ . The r e s i s t i v e component i s given by J 2 r = G 2 d V2 The above considerations lead to the small s i g n a l equivalent c i r c u i t shown i n F i g . 5.14 91 hr R ' d j C J3r J2r R2d 1q C Id J 3q C 3d F i g . 5.14- Small ac s i g n a l equivalent c i r c u i t , The equivalent s e r i e s resistance and capacitance of the above c i r c u i t , n eglecting C, are given by 2 2 2 2 . R. 0 + ai . T- R O J + oi T„RT , s ~ . , 2, 2 , 2> . 4 2 2 K 2 d 1 + 0) Ct^ + T ^ ) + 0) T ^ T ^ (5.62) .-1 _ "'^ Ad + T 3 R 3 d ) + MS T3 ( T3 Rld + T l R 3 d ) "s • . 2. 2 . 2. , 4 2 2 1 + 0) ( l ^ + T 3 ) + 0) T 3 (5.63) where T l " R l d C l d T 3 ~ R 3 d C3d R l d + R 3 d 92 The s e r i e s equivalent resistance tends to a constant value as the frequency tends to zero or i n f i n i t y : Lim R (co) = R. , + R 0 , + R 0 J (5.64) s Id 2d i d or»-o Lim R ".(to) = R„, (5.65) s 2. d and generally, we have R (o) V R (CO) V R (») (5.66) s s s The present model allows for a negative (see F i g . 3.11). Thus, the sample could e x h i b i t NDC over a range of frequencies. The magnitude of the capacitance C w i l l determine the upper frequency l i m i t . 5.5 Formation of Current Filaments Due to Radial D i f f u s i o n 73 Ridley has shown that bulk diodes e x h i b i t i n g CCNR c h a r a c t e r i s t i c s form current filaments i n an attempt to achieve e l e c t r i c a l s t a b i l i t y . Current 54 filaments have been observed experimentally i n - t h i n f i l m chalcogenide diodes The filament shape depends on the actual mechanisms involved, e.g., Joule 74 heating (Sec. 5.2), r a d i a l d i f f u s i o n , r a d i a l d r i f t due to r a d i a l f i e l d and space charge e f f e c t s . The problem of filament formation i s a d i f f i c u l t one and only an approximate analysis may be p o s s i b l e . In t h i s s e c t i o n , an approximate treatment of current filament due to r a d i a l d i f f u s i o n w i l l be presented. Eqs. (5.29) and (5.33)-(5.35) may be w r i t t e n i n the normalized form as V.-J = U (5.67) n VvJ * -U (5.68) P 93 J = -C*vn + En (5.69) n 1 J = C'Vp.+ Ep (5.70) P 1 V-E = n - p (5.71) Where C| i s the r a d i a l d i f f u s i o n constant, n e g l e c t i n g the l o n g i t u d i n a l d i f f u s i o n . The above set of equations can be combined to y i e l d a d i f f e r e n t i a l equation d e s c r i b i n g the current d i s t r i b u t i o n i n the r a d i a l d i r e c t i o n ( i . e . , the f i l a m e n t shape). To o b t a i n t h i s equation, s u b t r a c t Eq. (5.68) from Eq. (5.67) to get V - ( J - J ) = 2U n p S u b s t i t u t i n g from Eqs. (5.69) and (5.70) and assuming that J = (n+p)E, we get Now, i f and and s i n c e we get -CJ_ V 2 ( | ) + V-(E(n-p)) = 2U (5.72) J - J ( r ) E - E(x) V = V> + V' x r V 2(J/E) = J V*(l / E ) + (1/E)V 2(J) (5.73) S u b s t i t u t i n g Eq. (5.73) i n Eq. (5.72), and i n t e g r a t i n g between the sample ends ( i . e . , 0.0 and 1.0) assuming that n(0) << p(0) and p ( l ) « n ( l ) and using Eq. (5.54) f o r U, we get V 2 J + JiU1 V 2 ( l / E ) d x +'|r( / 1 ( f ( E ) / E ) d x - l ) ] / ( / 1 ( l / E ) d x ) } = 0 (5.74) r o x l 0 Since the term i n s i d e the bracket could be a p o s i t i v e or negative quantity we may write Eq. (5.74) as Cj V*J = ( J / T L ) - J / T 2 ) (5.75) A ^ = - J / T o $ r $ r (5.76a) - c A T / *. • elsewhere (5.76b) - J/x + To determine the filament shape, Eq. (3.74) may be solved with the boundary conditions (V J ) = 0 and (V J ) -> 0 (5.77) r r=o r r-*" However, only approximate expressions for the functions involved may be determined. In the remaining part of t h i s s e c t i o n , we w i l l assume T_-and T are functions of the current density J . Rectangular co-ordinates w i l l T" be used. Table 5.2 has been obtained for preaasumed current dependence of both Tj and T + . The complete filament shape could be determined by matching current d e n s i t i e s and t h e i r d e r i v a t i v e s at y=y . Several features of current • c filaments are shown i n F i g . 5.15 which has been computed for the case i n which both T_ and T+- are constant. The current density d i s t r i b u t i o n i s given by J x ( y ) = J Q c o s ( y / v ^ J J ) o $ y ^ y c J x ( y ) = ( J Q f/2)exp((yc~y)/ /C^B) elsewhere With tan ( y c / / C ] B ) = 1 Solution of the equation dy 2 » T * ' ' ;'Vd.y-y-0 = 0 Sol u t i o n of the equation d J _ J , d J s 2 " C ' T * J Mv ; 0 d y 1 y - * » J(y) = J q cos(y// Cj&) J(y) = J b e x p ( - ( y - y c ) / / q ? ) 3 / ^ J /(M/2+l)C1'g j- o 1 J(y) j M+2 ^K+2 o ^ J(y) 11+(7-7 c ) J b 7 2(M//(2M+4)C^B)] 2/M 8J/exp(oJ) J(y) = J Q + log e(l+/l-exp(a(J-J O)) -y//2^Cj? sec """(expC"! J b))-sec _ 1(exp(-| J(y)) = /a/2C|8 (y - y c> a, 8 and M are constants. J q and are the values of current d e n s i t i e s at y=0 and y=y c r e s p e c t i v e l y . TABLE 5.2 A n a l y t i c a l Expressions f or Filament Shape J/Jo 1.0 0.5 0 0.5 1.0 Y/nV^ F i g . 5.15 S p a t i a l v a r i a t i o n of current density through the current filament. I t i s apparent that the current density i s almost constant around the filament centre and then drops r a p i d l y to a diminishing value w i t h i n a few c h a r a c t e r i s t i c lengths (/cjT), and that the width of the current filament i s proportional to the r a d i a l d i f f u s i o n constant. 5.6 Memory State Formation Experimental r e s u l t s i n d i c a t e that a memory state can be r e a l i z e d by passing high current density through the sample i n the low r e s i s t i v i t y state so that a c r i t i c a l current density i s exceeded. This c r i t i c a l current density depends on the composition, sample thickness, ambient temperature and perhaps other parameters. In order to explain the memory state, which i s stable under zero 54 ' b i a s , Pearson has postulated some kind of semipermanent change i n the sample str u c t u r e . I t seems reasonable to assume that the passage of high current, e s p e c i a l l y i f the conduction current concentrates i n a narrow filament, r e s u l t s i n enough Joule heating to allow s t r u c t u r a l changes (e.g., phase separation or amorphous to c r y s t a l l i n e transformation). I f the new phases are of low r e s i s t i v i t y and tend to form a continuous channel ( i n the p o s i t i o n of the current filament) between the two metal electrodes, the low resistance channel that had been formed would be stable at zero b i a s . Thus, the conducting state w i l l p e r s i s t without noticeable decay. The re-establishment of the high r e s i s t a n c e state by passing a high current pulse i n the memory state could be explained by p o s t u l a t i n g that the high pulse w i l l r e s u l t i n fusion of some spots i n the conducting channel that had been formed. Now, i f the current ceases abruptly, these fused spots would be quenched to r e i n s t a t e the amorphous high r e s i s t a n c e state i n the sample. On the other hand, i f the current i s decreased slowly, the slow cooling of the fused spots would r e s u l t i n r e c r y s t a l l i z a t i o n or phase Separation to reform the continuous conducting channel and thus the sample would remain i n the memory state. Thus, memory switching could be considered as due to s t r u c t u r a l changes brought about by high e l e c t r i c f i e l d s and Joule heating i n filaments. 5.7 Summary Although sample s e l f - h e a t i n g could account f o r delay times observed experimentally, the fa s t switching i n t h i n chalcogenide semiconductor films i s b e l i e v e d to be e l e c t r o n i c i n nature. Thus, s e l f - h e a t i n g serves only to i n i t i a t e an e l e c t r o n i c switching mechanism. A model for CCNR due to space charge formation i s proposed. The i n j e c t e d current c a r r i e r s could develop a space charge b a r r i e r near the electrode by impact i o n i z a t i o n . The concept of p o s i t i o n dependent generation-recombination rate i s discussed. The p o s i t i o n dependent processes could represent the actual p h y s i c a l s i t u a t i o n . The computed r e s u l t s , using the proposed model, show many features which are i n agreement with the experimental r e s u l t s , e.g., the dependence of threshold voltage on model parameters and the constancy of the sustaining voltage. Conduction through diodes e x h i b i t i n g CCNR c h a r a c t e r i s t i c s i s of the filamentary type. A d i f f e r e n t i a l equation i s derived to describe the filament shape. The memory state can be explained i n terms of a phase change mechanism due to excessive heating which could produce a hig h l y conducting channel. To destroy the memory sta t e , the conducting channel must be i n -terrupted at some spot. • 6. Filamentary Breakdown i n Thin Anodic Films 6.1 Introduction D i e l e c t r i c breakdown could occur i n t h i n anodic films before CCNR 57 c h a r a c t e r i s t i c s can be observed i f the the forming process described i n Sec. 5.1 has not been performed. D i e l e c t r i c breakdown i s often c l a s s i f i e d 75 76 as e l e c t r i c ( i n t r i n s i c or avalanche) or thermal breakdown ' . This i s a loose c l a s s i f i c a t i o n and i t corresponds to the t h e o r e t i c a l solutions of the problem of determining the breakdown strength. The breakdown event i n t h i n 77 f i l m s seems to occur i n two stages, namely : (1) the establishment of a conducting channel between the electrodes and (2) the discharge of the sample's stored energy through t h i s channel. The second stage can be explained i n terms of the heating and evaporation of the d i e l e c t r i c and the associated counter electrodes, but the main problem i s i n i d e n t i f y i n g and i n t e r p r e t i n g the conducting channel formative processes. I t seems probable that more than one process i s operative i n developing the conducting channel. The p r i n c i p a l theories of breakdown have concentrated only on the in c e p t i o n of breakdown. Experimental studies using samples with s e l f - h e a t i n g breakdown provide a b e t t e r opportunity for understanding the dest r u c t i v e phase and a p o s s i b i l i t y of d i s t i n g u i s h i n g between f i l m e l e c t r i c a l properties governed by weak spots or bulk conditions. The present study i s concerned mainly with the breakdown i n t h i n oxide f i l m s prepared a n o d i c a l l y . I t i s confirmed experimentally that filamentary conduction takes place j u s t before the onset of breakdown, but i t i s rather d i f f i c u l t to i d e n t i f y the process responsible. Some of the po s s i b l e p h y s i c a l processes (electron avalanche, c o l l e c t i v e e l e c t r o n , i o n i c transport and thermal runaway) are discussed. The conditions for thermal runaway of an 100 incomplete channel are i n v e s t i g a t e d . The discharge of a channel i s observed e x p e r i m e n t a l l y and the r a t e of v o l t a g e c o l l a p s e i s accounted f o r i n terms of the sample capacitance d i s c h a r g i n g through the channel. 6.2 E l e c t r i c Breakdown Theories E l e c t r i c breakdown theories'^'76,78 e i t h e r of the avalanche or c o l l e c t i v e e l e c t r o n type. C o l l e c t i v e e l e c t r o n t h e o r i e s assume that breakdown i s i n t r i n s i c , i . e . , independent of cathode m a t e r i a l and sample t h i c k n e s s . They assume that breakdown occurs when the r a t e of g a i n of energy by conduction e l e c t r o n s from the a p p l i e d f i e l d , A, i s more than the r a t e of l o s s of energy, B, which may be due to electron-phonon i n t e r a c t i o n s and, i n an imperfect c r y s t a l , by c o l l i s i o n s w i t h trapped e l e c t r o n s and l a t t i c e defects, i . e . , A(E, e, T O ) * B(e, T q ) (6.1) where E i s the e l e c t r i c f i e l d . e i s the e l e c t r o n energy. T i s the ambient ( l a t t i c e ) temperature, o Thus no s t a b l e conduction e l e c t r o n d i s t r i b u t i o n can e x i s t when the e l e c t r i c f i e l d exceeds a c r i t i c a l v a l u e ; and when t h i s i s exceeded the conduction e l e c t r o n d e n s i t y i n c r e a s e s c a t a s t r o p h i c a l l y . To be able to c a l c u l a t e the breakdown s t r e n g t h , a c r i t i c a l f i e l d s t r e n g t h c r i t e r i o n i s needed. F r o h l i c h proposed that the c r i t i c a l f i e l d s t r e n g t h , E^, i s the h i g h e s t at which balance is p o s s i b l e , i . e . , A ( E c , 1, T q) = B ( I , T q ) (6.2) where I i s the i o n i z a t i o n energy. This i s known as the h i g h energy c r i t e r i o n in c o n t r a s t to the low energy c r i t e r i o n proposed by von H i p p e l and developed by C a l l e n . It- i d e n t i f i e s the e l e c t r o n energy E of Eq. (6.1) w i t h e', 101 that energy f o r which B(e, T ) i s a maximum with respect to E . Thus, the c r i t i c a l f i e l d strength i s given by A( E c , e', T o) - B ( e ' , T o) (6.3) The high energy c r i t e r i o n y i e l d s values of e l e c t r i c strength equal to about. 50% of the corresponding low energy values. Avalanche theories assume that the loss of i n s u l a t i n g properties i s caused by a large increase i n the number of conduction electrons s u f f i c i e n t l y large to destroy the stru c t u r e . The i n i t i a t i o n of e l e c t r o n avalanche may be due to f i e l d emission, by quantum tunneling from valence to conduction band (Zener e f f e c t ) , or by c o l l i s i o n i o n i z a t i o n . The presence of space charge and cathode i n j e c t i o n may a f f e c t the avalanche development. The s i n g l e e l e c t r o n theory, or the f o r t y generations theory, developed by Se i t z has been used frequently f o r the i n t e r p r e t a t i o n of experimental r e s u l t s . It assumes that a s i n g l e e l e c t r o n at the cathode s t a r t s an avalanche by impact i o n i z a t i o n . S e i t z determined the c r i t i c a l avalanche s i z e at the anode considering the energy t r a n s f e r to the l a t t i c e , and concluded that melting 12 would r e s u l t i f the avalanche contained 10 ele c t r o n s , i . e . , an ele c t r o n must take part i n about 40 i o n i z i n g c o l l i s i o n s i n tr a v e r s i n g the specimen, i r r e s p e c t i v e of i t s thickness. The p r o b a b i l i t y , P, that the e l e c t r o n energy increases from average energy, e , to the i o n i z a t i o n energy, e T, assuming no phonon s c a t t e r i n g , i s given by e I P = exp<- / ±fe ) _ 1 de) (6.4) av T dt where T i s the mean time between two i n t e r a c t i o n s . Since f o r an ele c t r o n not undergoing c o l l i s i o n s , de /„ x1/2 eE e v dt = ( 2 w e ) < 6 - 5 ) Eq. (6.4) can be reduced to P = exp(-H/E) (6.6) where v 2e av e x and the e l e c t r i c f i e l d strength i s given by E = ^ — r r i r- (6.8) c ln(d/E cyxY) where d i s the f i l m thickness, y i s the average m o b i l i t y . y i s the c r i t i c a l number of generation (= 40). The growth of the avalanche i s a chance event, i t depends on the p r o b a b i l i t y P^ o f . i n j e c t i o n of electrons at the cathode and the p r o b a b i l i t y P 2 of an avalanche growing i n excess of the s i z e required f o r destruction. Thus, the s t a t i s t i c a l time l a g i s given by T g - ( P ^ r 1 (6.9) which decreases r a p i d l y when the applied f i e l d exceeds the c r i t i c a l f i e l d strength. 79 O'Dwyer calculated the f i e l d produced by p o s i t i v e charges l e f t behind i n the i n s u l a t o r by the ele c t r o n avalanche. His c a l c u l a t i o n s have 13 shown that the f i e l d opposing the applied f i e l d would be 10 Vr/m f o r a des-12 t r u c t i v e avalanche containing 10 electrons, spread over an area of about -9 8 10 m, while the d i e l e c t r i c breakdown occurs at about 10 V/m. In an attempt to avoid such d i f f i c u l t y , O'Dwyer has developed a space charge modified f i e l d emission theory. The s t a r t i n g point of h i s theory i s again Eq. (6.6) but he imposed the current c o n t i n u i t y condition which r e s u l t s i n a p o s i t i v e space 103 charge region near the cathode and thus a dependence of the d i e l e c t r i c strength on the sample thickness. I t should be noted here, that the current c o n t i n u i t y equation need not be sustained i n the unidimensional con-f i g u r a t i o n which O'Dwyer used, e s p e c i a l l y when a conducting channel i s set up through the d i e l e c t r i c f i l m . The e f f e c t of cathode i n j e c t i o n has been considered by F o r l a n i and 80 Minnaja . They assumed that the current i n j e c t i o n at the cathode d i e l e c t r i c i n t e r f a c e i s governed by Fowler-Nordheim tunneling mechanism. The i n j e c t e d current, 1^, increases by c o l l i s i o n i o n i z a t i o n through the d i e l e c t r i c to a value at the anode, I , given by 3-I. - I exp a(E)d (6.10) A C 3/2 = AE 2 exp(- + a ( E ) d ) (6.11) where a(E) i s the i o n i z a t i o n rate. <f> i s the cathode work function. A and B are constants. The c r i t i c a l f i e l d strength w i l l be given by the zero of the exponent, I , E " JJ J. 3/ 2 E = • (6.12) c a(E )d c And for the case where the f i e l d strength i s so high that the l a t t i c e v i b r a t i o n has no e f f e c t on the process of c o l l i s i o n i o n i z a t i o n , the rate of c o l l i s i o n i o n i z a t i o n i s thus the r e c i p r o c a l of the distance t r a v e l l e d by a free e l e c t r o n i n a t t a i n i n g energy I, i . e . , a(E) = ^ (6.13) 104 And thus, 4 (2m) 1/2 (6.14) Where "h i s Planck's constant. A s i m i l a r expression with d i f f e r e n t thickness dependence can be derived by taking the i n j e c t i o n current to be governed by Schottky thermionic emission. seems to be able to account f o r , even q u a l i t a t i v e l y , the experimental r e s u l t s . Each theory disagrees with one or more of the observed dependences on experi-mental conditions and sample parameters such as sample thickness and ambient temperature dependence and possible electrode dependence of breakdown strength. I t seems more probable that more than one of the above mechanisms i s operative at a time, so that the actual mechanism i s more complex and changes considerably with changes i n the condition of the experiment. 6.3 Breakdown Tests: Experimental Procedure 6.3.1 Sample Preparation Ta20j., A^O^, Nb£^5 a n d T*^2 c a P a c i t o r s 2 mm i n diameter were pre-pared by the techniques summarized i n Table 6.1. To ensure s e l f - h e a l i n g break-down the metal counterelectrode thickness was, i n a l l cases, l e s s than 1000 A such.that i t evaporates at the breakdown s i t e s before sample destruction occurs. Ta^O^ and M^^ O,. f i l m thicknesses were estimated using a model 14-Cary spectro-photometer making no allowance for the d i f f e r e n c e i n substrate. ^ 2 ^ 3 a n d f i l m thicknesses were estimated from bridge measurements assuming = 8.8 f o r 2 A^O-j and e r = 50 for H O 2 . A Sloan angstroraeter was used to estimate counter-electrode thickness. F i n a l l y , i t should be pointed out that no one of the above theories o Ta/Ta 20 5/M A l / A l 2 0 3 / M Nb/Nb205/M T i / T i 0 2 / M Base Metal 1. Used metal 2. Metal surface preparation sputtered Ta Fi l m Evaporated A l Fi l m * Nb high p u r i t y sheet chemically polished i n . an HF-HN03 s o l u t i o n (3:1 by value %) T i high p u r i t y sheet chemically polished i n an HN0o H„S0,-HF-11 0 3 2 4 2 Sol u t i o n (2:1:1:1. by value %) D i e l e c t r i c F i l m GrOwth 3. Anodizing e l e c t r o l y t e 4. current density 5. F i n a l voltage i s kept constant f o r 0.5%(by va l u e ) H 2 S 0 4 • 2 0.5 roA/ cm one hour 5%(by weight)H PO^ s o l u t i o n concents-rated with concent-rated NH .OH 0.5 mA/cm 20 minutes 0.5%(by volume)H„S0, 2 4 2 0.5 mA/cm one hour Saturated ammonium borate i n ethylene g l y c o l 2 0.5 mA/cm one hour Counterelectrodes 6. Counterelectrode preparation tech- . nique ** evaporated Au, A l , In evaporated Au,Al evaporated Au evaporated Au * Evaporation was c a r r i e d out at 10 t o r r or l e s s using conventional b e l l j a r system Veeco -(400) ** Counterelectrode areas were defined by using photoetched b e r y l l i u m copper masks. Table 6.1 Sample Preparation techniques i—• o Cn 106 6.3.2 E l e c t r i c a l Measurements Prebreakdown conduction currents were measured using a Ke i t h l e y type 417 high speed picoammeter. Breakdown strength was measured using a ramp wave form from the sweep c i r c u i t of a Tektronix model 515A o s c i l l o s c o p e . The ramp rate was adjusted such that the applied e l e c t r i c f i e l d increased by about 0.5 MV/cm sec. When the breakdown event took place, the breakdown pulse generated by the abrupt change i n the sample voltage was amplified, shaped and fed i n t o the gate terminal of a s i l i c o n c o n t r o l l e d r e c t i f i e r (SCR) which, on tr i g g e r i n g , short c i r c u i t e d the sample. The SCR remained i n the high conductivity state u n t i l the applied voltage dropped to zero. Then another ramp was applied to the sample. Voltage collapse was observed during the breakdown event, on applying a s i n g l e pulse generated by an HP model 214A pulse generator, with a Tektronix type 581 o s c i l l o s c o p e and photographed with P o l a r o i d type 410 f i l m . Most measurements were c a r r i e d out i n a i r at room temperature with the sample i n an e l e c t r i c a l l y shielded dark chamber. For temperature dependence measurements, the temperature was c o n t r o l l e d by pl a c i n g the sample i n a Statham SD6 oven which allowed f o r a temperature range between -40°C and 150°C. 6 .4 Breakdown Tests: Experimental Results arid Discussion 6.4.1 Prebreakdown Conduction Dc conduction currents through amorphous d i e l e c t r i c films f o r 4 e l e c t r i c f i e l d s i n excess of 10 V/m often approximate to the form J = C exp(-(<f> - B E 1 / 2 ) / k T ) (6.15) Where J i s the current density. E i s the average e l e c t r i c f i e l d , k i s Boltzmann's constant. C, <j> and B are constants. 107 Fi g s . 6.1 and 6.2 show Schottky p l o t s for la/la^O^/kn samples and the temperature dependence of the dc conduction current r e s p e c t i v e l y . Even though these samples obey Eq. (6.15) up to f i e l d strengths close to the breakdown strength, the values of C, d> and 8 depend strongly on the applied f i e l d p o l a r i t y . The observed r e c t i f i c a t i o n can be a t t r i b u t e d to the r e l a t i v e ease of e l e c t r o n i n j e c t i o n when the tantalum electrode i s negative. Apart from the d i f f e r e n c e between the two metal work functions, the presence of io n i z e d p o s i t i v e defects near the tantalum electrode, which can be due to excess tantalum ions at the metal oxide i n t e r f a c e , may enhance current i n j e c t i o n when the tantalum electrode i s negative. The energy band diagram given i n F i g . 6.3 shows c l e a r l y that the energy b a r r i e r at the cathode i s lowered only when the p o s i t i v e l y charged defects are near the cathode. The agreement between experimental r e s u l t s and Eq. (6.15) i n d i c a t e s that the conduction process i s a thermally activ a t e d e l e c t r o n i c process rather than an i o n i c process which, i f present, would e x h i b i t an exponential voltage-current c h a r a c t e r i s t i c . Measurements of sample current versus ramp voltage (or, time) up to the point of the breakdown are shown i n F i g . 6.4. Such currents are given by I » I + c'|r (6.16) c dt Where I i s the conduction current, c C i s the sample capacitance. V i s the voltage. For a ramp voltage, the charging current i s constant while the conduction current v a r i e s with the voltage magnitude. Current r e c t i f i c a t i o n i s again c l e a r i n F i g . 6.4; the r a t i o between the conduction current at the onset 108 (og}Q(J/AMP cm'2)' -8--10. • • • i . / J A d=1050 A / / 9 = 1820 A / 9 A / T ® / / / Jo-i i "' i — i i 1 i , -10. * ^ -8 I I I \Z^,VOLTY2 F i g . 6.1 Schottky p l o t f o r Ta o0_ films Fig. 6.2 Temperature dependence of conduction current: applied voltage- 25V; Film thickness (Ta.,0,.) - 1820 A) F i g . 6 . 3 Energy band diagram for a met a l / d i e l e c t r i c / m e t a l showing the e f f e c t of a p o s i t i v e ion at a depth of a. The d i f f e r e n t b a r r i e r shapes are for d i f f e r e n t impact radius r (Ref. 81). 0-4. Ta + i i 1 m AMP cm'2. J r a -120 40 r 2 1 4. 40 120 • VOLTS 2 - ju AMP cm Fig. 6 . 4 Current-voltage c h a r a c t e r i s t i c s of a oTa 20_ f i l m using an applied ramp voltage ( f i l m thickness « 2050 A) 110 of breakdown when the tantalum electrode i s negative to that when i t i s 3 p o s i t i v e i s about 10 . F i g . 6.4 also i n d i c a t e s that the t r a n s i t i o n to the breakdown condition occurs without a current precursor. 6.4.2 Filamentary Breakdown The following experiments performed on s e l f - h e a l i n g samples were intended to demonstrate the existence of filamentary breakdown i n anodic oxide f i l m s . : 6.4.2.1 O p t i c a l Microsope Observations By using s e l f - h e a l i n g samples a d i s t i n c t i o n can be made as to whether the destruction of a sample occurs over the whole of i t s area or i n a narrow filamentary c h a n n e l , since the t h i n metal electrode evaporates at the breakdown s i t e s . F i g . 6.5 shows c l e a r l y that breakdown occurs over a l o c a l i z e d s i t e of the tested sample. Breakdown patterns as shown i n F i g . 6.5 contain a c e n t r a l region of complete destruction of the three layers of the sample surrounded by a region of heavy, but l e s s e r , damage. The area evaporated from the metal electrode was concentric with the c e n t r a l region and i t s spread depended on 'the metal f i l m thickness. Single channel breakdown can be observed only when the measuring c i r c u i t time constant i s l a r g e r than the duration of the breakdown event, v i r t u a l l y no energy i s supplied by the external c i r c u i t . On the other hand destruction over a l a r g e r area than that of a s i n g l e channel can be observed i f the voltage source i s allowed to supply energy to the sample during breakdown. This breakdown i s pos s i b l y triggered by a s i n g l e channel breakdown. As K l e i n and 82 Gafni noted i n t h e i r studies on s i l i c o n oxide f i l m s , there may be two modes of propagation. In the f i r s t mode (Fig.. 6.6A^single channel breakdowns occur at adjacent s i t e s because the temperature r i s e and mechanical damage at the (A) (B) F i g . 6.6 Extended breakdown patterns (lOy m/div) previous s i t e aid the breakdown at the new s i t e when s u f f i c i e n t energy i s supplied from the supply. In the second mode (Fig. 6.6B), an arc destroys the upper electrode, the arc burning for as long as the supply can maintain i t . 6.4.2.2 Observation of Voltage Collapse The p o s s i b i l i t y that a sample can be recharged a f t e r the occurrence of breakdown events i s another i n d i c a t i o n that breakdown i s a filamentary type. This can e a s i l y be observed on an o s c i l l o s c o p e . T y p i c a l voltage-time oscillograms on the- a p p l i c a t i o n of s i n g l e rectangular pulses are presented i n F i g . 6.7 f o r one sample. These o s c i l l o -grams show c l e a r l y that the rate of breakdown occurence increases with i n -creasing pulse magnitude and that a l l breakdown events s t a r t when the voltage magnitude across the sample reaches a w e l l defined voltage, V^, confirming that the observed breakdown strength i s that of the bulk d i e l e c t r i c rather than of weak spots. I t also shows that breakdown terminates at another defined voltage, V . , ( f o r p o s s i b l e explanation see Sec. 6.6.2). Thus, the energy mm . dissipated i n each breakdown event i s given by e = |c(V 2 - V 2 ) (6.17) l b mm provided that no energy i s supplied by the external source. The voltage waveform on a s i n g l e breakdown event i s shown i n F i g . 6.8i The oscillogram shows that the t r a n s i t i o n from prebreakdown to breakdown con-duction occurs without any current precursors. The current density at the onset 6 2 of breakdown i s as high as 10 Amp/cm and i t s t a r t s to decrease as the threshold f o r the. cessation of breakdown i s approached. I t was found that, i n a l l cases, the voltage collapse takes place i n a time period l e s s than 200 nanoseconds. F i g . 6.7 Voltage waveform across a sample. The number of breakdown events increase as the magnitude of the applied pulse increases. H o r i -zontal scale 10~5 sec/div and v e r t i c a l scale 15 V/div. I 1 1 1 1 1 1 I I 1 M i l 1 • ! 1 l i l t url 1 1 -M i l 1 1 I r 1 1 — • ' 1 111V 1111 | | | | " t i l l i i • I 1 1 I f l i t ! I I 1 f I I I I H F i g . 6.8 Voltage collapse on sin g l e breakdown event. Horizontal scale 2 x IO"? sec/div and v e r t i c a l scale 15 V/div. 114 6.4.2.3 Possible P h y s i c a l Mechanisms for Channel Formation In view of the facts that breakdown occurs at l o c a l i z e d spots, that the rate of occurrence increases with increasing applied voltage and that thermal breakdown can not account f o r f a s t breakdown events i n the case of high r e s i s t i v i t y materials (Sec. 6.5), the breakdown must be associated with e l e c t r i c processes which can concentrate the conduction current i n t o a narrow filament. The concentration of power d i s s i p a t i o n within a narrow region could lead to instantaneous breakdown or, at l e a s t i n i t i a t e a thermal runaway which would eventually lead to a breakdown event. Some of the p o s s i b l e e l e c t r i c processes responsible for channel formation w i l l now be discussed, (a) E l e c t r o n i c Avalanche The formative processes could be an e l e c t r o n i c avalanche, sustained po s s i b l y by f i e l d emission or Schottky emission at the c a t h o d e / d i e l e c t r i c f i l m i n t e r f a c e . Due to v a r i a t i o n s i n the electron e m i s s i v i t y at the cathode, i t seems probable that one p a r t i c u l a r s i t e w i l l eventually dominate so that a channel of high conductance w i l l be formed. I f the e l e c t r o n i c avalanche i s not s u f f i c i e n t to cause instantaneous breakdown, i t could nevertheless r e s u l t i n a temperature increase of a few hundred degrees centigrade i n the channel. A simple analysis based on the assumption that the channel i s a uniform c y l i n d e r of c r o s s - s e c t i o n a l area A^ and that no heat conducts away from the channel, gives for the temperature r i s e AT the following r e l a t i o n AT = T, - T = ^ ~ - (6.18) 1 o cpA . c where n i s the number of electrons i n the avalanche, c and p are the f i l m s p e c i f i c heat and density r e s p e c t i v e l y . This temperature r i s e could i n i t i a t e thermal runaway and breakdown would occur a f t e r a delay time which would depend on sample and channel parameters. 115 (b) CCNR C h a r a c t e r i s t i c s Any diode e x h i b i t i n g CCNR c h a r a c t e r i s t i c s tends to form a current filament i n an attempt to reach e l e c t r i c a l s t a b i l i t y . The excessive Joule heating i n the channel could i n i t i a t e a breakdown event (Chapter 5). (c) Molecular D i s s o c i a t i o n A high applied f i e l d approaching the formation f i e l d could disrupt 83 84 the chemical bonds between f i l m constituents ' . The product of d i s s o c i a t i o n at a s t a r t i n g point w i l l help i n forming the channel: the produced ions can cause f i e l d d i s t o r t i o n and the presence of energetic ions and electrons could break more bonds and thus, the accumulative process could f i n a l l y set up a conducting channel between the two electrodes. The channel temperature w i l l depend on the t o t a l charge passed and the heat produced due to the breaking of chemical bonds. (d) L o c a l i z e d Defects In order to show that the presence of a defect i n s i d e the f i l m may lead to a high l y conducting channel, consider F i g . 6.9 i n which D represents a defect, presumably of low r e s i s t i v i t y . (a) (b) F i g . 6.9 Channel development due to a l o c a l i z e d defect. The concentration of conduction current i n the weak spot can expand the low resistance region by Joule heating (Fig. 6.9b). The breakdown would occur when the maximum temperature i n the channel reaches some c r i t i c a l value. 6.4.3 Breakdown Strength of Ta^O^ Films The breakdown voltage, , i s defined as the average breakdown voltage measured as out l i n e d i n Sec. 6.3.2, a f t e r removing a l l weak spots i n the f i l m . The breakdown f i e l d strength, , i s then given by = V b/d (6.19)' 6.4.3.1 Thickness Dependence of Breakdown Strength The knowledge of the thickness dependence of breakdown strength could be of considerable importance i n determining the breakdown mechanism, however i t does not enable a d e f i n i t e conclusion to be drawn, e.g., i n t r i n s i c and impulse breakdown should be thickness independent while avalanche and steady state thermal breakdown would be dependent on thickness. Experimental r e s u l t s presented i n F i g . 6.10 show that Ta/Ta20,_/Au samples could withstand a f i e l d strength very close to the formation f i e l d when the tantalum electrode i s p o s i t i v e . This gives an average breakdown strength, almost independent of d i e l e c t r i c f i l m thickness, of the order of 5.2 MV/cm. When the SCR c i r c u i t was removed and the applied f i e l d was allowed to increase a few percent above the measured strength, a dest r u c t i v e breakdown was observed i n a l l cases. The fact that breakdown strength i s almost the same as the formation f i e l d suggests a possible existence of a r e l a t i o n between the breakdown mechanism and i o n i c transport. On the other hand, cathodic breakdown strength ( i . e . , when the tantalum electrode i s negative) \ 117 160 o 60 Ta - POSITIVE VOLTAGE ELECTRIC FIELD 2400 FILM THICKNESS, A F i g . 6.10 Thickness dependence of breakdown strength. \8 N — j kl 4 £ o o — i kj 0 -2 -3 -4 l°9w(Tp/sec) F i g . 6.11 Breakdown dependence on pulse width. i s also almost independent of d i e l e c t r i c f i l m thickness but i t drops to an average value of the order of 3.1 MV/cm. This r e s u l t , s i m i l a r to the r e c t i -f i c a t i o n phenomena discussed i n Sec. 6.4.1 can be explained i n terms of the presence of p o s i t i v e charged defects near the tantalum electrode whose presence, i n add i t i o n to enhancing current i n j e c t i o n , could reduce the e l e c t r i c f i e l d strength from a high value at the tantalum/ d i e l e c t r i c f i l m i n t e r f a c e (presumably, close to the formation f i e l d ) to a lower value i n the bulk such that the average f i e l d i s appreciably lower than the maximum f i e l d strength i n s i d e the f i l m . I f t h i s were true, one would expect that breakdown events would s t a r t near the tantalum electrode where the e l e c t r i c f i e l d i s a maximum. . _/ 6.4.3.2 Breakdown Dependence on Pulse Width The breakdown strength of Ta/Ta20,-/Au samples was also determined by applying pulses of f i x e d width while increasing the voltage magnitude u n t i l breakdown took place. The breakdown events were.observed o s c i l l o g r a p h i c a l l y . Voltage pulses were generated by a HP model 214A pulse generator which enables the generation of a s i n g l e pulse each time. F i g . 6.11 shows the breakdown voltages f o r d i f f e r e n t pulse durations. The decrease of breakdown strength with the pulse duration can be a t t r i b u t e d to space charge formation, near the tantalum electrode, whose density can be argued to increase with pulse duration r e s u l t i n g i n a decrease of breakdown strength (assuming that the formed charge i s p o s i t i v e due to the removal of electrons from n e u t r a l d e f e c t s ) . I t i s i n t e r e s t i n g to note that the cathodic breakdown strength increases more than the anodic breakdown strength. This suggests that the formation of p o s i t i v e space charge i s more important when the tantalum electrode i s biased negatively 119 When many breakdowns are observed on a sample by applying a long pulse or a dc voltage, the pulse duration, x , can be i d e n t i f i e d with the mean i n t e r v a l between breakdown events, x^, as x = x- = 1/R (6.20) p D where R i s the breakdown rate which decreases with decreasing voltage magnitude, and the delay time, x^, could p o s s i b l y be of the order of minutes or even hours (Sec. 6.5). 6.4.3.3 Temperature Dependence of Breakdown Strength The average breakdown strength measured by the ramp method as a function of the ambient temperature, c o n t r o l l e d by a Statham SC6 oven, i s presented i n F i g . 6.12. The experimental r e s u l t s can be f i t t e d to = A exp(AV/kT) (6.21) Where T i s the ambient temperature. k i s Boltzmann's constant. A and AV are constants. AV was found to be about 0.04 eV when the tantalum electrode was negative and about 0.024 eV when i t was p o s i t i v e . Temperature dependence s i m i l a r to that given by Eq. (6.21) can be 85 derived using a model proposed by F r b h l i c h f o r an amorphous d i e l e c t r i c containing traps over an energy range equal to 2AV below the conduction band. F r b h l i c h assumed that the free and trapped electrons are strongly coupled by i n t e r e l e c t r o n i c c o l l i s i o n s and that energy i s l o s t to l a t t i c e v i b r a t i o n s both by free and trapped electrons. Energy i s gained only by the free electrons and the breakdown strength i s that at which a d e f i n i t e l i m i t e d e l e c t r o n i c 2.25 F i g . 6.12 Temperature dependence of breakdown strength ( f i l m thickness = 2050 A) - 5 -4 -3 -2 (og1f) (J/AMP cm-2) F i g . 6.13 Breakdown dependence on the formation current density. temperature i s no longer p o s s i b l e . F r o h l i c h assumed that AV>>kT and AV should be p o l a r i t y independent: both these assumptions are inconsistent with the present experimental r e s u l t s . The observed temperature dependence may be explained i n terms of the ease of i o n i c motion as the temperature increases, and i f the i o n i c motion i s the i n i t i a t i n g mechanism for breakdown, one would expect a decrease of breakdown strength as temperature increases. 6.4.3.4' Breakdown Dependence on the Formation Current Density From the study of growth k i n e t i c s of t h i n anodic films (Appendix 2), i t i s known that f i l m thickness depends on the f i n a l voltage and the current density, i . e . , the e l e c t r i c f i e l d required to grow a f i l m depends on the current density. Such a dependence i s given by D = BV/(kT *-ln(J/J )) (6.22) o whe re J i s the current density. V" i s the f i n a l voltage. J and B are constants, o F i g . 6.13 shows the breakdown strength of several films grown on tantalum sputtered films to a f i n a l voltage of 100 v o l t s using d i f f e r e n t current d e n s i t i e s . Though each f i l m grows at a d i f f e r e n t f i e l d strength, a l l prepared films can withstand the forming f i e l d required for low current 2 density growth (a few uAmp/cm ). This r e s u l t , i n agreement with the r e s u l t s given i n Appendix 2, in d i c a t e s that some r e l a x a t i o n processes i n the f i l m structure take place a f t e r removing the formation voltage i n such a way that the prepared films have, sometime a f t e r t h e i r preparation, almost s i m i l a r c h a r a c t e r i s t i c s which are independent of the formation current density. 122 6.4.A Breakdown Dependence on D i e l e c t r i c Constant Breakdown dependence on both d i e l e c t r i c constant and melting point i s presented i n F i g . 6.14 for four d i f f e r e n t d i e l e c t r i c s grown a n o d i c a l l y on A l , Ta, Nb and T i as base metals. Even though i t seems that there i s no cl e a r r e l a t i o n between breakdown strength and the melting point, the break-down strength decreases as the d i e l e c t r i c constant increases, i . e . , • E b - F ( l / e r ) 86 where F i s an appropriate function. Young has recently derived a r e l a t i o n between i o n i c conduction and d i e l e c t r i c constant: B « e r ( e r - n 2 ) / n 2 (6.23) where n i s the r e f r a c t i v e index. Thus, the p o t e n t i a l b a r r i e r f o r i o n i c motion decreases with increasing d i e l e c t r i c constant r e s u l t i n g i n a lower breakdown strength. The above discussion confirms that there i s a pos s i b l e r e l a t i o n between i o n i c motion and breakdown mechanism. Thus, one may conclude that the i n i t i a t i n g mechanism for breakdown could be the i o n i c motion, but i t i s not excluded that thermal runaway may occur at some stages of the breakdown process. 6.4.5 Electrode E f f e c t s The metal electrode has manifold e f f e c t s on the breakdown strength of d i e l e c t r i c f i l m s . Some metals are more l i k e l y to penetrate into microfissures i n the d i e l e c t r i c f i l m during vacuum deposition. Thus, i f thick counter-electrodes ( i . e . , the samples are not s e l f - h e a l i n g ) are used the d i e l e c t r i c breakdown strength w i l l depend on the nature of the present microfissures and i t w i l l be lower than the true bulk breakdown strength. Even i f a l l weak spots and microfissures are removed from the d i e l e c t r i c f i l m , the thermal properties 123 1450 1700 1950 Tmj K F i g . 6.14 Breakdown dependence on d i e l e c t r i c constant and melting point ( d i e l e c t r i c s are A l ^ , T a ^ , Nb 0 5 and T i 0 5 ) . 8 COUNTERELECTRODE WORK FUNCTION, eV F i g . 6.15 Electrode e f f e c t on breakdown strength. (Metals used are Au, A l and In). of the counterelectrode metal could p o s s i b l y influence the breakdown strength e s p e c i a l l y i f thermal runaway took place at some stage of the breakdown process. Furthermore, the f i e l d emitting properties of the metal electrode, e s p e c i a l l y . i t s work function, x ^ i l l determine the i n j e c t e d current density when i t acts as a cathode and t h i s i n turn could a f f e c t the breakdown strength. F i g . 6. 15 shows the breakdown strength of Ta/Ta20,-/metal samples using three d i f f e r e n t metals f o r the counterelectrode. Though the cathodic breakdown strength does not depend on the metal counterelectrode, the anodic breakdown does. I t seems reasonable to assume that the f i e l d emitting properties of the metal counterelectrode are responsible f o r the observed increase i n breakdown strength as the metal work function increases. 6.5 On the P o s s i b i l i t y of Thermal Breakdown Thermal breakdown i s often thought to occur over the whole sample, 7 7 i . e . , complete sample destruction would occur on breakdown event. K l e i n suggested that d i s t i n c t i o n can be made between thermal and e l e c t r i c breakdown on the basis of whether the destruction of a sample occurs over the whole of 87 i t s area or i n a narrow filamentary channel. However, i t was suggested that complete sample destruction i s merely a r e s u l t of the t e s t c i r c u i t used and can, by s u i t a b l e c i r c u i t design, be prevented from taking place. Further-more, when t o t a l d estruction i s allowed to occur i t does so through a s e r i e s of consecutive breakdowns, each of which i s localized and s i m i l a r to s i n g l e -channel breakdowns that occur during e l e c t r i c breakdown. Further evidence was drawn f o r the non-thermal nature of these single-channel breakdown from measure-83 84 ments ' which i n d i c a t e d that the t r a n s i t i o n to breakdown conduction occurred very r a p i d l y without a current precursor, and that the breakdown voltage threshold was not very s e n s i t i v e to temperature or to pre-breakdown conductance. 12: Further evidence c i t e d was that the l i g h t emission during breakdown had the arc spectrum of.exc i t e d species derived from the i n s u l a t o r and electrode material, i . e . , t h i s luminescence was from the f i n a l destructive phase of breakdown when a plasma e x i s t s i n some region of the sample. The purpose of th i s s e c t i o n i s to show that, even though destruction may be of a filamentary nature i t i s s t i l l p o s s i b l e that breakdown could r e s u l t from thermal e f f e c t s . Due to v a r i a t i o n s i n ele c t r o n e m i s s i v i t y at the ca t h o d e / d i e l e c t r i c i n t e r f a c e and the poss i b l e occurrence of any of the processes discussed i n Sec. 6.4.2, i t seems probable that a p a r t i c u l a r s i t e w i l l eventually dominate forming a high conducting channel between the two metal electrodes. An e l e c t r o n i c charge w i l l be associated with the development of such a channel and i t could p o s s i b l y r a i s e the channel temperature from the ambient temperature, T q, to a higher temperature, T^. The i n i t i a l temperature r i s e , AT (=T^-T Q), depends on the true mechanisms involved. I f the i n i t i a l temperature r i s e i s s u f f i c i e n t l y high such that the voltage collapse across the sample would occur instantaneously, the breakdown would be termed e l e c t r i c i n type. On the other hand, i f the i n i t i a l temperature r i s e i n the channel was not s u f f i -c i e n t to cause breakdown, i t may be poss i b l e that a thermal runaway process could s t a r t that would eventually lead to sample destruction. Such breakdown would then be termed thermal i n nature. A model i s presented below that e x h i b i t s those l a t t e r c h a r a c t e r i s t i c s . 6.5.1 Basic Equations The passage of an e l e c t r o n i c charge r a i s e s the temperature of i t s channel to a value depending on the magnitude of the e l e c t r o n i c charge and on the applied f i e l d . The temperature r i s e could be appropriate to s t a r t thermal runaway w i t h i n the channel while the re s t of the sample remains at the ambient temperature, T^. The temperature d i s t r i b u t i o n w i t h i n the d i e l e c t r i c f i l m i s governed by the heat conduction equation^ 9. The e l e c t r i c input power per unit volume to the sample i s given by p. = a(E,T)E 2 where a i s the e l e c t r i c conductivity of the f i l m and E i s the applied f i e l d . Various forms for a(E,T) have been considered i n formulating breakdown theories^"*: the following equation i s a p p l i c a b l e to t h i n amorphous film s subject to high applied f i e l d s when e i t h e r Schottky emission or a Poole-Frenkel mechanism governs the e l e c t r o n i c conduction o(E,T) = a 0 expCCBE"1-^ - <Jj)/kT) (6.24) where O Q , 6, \\> are the temperature-independent constants for the material and k i s Boltzmann's constant. When the current i s c o n t r o l l e d by some mechanism i n v o l v i n g a combination of the Schottky and Poole-Frenkel e f f e c t s , the f i e l d 25 and temperature conductivxty dependencies become more complicated The d i e l e c t r i c f i l m i s considered to be c y l i n d r i c a l with the hot channel symmetrical around the Z-axis. Approximations s i m i l a r to those used i n Sec. 5.2, reduce the heat conduction equation to A + 1 U " 1F< T- T > ~ £ e 2 + pc-^ (6.25) „ 2 r 8r dK o K M St 3r ° -where 1/X i s the external thermal resistance of the sample and p, c, K r e f e r to the f i l m density, s p e c i f i c heat and thermal conductivity r e s p e c t i v e l y . The above equation has been solved numerically to i n v e s t i g a t e the growth of the current filament and the breakdown delay time, x^, defined as the time required for the hottest point i n the channel to reach a c r i t i c a l temperature, T , which i s taken as the d i e l e c t r i c melting point. 127 6.5.2 Results and Discussion I t i s assumed that breakdown w i l l occur i f the input power i s s u f f i c i e n t l y high that despite the heat loss from the hot channel the c e n t r a l p o s i t i o n of the channel can be ra i s e d to a c r i t i c a l temperature. The delay time, T ^ , between the i n i t i a l avalanche and the onset of breakdown w i l l depend on the f i l m parameters, the applied f i e l d , and i n the present model, on the i n i t i a l temperature r i s e and other channel parameters. The growth of the conducting channel has been i n v e s t i g a t e d numeri-c a l l y using Eq." (6.25) i n the f i n i t e d i f f e r e n c e form ( c e n t r a l differences f o r s p a t i a l d e r i v a t i v e s and forward d i f f e r e n c e s for time d e r i v a t i v e ) . A p l a u s i b l e form f o r the i n i t i a l temperature d i s t r i b u t i o n would seem to be given by T(r,0) = T Q + AT/(l+exp((r-r c)/<5)) (6.26) where AT, r and 6 are constants, c F i g . 6.16 shows how the channel current and the temperature of the h o t t est point (T(0,t)) change with time for d i f f e r e n t i n i t i a l temperatures. The culmination of any breakdown process i s a catastrophic increase i n temperature over some region of the sample. However, sample current need not increase c a t a s t r o p h i c a l l y . The current flowing through the sample can be thought of as c o n s i s t i n g of a component I flowing through the high conduc-t i v i t y channel and a component I flowing through the remainder of the sample which i s at a low temperature and therefore presents a higher r e s i s t i v i t y region. I- = I + I T c r = 2u / J ( r ) r d r + J A (6.27) c c r r Now, i f the channel area i s small compared with the sample area the c o n t r i b u t i o n log 1Q( TIME/sec) F i g . 6.16 Thermal runaway: ? c h a n n e l c u r r e n t and maximum temperature v a r i a t i o n versus times. (Constants used are x/d = 10 , a = 10 , 0 c = 4 x I O - 6 , <b = 0.6 eV, B/k = 0.2 eV, K = 0.5, r = 20A, " o O ' C 6 = 5A. Unless otherwise s t a t e d , M.K.S. u n i t s are used). M N3 of I to I T w i l l be s m a l l . Hence, I T w i l l tend to some l i m i t i n g value governed mainly by the current i n the low c o n d u c t i v i t y r e g i o n . On the other hand, as the channel area i n c r e a s e s I w i l l tend to dominate and thus c Thus, i t i s only f o r l a r g e values of the channel radius that.I^, i n c r e a s e s r a p i d l y . P r a c t i c a l l y , breakdown events are encountered which are not a s s o c i a t e d w i t h c a t a s t r o p h i c i n c r e a s e of c u r r e n t . 89 The r e s u l t s showed that the e f f e c t i v e channel r a d i u s decreases w i t h the passage of time. This suggests that the r a t e of heat l o s s from the channel edges exceeds that from the c e n t r a l p o r t i o n of the channel. However, i t would be expected that f o r a t h i n f i l m the heat l o s s normal to the s u r f a c e would tend to be more important than the r a d i a l flow p a r a l l e l to the s u r f a c e In f a c t , i t turns out that a good approximation to the delay times (x^ model) computed from the above model may be obtained by c a l c u l a t i n g T ( t ) at r = 0 from * f = P . - P £ (6.28) 2 i . e . , by p u t t i n g (V T ) r _ Q = 0 provided the channel r a d i u s i s not too s m a l l . When the channel r a d i u s becomes so s m a l l that the r a d i a l heat conduction can s e r i o u s l y a f f e c t the maximum temperature i n the channel, then the delay time to breakdown w i l l i n c r e a s e , r e s u l t i n g i n h i g h percentage e r r o r . A l s o , i t f o l l o w s that f o r a given i n i t i a l temperature r i s e (T(0,0)) there i s a lower l i m i t f o r the channel radius r c below which the proposed mechanism may not be p o s s i b l e because the input power can be d i s s i p a t e d n o n - d e s t r u c t i v e l y . This l i m i t i n g value of channel radius depends i n v e r s e l y on the i n i t i a l temperature r i s e s i n c e as the l a t t e r i n c r e a s e s so does the channel e l e c t r i c a l c o n d u c t i v i t y and hence, the power i n p u t . The delay time to breakdown, shown i n F i g . 6.17, has been computed 2 by the Riinge-Kutta method, using the approximation V T = 0. For the range of parameters considered relevant to d i e l e c t r i c films (e.g., Ta^O^ films have 1 13 14 2 2 o 1 pea Q - 10 - 10 V sec m K ) the delay time i s of the order of one second. That breakdown can a c t u a l l y take place a f t e r the elapse of times of 90 t h i s duration can be seen, e.g., from the data of K l e i n and Burstein f o r vapor-grown Si02 t h i n f i l m s . For breakdown to occur i n much shorter times the i n i t i a l temperature r i s e must increase so that T(0,0) tends to the c r i t i c a l temperature T , under which circumstances the breakdown would be no longer classed as thermal. K l e i n and Burstein r e a l i z e d t h i s i n t h e i r pulsed applied voltage experiments and found i t necessary to c a l l the breakdowns on short pulses e l e c t r i c . Further data from K l e i n 7 7 , t h i s time for the dc breakdown of Al/anodic Al^O^/Au structures shows that T^'S as long as 100 seconds are po s s i b l e at room temperatures. As would be expected, the time delay to breakdown decreased with r a i s i n g of e i t h e r the applied voltage ( i . e . , input power) or the ambient temperature. An i n t e r e s t i n g consequence of the present model i s that the break-down could occur at lower applied f i e l d s provided a s u i t a b l e temperature r i s e i n a channel occurs. The required f i e l d may be obtained by equating the power input to the power l o s s . Assuming the l a t t e r to be given by P , i . e . , 2 again neglecting V T, we get E 2 e x p ( ( 3 E 1 / 2 - ij,)/kT) = -A- (T-T n) (6.29) da Q 0 As P. increases exponentially with T and P l i n e a r l y with T, Eq: (6.29) could be s a t i s f i e d at two points. For the purpose of c a l c u l a t i n g the break-down f i e l d i t i s necessary to consider only the condition when the d e r i v a t i v e of the input power i s equal or f a s t e r than that of power l o s s , i . e . , 12.5 INITIAL TEMPERATURE RISE, °K F i g . 6.18 V a r i a t i o n of c r i t i c a l f i e l d with the i n i t i a l channel temperature r i s e f o r d i f f e r e n t power losses. 132 E 2 (!l!rg| ) exp((6E 1 / 2-^)/kT) > ^ — T 2 (6.30) F i g . 6.18 shows the c r i t i c a l f i e l d as a function of i n i t i a l temperature r i s e f o r d i f f e r e n t values of X/da^. As expected, the computed r e s u l t s show that the c r i t i c a l f i e l d decreases with in c r e a s i n g i n i t i a l temperature r i s e . . However, i t should be noted that the p r o b a b i l i t y of an avalanche occurring at low applied f i e l d i s very small. In conclusion of t h i s s e c t i o n , thermal runaway can follow a non-destructive e l e c t r i c process which sets up a highly conducting channel between the electrodes and thus the breakdown can occur i n a narrow channel, demonstrating that complete sample destruction does not have to r e s u l t before a breakdown event can be classed as thermal. Current runaway may not be observed when the channel i s s u f f i c i e n t l y small and breakdown can occur at f i e l d s lower than the normally accepted breakdown strength provided the i n i t i a l temperature r i s e caused by the avalanche i s s u f f i c i e n t l y high. For t y p i c a l constants of d i e l e c t r i c films of current i n t e r e s t , the present model can only account f o r delay times of the order of seconds; f o r much shorter delay times the breakdown i s presumably e l e c t r i c i n nature. 6.6 Breakdown Mechanism The study of successive breakdown events on applying a rectangular pulse of sui t a b l e magnitude ( F i g . 6.13) suggests that the breakdown process consists of two successive stages. F i r s t a conducting channel between the metal electrodes i s developed, and then the sample's stored energy i s discharged through the formed channel. Both stages w i l l now be discussed i n some d e t a i l . 133 6.6.1 Stage I; Formation of a Conducting Channel The f i r s t few breakdown events which are observed to occur at f i e l d strength le s s than the average breakdown strength can be s t a r t e d at some defects i n the prepared sample as discussed i n Sec. 6.4.2. The breakdown strength measured a f t e r removing a l l weak spots should be that of the bulk d i e l e c t r i c . The study presented i n Sec. 6.5 shows that thermal breakdown through a narrow channel i n high r e s i s t i v i t y materials would occur a f t e r appreciable delay times and i t can not generally account f o r the observed much shorter delay times. The independence of breakdown strength on f i l m thickness excludes the p o s s i b i l i t y that the avalanche processes are the channel forming processes. The breakdown strength dependence on the metal counterelectrode exludes the p o s s i b i l i t y of being a completely i n t r i n s i c breakdown. In view of the facts that there i s a strong dependence of breakdown strength on f i l m d i e l e c t r i c constants and that anodic breakdown occurs at a f i e l d strength very close to the forming f i e l d , i t may be reasonable to propose that the breakdown would be i n i t i a t e d by d i s r u p t i o n i n the chemical bonds as the applied f i e l d approaches the forming f i e l d . The product of molecular d i s s o c i a t i o n and the presence of energetic electrons could s t a r t an accumulative process which may form a highly conductive channel. The i n j e c t e d e l e c t r o n s , f i e l d d i s t o r t i o n and thermal runaway could a s s i s t i n the channel development. Cathodic breakdown could be due to the same process except that the presence of p o s i t i v e l y charged defects near the base metal causes a 13* d i s t o r t i o n i n 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 with i t s highest value near the cathode. Thus,, any molecular d i s s o c i a t i o n w i l l s t a r t near the cathode while the average f i e l d i s l e s s than the forming f i e l d . Furthermore, the d i s s o c i a t i o n processes are greatly a s s i s t e d by the higher i n j e c t e d current density at the c a t h o d e / d i e l e c t r i c i n t e r f a c e . The higher s e n s i t i v i t y of cathodic breakdown strength on ambient temperature and pulse width can be a t t r i b u t e d to the s e n s i t i v i t y of space charge development under the d i f f e r e n t , conditions. 6.6.2 Stage I I : Discharge of Sample's Stored Energy . As the conducting channel i s completely developed, i . e . , has attained the breakdown conditions, the sample's stored energy begins to d i s s i p a t e i n the channel. During the discharge, the sample can be represented by i t s capacitance, C, (assuming that i t does not vary during the discharge since the channel area i s a small p o r t i o n of the sample area) i n p a r a l l e l with the channel r e s i s t a n c e , R ( t ) , which varies with time. Thus, the equivalent discharge c i r c u i t can be represented as shown below, where V g i s the applied voltage and R i s the s e r i e s l i m i t i n g r e s i s t a n c e . The voltage across the sample", V(t) , i s thus given by V(t) = R ( t ) [ - ~ C^Jp 1] <6-31> s and the energy d i s s i p a t e d i n the channel i s given by e(t) = ft P ( t ) d t = / f c V 2 ( t ) / R ( t ) d t (6.32) o o This energy must be equated with the heat needed f o r material evaporation, the heat l o s t by r a d i a t i o n , the heat producing temperature increase around 91 the breakdown spot and the heat gained by the oxidation of the electrodes , i i e . , e(t) = H + H + H - H (6.33) e r s o The r a d i a t i o n losses are n e g l i g i b l e and so i s oxidation. Conduction losses were estimated by considering the short time temperature transients i n the material adjacent to the evaporated electrode and these were found to be a few percent of the heat energy required f o r material evaporation. Thus, i f q e i s the l a t e n t heat of vaporization of a given sample per unit area, one may w r i t e e(t) = q e [ A c ( 0 ) - A c ( t ) ] (6.34) Where ^ c ( t ) i s the cross s e c t i o n a l area of the channel at time t a f t e r s t a r t i n g the discharge. The channel resistance at time t i s given by R(t) = pd/A c(t) Thus, Eq. (6.34) can be w r i t t e n as e(t) = q epd[l/R(0) - 1/R(t)] = XR _ 1(0)[1 - R(0)/R(t)] (6.35) where 136 p i s the channel r e s i s t i v i t y . X i s a sample constant. F i g . 6.19 Computed voltage collapse and channel resistance during the discharge. The voltage collapse across the sample, V ( t ) , and the channel r e s i s t a n c e , R ( t ) , can be obtained by s o l v i n g Eqs. (6.31) and (6.35). The computed example (Fig.6.19) shows that the voltage collapse occurs i n a f r a c t i o n of a microsecond and that the rate of voltage decay decreases as the voltage across the sample (or the sample's stored energy) becomes s u f f i c i e n t l y low. These features are i n good agreement with the experimental r e s u l t s (Sec. 6.4.2). A f t e r the channel resistance becomes s u f f i c i e n t l y high ( i . e . , a main part of the channel has been evaporated), the sample w i l l be charged again from the external source u n t i l another breakdown event takes place. 7. CONCLUSIONS Space charge and high f i e l d e f f e c t s on some e l e c t r i c a l properties of t h i n amorphous films have been in v e s t i g a t e d . A theory of space charge c o n t r i b u t i o n to p o l a r i z a t i o n currents i n t h i n d i e l e c t r i c films has been proposed. The external discharge current on short c i r c u i t i n g a t h i n f i l m capacitor i s believed to consist of two components one due to d i e l e c t r i c p o l a r i z a t i o n and the other due to trapped space charge. The space charge contribution has been investigated using a model for a f i l m containing traps whose d e n s i t i e s vary exponentially with t h e i r binding energies The e f f e c t s of d i f f u s i o n , metal work function and the presence of p o s i t i v e space charge have been considered. The computed space charge p o l a r i z a t i o n currents follow a 1/t law when trapped electrons 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 . A time-independent current i s p o s s i b l e when there i s a p o s i t i v e space charge. The current magnitude i s almost independent of the i n i t i a l amount of trapped charge i n contrast to the l i n e a r d i e l e c t r i c p o l a r i z a t i o n current which varies linearly with the preapplied f i e l d . Experimental r e s u l t s on Ta/Ha^Q) ^ /ko. diodes seem to be consistent with the present model, so that space charge e f f e c t s are more important at low preapplied f i e l d s . This r e s u l t i s also confirmed by the apparent a p p l i c a b i l i t y of step response procedures at high preapplied f i e l d s . The theory of thermoluminescence and thermally stimulated currents has been extended to the case of traps with d i s t r i b u t e d binding energies to i n v e s t i g a t e the p o s s i b i l i t y of d i s t i n g u i s h i n g between d i s t r i b u t e d and d i s c r e t e trap l e v e l s . The r e s u l t s show that i t should be p o s s i b l e to d i s t i n g u i s h experi mentally between both cases. Apart from the c l e a r d i f f e r e n c e s i n glow curves, 138 the following experiments may be useful to confirm the nature of the trap d i s t r i b u t i o n : (1) d i f f e r e n t doses of o p t i c a l r a d i a t i o n could be used to obtain d i f f e r e n t amounts of trapped charges; (2) the frequency of o p t i c a l e x c i t a t i o n could be v a r i e d over a s u i t a b l e range of frequencies to allow c e r t a i n energy l e v e l s to be occupied by excited electrons. The predominant high f i e l d emission mechanism from a metal electrode can be e i t h e r tunneling or Schottky thermionic emission, depending on f i l m parameters, applied voltage and ambient temperature. In view of the f a s t tunneling time of e l e c t r o n through very t h i n f i l m s , MIM structures could be used f o r microwave detection. The dc and ac tunneling c h a r a c t e r i s t i c s of such detectors have been analysed. The r e s u l t s show that a bias equal to the anode work function ( i n v o l t s ) gives the maximum responsivity-band-width product of MIM detectors, and that the presence of i n v a r i a n t p o s i t i v e space charge increases the magnitude of t h i s maximum. High f i e l d switching i n t h i n amorphous films i s considered to be an e l e c t r o n i c process. However, sample s e l f - h e a t i n g , which could account for the delay times observed experimentally, precedes the switching event and serves to t r i g g e r i t . The delay time, due to Joule heating, has been shown to depend exponentially on the applied voltage, i . e . , aexp(-V/V o). A model for CCNR due to space charge b a r r i e r s has been developed. C a r r i e r i n j e c t i o n has been taken to occur e i t h e r by Fowler-Nordheim tunneling or Schottky thermionic emission. Injected c a r r i e r s could undergo a m u l t i p l i c a t i o n process by impact i o n i z a t i o n u n t i l a balance between generation and recombination rates i s achieved. The concept of p o s i t i o n dependent generation-recombination rates was discussed. The computed r e s u l t s show many features which are i n agreement with experimental r e s u l t s , e.g., the dependence of threshold voltage 13S on model parameters and the constancy of the sustaining voltage. The small ac equivalent c i r c u i t of the proposed model has been given. The formation of current filaments, to achieve e l e c t r i c a l s t a b i l i t y , has been formulated i n terms of r a d i a l d i f f u s i o n processes. Memory state formation was a t t r i b u t e d to phase change processes due to excessive heating. Filamentary breakdown has been observed i n anodic oxide f i l m s grown on Ta,' A l , Nb and T i . A l l prepared samples were able to withstand f i e l d strengths of the order of the forming f i e l d required for low current density 2 growth (a few uAmp/cm ) when the base metal was biased p o s i t i v e l y but only about 0.6-0.7 of the forming f i e l d when the base metal was biased negative. This d i f f e r e n c e may be a t t r i b u t e d to the presence of p o s i t i v e charged defects, presumably excess metal ions, near the base metal. The breakdown strength has been found to be independent of the anodizing current density, provided the measurements are taken some time a f t e r the end of the growth process. The strong dependence of breakdown strength on the f i l m d i e l e c t r i c constant gives further evidence f o r the existence of a r e l a t i o n between i o n i c motion and breakdown process. The metal counterelectrode has been found to a f f e c t the cathodic breakdown strength and not the anodic, breakdown strength. This can be explained i n terms of the di f f e r e n c e s between work functions which i n turn a f f e c t the i n j e c t e d current density at the cat h o d e / d i e l e c t r i c f i l m i n t e r f a c e . The observed breakdown events appear to consist of two successive stages, namely: (1) The formation of a highly conducting channel i n d i e l e c t r i c f i l m s : the forming could be i n i t i a t e d by dis r u p t i o n of the chemical bonds under the high applied f i e l d . The ions produced can cause f i e l d d i s t o r t i o n and the presence of energetic ions and electrons can break more bonds. The development of the channel i s achieved through successive i o n i c d i s s o c i a t i o n a s s i s t e d by the products of previous d i s s o c i a t i o n , energetic electrons, f i e l d d i s t o r t i o n and thermal runaway. (2) The discharge of the sample's stored energy through the formed channel This can be explained i n terms of heating and evaporation of the d i e l e c t r i c f i l m and the associated metal counterelectrode. The voltage collapse has been found experimentally to occur i n a time of le s s than 200 nanoseconds. APPENDIX 1. CALCULATION OF RELAXATION TIME SPECTRA IN AMORPHOUS FILMS Most r e l a x a t i o n processes i n amorphous f i l m s can not be described by an e x p o n e n t i a l f u n c t i o n w i t h s i n g l e r e l a x a t i o n time. Two approaches to a 92 phenomenological d e s c r i p t i o n can t h e r e f o r e be taken : (1) w r i t i n g of an e x p l i c i t nonexponential f u n c t i o n to describe the time v a r i a t i o n of a given property o r , (2) d e s c r i b i n g the behaviour by a sum.of simple e x p o n e n t i a l terms, reaching an i n t e g r a l i n the l i m i t . The l a t t e r approach i s the most common one. A procedure to c a l c u l a t e the r e l a x a t i o n time s p e c t r a from a given experimental data w i l l now be disc u s s e d . p r o p e r t y S of a group of processes having r e l a x a t i o n times i n the range -.dx around x. The t o t a l c o n t r i b u t i o n of a l l present processes, i f the super-p o s i t i o n p r i n c i p l e h o l d s , w i l l then be given by Let N(x)dx be the c o n t r i b u t i o n to an a r b i t r a r y t h e rmally a c t i v a t e d S - S = / N ( T ) dx S oo o ( A l . l ) where S i s the s t a t i c value of S s S i s the instantaneous value of S 00 The time dependence of the response, R ( t ) , a s s o c i a t e d w i t h S due to the a p p l i c a t i o n of a f i e l d E ( t ) may be w r i t t e n as R(t) = S E ( t ) + f% h(T)E(t-T)dT (A1.2) . Where (A1.3) But i f h(T) = 0 T < 0 142 Eq. (A 1.2) may be wr i t t e n as R(t) = S E(t) + f° h(T)E(t-T)dT (A1.4) oo o In p e r i o d i c f i e l d s , E(t) may be expressed as E(t) = e exp(joit) (A1.5) Su b s t i t u t i n g Eq. (A1.5) i n Eq. (A1.4), we get for the r e a l and imaginary part of S(o)) : S'(u)) = S + /" h(T) cos uT dT (A1.6a) S"(o>) = F h(T) s i n oiT dT (A1.6b) o with S(u) - s'(o>) - j S M (aj) (A1.6c) Substituting f o r h(T) from Eq. (A1.3) we get S'(o.)=S +/°°-^ 4 dt (A1.7a) 1+0) X s"(u>= C-^^rdT (A1-7b) 1+0) T The f u n c t i o n a l form of N(x) could be determined using Eqs. (Al.6) 93 and ( A l * 3 ) . Taking inverse Fourier transform of Eq. (A1.6a), we get h(T) = - /" (S'(u) - S ) cos o)T du (A1.8) Tr o °° s u b s t i t u t i n g v «= — i n Eq. (A1.3), we have T h(T) = f e " T v (N(l/v)/v)dv which i s a Laplace transform. Taking the inverse transform, we get for N(x) N(T) = • o r T r L Y t i 0 ° h ( T > e V T d T ^ ^ A 1 - 9 > 2ujx Y - J » V = 1 / T I t i s p o s s i b l e , i n p r i n c i p l e , to determine the f u n c t i o n a l form of N ( T ) from the experimental data using Eqs. (Ai.8) and (A1.9) successively. An a l t e r n a t i v e procedure i s to assume the form of N(x) which describes the ph y s i c a l s i t u a t i o n , and then to make a check between the calculated and the measured values of the property i n question. 143 APPENDIX 2. GROWTH AND IONIC CONDUCTIVITY OF FILMS Ionic conduction i n t h i n anodic films can be in v e s t i g a t e d using a small ac s i g n a l superimposed on the dc voltage. Ac measurements can be considered the most general measuring technique i n which steady state and transient measurements are only two s p e c i a l cases. In t h i s Appendix .growth and i o n i c conductivity of I^O,. fi l m s w i l l be investigated b r i e f l y using ac measurements. A2.1 Theories of Ionic Conduction Various theories have been developed to describe i o n i c conduction i n s o l i d s . The most important of these are (for a d e t a i l e d discussion see References 3, 86, 94): (a) The c l a s s i c a l theory of i o n i c conduction. (b) Frenkel defect theory. (c) The channel model. (d) The d i e l e c t r i c p o l a r i z a t i o n theory. Experimentally and t h e o r e t i c a l l y dc i o n i c conduction currents can be described by the expression J = J . exp(-W(E)) (A2.1) o The f u n c t i o n a l form of W(E) depends on the theory used. The above theories give f o r W(E) the following expressions r e s p e c t i v e l y (a) W(E) = (W - q a E)/kT (A2.2) (b) W(E) - ((W + W')/2 - q E(a + a')/2)/kT (A2.3) (c) W(E) = (W - v E 1 / 2 ) / k T (A2.4) (d) W(E) = W - a E - BE 2 (A2.5) e e where W and a are the a c t i v a t i o n energy at zero f i e l d and the h a l f jump distance of a defect. W* and a' are the a c t i v a t i o n energy at zero f i e l d and the h a l f jump distance f o r the production of a Frenkel defect. q i s the charge on the ions. ' • E i s the e f f e c t i v e f i e l d e = E + p/e o p i s the t o t a l p o l a r i z a t i o n a, 6, y a n ^ <5 are constants. A2.2 Sample Preparation Tantalum samples were cut from Fansteel capacitor grade metal 3 sheet and were about 0.9x0.4x0.12 cm with a tab. Tantalum surfaces were chemically polished i n an H^SO^, HNO^ and HF s o l u t i o n (5:2:1.5 by volume). The working areas were defined using e i t h e r epoxy r e s i n or black wax. The cathode was a large p l a t i n i z e d platinum electrode i n the form of a c y l i n d e r surrounding the sample. The c e l l was immersed i n a thermostatted bath of water and g l y c e r o l (temperature c o n t r o l b e t t e r than + 0.2°C). Samples were anodized i n 0.2 N sulphuric a c i d at room temperature using a constant current supply. A2.3 Measurements Fi l m growth was studied using small ac s i g n a l s ^ ' 9 * ' . Two current components, one i n phase and the other out of phase with the ac voltage, across the c e l l were monitored using two lock i n amplifiers (model HR8) incorporated i n the c i r c u i t shown i n F i g . A2.1. The dc formation. 1 current was supplied 14. to the c i r c u i t from a constant current supply v i a a toggle switch which also opened relay The ac s i g n a l to the c e l l \<ras provided by an o s c i l l a t o r (Wavetek 111) v i a an i s o l a t i n g transformer and voltage d i v i d e r . Lock i n amp l i f i e r s were used i n the external mode with the amplified voltage across the c e l l as t h e i r reference s i g n a l . The phase angle was adjusted to be 0° for one a m p l i f i e r and 90° for the other. The outputs of the two a m p l i f i e r s were monitored by the two channels of a Moseley 7100 BM recorder. F i g . A2.1 C i r c u i t emplying lock i n amplifiers to measure the f i l m impedance during f i l m grov/th Film thicknesses were determined from the minima i n the specular r e f l e c t i v i t y as a function of wave length using a Cary double beam spectro-photometer. Curves given on page 80 Reference 3 were used to obtain f i l m 146 thickness making no allowance for the d i f f e r e n c e i n the substrate. A2.4 Small Signal Film Impedance A d i e l e c t r i c f i l m may be represented by a capacitance, C p, i n p a r a l l e l with a resistance, Rp. Both components could be frequency dependent. During the f i l m growth, the presence of a dc i o n i c current contributes to both C p and R p by changing the f i l m s u s c e p t i b i l i t y and conductivity. Thus, t h e i r values could be d i f f e r e n t from those measured without a dc i o n i c current. The p a r a l l e l representation and i t s equivalent s e r i e s representation are given i n F i g . A2.2, where R g x t represent any external resistance, e.g., the e l e c t r o l y t e resistance, and R i s the resistance across which the lock i n o a m p l i f i e r measured the voltage drop. F i g . A2.2 P a r a l l e l and s e r i e s equivalent c i r c u i t s The normalized ac voltage measured by the lock i n a m p l i f i e r , using the s e r i e s equivalent c i r c u i t , i s given by 147 v ( O C R (R + r ) + jwC R . ° / Q\ s ° e x t s s o n — (e) = 2 — ( A 2 , 6 ) V i 1 + u)V (R + r ) Z s .ext s The r e s i s t i v e and c a p a c i t i v e components are given by ' v w 2C 2R (R + r ) A o ,nS J3 o ext s , A O n . v = — (0) = — (A2.7a) nr v. 2 „ / ( , _ S 2 I 1+u C (R +r ) s ext s . v 10C R v £-^(90) (A2.7b) nc v. - , 2_2,_. • , v2 l ' 1+w C (R +r ) s ext s The values of the f i l m capacitance and r e s i s t a n c e may be obtained from the r e l a t i o n s ^^ ( V " r ) 2 ) C = v I s (A2.8a) s nc (oR o and ' r = ) - R , (A2.8b) s uC v ext s nc The e x t e r n a l r e s i s t a n c e i s determined from the zero i n t e r c e p t of the t o t a l s e r i e s r e s i s t a n c e versus 1/w as to goes to i n f i n i t y . The p a r a l l e l e q u i v a l e n t components R p and C p can be obtained by the conversion C = C / ( l + 9 \ J (A2.9a) p s a ) 2 C 2 r 2 s s E0 = r s < 1 + -TTT> < A 2 ' 9 b > p s 2„2 2 . a) C r s s A2.5 Experimental Results and D i s c u s s i o n The measured values of 1/C and R during f i l m growth are given i n P P F i g . A2.3. The ac s i g n a l frequency was 100 Hz and i t s amplitude was 15 mV. The r e s u l t can be approximated to 148 CHARGE, COULOMB F i g . A2.3 Dependence of f i l m resistance and capacitance on the charge passed during anodization. -2 -51 i i I 5.4 5.8 6.2 6.6 E,W6 VOLT cm'1 F i g . A2.4 Log.- J-E c h a r a c t e r i s t i c s of grown f i l m s . ( t a f e l p l o t ) . 14S 1/C • = k±q • (A2.10) and R = k.t ( A 2 . l l ) p 2 where and k^ are constants, almost independent of the c u r r e n t d e n s i t y . T h e i r values are 20+1 Coulomb ^ micr o f a r a d ^ and 28+1 ohm sec ^ r e s p e c t i v e l y . R e l a t i o n (A2.10), w i t h the help of Faraday's law (daQ), shows that the f i l m d i e l e c t r i c constant i s independent of the c u r r e n t d e n s i t y w h i l e r e l a t i o n ( A 2 . l l ) indicates that f i l m c o n d u c t i v i t y during a n o d i z a t i o n i s p r o p o r t i o n a l to the current d e n s i t y , i . e . , 0 = k 3 J (A2.12) Such a dependence can be obtained by d i f f e r e n t i a t i n g r e l a t i o n (A2.1), k^ and W(E) are r e l a t e d by K 3 " dE As discussed i n Sec. A2.4, the measured values of C and R have P P c o n t r i b u t i o n s from both d i e l e c t r i c p r o p e r t i e s and i o n i c conduction of the d i e l e c t r i c f i l m . To estimate the d i e l e c t r i c c o n t r i b u t i o n only, i . e . , i f the e f f e c t of the i o n i c current vanishes, C and R should be measured sometime P P a f t e r removing the i o n i c c u r r e n t . The time p e r i o d must be s u f f i c i e n t to a l l o w the mobile i o n i c c a r r i e r s to r e l a x i n t o l a t t i c e s i t e s l e a d i n g to a decrease i n t h e i r c o n c e n t r a t i o n i n the f i l m , and thus, a decrease i n the i o n i c c o n d u c t i v i t v . C and R were measured about 15 hours a f t e r removing the i o n i c p p c u r r e n t . Even though no d i f f e r e n c e could be detected i n the value of f i l m capacitance, the f i l m r e s i s t a n c e increased to a much hi g h e r value F i g . A2.5 and i t can be a t t r i b u t e d only to the d i e l e c t r i c l o s s e s . The f i l m t h i c k n e s s i s p r o p o r t i o n a l to the amount of charge passed across the. oxide e l e c t r o l y t e during the growth time (Faraday's law) Thus, MEASURED DURING ANODTZA TION MEASURED AFTER 15 HOURS logJ0 (J/AMP cm-2) F i g . A2.5 Film resistance dependence on the forming current density iogJ0(f/Hz) F i g . A2.6 Frequency dependence of i o n i c c o n d u c t i v i t y the f i n a l t h i c k n e s s depends on both the f i n a l values of i o n i c current d e n s i t y and the a p p l i e d v o l t a g e . D i f f e r e n t f i l m s were grown up to f o r t y v o l t s using constant current d e n s i t i e s . Assuming that the e l e c t r i c f i e l d does not depend on f i l m t h i c k n e s s , i t s value was estimated by d i v i d i n g the f i n a l v o l t age by the f i n a l f i l m t h i c k n e s s measured by the spectrophotometric method. The t a f e l p l o t of the prepared f i l m s i s given i n F i g . A2.4. As a f i r s t approximation, i t can be f i t t e d to J = J q exp(gE) (A2.13) w i t h J q = 4.3 x 10 Amp cm 2 and 3 = 5.4 x 10 ^ V cm 3 I t has been known that ,. i f the p o t e n t i a l changes suddenly from a steady s t a t e value to another value g i v i n g a change of f i e l d from V^/d to V^/dy the i o n i c current changes suddenly to some new value and then g r a d u a l l y reaches the steady s t a t e value. This means that the value of 8 depends on the r a t e of change of the a p p l i e d s i g n a l and we may define 6 ^ T • d l n J s ~ L i m dE (A2.14a) o A T . d l n J p. = Lim • 1 dE . d h (A2.14b) — H» dt F i g . A2.6 shows the frequency dependence of k^ f o r a f i l m grown using current 2 d e n s i t y of 0.1 mAmp/cm . The observed frequency dependence of ac i o n i c c o n d u c t i v i t y could be explained i n terms of r e l a x a t i o n processes i n v o l v i n g i o n i c motion over p o t e n t i a l b a r r i e r s , (e.g., c r e a t i o n and a n n i h i l a t i o n of F r e n k e l d e f e c t s ) . These processes should be thermally-activated processes whose r e l a x a t i o n times can be expressed i n Arrhenius form T - T q exp(W(E)/kT) where W i s the f i e l d dependent a c t i v a t i o n energy arid-r i s an appropriate constant. For amorphous f i l m s , one would expect a spread i n the values of T which can a r i s e from a d i s t r i b u t i o n i n e i t h e r T or W, or both simultaneously. o ' J For every r e l a x a t i o n process, there i s an upper l i m i t f o r the s i g n a l frequency above which the process cannot respond to the applied s i g n a l . Now i f the frequency i s s u f f i c i e n t l y high, then only those processes which can be considered as instantaneous processes ( i . e . , T *»-0) w i l l contribute to the measured conductivity which can be r e l a t e d to 8^ through the r e l a t i o n Lim a (to) = a. = JB. (A2.15a) 1 1 to-*>° On the other hand, i f the frequency i s s u f f i c i e n t l y low to enable a l l the present processes to contribute to the measured conductivity, one can write Lim o(<o) = a = JB (A2.15b) s s to-> 0 Relations (A2.15a) and (A2.15b) are i n agreement with the experimental r e s u l t s presented. 153 BIBLIOGRAPHY 1. L.J. Maissel and R. Glang ( e d i t o r s ) , Handbook of Thin Film Technology, McGraw-Hill (1970). 2. K.L. Chopra, Thin Film Phenomena, McGraw-Hill (1969). 3. L. Young, Anodic Oxide Films, Academic Press (1961). 4. A.I. Gubanov, Quantum Ele c t r o n Theory of Amorphous Semiconductors, Con-sultant Bureau (1965). 5. A.K. Jonscher and P.A. 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