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UBC Theses and Dissertations

Thin oxide films: mechanisms of growth, dielectric and optical properties and applications in semiconductor… Smith, David J. 1981

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THIN OXIDE FILMS: MECHANISMS OF GROWTH, DIELECTRIC AND OPTICAL PROPERTIES AND APPLICATIONS IN SEMICONDUCTOR DEVICES by David J . Smith B.Sc. (Hon), University of B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (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 required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1931 © David J . Smith, 1931 In present ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the Un ivers i ty 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 fo r reference and study. I fur ther agree that permission for extensive copying of t h i s t h e s i s fo 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 or her representa t ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes is for f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion. Department of E l e c t r i c a l Engineering The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date September 21. 1981 n r . f i n /7Q ^  ABSTRACT Di e l e c t r i c films are important i n the fabrication of semiconductor devices. Thermally grown and CVD Si0 2, and CVD s i l i c o n n i t r i d e films are per-haps the most widely used d i e l e c t r i c materials at present. Future developments i n both Si and GaAs technology, however, would seem to require the introduction of both new materials and processes. Since i t i s a room temperature process, anodic oxidation i s attractive for some device applications. The mechanism by which the oxide f i l m grows, however, i s not well understood. Low temperature thermal oxidation of certain deposited metals i s also of inter e s t . For this thesis, anodic and thermally grown tantalum oxide films were investigated i n connection with their possible engineering applications. Part of the work described i n this thesis involved developing i n s t r u -mentation for computer-controlled experiments and data acquisition. This included development of a new balancing algorithm and control program for a computer-controlled ellipsometer. Careful attention was also given to c a l i b r a -tion of the ellipsometer and other equipment used i n this work. The o p t i c a l properties of tantalum oxide films grown i n d i l u t e I^SO^ were measured by i n s i t u ellipsometry. The ellipsometer data are consistent with a model i n which a nonabsorbing single-layer f i l m overlays a thin i n t e r -f a c i a l f i l m of variable thickness and refractive index. The i n t e r f a c i a l f i l m i s due to surface roughness and a f l u o r i n e - r i c h f i l m remaining from the surface preparation. The oxide f i l m thicknesses were compared with thicknesses deter-mined from the wavelengths of minimum r e f l e c t i v i t y . Anodic oxides are grown i n a thermally activated process i n which ions are transported across the f i l m under the influence of a large e l e c t r i c f i e l d . It i s argued that the observed history dependence of the current-field r e l a t i o n i i i s due to changes i n the structure of the oxide f i l m . I t i s then shown that equations can be developed which are equivalent to those of Dignam's current-driven d i e l e c t r i c polarization model but which are based on a more acceptable physical model involving these s t r u c t u r a l changes. The process of high f i e l d i o n i c conduction i n the oxide f i l m was investigated by stepped current and open c i r c u i t transient methods. The data were analyzed numerically and compared with data predicted by Dignam's equa-tions. The experimental and computed data are shown to be i n close agreement. However, small but possibly s i g n i f i c a n t anomalies were observed regarding the . l i n e a r i t y of the stepped current data and parameter values used i n f i t t i n g r e l a -tions to the experimental data. For possible device applications, tantalum films were sputtered onto s i l i c o n and thermally oxidized. The o p t i c a l properties of the oxide f i l m were investigated by ellipsometry. Current-voltage measurements of MIS capacitors indicate the present oxide films would be suitable as a second but not a f i r s t d i e l e c t r i c layer i n a s i l i c o n MIS device. The capacitance of the MIS capacitors depended strongly on the oxidation time and post-oxidation heat treatment. i i i TABLE OF CONTENTS P a 9 e ABSTRACT i i TABLE OF CONTENTS i v LIST OF ILLUSTRATIONS i x LIST OF TABLES x v i i ACKNOWLEDGEMENT xix 1. INTRODUCTION 1 2. IONIC CONDUCTION IN ANODIC OXIDES 4 1. Introduction 4 2. Properties of Ionic Conduction and Anodic Oxides 4 2.1 Overpotential and O v e r f i e l d 4 2.2 Oxide Formation 5 2.3 Dependence of J on E 7 2.4 Transient Ionic Current Response 8 2.5 Structural Properties of Anodic Tantalum Oxide 12 2.6 Evidence that Both Ions are Mobile 15 2.7 U l t r a v i o l e t E f f e c t s on Anodic Oxides 20 3. Theories of Ionic Conduction 21 3.1 C l a s s i c a l Theory of Ionic Conduction 21 3.2 High F i e l d Frenkel Defect Theory and 25 Related Theories 3.3 D i e l e c t r i c P o l a r i z a t i o n Theory 28 3.4 Evidence for Current Driven P o l a r i z a t i o n 31 4. Phenomenological Models for Ionic Conduction 32 4.1 Channel Model 32 4.2 Structural Model 34 4.3 Surface Charge Model 35 5. Conclusions 38 3. ELLIPSOMETRY 39 1. Introduction 39 2. P r i n c i p l e s of Ellipsometry 40 2.1 The Fresnel Reflection C o e f f i c i e n t s 40 2.2 Ellipsometry Equations 43 i v Page 2.3 Computation of the Reflectance Ratio 45 2 .4 Computed Results 47 2 . 4 . 1 Single Layer Films on a Substrate 47 2 . 4 . 2 Multiple Films on a Substrate 47 3 . The Ellipsometer 52 3.1 Apparatus 52 3.2 P r i n c i p l e of Operation 56 3 .3 The Ellipsometer Control Program 58 3 . 3 . 1 The Balancing Algorithm 58 3 . 3 . 2 The Control Program 61 3 .4 S e n s i t i v i t y of the Ellipsometer Balance 62 3 .5 Precision of Ellipsometer N u l l i n g 66 4 . Ellipsometer Alignment and C a l i b r a t i o n 68 4 .1 Response of the Photodetector 68 4 .2 Telescope Axes 69 4 .3 Alignment of the Laser Source 70 4 . 4 T i l t i n g of the Optical Elements 74 4 .5 The Angle of Incidence 74 4 . 6 P o l a r i z e r and Analyzer Azimuths 75 4 . 6 . 1 Method 1: Reflection from a 76 Fused Quartz Slab 4 . 6 . 2 Method 2: Reflection from an 78 Iconel S l i d e 4 . 6 . 3 Method 3: Reflection at the 79 Brewster angle 4 . 6 . 4 Method 4 : Two-zone Balances 80 4 . 6 . 5 Conclusions 82 4 . 7 The Compensator Azimuth .' 82 4 .8 Compensator Transmittance and Retardation Errors 83 4 . 9 Azimuth Errors and Component Imperfections 85 4 . 9 . 1 Azimuthal Errors 87 4 . 9 . 2 Compensator Imperfections 88 4 . 9 . 3 P o l a r i z e r and Analyzer Imperfections 89 4 . 9 . 4 C e l l Window Birefringence 90 4 . 9 . 5 Four-Zone Measurements 94 4 . 9 . 6 Two-Zone Measurements 94 4 . 9 . 7 One-Zone Measurements 95 4 . 9 . 8 Measurement of the Ellipsometer Errors 96 4 . 9 . 9 Relation between t 1 c , t 2 c and T c, A c 99 4 . 9 . 1 0 Some Conclusions Regarding the Method of 100 Azzam and Bashara 4 . OPTICAL AND DIELECTRIC PROPERTIES OF 101 ANODICALLY OXIDIZED TANTALUM 1. Introduction 101 v Page 2. Experimental Apparatus and Procedure 101 2.1 Anodization C e l l 101 2.2 Measurement of the C e l l Window Birefringence 104 2.3 Sample Preparation 107 2.4 Procedure 109 3. Ellipsometry 109 3.1 Tracking a Growing Oxide Film 109 3.2 Evidence for an Isotropic Nonabsorbing Single- 115 Layer Oxide Film 3.3 Dependence on the Surface F i n i s h 117 3.4 Single Layer Film Model 117 3.5 Ellipsometry on a Rough Surface 122 3.6 S e n s i t i v i t y to an I n t e r f a c i a l Film 125 3.7 I n t e r f a c i a l Film Model 126 3.8 Comparison with the Spectrophotometric Method 130 3.9 D i f f e r e n t i a l E l e c t r i c F i e l d 134 4. Small Signal Capacitance 137 4.1 Theory 137 4.2 Method 137 4.3 Results and Discussion 139 4.3.1 E l e c t r o l y t e Resistance 139 4.3.2 Loss Tangent 141 4.3.3 Reversible Pot e n t i a l 141 4.3.4 Oxide P e r m i t t i v i t y 144 5. THE OPEN CIRCUIT TRANSIENT 147 1. Introduction 147 2. Experimental Procedure 148 2.1 Sample Preparation 148 2.2 Oxide Formation 148 2.3 Voltage Measurements 149 2.4 Time Measurements 151 2.5 Synchronization between Time and Voltage Measurements 152 2.6 Data A c q u i s i t i o n 152 3. Equipment Procedure 153 3.1 Voltmeter Response 153 3.2 A/D Converter Response 154 3.3 Experimental Simulation 155 4. Results and Data Analysis 158 4.1 Oxide P e r m i t t i v i t y 161 4.2 First-Order Analysis 161 v i Page 4.3 Second-Order Analysis 171 4.4 Dependence of the P e r m i t t i v i t y on F i e l d 174 4.5 Numerical D i f f e r e n t i a t i o n of the Open C i r c u i t 177 Transient E(t) Data 5. Discussion 179 6. THE STEPPED CURRENT TRANSIENT 182 1. Introduction 182 2. Experimental Procedure 182 2.1 Sample Preparation 182 2.2 Oxide Formation 182 2.3 Voltage Measurements 183 2.4 Current Source 183 2.5 Experimental Control and Data Acquisition 185 3. Experimental Tests and Simulations 186 3.1 Current Source Response 186 3.2 Experimental Simulation 187 4. Results and Analysis 187 4.1 Voltage-to-Field Conversions 191 4.2 P e r m i t t i v i t y of the Oxide 195 4.3 Peak F i e l d Data 195 4.3.1 First-Order Analysis 195 4.3.2 Temperature Dependence 199 4.3.3 Comparison with the Open C i r c u i t Transient 202 4.3.4 Comparison with Previously Published Data 204 4.3.5 The Excess F i e l d 204 7. COMPARISON WITH THEORY 208 1. Introduction 208 2. Mathematical Framework 208 2.1 The J(E) Relation 208 2.2 The Ideal Stepped F i e l d Transient 210 2.3 Comments on the Stepped F i e l d Transient 212 2.4 Comparison of Steady State and Stepped F i e l d Parameters 215 2.5 Numerical Calc u l a t i o n of Stepped Current and 216 Open C i r c u i t Transients 3. Stepped Current Transient 217 3.1 Comparison with Experimental Data 217 3.2 Predicted Change i n P o l a r i z a t i o n 222 v i i P a g e 4. Open C i r c u i t Transient 224 4.1 Comparison with Experimental Data 224 4.2 Predicted Change i n P o l a r i z a t i o n 229 5. Discussion 229 8. OPTICAL AND ELECTRICAL PROPERTIES OF THERMAL TANTALUM 233 OXIDE FILMS ON SILICON 1. Introduction 233 2. Some Recent Applications of T a 2 0 5 Films 233 3. Experimental Procedure 234 3.1 Sample Preparation 234 3.2 Sputtering 235 3.3 Oxidation 236 3.4 Ellipsometry 236 4. Results 238 4.1 Time of Oxidation 238 4.2 Model for Optical Properties 238 4.3 Discussion of the Optical Model 245 4.4 E l e c t r i c a l Properties 247 4.4.1 Current-Voltage Dependence 247 4.4.2 Capacitance-Voltage Dependence 252 5. Summary 255 9. CONCLUSIONS 256 1. Suggestions for Further Research 258 REFERENCES 259 APPENDIX A - DERIVATION OF THE ELLIPSOMETER EQUATION 267 APPENDIX B - COMPUTATION OF T c AND . 271 APPENDIX C - THE ELLIPSOMETER BALANCING ALGORITHM 272 APPENDIX D - THE PDP8/E COMPUTER SYSTEM 274 D.l Computer Configuration 274 D.2 The Ellipsometer Control Program 276 D.3 User's Guide to the Ellipsometer 281 v Control Program APPENDIX E - User's Guide to ELLIPS 324 v i i i LIST OF ILLUSTRATIONS Page Fi g . 2.1 Constant f i e l d transients (due to Cornish 1972). 10 F i g . 2.2(a) Cross-linking of chains of pentagonal bipyramids 14 i n L-Ta 20 5 (due to Roth and Stephenson 1971). 14 F i g . 2.2(b) The "herring-bone" structure of L-Ta 20 5 due to f o l d i n g of the otherwise s t r a i g h t chains. D i s t o r t i o n planes are created at the f o l d s . F i g . 2.3 Assumed p o t e n t i a l energy of an ion versus distance 22 with ( ) and without ( ) an applied e l e c t r i c f i e l d . F i g . 2.4 / The d i r e c t i o n of current flow and e l e c t r i c f i e l d s 22 i n the surface charge model of deWit et a l (1979) . F i g . 3.1 Reflection and r e f r a c t i o n at an i n t e r f a c e . 41 F i g . 3.2 Reflection from a film-covered surface. 41 F i g . 3.3 Computed ellipsometer curves for absorbing and 48 nonabsorbing films on Ta. F i g . 3.4 The e f f e c t of a 3 nm thick i n t e r f a c i a l f i l m on the 48 computed ellipsometer curve. F i g . 3.5 The computed curve for a 2 layer structure with 50 d(outer) = d(inner). The o p t i c a l constants are nm=1.334, ns=2.46-j2.2.573, 0 i=62.77° and X=632.8 nm. F i g . 3.6 The same conditions as i n F i g . ,3.5 but with the 50 f i l m indices reversed. F i g . 3.7 Configuration of the Rudolph ellipsometer used i n 53 the present work. Fi g . 3.8 A t y p i c a l photodetector signal recorded by the 59 computer. F i g . 3.9 The calculated r e l a t i v e s e n s i t i v i t i e s S A and 64 Sp for r e f l e c t i o n from Ta 20 5/Ta at 0^62.77° and X=632.8 nm. The r e f r a c t i v e indices were taken as nm=1.334, nf=2.195, and n s=2.46-j2.573. F i g . 3.10 The error functions F A and Fp as defined by 65 Merkt (1981). Page Fig. 3.11(a) The variation of the photodetector signal with 71 analyzer azimuth after i n i t i a l alignment of the laser and with circularly polarized light incident on the analyzer. Fig. 3.11(b) The variation of the photodetector signal after 71 realigning the laser with a third aperture in the analyzer arm. Fig. 3.12 The path of the centre of the light beam incident 73 on the photodetector pinhole with (a) a well aligned and (b) a poorly aligned laser as the analyzer is rotated through 360 degrees. Fig. 3.13 Zero correction for the angle of incidence scale. 73 Fig. 3.14 Zero correction to the polarizer and analyzer 77 scales using reflection from a fused quartz slab. Fig. 3.15 Zero correction to the polarizer and analyzer 77 scales using reflection from a Iconel slide at Gi=65. Fig. 3.16 Zero correction to the polarizer and analyzer 81 scales by the method of Azzam and Bashara. Fig. 3.17 Zero correction to azimuth scale of the 81 compensator. Fig. 4.1 Side and top view of the anodization c e l l used in 102 the present work. C=calomel electrode; Cl,C2=cooling c o i l ; T=thermometer; TP=thermistor probe; Gl,G2=gas bubbling tubes; H=150 W heater; S=stirring propellor; El,E2,E3=Pt/H electrodes. Fig. 4.2 The change in the polarizer balance angle due to 106 natural (0) and stress-induced (+) birefringence of the c e l l window. Fig. 4.3 Typical i|>(t) and A(t) data for an oxide film 111 growing at 0.66 A/m2. Fig. 4.4 The data of Fig. 4.3 shown on an enlarged scale. I l l Fig. 4.5 The ellipsometer data for an oxide film growing 113 0.66 A/m2. The solid line is the computed ellipsometer curve f i t t e d to the zero-field data. x F i g . 4.6 Z e r o - f i e l d ellipsometer data obtained at i n t e r v a l s during the constant current anodization of Ta. The s o l i d l i n e i s the computed curve f o r the i n t e r f a c i a l f i l m model (Section 3.7). F i g . 4.7 Z e r o - f i e l d ellipsometer data f o r two d i f f e r e n t . The s o l i d l i n e i s the computed curve for the i n t e r f a c i a l f i l m model (Section 3.7). Fi g . 4.8 Z e r o - f i e l d ellipsometer data f o r a t y p i c a l sample. The s o l i d l i n e was computed for the r e f r a c t i v e indices reported by Cornish (1972). F i g . 4.9 The e f f e c t of a 3 nm thick i n t e r f a c i a l f i l m on the computed ellipsometer curve for a single layer f i l m . F i g . 4 10 Representing a rough interface region by a t r a n s i t i o n layer with r e f r a c t i v e index N e and thickness D e. F i g . 4.11 Rp and Rg (for nm=1.334, nf=2.195, n s=2.46-j2.573, 0 i=62.77° and X=632.8 nm) plot t e d i n the complex plane as a function of f i l m thickness. F i g . 4.12 The data of F i g . 4.8 and the computed curve f o r the i n t e r f a c i a l f i l m model. Fi g . 4.13 Data for sample OCT1980 measured i n a i r at several angles of incidence. The contours of constant f i l m thickness and r e f r a c t i v e index were computed using the o p t i c a l constants given i n Table 4.2. F i g . 4.14 Wavelengths of minimum r e f l e c t i v i t y at 0^=11° versus Ta 20^ f i l m thickness (Young 1958). F i g . 4.15 The spectrophotometer trace of a t h i r d order minimum i n r e f l e c t i v i t y ( i . e . maximum i n apparent absorption). F i g . 4.16 The f i l m thickness measured by ellipsometry at i n t e r v a l s during the oxide growth at 0.66 A/m2. The f i l m thickness does not include the thickness of the i n t e r f a c i a l f i l m . F i g . 4.17 Representation of the anodic oxide by p a r a l l e l RpCp and series R SC S combinations. R c i s the resistance of the e l e c t r o l y t e . F i g . 4.18 A Wien capacitance bridge. x i F i g . 4.19 F i g . 4.20 F i g . 4.21 F i g . 5.1 F i g . 5.2 Fi g . 5.3 Fi g . 5 . 4 F i g . 5.5 Fi g . 5.6 Fig . 5.7 Fi g . 5.8 Fi g . 5.9 Fi g . 5.10 Fi g . 5.11 F i g . 5.12 Fi g . 5.13 The series resistance of the c e l l p l o t t e d as a function of measuring frequency. The inverse of the small s i g n a l capacitance of the oxide f i l m , meausred at 1 kHz, as a function of formation voltage. The capacitance data of F i g . 4.20 plotted against the oxide f i l m thickness measured by ellipsometry. Apparatus for constant current anodication and measurement of the anodie voltage. Timing diagram for the 10 kHz c r y s t a l clock. The times are given i n microseconds. V(t) data for a 1 pF capacitor discharing through a 250 k r e s i s t o r . The derivative (dV/dt) of the data p l o t t e d on F i g . 5.3. Typical open c i r c u i t transient V(t) data for an oxide f i l m grown to 15 v o l t s . E(t) data for oxide films grown under the same conditions as for F i g . 5.5. V(t) data during the i n i t i a l stages of an open c i r c u i t self-discharge. F i r s t - o r d e r analysis of the E(t) data plotted i n Fi g . 5.6. F i r s t - o r d e r analysis of E(t) data. Var i a t i o n of the optimum value of T with the time range of the f i r s t order a n a l y s i s . V a r i a t i o n of a with the time range of the f i r s t -order a n a l y s i s . V a r i a t i o n of e r with the time range of the f i r s t - o r d e r a n a l y s i s . The dependence of the i n i t i a l slope dlogJ/dE of the open c i r c u i t transient on the steady state f i e l d . x i i Page Fi g . 5.14 An exaggerated representation of the apparent non- 170 l i n e a r logJ-E r e l a t i o n for the open c i r c u i t s e l f -discharge. The dashed l i n e s represent the l i n e a r logJ-E r e l a t i o n s assumed i n the f i r s t - o r d e r analysis and t h e i r dependence on the extent of the transient included i n the a n a l y s i s . F i g . 5.15 Experimental E(t) data and the r e l a t i o n f i t t e d to 176 the dat by parameter optimization. F i g . 5.16 The deviation between the data i n F i g . 5.15 and 176 the f i t t e d r e l a t i o n . F i g . 5.17 Typical J(E) data calculated from the E(t) data by 178 numerical d i f f e r e n t i a t i o n . The steady state l i n e was calculated from Young's (1960) data. Fi g . 6.1 Schematic representation of the Northeast 184 S c i e n t i f i c model RI-233 current source. F i g . 6.2 V(t) data for a series LCR c i r c u i t subjected to a 188 voltage step. The dashed l i n e was calculated using the values for R, L, and C given i n the f i g u r e . F i g . 6.3 Typical V(Q) data f o r a stepped current t r a n s i e n t . 189 Fi g . 6.4 Typical charging current. 189 F i g . 6.5 E(,Q) data for 3 stepped current transients 193 recorded for the same and current r a t i o ^2/^1" Fi g . 6.6 E(Q) data for the i n i t i a l stages of a stepped 194 current transient. F i g . 6.7 J 2 ~ E p d a t a f ° r a series of stepped current 197 transients with Jj(A) = 3 A/m2, J ^ B ) = 1 A/m2, J 2(C) = 0.3 A/m2, and J^D) = 0.08 A/m2. F i g . 6.8 The slope of data set D i n F i g . 6.7 calculated by 198 f i r s t divided differences. F i g . 6.9 The temperature dependence of the slope of the 200 logJ 2-Ep data. The diagonal l i n e of p o s i t i v e slope shows the temperature dependence that might be expected on elementary theories. F i g . 6.10 The dependence of the slope of the logJ 2~Ep on 201 the i n i t i a l steady state f i e l d . x i i i Page F i g . 6.11 J 2 ( E m ) data for stepped current transients with 203 J2/'J1 < ^' ^ e ° P e n c i r c u i t t ransient J(E) l i n e represents the experimental data obtained for J s = 3 A/m2. F i g . 6.12 The dependence of the slope of the J 2 ~ E p d a t a 205 (•) on the formation current density. The data reported by Young (1961b) for the stepped f i e l d (J -»• V) and stepped current ( J ^ •*• J 2 ) transients are reproduced for comparison. Fi g . 6.13 The dependence of the excess f i e l d on J j and the 206 current r a t i o J 2 / J ^ . F i g . 7.1 Calculated stepped f i e l d logJ-E r e l a t i o n s . 214 Fig . 7.2 Experimental (+) and computed (—) E(Q) data f o r 218 stepped current transients with the same but d i f f e r e n t current r a t i o s J ^ J ^ . F i g . 7.3 Experimental and computed J 2 ~ E p ^ a t a ' 219 F i g . 7.4 The slope of the experimental J 2 ~ E p d a t a a n d ^ e 221 slope predicted by the phenomenological equations. F i g . 7.5 Dependence of the change i n t o t a l p o l a r i z a t i o n on 223 and current r a t i o J 2 / J ^ . F i g . 7.6 Experimental (+) and computed (—) E(t) data f or 225 an open c i r c u i t self-discharge. F i g . 7.7 Experimental (+) and computed (—) E(t) data for 3 227 open c i r c u i t transients. Fi g . 7.8 J(E) data calculated by numerical d i f f e r e n t i a t i o n 228 of the experimental E(t) data. The l i n e drawn through the data points i s the J(E) r e l a t i o n calculated for the i d e a l stepped f i e l d t r a n s i e n t . F i g . 7.9 Change i n t o t a l p o l a r i z a t i o n during the s e l f - 230 discharge. F i g . 8.1 Dependence of the Ta f i l m thickness on the 237 sputtering time. F i g . 8.2 The deposition rate as a function of the 237 sputtering power. F i g . 8.3 \J> and A measured at i n t e r v a l s during thermal 239 oxidation of a 40 nm thick Ta f i l m on s i l i c o n . x i v Page F i g . 8.4 Ellipsometry data f or several completely oxidized 240 fil m s . 0 = f i r s t cycle, • = second cycle. The s o l i d l i n e was computed f o r the i n t e r f a c i a l f i l m model with n f = 2.22 and n s = 3.87-J0.025. Bulk oxide thicknesses (nm) are indicated f o r the f i r s t c ycle. F i g . 8.5 The data of F i g . 8.4 with computed curves f or 241 single layer homogeneous films and n s = 3.87-jO.025. F i g . 8.6(a) Va r i a t i o n of r e f r a c t i v e index with distance from 243 the Si-oxide i n t e r f a c e . F i g . 8.6(b) The calculated mole f r a c t i o n of S i 0 2 i n the 243 inter f a c e region. F i g . 8.7 Enlarged parts of F i g . 8.4 with computed curves 244 for n f = 2.20, 2.2, and 2.24 ( i n t e r f a c i a l f i l m model). F i g . 8.8 Etchback data f or i n i t i a l oxide thicknesses of 246 130 nm (0) and 220 nm (•). Bulk oxide thicknesses (nm) are indicated for the f i r s t cycle ( i n t e r f a c i a l layer model). Fi g . 8.9 J(E) data for two MIS capacitors on the same p- 249 type substrate ((111) o r i e n t a t i o n , 0.06 ohm-m). The Ta f i l m (70 nm thick) was oxidized for 2.5 h giving an oxide thickness of 2.3 nm. Fig . 8.10 The e f f e c t of the length of oxidation on the 250 conductivity of the oxide f i l m s . For an i n i t i a l Ta f i l m thickness of 40 nm on p-type substrates ((111) o r i e n t a t i o n , 0.06 ohm-ra) the oxidation times were 1 h (+), 1.5 h (O), and 3 h (•). The data reported by Kaplan et a l for CVD T a 2 0 5 on p-type s i l i c o n are reproduced for comparison. F i g . 8.11 Capacitance as a function of bias voltage f or the 254 MIS capacitor used to obtain curve A i n F i g . 8.10. Fi g . 8.12 C(V) data f or a 40 nm thick Ta f i l m on p-type 254 s i l i c o n ((111) o r i e n t a t i o n , 0.06 ohm-m) oxidized for 3 h. Measurement frequencies were 5 kHz (•) and 500 kHz (0). F i g . A . l A PCSA ellipsometer configuration. 268 Fi g . A.2(a) P o l a r i z e r and compensator azimuths measured from 268 the plane of incidence. xv Page F i g . A.2(b) Analyzer azimuth measured from the plane of 268 incidence. F i g . D.l Configuration of the PDP8/E computer system. 275 F i g . D.2 Schematic of the photodetector and variable gain 277 amplifier interfaced to the computer. > x v i LIST OF TABLES 2.1 Oxygen Ion Transport Numbers for Anodic Tantalum Oxide Grown i n 0.1 M ^SO^ (due to Pringle, quoted by Dell'Oca, Pulfrey and Young 1971). 3.1 Relation between the Zones Defined by McCrackin et a l (1963) and the Azimuths of the Compensator and Analyzer. 3.2 Ideal Relations between P and A and the Ellipsometer Angles ty and A. 3.3 P o l a r i z e r and Analyzer Balance Azimuths for Reflection from an Iconel Slide at an Angle of Incidence of 70. 3.4 Reference Azimuths as a Function of the Angle of Incidence. 3.5 Var i a t i o n of the Error with the Angle of Incidence f o r Reflection from an Iconel S l i d e . 3.6 ty and A Corrected to First-Order (after Azzam and Bashara 1971). 3.7 N u l l i n g Azimuthal Angles as Functions of ty and A (after Azzam and Bashara 1971). 3 .8 Summary of Measured Ellipsometer Correction Factors. 4.1 Previously Reported Values for the Refractive Indices of Anodic T a 2 0 5 and Bulk Tantalum. 4.2 Summary of Results f o r the I n t e r f a c i a l Film Model. 4.3 Oxide Film Thicknesses Measured by Spectrophotometry and Ellipsometry and Calculated from the Steady State Data Reported by Young (1960). 4.4 Current-Field Data and Relative P e r m i t t i v i t y of Anodic T a 2 0 5 . 5.1 Reported Results f or the Second-Order Analysis of the Open C i r c u i t Transients for Ta, Nb, and W. 5.2 Quadratic C o e f f i c i e n t s i n the F i t t e d Relation l o g ( J / J Q ) = (5q/kT) (ctE-gE 2) for the Open C i r c u i t Transient. 6.1 Values of the Peak F i e l d as Measured with the A/D Converter and Storage Oscilloscope. 7.1 Values of Parameters used i n Ca l c u l a t i n g J(E) Data for T a 2 0 5 . x v i i Page 7.2 Parameter Values used i n F i t t i n g Stepped Current Transient 220 J(E) Data. 7.3 Values of Parameters Xi a n < a B i used i n F i t t i n g the Open 224 C i r c u i t Transient Data. D.l The Ellipsometer Control Program. 278 D.2 The Overlay Structure for the Ellipsometer Control Program. 280 ) x v i i i ACKNOWLEDGEMENT I would l i k e to thank my supervisor Dr. L. Young for h i s encourage-ment and guidance during the course of t h i s research. I wish to express my appreciation to Ms. G. Hrehorka for typing t h i s t h e s i s , Mr. J . Stuber and Mr. D. Fletcher for t h e i r assistance i n the machine shop, Mr. A. MacKenzie for drawing some of the graphs, and Dr. H.K. Tsoi and Mr. D. Wong for a s s i s -tance with the MOS measurements. The Natural Sciences and Engineering Research Council of Canada (Grant No. A3392 and scholarship awarded 1979-1980) and the Unive r s i t y of B r i t i s h Columbia (graduate fellowship 1977-1978) are g r a t e f u l l y acknowledged for t h e i r f i n a n c i a l support. xix 1 I. INTRODUCTION Thin d i e l e c t r i c f i l m s are of fundamental Importance to the f a b r i c a -t i o n of semiconductor devices. To date, thermally grown s i l i c o n dioxide has been the most widely used d i e l e c t r i c material in the f a b r i c a t i o n of i n t e -grated c i r c u i t s . In f a c t , the properties of Si02 have formed the basis for much of the present semiconductor technology. Doping of selected areas on a s i l i c o n substrate i s made possible by the a b i l i t y of the oxide to form a bar r i e r to most contaminants. In addition, the excellent e l e c t r i c a l and in t e r f a c e properties exhibited by the oxide have enabled i t to be used as a th i n f i l m insulator i n a wide v a r i e t y of MOS devices (Penney and Lau 1972). In recent years, however, other d i e l e c t r i c s have been used increasingly in specialized a p p l i c a t i o n s . For example, s i l i c o n n i t r i d e , a material once thought to be the solution to the sodium d r i f t problem in MOS devices (Sze 1969), i s now used in MNOS EAROMs and MOSFETs and for l o c a l c o n t r o l of ther-mal oxidation of s i l i c o n (Milnes 1980). More recently, anodic processing has become an important method of obtaining i n s u l a t i n g f i l m s between m u l t i l e v e l m e t a l l i z a t i o n s in s i l i c o n integrated c i r c u i t s (Schwartz and P l a t t e r 1976a,b). Since the oxide grows outward from the exposed surfaces of the metal i n t e r -connect l i n e s (usually aluminum metal), i t does not present the coverage pro-blems that a deposited i n s u l a t i n g f i l m might have. Anodic oxidation of GaAs i s also of s p e c i a l i n t e r e s t since a s u i t a b l e oxide cannot be grown on t h i s material by thermal oxidation as used in s i l i c o n processing (Hasegawa, Forward, and Hartnagel 1975, Kohn and Hartnagel 1978). Despite the increasing use of anodic oxides i n device a p p l i c a t i o n s , the high f i e l d ionic conduction mechanism by which the oxide f i l m grows i s not f u l l y understood. It i s known} 2 however, that the properties of the oxide f i l m are affected by the anodiza-t i o n conditions. Therefore, i t would seem reasonable to expect that a more complete understanding of the growth processes would have d i r e c t a p p l i c a t i o n to the p r a c t i c a l use of these f i l m s . Much of what i s known about high f i e l d ionic conduction in anodic oxides has been obtained from studies involving metals such as tantalum, niobium, and aluminum (Young 1961a). The present understanding of ionic con-duction i s b r i e f l y reviewed in chapter 2 of t h i s t h e s i s . Anodic oxides formed on tantalum in d i l u t e acid solutions were chosen for the present i n v e s t i g a -t i o n for several reasons. F i r s t , the metal gives highly accurate and repro-ducible r e s u l t s for both the properties and thickness of the oxide f i l m , which can then be compared with previously reported r e s u l t s for t h i s metal. Secondly, the values of parameters used in some models for ioni c conduction can be deduced from the previously reported r e s u l t s . Therefore, the r e s u l t s predicted by the models can be calculated and then compared with the experi-mental r e s u l t s obtained in the present study. Such comparisons between t h e o r e t i c a l and experimental r e s u l t s are e s s e n t i a l in t e s t i n g the v a l i d i t y of any model for ionic conduction. L a s t l y , and quite apart from any considera-t i o n s regarding the s u i t a b i l i t y of the oxide for an i n v e s t i g a t i o n of ioni c conduction, anodic tantalum oxide i s of considerable technical importance in the f a b r i c a t i o n of e l e c t r o l y t i c capacitors and t h i n f i l m RC c i r c u i t s (Westwood, Waterhouse, and Wilcox 1975). A computer-controlled ellipsometer was used to make in s i t u measurements of the o p t i c a l properties of the tantalum oxide f i l m . A new ellipsometer balancing algorithm and computer program which were developed to c o n t r o l the ellipsometer, are discussed in chapter 3. The r e s u l t s of an 3 extensive c a l i b r a t i o n procedure using the new con t r o l program, as well as a br i e f review of the p r i n c i p l e s of ellipsometry are also presented. The r e s u l t s of several sets of ellipsometer and capacitance measurements of the anodic oxide are given in chapter 4. The his t o r y dependence of the ionic con-d u c t i v i t y during the growth of the anodic oxide was also investigated. Using computer-controlled data a c q u i s i t i o n , the so-called open c i r c u i t and stepped current transients were recorded and then compared to previously reported data for steady state growth and so-called stepped f i e l d transient conditions. The r e s u l t s and analyses of the data are presented i n chapters 5 and 6 and are then compared in chapter 7 with the r e s u l t s predicted by Dignam's d i e l e c t r i c p o l a r i z a t i o n model for ionic conduction. Some conclusions are then drawn r e -garding the v a l i d i t y of t h i s model. F i n a l l y , the s u i t a b i l i t y of tantalum oxide f i l m s for s i l i c o n device f a b r i c a t i o n i s investigated in chapter 8. At the end of t h i s t h e s i s , some conclusions are drawn regarding the present understanding of high f i e l d ionic conduction and the s u i t a b i l i t y of a l t e r n a -t i v e d i e l e c t r i c s , p a r t i c u l a r l y tantalum oxide f i l m s , for s i l i c o n processing. I I . IONIC CONDUCTION IN ANODIC OXIDES 1. Introduction When c e r t a i n metals or semiconductors are immersed i n solutions of suit a b l e e l e c t r o l y t e s and anodically p o l a r i z e d , the r e s u l t i n g e l e c t r o s t a t i c f i e l d established i n the e x i s t i n g oxide a s s i s t s the transport of metal and oxygen ions through the f i l m , thereby causing the continued growth of the 8 9 oxide. T y p i c a l l y , an e l e c t r i c f i e l d i n the range 10 -10 V/m i s required f o r an appreciable oxide growth rate. Thus, the growth of anodic oxides i s e s s e n t i a l l y a problem of high f i e l d i o n i c conduction, complicated by the transfer processes which occur at the metal/oxide and o x i d e / e l e c t r o l y t e i n t e r faces. Some of the features of high f i e l d i o n i c conduction i n anodic oxides are b r i e f l y reviewed i n the following sections of t h i s chapter. The proper-t i e s of anodic tantalum oxide are p a r t i c u l a r l y emphasized because of the large amount of accurate and reproducible information that has been published for t h i s oxide. In the l a t t e r half of t h i s chapter, several models which hav been proposed for the ionic conduction process are discussed. The v a l i d i t y o each of these models i s tested by i t s a b i l i t y to describe the observed be-haviour of ionic conductivity under various oxide growth conditions. 2. Properties of Ionic Conduction and Anodic Oxides 2 .1 Overpot-ential and Over f i e l d The oxide overpotential i s defined by V = V - V (1) m rev where V i s the p o t e n t i a l of the oxide electrode measured with respect to a m c r e v e r s i b l e reference electrode (corrected f o r the ohmic p o t e n t i a l d i f f e r e n c e i n the solution) and the r e v e r s i b l e p o t e n t i a l V i s the p o t e n t i a l which r rev r would e x i s t between the two electrodes i f the oxide electrode were in equi-l i b r i u m . The r e v e r s i b l e p o t e n t i a l of the oxide-covered tantalum electrode i n the present study, calculated from thermodynamic data, i s -0.83 v o l t s with J 5 respect to a r e v e r s i b l e hydrogen electrode (Latimer 1952). The hydrogen electrode can be approximated experimentally by a p l a t i n i z e d platinum e l e c -trode i n a s o l u t i o n saturated with hydrogen. The r e v e r s i b l e p o t e n t i a l of the tantalum electrode with respect to some other reference electrode, such as a calomel electrode, i s obtained by subtracting from the above value the p o t e n t i a l of the reference electrode with respect to the hydrogen electrode. By analogy with the overpotential, Young (1957) defined an o v e r f i e l d E = V/D where D i s the oxide thickness. Even neglecting a possible space charge e f f e c t , the o v e r f i e l d i s not s t r i c t l y equal to the f i e l d i n the oxide since V includes p o t e n t i a l differences which might occur at the metal/oxide and o x i d e / e l e c t r o l y t e i n t e r f a c e s . For anodic tantalum oxide, however, these p o t e n t i a l d i f f e r e n c e s are l e s s than 0.1 v o l t over a large range of current d e n s i t i e s (e.g. Seijka, Nadai, and Amsel 1971). Therefore, with f i l m s thicker than about 20 nm (for which space e f f e c t s appear to be n e g l i g i b l e ) , the over-f i e l d has usually been taken to be the mean e l e c t r i c f i e l d i n the oxide. L i t t l e information has been published for thinner oxide f i l m s . For tantalum, the d i f f e r e n t i a l f i e l d AV/AD for a given formation current density and temperature i s independent of the oxide,thickness and, within an experimental error of ^1%, equals the o v e r f i e l d . 2.2 Oxide Formation The increase i n thickness D of an anodic oxide due to the passage of an amount of charge Q through the oxide i s given by Faraday's law as D = QMn/zyAFp (2) where A i s the area, p i s the oxide density, and zy i s the number of Faraday's, F, of charge which must be passed to produce one mole of oxide of formula Me 0 and molecular weight M. The current e f f i c i e n c y n, defined as the x y f r a c t i o n of charge which contributes to oxide growth rather than side reactions, e l e c t r o n i c leakage, or metal l o s s to the e l e c t r o l y t e , may be 6 estimated from weight gain of the oxide and oxygen uptake. The estimated e f f i c i e n c y i s a f f e c t e d by e l e c t r o l y t e i n c o r p o r a t i o n i n t o the oxide and by an e f f e c t i v e area which i s d i f f e r e n t from the measured area. For tantalum oxide f i l m s formed i n d i l u t e ^SO^ o n c n e m : L C a l l y or e l e c t r o c h e m i c a l l y p o l i s h e d t a n -talum, the current e f f i c i e n c y approaches u n i t y (Young 1961). Mechanically prepared tantalum u s u a l l y gives lower e f f i c i e n c i e s due mainly to oxygen evo l u -t i o n at scratches i n the metal surface. Experimental data f o r the anodization at constant current d e n s i t y J of metals such as tantalum (Young 1960), niobium (Young and Zobel 1966), and aluminum (Dignam 1965a) have shown that a constant f i e l d E, w i t h i n an e r r o r on the order of vL%, i s e v e n t u a l l y e s t a b l i s h e d i n the oxide independent of voltage, at l e a s t f o r f i l m s t h i c k e r than about 20 nm. This constant J-E c o n d i t i o n at constant temperature i s r e f e r r e d to as the steady s t a t e . Dignam (1978) has r e c e n t l y asserted that the e r r o r i n E i s s u f f i c i e n t l y l a r g e f o r the data to be c o n s i s t e n t w i t h a model i n which E decreases w i t h i n c r e a s i n g f i l m t h i c k n e s s f o r oxide growth at constant J (see Section 2.3). In e i t h e r case, the r a t e of increase i n voltage across the oxide i s given by dV = dV dD = Mn dt dD dt 2yFp 3 K } dV Therefore, the constancy or near-constancy of — may be used to check that the a n o d i z a t i o n i s proceeding normally. Oxide formation at constant voltage i s not a steady s t a t e process s i n c e , as the oxide grows, the f i e l d i n the oxide decreases, thereby reducing the i o n i c current and thus the r a t e of oxide growth. The current e v e n t u a l l y becomes s u f f i c i e n t l y small that e l e c t r o n i c leakage predominates and oxide growth stops. A combination of formation at constant current and constant v o l t a g e i s o f t e n used i n component f a b r i c a t i o n . I t i s g e n e r a l l y b e l i e v e d that a constant v o l t a g e "soak" somehow r e p a i r s s t r u c t u r a l f a u l t s i n the 7 oxide, thereby improving i t s e l e c t r i c a l properties (Young 1961a). 2.3 Dependence of J on E Experimental data f o r the steady state dependence of J on E f o r several metals have been f i t t e d by the expression J = J q exp - W(E)/kT (4) where J q i s a constant and W(E) i s a field-dependent a c t i v a t i o n energy. The a c t i v a t i o n energy was o r i g i n a l l y thought to be a l i n e a r function of the f i e l d (Verwey 1935, Vermilyea 1955) i . e . W(E) = Wq - XE (5) where and A are constants. Young (1960) l a t e r demonstrated for tantalum anodized in d i l u t e I^SO^ that W(E) has, i n f a c t , a nonlinear dependence on f i e l d . Expanding W(E) as a Taylor series and neglecting high order terms beyond the quadratic, the dependence of W(E) on f i e l d was described by W(E) = W - ctE + BE 2 (6) o where W^ , a, and B are constants. Young and Zobel (1966) l a t e r showed that the steady state data could be represented almost as well by the r e l a t i o n W(E) = W - Y E 2 (7) o where y i s a constant. Relations of the form given by eqns. 6 and 7 are necessary to describe the temperature dependence of the J-E r e l a t i o n and have been shown to be v a l i d f o r temperatures up to 250°C (Dreiner and Tripp 1970). Anodization of tantalum i n e l e c t r o l y t e s giving s i g n i f i c a n t incorporation of e l e c t r o l y t e ( i . e . H^O^ (Dell'Oca and Young 1970), concentrated H 2S0^ (Young I960)) increases the n o n l i n e a r i t y of the W(E) r e l a t i o n . Although (6) has been shown to give a better f i t to the data than (7) (Dignam and Gibbs 1969), Dell'Oca and Young (1970) noted that the accuracy of the f i t cannot be used as a c r i t e r i o n f o r s e l e c t i n g one g r e a t l y s i m p l i f i e d model over 8 another. The log J-E r e l a t i o n s f o r aluminum (Dignam, Goad, and Sole 1965, Dignam and Goad 1966) and niobium (Young and Zobel 1966) have also been reported to be nonlinear. Other metals such as bismuth (Masing and Young 1962) and zirconium (Hopper, Wright and DeSmet 1977) have been reported to give a l i n e a r log J-E r e l a t i o n but t h i s may be due to l i m i t e d accuracy of the measurements. Recently, Dignam (1979) proposed a model in which the ionic current i s described by J = J q X exp - W(E)/kT (8) where X i s the f i l m thickness and W(E) i s given by eqn. 6. As evidence i n support of (8), Dignam compared the E-V r e l a t i o n calculated from (8) and some assumptions regarding the rate of change of E with X, with two sets of Young's (1958) steady state E-V data for T a ^ f i l m s grown i n d i l u t e H^SO^. Although the calculated data did not agree with one set of experimental data Dignam concluded that there was s u f f i c i e n t evidence in support of the f i e l d decreasing with increasing fi^m thickness that eqn. 8 could not be j u s t i f i a b l y r e j e c t e d . Additional evidence was given by the dV/dt data of Dignam and Ryan (1963) for the constant current anodization of aluminum. Although dV/dt decreased with time during the oxide growth, as expected i f E decreased with increasing X, Dignam admitted that other explanations are possible. Therefore, i t i s not clear at the moment whether eqn.8 i s a correct d e s c r i p t i o n of the J-E r e l a t i o n . 2.4 Transient Ionic Current Response If an anodic oxide i s grown at constant current density to some 9 voltage V and then held constant at V, the f i e l d i n the oxide decreases continuously from i t s i n i t i a l steady state value due to oxide growth. The decrease i n f i e l d strength i s slow enough, however, that at a l l new f i e l d s E' the ionic current density i s given by the steady state r e l a t i o n (Young and Zobel 1966). In other words, the operating point f o r the anodization moves along the steady state l i n e , defined by log J plotted against E, during the constant voltage phase of the anodization. On the other hand, the ionic current and f i e l d do not immediately adopt t h e i r new steady state values when the anodization conditions are r a p i d l y changed. Instead, the h i s t o r y dependence of the oxide e x h i b i t s i t s e l f in a transient response of the current and f i e l d to the changed c e l l conditions before allowing these quantities to s e t t l e to t h e i r new steady state values. For example, i f the f i e l d i s forced to decrease to a new f i e l d E' as r a p i d l y as possible s t a r t i n g from a steady state condition, the instan-taneous ionic current density at E' i s larger than the corresponding steady state value. When plotted as log J vs E, the locus of several such c u r r e n t - f i e l d points having the same steady state s t a r t i n g point define a so-called stepped f i e l d transient l i n e . The ionic current density at the new f i e l d E' can then be thought of as being reached by traversing t h i s l i n e . In a s i m i l a r experiment, r a p i d l y increasing the applied current from an i n i t i a l steady state value to a new value J causes the f i e l d to go through a peak value E^ before i t relaxes toward i t s new steady-state value E^. A u s e f u l i n t e r p r e t a t i o n of t h i s so-called stepped current transient i s to consider J„ and E as a " f i r s t - o r d e r " approximation to stepped f i e l d 2 P conditions (Dewald 1957). The so-called open c i r c u i t transient i s a spe c i a l case of the stepped current transient i n which the applied current i s reduced I I -I 1 1 — 1 0 5 10 15 20 25 30 T I M E / S E C F i g . 2.1 Constant f i e l d transients (due to Cornish 1972). 11 to zero ( i . e . the external c i r c u i t i s opened) and the capacitor formed by the metal/oxide/electrolyte i s allowed to self-discharge by io n i c conduction. These transients w i l l be discussed i n greater d e t a i l i n chapters 5 - 7 . The transient responses of J and E have been suggested by many authors as being due to a transport mechanism involving defects whose concentration i s a function of the f i e l d but changes sluggishly with sudden changes i n the f i e l d (see section 3.2). Vermilyea (1957a),using stepped f i e l d methods, investigated the ef f e c t of annealing on these presumed defects. Young (1964) reported on s i m i l a r experiments i n which the oxide was formed at constant J and then held at zero f i e l d f o r a time t . Upon reapplying the formation f i e l d E, i t was observed that the instantaneous current was l e s s than the formation current and that the d i f f e r e n c e increased with increasing t. These r e s u l t s are possibly r e l a t e d to the observations that the small signal capacitance and losses of the oxide decays during the period at zero f i e l d (Young 1956) and that the audio frequency losses are independent of frequency, which suggests a range of r e l a x a t i o n times (Young 1961a). In another experiment (Young 1961b), the concentration of defects was presumably reduced by immersing the oxide-covered electrode i n b o i l i n g water f o r several minutes. When the electrode was returned to the anodization c e l l and a f i e l d , much greater than the formation field,was applied the ionic current b u i l t up to a maximum in an accelerating manner. Figure 2.1 shows t y p i c a l data recorded by Cornish (1972) for anodic tantalum oxide i n d i l u t e IL^SO^. Similar r e s u l t s have been reported by Dignam and Ryan (1968b) for aluminum. These so-called constant f i e l d t r a n s i e n t s are s i g n i f i c a n t i n that they demonstrate that the io n i c conduction process depends on io n i c current as well as e l e c t r i c f i e l d and temperature. 12 Several theories have been proposed to describe the above transient phenomena. Some of these w i l l be reviewed i n l a t e r sections of t h i s chapter. 2.5 St r u c t u r a l Properties of Anodic Tantalum Oxide The foregoing discussion i s general to anodic oxides. In the remainder of t h i s section, we w i l l be concerned p r i m a r i l y with the properties of anodic tantalum oxide, although some of i t s properties are si m i l a r to those of other valve metals. Tantalum oxide f i l m s grown under normal conditions are s t o i c h i o -metric Ta^O^ (Knausenberger and Tauber 1973, Silverman and Schwartz 1974) and have an amorphous or g l a s s - l i k e structure (Vermilyea 1957c). However, prolonged high voltage p o l a r i z a t i o n i n solu t i o n can cause the oxide to undergo a so-called f i e l d r e c r y s t a l l i z a t i o n process i n which new c r y s t a l l i n e oxide di s p l a c e s the e x i s t i n g amorphous oxide (Vermilyea 1957d). Detached flakes of amorphous oxide can be c r y s t a l l i z e d by thermal treatment (Dell'Oca 1969). The oxide i s also remarkably d u c t i l e ; when stretched or r o l l e d , i t thins considerably but, for c e r t a i n metal preparations, remains f i r m l y attached to the metal (Dunn 1968, Propp and Young 1979). Even on r e l a t i v e l y rough surfaces the oxide thickness, to a f i r s t approximation, i s uniform over the area, as indicated by the uniformity of the thickness-dependent interference colours. The remarkable p l a s t i c i t y of the oxide can, i n some cases, support smoothing of the oxide surface. The amorphous and, p a r t i c u l a r l y , the r e c r y s t a l l i z e d oxide are chemically r e s i s t a n t to acids, being attacked appreciably only by HF. The etch rate depends on the formation current density, the annealing h i s t o r y of the oxide f i l m (Vermilyea 1957c), and on the e l e c t r o l y t e used i n the 13 a n o d i z a t i o n (Vermilyea 1954). Although anodic Ta^O^ i s amorphous some degree of short range order would be expected. Therefore, the s t r u c t u r a l p r o p e r t i e s of the c r y s -t a l l i n e oxide may have some a p p l i c a t i o n to the anodic oxide. Moreover, i n view of the s t r u c t u r a l f l e x i b i l i t y of the oxide, i t i s l i k e l y that the s t r u c t u r e i s a f a c t o r i n the i o n i c c o n d u c t i v i t y . Roth and Stephenson (1971, 1976) regard the "semiamorphous" form of Ta^O^, which i s produced by heating tantalum i n oxygen at 600°C, as co n t a i n i n g chains of pentagonal bypyramids w i t h oxygen atoms at the corners and tantalum atoms i n the center. These chains are of v a r i o u s lengths and are j o i n e d randomly by sharing corners. Presumably the anodic oxide f i l m s are constructed according to a s i m i l a r p r e s c r i p t i o n . On prolonged heating, the thermal oxide i s converted i n t o L-Ta20,. whose s t r u c t u r e i n i t s i d e a l form seems to be, according to Roth and Stephenson, roughly as f o l l o w s . F i r s t , chains of edge-sharing pentagonal bipyramids, arranged so that adjacent pentagons point i n opposite d i r e c t i o n s , a t t a c h to each other by sharing corners ( f i g u r e 2.2(a)). Octahedral s i t e s (which c o n t a i n tantalum atoms) are created between adjacent chains thus l i n k e d . I n an e f f e c t which reduces the oxygen-metal r a t i o from 2.667:1, the otherwise s t r a i g h t chains are fo l d e d so that some p a i r s of octahedra share a corner ( f i g u r e 2 .2(b)). A sequence of s t r u c t u r e s w i t h d i f f e r e n t oxygen-metal r a t i o s may be formed by f o l d i n g chains w i t h d i f f e r e n t numbers of pentagons. In the r e a l , as opposed to i d e a l , s t r u c t u r e , oxygen atoms at the f o l d s shared between two pentagonal bipyramids and one octahedron may be absent, r e s u l t i n g i n d i s t o r t e d o c t a h e d r a l c o o r d i n a t i o n f o r a l l metal atoms surrounding the "vacancy". 14 F i g . 2.2(a) C r o s s - l i n k i n g o f c h a i n s of p e n t a g o n a l b i p y r a m i d s i n L-Ta-CV (due to Roth and Stephenson 1971). B 1 0 ° 2 6 F i g . 2.2(b) The " h e r r i n g - b o n e " s t r u c t u r e of L-Ta 2C> 5 due t o f o l d i n g of o t h e r w i s e s t r a i g h t c h a i n s . D i s t o r t i o n p l a n e s a re c r e a t e d at the f o l d s . 15 When the amorphous form i s annealed, a slow evolution to the most stable form of L-Ta20,. occurs. Roth and Stephenson i d e n t i f i e d three important processes. F i r s t , the lengths of the straight portions of the folded chains decreases. Second, the number of "distortion"planes decreases (one way to define the d i s t o r t i o n plane i s as the plane on which l i e the metal atoms whose coordination i s decreased from 7 to 6). Third, the d i s t o r t i o n planes tend to p a i r . S t r u c t u r a l f l e x i b i l i t y of the type outlined by Roth and Stephenson seems very adequate to account for the transient J and E behavior outlined i n the previous section. The atomic arrangement surrounding the atoms of reduced coordination seems a possible candidate as a defect l i k e l y to be involved in the ion transport process. An important point i s that such defects appear to i n t e r a c t , as indicated by the s t r u c t u r a l changes on annealing. Both the concentration and degree of order of the defects could be involved i n the h i s t o r y e f f e c t s . A l l p r e s s (1969) and Young and Zobel (1966) seem to have envisaged a similar s i t u a t i o n i n which "extra" ions are trapped in "tunnels" i n the oxide structure. Such tunnels in c r y s t a l l i n e Nb^C^q ( s :' i n^ a r ^ n structure to Ta^O^) have been resolved r e c e n t l y by high r e s o l u t i o n electron microscopy (Horiuchi, Matsui, and Bando 1976). 2.6 Evidence that Both Ions are Mobile In the early studies of anodic oxidation i t was generally assumed that only one ion (usually the metal ion) was involved i n ion transport through the growing oxide f i l m . This r e l a t i v e l y simple p i c t u r e has since been complicated by the r e s u l t s of marker experiments which show for many oxides that both metal and oxygen ions are mobile. In the'.se marker 16 experiments, the growth of the oxide i s measured r e l a t i v e to the p o s i t i o n of an i n e r t marker layer within the oxide. Assuming that the marker atoms are immobile, the increase i n thickness between the marker layer and one of the inte r f a c e s represents the contribution to growth of the ion responsible for growth at that i n t e r f a c e . Presumably, i f the oxide i s a network, the marker atoms remain at f i x e d p o s i t i o n s r e l a t i v e to the network even though growth processes may displace the normal oxide. The ion transport number, defined as the f r a c t i o n of i o n i c current c a r r i e d by the ion i n question, i s then given by the f r a c t i o n of oxide growth due to the ion, provided the oxide i s b a s i c a l l y the same on either side of marker layer. In a s e r i e s of marker layer experiments with valve metals, Davies, Domeij, Pringle, and Brown (1965) implanted radioisotopes of i n e r t gases into the metal or e x i s t i n g oxide and, during subsequent oxide growth, followed the p o s i t i o n of t h i s marker layer by a or 3-ray spectroscopy. I t was found for Nb, Ta, A l , and W that both metal and oxygen ions are comparably mobile and that new oxide i s formed simultaneously at the metal/oxide and oxide/ e l e c t r o l y t e i n t e r f a c e s . Formation of fresh oxide within the e x i s t i n g oxide, as suggested by Cheseldine (1964), was not observed. The r e s u l t s f or Al were l a t e r confirmed by Brown and Mackintosh (1973). By contrast, Domeij et a l found with Zr and Hf and Mackintosh and Plattner (1977) with S i that nearly a l l of the oxide growth i s due to mobile oxygen ions. P r i n g l e (1972, 1973a) combined a chemical sectioning technique with precise o p t i c a l thickness measurements to obtain the concentration p r o f i l e of the r a d i o -isotopes implanted into anodic Ta^O^. The transport numbers given i n Table 2.1 were determined from the mean posit i o n s of the p r o f i l e s measured before and a f t e r oxide growth. 17 Table 2.1 Oxygen Ion Transport Numbers for Anodic Tantalum Oxide Grown in 0.1 M H SO^ (due to Pringle, quoted by D e l l Oca, Pulfrey, and Young 1971). 0°C 25°C 50°C 75°C 95°C 100 A/m2 0.712 0.726 0.741 0.756 0.770 10 A/m2 0.729 0.744 0.763 0.784 0.803 1 A/m2 0.759 0.767 0.793 0.817 0.841 E l e c t r o l y t e incorporation has also been used to determine ion transport numbers. Assuming the incorporated species are immobile, then, at constant formation current density, the metal ion transport number i s given by the f r a c t i o n of oxide in which incorporation has occurred. Randall, Bernard, and Wilkinson (1965) and Randall (1975) used a r a d i o -es a c t i v e l y l a b e l l e d e l e c t r o l y t e to measure the width of the layer containing incorporated e l e c t r o l y t e . A metal ion transport number of 0.51 was found for tantalum anodized at 10 A/m2 i n 0.001 M H 3P0^ at 25°C. The transport number was reported to increase with e l e c t r o l y t e concentration, current density, and decreasing temperature, i n general agreement with the r e s u l t s reported by P r i n g l e . Dell'Oca and Young (1969, 1970a), using ellipsometry, confirmed the double l a y e r structure of tantalum oxide f i l m s grown i n d i l u t e H^PO^ and the transport number reported by Randall et a l . For T a 2 0 5 f i l m s grown at 10 A/m2 in. 0.1 M l^SO^ at 25°C, Randall et a l found a metal ion transport number of 0.48. Although the ion implantation and e l e c t r o l y t e incorporation methods give c o n f l i c t i n g r e s u l t s f or the transport numbers, the s i m i l a r 18 dependence of these numbers on current density and temperature seems to indic a t e that both methods are a measure of the same process. The disagreement i s l i k e l y due to motion of the supposedly immobile marker atoms. P r i n g l e (1972) found that the concentration p r o f i l e of the marker atoms broadened with oxide growth. The broadening increased with decreasing mass of the marker atom but the center of the p r o f i l e was found at the same point in the oxide (for a given f i l m thickness) independent of the mass of the marker atom. Prin g l e postulated that the marker atoms are subjected to a form of Brownian motion by c o l l i s i o n s with the migrating tantalum and oxygen ions. In the e l e c t r o l y t e incorporation experiments, on the other hand, the sharp boundaries found between layers of high and low concentration of incorporated species seem to indi c a t e that the species are immobile. -3 However, i f the species are incorporated as anions (PO^ for oxide growth in H^PO^) the charge i s of the r i g h t sign to have the species migrate toward the metal/oxide i n t e r f a c e , thereby increasing the apparent metal ion transport number. Such behavior has been observed with implanted anions (Brown and Mackintosh 1973) although the ion transport depends on the type of impurity ion (Pawel, Pemsler, and Evans 1972). Randall et a l (1965) and Dell'Oca (1969) concluded that i t i s reasonable to expect an incorporated anion, which presumably bonds with atoms i n the oxide, to be l e s s mobile than an implanted atom and that, therefore, the transport numbers found by the e l e c t r o l y t e incorporation method are the correct r e s u l t s . The above-mentioned marker experiments provide information regarding only how much the metal and oxygen atoms move during oxide growth. To obtain information about how the atoms move, is o t o p i c tracer experiments 18 are required. Tracer experiments using 0 (Amsel and Samuel 1962, Seij.ka et a l 1971, P r i n g l e 1973b) and Ta (Pringle 1978) have shown that the order of both tantalum and oxygen atoms i s conserved, apart from a small d i f f u s i v e mixing. That i s , tantalum atoms o r i g i n a l l y at the surface of the bare metal appear at the surface of the oxide. S i m i l a r l y oxygen atoms used to form the f i r s t few monolayers of oxide adjacent to the metal surface are always found at the metal/oxide i n t e r f a c e . Thus, i t appears that charge transfer in the growing oxide involves simultaneous movement of oxygen and tantalum atoms over short distances in a way s i m i l a r to vacancy migration. A ser i e s of such events across the oxide would transfer charge from one side of the oxide to the other. The foregoing discussion applies only to anodic oxides on bulk or si n g l e f i l m substrates. The e f f e c t of one metal on the anodization of another, as i n an a l l o y , i s usually d i f f i c u l t to d i s t i n g u i s h . P e r r i e r e , Rigo, and Seijka (1978a, b, c) overcame t h i s problem by studying the anodization of superimposed layers of d i f f e r e n t metals. I n t e r e s t i n g l y , the metal and oxygen orders were found to be conserved when the layers were superimposed i n one way but, when superimposed i n the opposite way, the order was p a r t i a l l y inverted. An explanation recently proposed by P r i n g l e (1980 a, b) involves the r e l a t i v e r esistance of the two oxide layers to i o n i c conduction. When the superimposed oxide i s more r e s i s t i v e to i o n i c conduction than the substrate oxide, i t was proposed that fingers of the substrate oxide force t h e i r way toward the surface in a manner analagous to the Rayleigh-Taylor e f f e c t in superimposed l i q u i d l a y e r s . According to Pr i n g l e ' s theory (and consistent with experimental r e s u l t s ) a layer of pure substrate metal oxide can, in some cases, form on top of the o r i g i n a l surface oxide. The theory may also be u s e f u l i n explaining the anodic 20 oxidation of compound semiconductors, p a r t i c u l a r l y when a metal f i l m such as aluminum has been f i r s t deposited.on the semiconductor. I t should be noted that P r i n g l e has stated h i s theory applies only to oxides grown by anodic oxidation. No penetration of substrate metal oxide into the surface oxide has been found (so far) for thermally grown oxides. 2.7 U l t r a v i o l e t E f f e c t s on Anodic Oxides An a d d i t i o n a l factor which should be considered i n any discussion of anodic oxide f i l m s i s the e f f e c t of u l t r a v i o l e t (UV) r a d i a t i o n on the properties of the oxide f i l m . It i s known from the early work of Bray, Jacobs, and Young (1959) and Vermilyea (1957b), among others, that UV r a d i a t i o n causes a large photocurrent to flow in anodic tantalum oxide at f i e l d strengths too low to support normal oxidation.(see Young 1961a for references to e a r l i e r work). The photocurrent c o n s i s t s of a large e l e c t r o n i c current which appears i n s t a n t l y on applying the UV r a d i a t i o n and decays over a period of hours, and a "secondary" or dark current which increases from i t s i n i t i a l l y low value, usually a f t e r an incubation period of 10 -20 minutes, and then decays a f t e r reaching a maximum value. This dark current i s measured by p e r i o d i c a l l y i n t e r r u p t i n g the UV r a d i a t i o n . During the incubation period, the properties of the oxide f i l m are slowly modified, as indicated by the changing etch rate of the oxide by d i l u t e HF (Vermilyea 1957b). For short exposures to UV l i g h t , the change i n the f i l m i s almost e n t i r e l y r e v e r s i b l e (Cornish 1972). However, once the dark current begins to b u i l d up the changes i n the oxide become permanent and noticeable by the changes i n the interference color of the oxide. That oxide growth a c t u a l l y occurs during the increase i n dark current was o r i g i n a l l y confirmed by an increase i n weight of the oxide. More recently, ellipsometer measurements 21 (Cornish 1972, Dell'Oca 1969) have shown that both the th i c k n e s s and r e f r a c t i v e index of the oxide f i l m change. The data a l s o showed that during the i n c u b a t i o n p e r i o d , the o r i g i n a l homogeneous oxide i s somehow transformed i n t o a double l a y e r s t r u c t u r e w i t h n e i t h e r of the l a y e r i n d i c e s equal to the r e f r a c t i v e index of the o r i g i n a l f i l m . Cornish (1972) suggested that s t r u c -t u r a l changes r e s u l t i n g i n a reduced f i l m d e n s i t y or i n c o r p o r a t i o n of water or s u l f a t e ions during the incu b a t i o n period could account f o r the change i n r e f r a c t i v e index. The foregoing d i s c u s s i o n suggests an interdependence between i o n i c and e l e c t r o n i c processes i n the growth of anodic oxides. However, present models f o r i o n i c conduction are based on the assumption that the two processes are independent. 3. Theories of I o n i c Conduction Several t h e o r i e s have been proposed f o r i o n i c conduction i n anodic oxides. Some of these w i l l be b r i e f l y reviewed i n the next s e c t i o n s of t h i s chapter. 3.1 C l a s s i c a l Theory of I o n i c Conduction The c l a s s i c a l theory of i o n i c conduction d e s c r i b e s the motion of a low c o n c e n t r a t i o n of n o n i n t e r a c t i n g point d e f e c t s i n a c r y s t a l l i n e s o l i d . The d e f e c t s may be vacant l a t t i c e s i t e s , i n t e r s t i t i a l i o n s , or a combination of the two. Since the d e f e c t s move by s i m i l a r mechanisms only one type of de f e c t , say an i n t e r s t i t i a l c a t i o n , need be considered i n d e r i v i n g a r e l a t i o n f o r the current d e n s i t y . The tr a n s p o r t of the defect w i t h i n the s o l i d i s p i c t u r e d as being c o n t r o l l e d by a s e r i e s of p o t e n t i a l energy b a r r i e r s of the type depicted i n f i g u r e 2.3. For s i m p l i c i t y , the p o t e n t i a l energy b a r r i e r 22 METAL OXIDE BOUNDARY LAYER BULK J Z ~ Jn I —E F i g . 2 . 4 The d i r e c t i o n of c u r r e n t f low and e l e c t r i c f i e l d s i n the s u r f a c e charge model o f deWit et a l ( 1 9 7 9 ) . 23 i s taken to be p e r i o d i c and time independent. Consider an i n t e r s t i t i a l i o n w i t h charge q o s c i l l a t i n g w i t h frequency v about a minimum i n the p o t e n t i a l energy. To jump a di s t a n c e 2a to an adjacent minimum, the i o n must, i n the absence of an e x t e r n a l f i e l d , overcome an a c t i v a t i o n b a r r i e r of energy W. The p r o b a b i l i t y that the i o n has s u f f i c i e n t energy to overcome the b a r r i e r i s given by exp (-W/kT). I t i s assumed t h a t , having jumped to the adjacent s i t e , the i o n i s immediately d e a c t i v a t e d . A p p l i c a t i o n of an e l e c t r i c f i e l d E reduces the height of the a c t i v a t i o n b a r r i e r by an amount qaE where, f o r s i m p l i c i t y , i t i s assumed that the i o n jumps only p a r a l l e l to the a p p l i e d f i e l d . The p r o b a b i l i t y of ion transport over the b a r r i e r i n v attempts per second i s P(E) = v exp - [W-qaE]/kT Therefore, the current d e n s i t y J ( x ) at a point XQ ^_ XQ + 2a i s given by J(x) = J f ( x ) - J r ( x ) = 2aqvn( X)P(E) - 2aqv(n( X) + 2 a ^ ) P ( - E ) (9) A where J^(x ) i s the forward current d e n s i t y due to an ion d e n s i t y n ( X ) t r a v e l l i n g a d i s t a n c e 2a i n the d i r e c t i o n of the f i e l d . The reverse current J^Cx) i n c l u d e s the e f f e c t of the co n c e n t r a t i o n gradient i n the ion d e n s i t y . In the " h i g h - f i e l d " approximation where qaE>>kT, the reverse current i s n e g l i g i b l y small compared to the forward c u r r e n t . Therefore, the fundamental equation of high f i e l d i o n i c conduction i s J = 2aqvn exp-[W-qaE]/kT = J exp-W(E)/kT (10) o This r e l a t i o n holds i n general f o r the motion of an ion over any p o t e n t i a l energy b a r r i e r . The b a s i c model may be extended to i n c l u d e c o l l e c t i v e motion of nearby ions and m u l t i p l e jumps. 24 Verwey (1935) was the f i r s t to apply eqn. 10 to anodic oxides. With amorphous oxides, the equation should be treated as a f i r s t approximation since with these films a range of jump distances and a c t i v a -t i o n energies i s expected (Winkel, P i s t o r i u s , and van Geel 1958, Young 1959). In addition, the d i s t i n c t i o n between l a t t i c e and i n t e r s t i t i a l s i t e s in an amorphous oxide may not be v a l i d . Several models for i o n i c conduction i n amorphous oxides have nevertheless been developed based on eqn. 10 and assumptions regarding rate c o n t r o l l i n g processes, the naturex>f the e l e c t r i c f i e l d , space charge, and the creation and a n n i h i l a t i o n of defects. Some of these models w i l l be discussed in l a t e r sections. The observation that both metal and oxygen atoms are mobile with transport numbers almost independent of the f i e l d i s d i f f i c u l t to explain in terms of the c l a s s i c a l model. An equation of the type given by (10) can be written for the metal and oxygen ion currents, with the e l e c t r o -n e u t r a l i t y c o n d i t i o n , ^ apparently required by the lack of dependence of f i e l d on thickness at constant current, written Space charge density = q^ n^ + ^2n2 + c o n s t a n t = 0 Such a double i n j e c t i o n s i t u a t i o n could produce the observed transi e n t s . However, accidental s i m i l a r i t y of parameters i s required for transport numbers nearly independent of f i e l d , or so i t i s usually believed. Most speculations have, therefore, involved some kind of center, possibly mobile, in which transport processes of both ions are linked. Following on the e a r l i e r suggestion of Vermilyea (1957c) that changes i n the l o c a l "'"Fromhold has used a method of dif f e r e n c e equations to numerically investigate space-charge and concentration-gradient e f f e c t s on the k i n e t i c s of oxide growth. The r e s u l t s of these extensive investigations are summarized i n a recent p u b l i c a t i o n (Fromhold 1976). 25 c o n f i g u r a t i o n of the glassy s t r u c t u r e occur on annealing, Young and Smith (1979a) r e c e n t l y suggested that such a center i s l i k e l y a s s o c i a t e d w i t h s i t e s of reduced c o o r d i n a t i o n i n the oxide (see s e c t i o n 2.5). Dignara (1965a) and Taylor and Dignam (1973a) have asser t e d that the center i s r e l a t e d to d i e l e c t r i c p o l a r i z a t i o n processes. 3.2 High F i e l d Frenkel Defect Theory and Related Theories The Frenkel defect theory of Bean, F i s h e r , and Vermilyea (1956) was proposed as an extension of the c l a s s i c a l theory of i o n i c c o n d u c t i v i t y . The b a s i c idea of the theory i s that the h i g h e l e c t r i c f i e l d creates Frenkel defects by p u l l i n g metal atoms from l a t t i c e or network s i t e s i n t o i n t e r s t i t i a l l o c a t i o n s . The c a t i o n vacancies are assumed to be immobile and to act as n e g a t i v e l y charged trapping centers f o r the mobile i n t e r s t i t i a l c a t i o n s . The production of Frenkel defects i s assumed to be a f i e l d - a s s i s t e d thermally a c t i v a t e d process, as i n s e c t i o n 3.1, w i t h a generation r a t e given by where N = c o n c e n t r a t i o n of l a t t i c e s i t e s , m = c o n c e n t r a t i o n of vacant c a t i o n s i t e s , and and a^ are, r e s p e c t i v e l y , the zero f i e l d a c t i v a t i o n energy and a c t i v a t i o n d i s t a n c e f o r generation of an i n t e r s t i t i a l i o n . The recombination r a t e at i o n i c current J i s assumed to be given by Jam where a i s the capture c r o s s - s e c t i o n of a vacant c a t i o n s i t e . Therefore, the d i f f e r e n c e s between generation and recombination r a t e s i s Assuming from the e l e c t r o n e u t r a l i t y c o n d i t i o n that the concentrations of (N-m)v 1 exp-tWj-qa^l/kT (11) dm dt (12) c a t i o n vacancies and i n t e r s t i t i a l c a t i o n s are equal ( i . e . m = n ) , eqn. 10 gives the i o n i c current as 26 J = 2a 2 v 2m exp-[W 2-qa 2E]/kT (13) where W2 and a 2 are, r e s p e c t i v e l y , the zero f i e l d a c t i v a t i o n energy and a c t i v a t i o n d i s t a n c e f o r the t r a n s p o r t of i n t e r s t i t i a l ions. S u b s t i t u t i n g (13) i n t o (12) and assuming N>>m, the steady s t a t e i o n i c current (-7™- =0) i s dt given by J = J Qexp-[W-qaE]/kt (14) where W = ^1 ^2 and a = a l + a2 are, r e s p e c t i v e l y , the average zero f i e l d 2 2 a c t i v a t i o n energy and a c t i v a t i o n d i s t a n c e f o r generating an i n t e r s t i t i a l i o n and then moving i t from one i n t e r s t i t i a l s i t e to the next. Note that steady s t a t e measurements cannot separate e f f e c t s due to generation of i n t e r s t i t i a l ions and the m o b i l i t y of these ions. However, under stepped f i e l d c o n d i t i o n s ( i . e . stepping from a steady s t a t e f i e l d E^ to a new f i e l d E 2 such that Am = 0), the t r a n s i e n t i o n i c current i s given by J t = J o'exp-[W 2 - q a 2 E 2 ] / k t . (15) where W2 and a 2 are defined above and J ' i s a constant. Vermilyea (1957a) and Dewald (1957) were the f i r s t to analyze i o n i c current t r a n s i e n t s i n terms of the Frenkel defect model. The simple Frenkel defect model presented above has been modified by s e v e r a l authors to account f o r the steady s t a t e dependence of l o g J on E. In t h e i r o r i g i n a l paper, Bean et a l (1956) p o s t u l a t e d a r a t h e r complex dependence of a c t i v a t i o n d i s t a n c e on f i e l d to e x p l a i n the apparent f i e l d -dependent m o b i l i t y of the i n t e r s t i t i a l ions. L a t e r , Young (1960) presented a t h e o r e t i c a l argument i n which the quadratic dependence of the a c t i v a t i o n energy on f i e l d was a consequence of the amorphous nature of the oxide. I t was l a t e r pointed out (Young 1963) that e l e c t r o s t r i c t i o n , an e f f e c t p r o p o r t i o n a l to E , could account i n part f o r the quadratic f i e l d dependence. An a l t e r n a t i v e e x p l a nation i n v o l v e s a f i e l d dependent a c t i v a t i o n d i s t a n c e ( C h r i s t o v and Ikonopisov 1969). As a consequence of the n o n l i n e a r l o g J-E r e l a t i o n f o r the steady s t a t e , a quadratic f i e l d term must a l s o be included i n eqn. 15 f o r the stepped f i e l d t r a n s i e n t (Dignam and Ryan 1968). Dignam (1964) a l s o i n v e s t i g a t e d the e f f e c t of the form of the p o t e n t i a l energy f u n c t i o n on the J-E r e l a t i o n p r e d i c t e d by the defect theory. He concluded that a Morse f u n c t i o n f o r the p o t e n t i a l energy 2 i . e . P.E. = W (l-exp (x /x )) where W and X a r e constants and X i s the d i s t a n c e from the o o / ' center of the trap gave r e s u l t s f o r W(E) that agreed w i t h the steady s t a t e data f o r tantalum, niobium, and aluminum. This model was l a t e r found to give a poorer f i t to the data than e i t h e r a quadratic or cubic f i e l d dependence i n eqn. 14 (Dignam and Gibbs 1969). A d d i t i o n a l l y , the Morse f u n c t i o n , w i t h i t s zero f i e l d maximum at i n f i n i t y , bears l i t t l e resemblance to the p e r i o d i c p o t e n t i a l energy f u n c t i o n o r i g i n a l l y envisaged f o r the Frenkel defect model. Young and Zobel (1966) pointed out that the Morse f u n c t i o n i s more a p p l i c a b l e to the channel model f o r i o n i c conduction (discussed i n s e c t i o n 4.1). F i n a l l y , Ord, Hopper, and Wang (1972) proposed an e f f e c t i v e - f i e l d model to account f o r the no n l i n e a r dependence of l o g J on E. According to the model, the mean f i e l d E i n eqn. 15 i s replaced by an e f f e c t i v e f i e l d E e which i s assumed to be p r o p o r t i o n a l to the product of the mean f i e l d and the p e r m i t t i v i t y , e, of the oxide f i l m . Further assuming that i s a l i n e a r f u n c t i o n of the f i e l d E i . e . e - K (1-yE) where K Q and y are constants gives the r e q u i r e d 28 quadratic dependence of the a c t i v a t i o n energy on the f i e l d W(E) = Wq - qaK 0E + qaK 0yE 2 (16) i n the steady s t a t e . More r e c e n t l y , Ord and Lushiku (1979) have assumed a n o n l i n e a r e-E r e l a t i o n . This model w i l l be discussed i n greater d e t a i l i n Chapter 5 regarding the open c i r c u i t t r a n s i e n t . I t should be noted that none of the above models takes i n t o account the e f f e c t of the two l a y e r s t r u c t u r e of anodic oxide f i l m s , due to e l e c t r o l y t e i n c o r p o r a t i o n , on the apparent J-E r e l a t i o n . 3.3 D i e l e c t r i c P o l a r i z a t i o n Theory Although the Frenkel defect model i s conc e p t u a l l y appealing, a major f a u l t w i t h the model i s that i t does not account f o r the constant f i e l d t r a n s i e n t ( s e c t i o n 2.4). Young (1961b) pointed out that i f a f i e l d E i s suddenly a p p l i e d to a f i l m w i t h a low i n i t i a l c o n centration of Frenkel d e f e c t s , the defect model p r e d i c t s the i o n i c current to increase most r a p i d l y at t = 0 +. Instead, Young observed, a f t e r an i n i t i a l charging 2 c u r r e n t , that the current increases at a r a t e p r o p o r t i o n a l to J before reaching a maximum cur r e n t . Therefore, a p a r t i a l l y c u r r e n t - c o n t r o l l e d model i s r e q u i r e d . At the time, Young suggested that a cascade or "bimolecular" process i n v o l v i n g focussed "energy packets" may be r e s p o n s i b l e f o r the a c c e l e r a t i n g nature of the i o n i c current during the t r a n s i e n t . The d i e l e c t r i c p o l a r i z a t i o n theory of Dignam (1965a) seems to have been i n s p i r e d by the r e s u l t s f o r the constant f i e l d t r a n s i e n t . Although the theory has undergone s e v e r a l r e v i s i o n s (Dignam and Ryan 1968a, Taylor and Dignam 1973a), the b a s i c p i c t u r e of the model i s unchanged. In the l a t e s t v e r s i o n , Taylor and Dignam (1973a) po s t u l a t e d that defect 29 i n j e c t i o n at the o x i d e / e l e c t r o l y t e i n t e r f a c e c o n s t i t u t e s the rate, con-t r o l l i n g step f o r i o n i c conduction through the oxide. As i n other defect models, the i o n i c current i s taken to be given by J = J exp-[W -ctE, + gE 2 ] / k T (17) o a d where i t i s assumed that the f i e l d a s s i s t i n g defect i n j e c t i o n i s the f i e l d i n the double l a y e r E^. Assuming that the surface 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 , the f i e l d s t r e n g t h i n the f i l m , E, i s r e l a t e d to th a t i n the double l a y e r , E^, through the equation E d = T J I T (%E + P) (18) where K i s the d i e l e c t r i c constant of the double l a y e r and e i s the d o p e r m i t t i v i t y of f r e e space. D i v i d i n g the p o l a r i z a t i o n , P, i n t o a " f a s t " component P^ = e Q X E , which always f o l l o w s the f i e l d , and one or more "slow" components P, , one can w r i t e K. E d " T T " ( E o K i E + E k V ( 1 9 > o d where = 1 + i-s t n e "dynamic" d i e l e c t r i c constant of the oxide ( i n p r a c t i c e the value at 1 kHz). Under steady s t a t e c o n d i t i o n s ( i . e . P^ = e oX^E), eqn. 19 reduces to E = K s E (20) where K = 1 + X - t + Xo = • • • i s the " s t a t i c " d i e l e c t r i c constant, s 1 2 At low current d e n s i t y , the time dependence of the slow p o l a r i z a -t i o n i s p o s t u l a t e d to i n v o l v e a conventional thermally a c t i v a t e d Debye-type process described by dP _J^_ 30 where i s the relaxation time associated with P^ and i s the e l e c t r i c -3 2 s u s c e p t i b i l i t y . At high J (in p r a c t i c e greater than ^ 10 A/m ) experimental r e s u l t s require a current driven process, i n which case the slow p o l a r i z a t i o n i s divided into a d i f f e r e n t set of components P whose time dependence i s given by dP. . = B. J(e X-E - P.) (22) t j o A j 2 where B^  i s a constant, independent of temperature (Cornish 1972). Both (21) and (22) s a t i s f y the steady state requirement than P^ = e^x^E. From 2 eqns. 17, 18, and 22 i t can also be shown that dJ/dt a J . The o v e r a l l time dependence of the p o l a r i z a t i o n i s therefore given by 4 r = e X, 4 f + I, ~~ (e Xi E - P, ) + £. B. j(e X-E - P.) (23) dt o 1 dt k o k k J J o j j where, depending on the current conditions, only one type of slow p o l a r i z a t i o n i s used. It should be noted that EX. = EX and EP. = EP, • J i k . j , k J k J k The basic idea of the current driven process i s that mobile ions or complexes sweep out a volume of influence i n which the approach to equilibrium p o l a r i z a t i o n corresponding to the instantaneous f i e l d i s expedited by the passage of i o n i c current. The nature of a p o l a r i z a t i o n process which would behave i n t h i s way i s not c l e a r . Although several s p e c i a l components of P are postulated, with d i f f e r e n t B^, and x^> usually only one or, at most, two terms have been used. This rather a r b i t r a r y feature of the p o l a r i z a t i o n model has lead to some controversy regarding the v a l i d i t y of the values derived for the 31 s t a t i c p e r m i t t i v i t y of the oxide. Using Dewald's (1957) stepped current data f o r the Ta^O^ and two slow p o l a r i z a t i o n terms, Goad and Dignam (1972) obtained the value K g = 80 compared to = 27.6 (Young 1958). Dell'Oca, P u l f r e y and Young (1971) estimated the s t a t i c d i e l e c t r i c constant should be only -5% greater than based on small s i g n a l capacitance measurements. However, Taylor and Dignam (1973b) claimed that the time constant of the slow p o l a r i z a t i o n i s too l a r g e to a f f e c t the small s i g n a l capacitance measurements and t h a t , t h e r e f o r e , the s u r p r i s i n g l y l a r g e value f o r K i s s c o r r e c t . 3.4 Evidence f o r Current Driven P o l a r i z a t i o n Since the stepped current and constant f i e l d t r a n s i e n t s do not provide d i r e c t evidence f o r c u r r e n t - d r i v e n p o l a r i z a t i o n , Taylor and Dignam (1973b) reported experiments l i k e the f o l l o w i n g w i t h which they claimed to have obtained such evidence. A preformed oxide held i n i t i a l l y at v o l t a g e was pulsed to a higher voltage such t h a t , i n a s e r i e s of experiments at d i f f e r e n t V^, the d i f f e r e n c e AV = - was constant. The width of the pulse was v a r i e d from 10 to 26 msec. The i o n i c current f l o w i n g during the pulse supposedly p o l a r i z e d the oxide and was assumed to be n e g l i g i b l e a f t e r the pulse. Thus, the p o l a r i z a t i o n returned to i t s o r i g i n a l value by only a thermally a c t i v a t e d process. The t o t a l amount of charge passed during the voltage pulse was measured and denoted Q . The t o t a l charge Q a s s o c i a t e d w i t h the p o l a r i z a t i o n was obtained by i n t e g r a t i n g the current f l o w i n g f o r a per i o d of 0.2 sec a f t e r reducing the v o l t a g e to V^. The net charge flow defined as Qf - Q + + Q_ was taken as a s s o c i a t e d w i t h i o n i c current flow. I t should be noted that 32 Q + and Q are p o s i t i v e and negative q u a n t i t i e s r e s p e c t i v e l y . The charge Q_ was found to be a l i n e a r f u n c t i o n of Q^ . This was taken as i n d i c a t i n g that more d i e l e c t r i c p o l a r i z a t i o n was produced when more i o n i c charge was passed. With some approximations, the theory w i t h known B 2 and terms q u a n t i t a t i v e l y p r e d i c t e d dQ_/dQ^ (about 0.25). However, f o r a c e r t a i n mode of applying the p u l s e s , the i n i t i a l dQ_/dQ^ was much higher. A l s o , the period of i n t e g r a t i o n f o r Q must a f f e c t the agreement; i n some cases the e l e c t r o d e s y i e l d e d current on r e t u r n i n g to f o r periods much longer than 0.2 sec. F i n a l l y , the assumption that n e g l i g i b l e i o n i c current flowed a f t e r the voltage pulse was a p p l i e d i s c r i t i c a l to the a n a l y s i s but no evidence was given that t h i s was i n f a c t the case. 4. Phenomenological Models f o r Io n i c Conduction Dignam has asserted that the success of the equations f o r the d i e l e c t r i c p o l a r i z a t i o n model i n d e s c r i b i n g i o n i c current t r a n s i e n t s confirms the model. However, i t may be argued (Young and Smith 1979a) that the d e r i v a t i o n of these equations owes so much to phenomenological c o n s i d e r a -t i o n s that they should be regarded as e m p i r i c a l . In f a c t , mathematically equivalent equations (as regards i o n i c c urrent p r e d i c t i o n s ) can be de r i v e d on the b a s i s of a d i f f e r e n t model. Three such models which have been r e c e n t l y proposed are reviewed below. 4.1 Channel Model h To e x p l a i n t h e i r e m p i r i c a l r e l a t i o n W(E) = W-yE f o r the steady s t a t e , Young and Zobel (1966) proposed a channel model f o r i o n i c conduction i n which ions pass through the r e l a t i v e l y open s t r u c t u r e of the oxide u n t i l trapped by coulombic a t t r a c t i o n i n randomly d i s t r i b u t e d pockets of 33 excess (opposite) charge. Assuming the t r a p s are s u f f i c i e n t l y f a r a p a r t , the p o t e n t i a l energy at a d i s t a n c e x from the center of the t r a p w i t h an a p p l i e d e l e c t r i c f i e l d E i s given by P.E = A/x - qEx ( 2 4 ) where A i s a constant. Following the d e r i v a t i o n of the Poole-Frenkel law f o r high f i e l d emission of e l e c t r o n s from t r a p s , the steady s t a t e current d e n s i t y i s given by J = J oexp-[W-yE 2]/kT (25) where J , W, and y are constants. I t was noted that the constant Y i s not o expected to have the value q2cL^ / ( T e ) (M.K.S.) p r e d i c t e d by the above treatment ( q 2 = charge on the t r a p , q^ = charge on the i o n , e = p e r m i t t i v i t y ) s i n c e the co n c e n t r a t i o n of traps would be expected to change w i t h the f i e l d . L a t e r , Young (1972) pointed out that the two l a y e r s t r u c t u r e of the f i l m and a range of trap e f f i c i e n c i e s may complicate the model. According to the channel model i o n i c current t r a n s i e n t s are due to changing co n c e n t r a t i o n s of trapped ions and to b u i l d i n g of queues at blockages i n the channels. A s i m i l a r idea was proposed by Young (1960). Thus, the constant f i e l d t r a n s i e n t i s due to an a v a l a n c h e - l i k e momentum t r a n s f e r process i n which the channels are unblocked by r a p i d l y moving ions. Young (1972) l a t e r provided a mathematical d e s c r i p t i o n f o r t h i s process. I t was assumed that the current d e n s i t y J i s p r o p o r t i o n a l to the f r a c t i o n 6 of unblocked channels J = B(E,T)0 and that the time dependence of 6 i s given by 4* = B J 2 ( l - 6 ) dt where B i s a constant. This scheme was found to reproduce the main 34 f e a t u r e s of the constant f i e l d t r a n s i e n t although i t was suggested a range of channel parameters should be used to improve the f i t . 4.2 S t r u c t u r a l Model Since the s t r u c t u r e of the anodic oxide i s apparently so f l e x i b l e , Young and Smith (1979a) proposed a model i n which the h i s t o r y e f f e c t s are due to s t r u c t u r a l changes of the type o u t l i n e d i n s e c t i o n 2.5. The model i s b r i e f l y summarized as f o l l o w s . Assuming a c o n c e n t r a t i o n of N' of d e f e c t s or centres, the usual high f i e l d dependence of J on E i s given by J = J N'exp-[W-!-aE+BE2l/kT (27) o where i s a constant or p o s s i b l y dependent on a low power of T, and w' i s the zero f i e l d a c t i v a t i o n energy f o r mobile species hopping from one defect s i t e to the next. To reproduce experimental r e s u l t s , the steady s t a t e dependence of N' on E i s given by N' = N exp-[W" -yE + 6E 2]/kT (28) o where N i s a constant and W" , y, and 6 are a s s o c i a t e d w i t h the production o of d e f e c t s . Introducing the dimensionless q u a n t i t y N and the t o t a l zero f i e l d a c t i v a t i o n energy W defined by N = (N'/N o)exp[W" /kT] (29) W = w' + w" and s u b s t i t u t i n g i n t o (27) gives J = J q N exp-[W-aE + 3E 2]/kT (30) where J = J 1 N o o o To o b t a i n an expression f o r dN/dt, the observed dependence of 2 dJ/dt on J at low N and constant E and the s a t u r a t i o n of N at i t s steady s t a t e value must be reproduced. 35 As discussed i n s e c t i o n 2.5, the " d e f e c t s " are expected to i n t e r a c t . The s t r a i n a s s o c i a t e d w i t h an e x i s t i n g defect i s p o s t u l a t e d to favour the formation of a new defect nearby. Assuming the a c t i v a t i o n of one of these defects t r i g g e r s an i n t e r a c t i o n w i t h a nearby d e f e c t , the r a t e of production of new defects w i l l be p r o p o r t i o n a l to both N and J . The equation 4 ^ = CNJ[(yE - 6E 2)/kT - £nN] (31) at where C i s a constant, s a t i s f i e s both requirements l i s t e d above. As w i t h Dignam's equations i t may be necessary to introduce f u r t h e r s t r u c t u r a l parameters to d e s c r i b e the e f f e c t of s t r u c t u r a l v a r i a t i o n w i t h i o n i c conduction. For example, the zero f i e l d a c t i v a t i o n energy might be expected to change w i t h changes i n s t r u c t u r e . The s t r u c t u r a l model was r e c e n t l y used to n u m e r i c a l l y c a l c u l a t e the amount of s t r u c t u r a l r e l a x a t i o n which might occur during a t y p i c a l open c i r c u i t t r a n s i e n t (Young and Smith 1979b). 4.3 Surface Charge Model de Wit, Wijenberg, and Crevecoeur (1979) a l s o were not s a t i s f i e d w i t h the evidence f o r the p o l a r i z a t i o n processes p o s t u l a t e d by Dignam. Instead, they proposed a model i n which the observed h i s t o r y e f f e c t s are due to a surface charge l a y e r . Both Young (1972) and Dignam (1972) have b r i e f l y mentioned the p o s s i b i l i t y of such an approach. The e s s e n t i a l features of the model are that both p o s i t i v e and negative ions are mobile i n the bulk of the oxide but that only p o s i t i v e charge can pass through a t h i n boundary l a y e r at the metal/oxide i n t e r f a c e ( f i g u r e 2.4). The t o t a l current J , equal to the p o s i t i v e current i n the boundary l a y e r , i s assumed to be 36 e x p o n e n t i a l l y dependent on the l o c a l f i e l d i n the boundary l a y e r . The f i e l d E i n the bulk of the oxide i s r e l a t e d to EL by E T = E + a lz (32) ^ n where a i s the negative charge d e n s i t y at the i n t e r f a c e between the boundary l a y e r and the bulk and the p e r m i t t i v i t y e i s assumed to be the same on e i t h e r s i d e of the i n t e r f a c e . Therefore, the i o n i c current i s given by J = J exp-[W-B (E + a /e)]/kT (33) o p n where i s a constant. The i n t e n t i o n , then, i s to a s c r i b e the h i s t o r y e f f e c t s to changes i n a . That i s , i s assumed to increase w i t h the steady s t a t e f i e l d but respond slowly to sudden changes i n the f i e l d . The time dependence of a i s determined by the r a t e at which p o s i t i v e charges are captured by negative charges at the i n t e r f a c e . Assuming the p r o b a b i l i t y of such an event i s given by o^w where w i s the capture c r o s s - s e c t i o n per negative charge, the p o s i t i v e charge flow i n the bulk i s given by J = J (1 - a w) (34) P n and the r a t e of change of a by d°n J - a wj (35) d T = n n where J i s the current i n the bulk due to the n e g a t i v e l y charged oxygen n ions. I t i s f u r t h e r assumed that the p o s i t i v e and negative charge flows are coupled by some process at the o x i d e / e l e c t r o l y t e i n t e r f a c e so that one can w r i t e J / J = gE (36) n P where g i s a constant. That i s , the p r o b a b i l i t y that a negative i o n enters the oxide from the e l e c t r o l y t e i s determined both by the f i e l d and the a r r i v a l of a p o s i t i v e i o n at t h i s i n t e r f a c e . Combining eqns. 34, 35 and 36 37 gives the time dependence of as da •j^ = J ( g E ( l - a n ) - anw) (37) Note that (37) describes a current d r i v e n process as r e q u i r e d by experiment. In the steady s t a t e (da n/dt = 0 ) , a^  i s given by a = g E / ( l + gE)w (38) n The steady s t a t e q u a n t i t y a^ w i s simply the oxygen ion tr a n s p o r t number. S u b s t i t u t i n g (38) i n t o (33) gives the steady s t a t e J-E r e l a t i o n as J = J oexp[-W + B p E ( l + ^ ( l ^ E J ^ / k T <39> I f the argument of the exponent i s expanded about the steady s t a t e f i e l d E, a negative quadratic term i n E i s obtained, i n agreement w i t h experimental r e s u l t s . The surface charge model of de Wit et a l i s apparently the only one that e x p l i c i t l y a l l o w s f o r the m o b i l i t y of both ions. I n t e r e s t i n g l y , when the metal ion current i s n e g l i g i b l e compared to the oxygen ion current ( i . e . gE<<l), eqn. 39 reduces to a l i n e a r l o g J-E r e l a t i o n J = J Qexpt-W + B E ( l +g/ew)]/kT (40) This may e x p l a i n the observation of a l i n e a r l o g J-E r e l a t i o n f o r metal ^ oxides such as ZvO^ i n which the current i s due e n t i r e l y to oxygen ions. As regards the parameters of the model, de Wit et a l concluded from an a n a l y s i s of impedance data f o r aluminum oxide f i l m s , that the model gives reasonable r e s u l t s f o r w, a, and oxygen i o n transport number. I t should be noted, however, that the model does not reproduce the dependence of the stepped f i e l d J-E r e l a t i o n on the i n i t i a l steady s t a t e c o n d i t i o n s . Nevertheless, the model i s worth c o n s i d e r i n g as an a l t e r n a t i v e to the 38 d i e l e c t r i c p o l a r i z a t i o n model of Dignam. 5. Conclusions Some conclusions about the present understanding of the growth of anodic oxides, p a r t i c u l a r l y as they apply to la^O^, are b r i e f l y as follows. The oxide grows simultaneously at both the metal/oxide and ox i d e / e l e c t r o l y t e i n t e r f a c e s due to the transport of metal and oxygen ions through the f i l m . As a r e s u l t of e l e c t r o l y t e incorporation, the oxide f i l m i s a two layer structure with the differences in e l e c t r i c a l and o p t i c a l properties between the two layers depending on the type and concentration of e l e c t r o l y t e as well as the conditions of oxide formation. Ionic conduction through the f i l m i s a f i e l d - a s s i s t e d thermally activated process with the empirical 2 r e l a t i o n log J^CaE-BE )/kT for oxide growth at constant current density. When the anodization conditons are changed from one steady state to another, ionic current transients are generally observed. The constant f i e l d transient i s of p a r t i c u l a r importance as i t indicates that the transient response involves a current-driven process. Although the d i e l e c t r i c p o l a r i z a t i o n theory of Dignam has succ e s s f u l l y accounted f o r the constant f i e l d transient as well as other transients, i t s success may owe more to phenomenological considerations than i t s d e s c r i p t i o n of the physical processes involved i n io n i c conduction. F i n a l l y , i o n i c and e l e c t r o n i c pro-cesses are somehow rel a t e d , as indicated by photostimulated oxide growth under UV r a d i a t i o n . To date, no theory has been proposed to adequately explain t h i s phenomenon. 39 I I I . ELLIPSOMETRY 1. Introduction The technique of ellipsometry i s concerned with the measurement of the changes in the state of p o l a r i z a t i o n of l i g h t upon r e f l e c t i o n from a surface. These changes can be r e l a t e d to the physical properties of the surface and, thus, for film-covered surfaces, provide information about both the f i l m and substrate. Although other o p t i c a l techniques such as interferometry and index matching give s i m i l a r information, ellipsometry has the advantage of simultaneously and accurately measuring the r e f r a c t i v e index and thickness of the f i l m as well as being very s e n s i t i v e to small changes in the o p t i c a l properties of the f i l m and substrate. The method i s nondestructive and can be used for in s i t u measurements i n any transparent-ambient system. These features make ellipsometry almost i d e a l l y suited to the present study of anodic tantalum oxide f i l m s . The p r i n c i p l e s of ellipsometry are b r i e f l y reviewed in the next sections of t h i s chapter. 7 Then, a f t e r a b r i e f d e s c r i p t i o n of the computer-controlled ellipsometer used i n these studies, the r e s u l t s of an extensive c a l i b r a t i o n procedure are discussed. C a l i b r a t i o n procedures of the type described in t h i s chapter are e s s e n t i a l to maximize the accuracy of the ellipsometer measurements. TThe t h e o r e t i c a l aspects of ellipsometry have been reviewed extensively by Azzam and Bashara (1977) and Aspnes (1976). Also, the proceedings of four conferences on ellipsometry have been published (Muller, Azzam and Aspnes 1980; Bashara and Azzam 1976; Bashara, Buckman and H a l l 1969; Passaglia, Stromberg and Kruger 1964). 40 2. P r i n c i p l e s of Ellipsometry 2 . 1 The Fresnel R e f l e c t i o n C o e f f i c i e n t s Consider a monochromatic e l l i p t i c a l l y p o l a r i z e d plane wave incident on the i n t e r f a c e between two media (figure 3 . 1 ) , each of which is characterized by a complex index of r e f r a c t i o n N = n - j k where n i s the r e f r a c t i v e index, k the absorption c o e f f i c i e n t , and j = J-L. The e l l i p t i c a l l y polarized wave can be resolved into two mutually orthogonal plane waves, one with i t s e l e c t r i c f i e l d vector in the plane of incidence and the other with i t s e l e c t r i c f i e l d vector perpendicular to t h i s plane. These two d i r e c t i o n s are denoted p and s respectively. Following the Mueller-Nebraska convention (Muller 1 9 6 9 ) , the e l e c t r i c f i e l d s of the p-and s-polarized waves i n an i s o t r o p i c medium i are written using the expj(wt + 6) time/phase dependence as Ep 0(Z,t) = Re{E p oexpj (cot - 2TTN.Z/A 0 + 6 p 0 ) } E S 0 ( Z , t ) = Re{E expj (a>t - 2TTN Z / A Q + & S O ) } where the waves are taken to be propagating i n the z d i r e c t i o n and A q i s the vacuum wavelength. The p o l a r i z a t i o n i s defined as being right-handed when the t i p of the e l e c t r i c f i e l d vector at constant Z traces an e l l i p s e i n a clockwise sense as the viewer faces the l i g h t source. Also, the phase di f f e r e n c e A = 6 ™ - 6 0 ^ i s c h a r a c t e r i s t i c of the p o l a r i z a t i o n . Q pu £aU The incident wave i s r e f l e c t e d and transmitted by the i n t e r f a c e . The so l u t i o n to Maxwell's equations, subject to boundary conditions on the phase and amplitude of the e l e c t r i c f i e l d s (Jackson 1 9 6 2 , Born and Wolf 1 9 7 5 ) gives, among other r e s u l t s , S n ell's law N o s i n 0 o = N^sinG-^ and the Fresnel complex-amplitude r e f l e c t i o n and transmission c o e f f i c i e n t s f or p- and F i g . 3.1 R e f l e c t i o n and r e f r a c t i o n at an i n t e r f a c e . N 0 6 o _ 9° N, t N 2 F i g . 3 .2 R e f l e c t i o n from a f i l m - c o v e r e d s u r f a c e . 42 s - p o l a r i z e d l i g h t defined by E N, cos 6 - N cos 9, r N rp _ _ i o o 1 _ p ( l a ) E. IP Ni cos 0 + N cos 8 n  x o o 1 E N cos G - N n cos 0i r . rs _ _o o L ±_ _ r ( l b ) E. N cos 6 + N-. cos e 1 ~ S i s o o 1 1 E^ 2N cos G tp o o _ t (Ic) E. N, cos 6 + N cos 6 i p 1 o o 1 E_ 2N cos 6 ts _ O o E. N cos 6 + N cos 6 xp o o 1 1 P Z s (Id) where N Q , N^, 6 q, and 8^ are defined i n f i g u r e 3.1 and i , r , and t denote, r e s p e c t i v e l y , the i n c i d e n t , r e f l e c t e d , and tran s m i t t e d waves. The r e l a t i v e magnetic p e r m e a b i l i t y of both media i s assumed to be u n i t y . I t can be shown from eqns. 1(a) - (d) that the r e f l e c t i o n and transm i s s i o n c o e f f i c i e n t s are r e l a t e d by o l l o 2 t t + r = 1 o l l o o l where the sequence of s u b s c r i p t s i n d i c a t e s the d i r e c t i o n of propagation I I 2 I I 2 from one medium to the next. The q u a n t i t i e s | t 0 ^ | and | r 0 ^ | are defined as the t r a n s m i s s i v i t y and r e f l e c t i v i t y , r e s p e c t i v e l y , of the i n t e r f a c e (Born and Wolf 1975). The Fr e s n e l equations a l s o show that when the i n c i d e n t wave i s p o l a r i z e d i n any s t a t e other than a p- or s - l i n e a r s t a t e , the p o l a r i z a t i o n s t a t e w i l l be changed by r e f l e c t i o n or tr a n s m i s s i o n at an i n t e r f a c e . 43 The r e f l e c t i o n and tr a n s m i s s i o n c o e f f i c i e n t s are, i n general, complex q u a n t i t i e s and can be w r i t t e n i n complex exponential n o t a t i o n as r = | r | expj 6^ t = | t | expj 6 where | r | and | t | are the amplitudes of r e f l e c t i o n and tr a n s m i s s i o n and 6 and 6 are the corresponding phase s h i f t s . I t should perhaps be noted that the r e f l e c t i o n c o e f f i c i e n t s r and r as defined by 1(a) and (b) have a b u i l t p s -i n phase d i f f e r e n c e of 180°. At normal incidence on an i s o t r o p i c medium r and r should be i n d i s t i n g u i s h a b l e . Instead, we have from the above p s equations the r e l a t i o n rp = - r g o r , i n terms of exponential n o t a t i o n r Ix = expj IT P s 2.2 E l l i p s o m e t r y Equations An e l l i p s o m e t e r measures the r e f l e c t a n c e r a t i o p = R /R of an p s o p t i c a l system, where R and R are the t o t a l r e f l e c t a n c e s of the system to p s p and s l i g h t , p i s u s u a l l y w r i t t e n i n complex exponential n o t a t i o n p = tan ¥ expjA (2) where tanY i s the r e l a t i v e amplitude a t t e n u a t i o n and A the r e l a t i v e phase change on r e f l e c t i o n of p and s l i g h t . Thus, one e l l i p s o m e t e r measurement of p gives s i n g l e values f o r Y and A. The r e l a t i o n between the measured q u a n t i t i e s and ¥ and A i s discussed i n Appendix A. The r e f l e c t a n c e s R^ and R g depend on the p r o p e r t i e s of the o p t i c a l system being s t u d i e d . For r e f l e c t i o n from a bare s u b s t r a t e , R^ and R are given by the F r e s n e l r e f l e c t i o n c o e f f i c i e n t s r and r . However, s 6 J p s R and R are complicated f u n c t i o n s of the r e f l e c t i o n c o e f f i c i e n t s ' f o r p s r e f l e c t i o n from a sub s t r a t e w i t h one or more surface f i l m s . For example, 44 the r e f l e c t a n c e R f o r r e f l e c t i o n from an i s o t r o p i c f i l m on an i s o t r o p i c substrate ( f i g u r e 3.2) i s given by R u = ( r 0 1 + r 1 2 U e X p " J 2 6 ) / ( 1 + r 0 l r l 2 U e X p " j 2 6 ) 0 ) 6 = 2TTN., cos B.dJX 1 1 1 o where u i n d i c a t e s the p o l a r i z a t i o n and r . . i s the Fr e s n e l r e f l e c t i o n c o e f f i c i e n t f o r the i n t e r f a c e between media i and j . S i m i l a r but more complicated expressions can be derived f o r r e f l e c t i o n from a m u l t i l a y e r f i l m s t r u c t u r e on a sub s t r a t e . The r e f l e c t a n c e r a t i o can be c a l c u l a t e d d i r e c t l y given the o p t i c a l constants of the f i l m and substrate and the f i l m t h i c k n e s s . However, the inver s e problem of determining the o p t i c a l constants and f i l m t h i c k n e s s given p i s more d i f f i c u l t because g e n e r a l l y i t i s not p o s s i b l e to ob t a i n a n a l y t i c expressions f o r these parameters i n terms of the measured q u a n t i t i e s A and In p r i n c i p l e , one e l l i p s o m e t e r measurement of A and V i s s u f f i c i e n t to solve f o r two unknowns i n the f i l m and s u b s t r a t e . Curve f i t t i n g or parameter o p t i m i z a t i o n methods i n v o l v i n g s e v e r a l e l l i p s o m e t e r measurements are required to solve f o r more than two unknowns. The measurements can be made as a f u n c t i o n of angle of inci d e n c e , r e f r a c t i v e index of the immersion medium, wavelength, or f i l m t h i c k n e s s . The present study of anodic T^O^ f i l m s i s an almost i d e a l a p p l i c a t i o n of e l l i p s o m e t r y i n that f i l m s of uniform th i c k n e s s can be measured i n s i t u i n the forming e l e c t r o l y t e and these measurements can be made over a range of s e v e r a l hundred nanometers i n f i l m t h i c k n e s s . Various methods have been used f o r d i s p l a y i n g the inf o r m a t i o n provided by the e l l i p s o m e t e r measurements. In m u l t i p l e wavelength or 45 spectroscopic e l l i p s o m e t r y , u s u a l l y cosA and tanY are p l o t t e d as a f u n c t i o n of the wavelength (Aspnes 1976). For measurements at a s i n g l e wavelength, the r e a l and imaginary components of p can be p l o t t e d against one another. However, the more usual method, and the one used i n the present study, i s to p l o t y against A. 2.3 Computation of the Reflectance Ratio The t o t a l r e f l e c t i o n c o e f f i c i e n t R^ f o r a system of £ homogeneous i s o t r o p i c l a y e r s on a substrate can be obtained by m a t r i x methods ( H a y f i e l d and White 1963). The e l e c t r i c f i e l d s of the i n c i d e n t wave ( E ^ ) , r e f l e c t e d wave (E ), and r e f r a c t e d wave i n the metal (E ) are r e l a t e d by the m a t r i x r m J equation I TT k=l m 1 r E m m rm 1 0 (4) TT k=l Z l l -Z12 Z21 Z22 E m 0 where A = k expj 6 k r k expj 6 k r k exp - j 6 k exp - j<S, 6 k = 2TTN COS 6 d /A i s the phase change of the l i g h t on c r o s s i n g the k t h K. K. K O l a y e r of t h i c k n e s s d^, and r ^ i s the F r e s n e l r e f l e c t i o n c o e f f i c i e n t f o r the i n t e r f a c e between l a y e r s k-1 and k. Equation 4 i s general to both p and s l i g h t , w i t h the t o t a l r e f l e c t i o n c o e f f i c i e n t R being given by R u " Z21 / Z 1 1 (5) For a s i n g l e f i l m / s u b s t r a t e system, i t i s easy to show that eqns. 4 and 5 give the same r e s u l t f o r as eqn. 3. Therefore, i f the appropriate values f o r r, and 6 are s u b s t i t u t e d i n t o eqn. A and R and R c a l c u l a t e d , the K K p S e l l i p s o m e t e r angles A and ¥ are given by * = t a n _ 1 | R /R | A = t a n " 1 {Im(R /R )/Re(R /R )} p s p s p s Although the above matri x method may i n v o l v e more computations than would be req u i r e d i f an a n a l y t i c expression f o r R were used (as i n eqn. 3), the matrix c a l c u l a t i o n s are more e a s i l y adapted to use on a d i g i t a l computer, p a r t i c u l a r l y when the o p t i c a l system being modelled has a l a r g e number of surface f i l m s . A computer program was w r i t t e n f o r the Amdahl 470 at the UBC Computing Centre to c a l c u l a t e ¥ and A f o r any system of m u l t i p l e i s o t r o p i c f i l m s (up to 50 l a y e r s ) on a substrate and p l o t the (4*,A) contours of constant r e f r a c t i v e index and th i c k n e s s on a Tektronix graphics t e r m i n a l . These computed curves could then be compared w i t h experimental data. By appropriate i n t e r a c t i o n With the program, the o p t i c a l model could be adjusted to o b t a i n the best v i s u a l f i t to the data. The f i n a l graph could then be output to a p l o t t e r at the Computing Centre. The program was t e s t e d by p l o t t i n g the (¥,A) curve f o r a s i n g l e l a y e r f i l m on a substrate and then comparing t h i s curve w i t h the one obtained when the f i l m i s d i v i d e d i n t o s e v e r a l l a y e r s , each w i t h the same r e f r a c t i v e index as the o r i g i n a l f i l m . This procedure was repeated w i t h one of the l a y e r s being s t r o n g l y absorbing but having zero t h i c k n e s s . As expected, there was no d i f f e r e n c e between any of the computed curves. The computed data a l s o were i n agreement w i t h the data given by the McCrackin (1969) program obtained from the N a t i o n a l Bureau of Standards. 47 2.4 Computed Results 2 . 4 . 1 Single Layer Films on a Substrate Figure 3.3 shows computed values of ¥ plotted against A as a function of f i l m thickness for a si n g l e layer tantalum oxide f i l m on tantalum with f i l m and substrate indices = 2 . 1 9 5 and N G = 2 . 4 6 - J 2 . 5 7 3 , r e s p e c t i v e l y as reported by Cornish ( 1 9 7 2 ) . To be consistent with the experimental r e s u l t s presented i n chapter 4, the angle of incidence i s 6 2 . 7 7 ° , the wavelength i s 6 3 2 . 8 nm, and the r e f r a c t i v e index of the medium i s n = 1 . 3 3 4 . The (Y.A) data s t a r t at the point marked 0 nm and, with m increasing f i l m thickness, trace out the s o l i d l i n e i n the d i r e c t i o n indicated, eventually returning to the s t a r t i n g point. Although i n t h i s example the I'-A curve i s closed, i t i s possible i n other cases (e.g. for other o p t i c a l constants, wavelengths, etc.) to have the computed curve cross the y-axis and then s t a r t over again at A = 3 6 0 ° . The computed data cycle because the t o t a l r e f l e c t i o n c o e f f i c i e n t R for a single layer/substrate system (given by eqn. 3 ) returns to i t s zero f i l m thickness value when the phase change 6 = 2TTN^ cos Qrdr/\ i s 1 8 0 ° or a multiple of 1 8 0 ° . The c y c l e ^ ° f f f o thickness d = A / 2 N ^ cos 6,. increases with increasing wavelength, cyc l e o f f angle of incidence, and r e f r a c t i v e index of the ambient medium. The other l i n e i n f i g u r e 3.3 shows the e f f e c t of an absorbing f i l m ( N = 2 . 1 9 5 - j 0 . 0 7 5 ) on the ¥-A curve. Because of attenuation of the l i g h t by the f i l m , the f-A curve tends to lower values of Y on succeeding cycles and eventually converges to a point c h a r a c t e r i s t i c of an i n f i n i t e l y t h i c k absorbing f i l m . 2 . 4 . 2 Multiple Films on a Substrate The number of p o s s i b i l i t i e s f o r the Y-A curve increases r a p i d l y F i g . 3.3 Computed e l l i p s o m e t e r c u r v e s f o r a b s o r b i n g and non a b s o r b i n g f i l m s on Ta. 15 40 65 90 PSI / DEGREES F i g . 3-4 The e f f e c t of a 3 nm t h i c k i n t e r -f a c i a l f i l m on the computed e l l i p s o m e t e r c u r v e . 49 w i t h the number of f i l m l a y e r s on the substrate. One of the more simple cases to consider i s that of an i n t e r f a c i a l f i l m of constant t h i c k n e s s sandwiched between a surface f i l m and the s u b s t r a t e . Figure 3.4 shows the e f f e c t of a 3 nm t h i c k i n t e r f a c i a l f i l m w i t h r e f r a c t i v e index N. = 1.54-il.55 l J on the computed ¥-A curve f o r a growing nonabsorbing surface f i l m w i t h the same r e f r a c t i v e index as used i n f i g u r e 3.3. For comparison, the computed curve f o r the nonabsorbing s i n g l e l a y e r f i l m i s p l o t t e d as the dashed l i n e . As w i t h the s i n g l e l a y e r f i l m , the computed f - A curve f o r the i n t e r f a c i a l f i l m system c y c l e s w i t h i n c r e a s i n g t h i c k n e s s of the surface f i l m . This i s to be expected s i n c e the changes i n r e f l e c t i v i t y (and thus y and A) of the system are due e n t i r e l y to the changing t h i c k n e s s of the surface f i l m , w i t h the c y c l e t h i c k n e s s being the same as i n the previous s e c t i o n . However, the computed curves f o r the two systems are d i s t i n c t l y d i f f e r e n t . Although both curves are i n r e l a t i v e l y c l o s e agreement f o r small values of ^, the curve f o r the i n t e r f a c i a l f i l m model diverges from that of the s i n g l e l a y e r f i l m at l a r g e values of V. The r e l a t i o n between the two curves depends on the o p t i c a l constants and t h i c k n e s s of the i n t e r f a c i a l f i l m . These r e s u l t s show, however, that the e l l i p s o m e t e r i s capable of d e t e c t i n g a very t h i n i n t e r f a c i a l f i l m . Another simple case to consider i s that of two nonabsorbing l a y e r s each of which grows simultaneously at a constant r a t e . Figure 3.5 shows the computed f-A curve f o r the case of two such l a y e r s w i t h N^<N2 where the s u b s c r i p t s denote l a y e r s adjacent to the ambient medium and s u b s t r a t e r e s p e c t i v e l y . The i n d i c e s and are taken.to d i f f e r only s l i g h t l y and the l a y e r thicknesses to be equal. In t h i s example, two complete c y c l e s , denoted by the s o l i d and dashed l i n e s , are r e q u i r e d before the curve e_i. O • ct-- J o - H CO *J p • o 0 • CD 120 DELTA / DEGREES 170 • 1 1 L 05 51 begins to retrace i t s original path. However, the recycling is only approximate since the f-A curve for the third cycle diverges from that of the f i r s t cycle at large values of f. That this behavior should occur is best understood by considering the total reflection coefficient R for the two layer model R = r Q 1 = r 1 2 exp-j26 1 + r ^ exp-j2(5 1 + 6,,) + r ^ r ^ r ^ exp-j262 ( g )  1 + r0l r12 e x P _ J 2 6 i + roi r23 e x P - J 2 ( 6 ] _ + 6 2 ) + rl2 r23 e x P - J 2 6 2 where 6 . = 2TTN. COS 6.d./A and the subscripts denote the interfaces, with 1 1 1 l o . 0 and 3 referring to the ambient medium and substrate respectively. Since the indices of the two layers differ only slightly and d^ = d 2 (i.e. 6^ = ^2^* R returns approximately to i t s zero film thickness value when 6^  + &^  = 360°. Thus, the computed curves have a tendency to recycle after every second cycle. This recycling w i l l not occur, however, i f either the indices or thicknesses of the two layers differ by a large amount. Figure 3.6 shows the effect on the computed ¥-A curve of interchanging the layer indices (i.e. N^ >N ) used to obtain figure 3.5. This curve differs from that shown in figure 3.5 in that the f i r s t cycle is traced out mostly to the inside of the second cycle curve. Thus, in addition to detecting slight differences in refractive index between two layers, the ellipsometer can also indicate quite clearly which of the layers has the larger refractive index. Other possi b i l i t i e s for the double layer structure include one or both layers being absorbing and the ratio of the thickness of the two layers being different from unity. In these cases, the complexity of the ellipsometer curves increases considerably over those discussed above. 52 3. The E l l i p s o m e t e r 3.1 Apparatus The Rudolph type 43603-200E n u l l e l l i p s o m e t e r used i n the present i n v e s t i g a t i o n s i s shown s c h e m a t i c a l l y i n f i g u r e 3.7. The e l l i p s o m e t e r c o n s i s t s of a s t a t i o n a r y arm, on which are mounted a l i g h t source, c o l l i m a t i n g o p t i c s , a l i n e a r p o l a r i z e r , and a phase r e t a r d e r , and a moveable arm on which are mounted a second l i n e a r p o l a r i z e r (commonly c a l l e d the analyzer) and a photodetector. This c o n f i g u r a t i o n i s commonly r e f e r r e d to as a PCSA e l l i p s o m e t e r . The moveable or analyzer arm can be r o t a t e d about the centre of the specimen t a b l e , a l l o w i n g the angle of incidence to be set to any angle from about 30° to 90° ( s t r a i g h t through p o s i t i o n ) . The angle of incidence i s read from a d i v i d e d c i r c l e mounted on the c e n t r a l a x i s of the e l l i p s o m e t e r . The sample t a b l e i t s e l f can be r o t a t e d about the a x i s , t i l t e d w i t h respect to the plane of incidence and moved i n any of 3 perpendicular d i r e c t i o n s . Apertures i n the p o l a r i z e r and analyzer arms d e f i n e the axes of the two telescopes and, thus, the plane of incidence. O r i g i n a l l y , two apertures were used i n the analyzer arm, one at the f r o n t of the telescope and the other j u s t ahead of the photodetector. Because of d i f f i c u l t i e s i n a l i g n i n g the l i g h t source,however, a t h i r d aperture was r e c e n t l y placed j u s t ahead of the analyzer c r y s t a l (see s e c t i o n 4.3). The apertures i n the -3 analyzer arm are about 10 m i n diameter. This small aperture s i z e i s important i n e s t a b l i s h i n g the plane of incidence and l i m i t i n g the acceptance angle f o r l i g h t r e f l e c t e d from a rough surface. The d i s t a n c e between the f i r s t and l a s t aperture i n the analyzer arm i s about 0.2 m. The l i g h t source i s a 1 mW He-Ne l a s e r ( A = 632.8 nm) p r o v i d i n g S A M P L E F i g . 3 .7 C o n f i g u r a t i o n o f the Rudolph e l l i p s o m e t e r used i n the p r e s e n t work. U> 54 randomly p o l a r i z e d l i g h t (see s e c t i o n 4.3). The c o l l i m a t i n g o p t i c s s l i g h t l y focus the l a s e r beam so t h a t , w i t h the two apertures i n the p o l a r i z e r arm stopped to t h e i r minimum s i z e , the beam diameter at the sample _3 t a b l e i s about 10 m. The beam divergence at t h i s point i s estimated to be l e s s than 0.1°. The l i n e a r p o l a r i z e r s are Glan-Thompson prisms. The e x t i n c t i o n r a t i o f o r these prisms, defined as the r a t i o of l i g h t i n t e n s i t y t r a n s m i t t e d perpendicular and p a r a l l e l to the trans m i s s i o n a x i s , i s given by the manufacturer as being on the order of 10 ^. The phase r e t a r d e r i s a B a b i n e t - S o l e i l compensator (Gaertner Corp.). The compensator can be adjusted by a micrometer attachment to give any r e l a t i v e phase r e t a r d a t i o n i n the range —IT/2 to 7rr/2 between l i g h t propagating along the f a s t and slow axes of the compensator. For t h i s i n v e s t i g a t i o n , the compensator i s set to act as a quarter wave p l a t e (see s e c t i o n 4.8). The p o l a r i z e r , compensator, and analyzer are mounted on d i v i d e d c i r c l e s which can be r o t a t e d about the telescope axes. R o t a t i o n angles or azimuths are measured w i t h respect to the t r a n s m i s s i o n axes of the p o l a r i z e r and analyzer and to the f a s t a x i s of the compensator. By convention, the azimuths are measured counterclockwise from the plane of inc i d e n c e , l o o k i n g toward the l i g h t source. In these experiments, the azimuth of the compensator i s set to C = 315.00° unless stated otherwise. The e l l i p s o m e t e r used i n these i n v e s t i g a t i o n s had been automated p r e v i o u s l y by a t t a c h i n g gear assemblies c o n t a i n i n g a n t i b a c k l a s h gears (W. M. Berg, Inc.) and stepping motor d r i v e s (I.M.C. Magnetic Corp., type 008-008) to the p o l a r i z e r and an a l y z e r , w i t h one motor step corresponding 55 to O.Ol" r o t a t i o n of the o p t i c a l element. Under load and without a c c e l e r a t i o n , the motors can be d r i v e n at a maximum r a t e of about 500-600 steps/second, depending on adjustments to the gears,before missing steps. In a d d i t i o n , the a n t i b a c k l a s h gears permitted the d i r e c t i o n of r o t a t i o n to be changed without l o s i n g steps. Absolute brush-type shaft encoders (Theta Instruments Co.) having a r e s o l u t i o n of 0.01° were used to measure the p o l a r i z e r and analyzer azimuths. A p h o t o m u l t i p l i e r preceded by a narrow band o p t i c a l f i l t e r , which allowed o r d i n a r y room l i g h t i l l u m i n a t i o n , was used as a detector i n the o r i g i n a l design. The stepping motors, shaft encoders, and p h o t o m u l t i p l i e r were i n t e r f a c e d to a PDP8/E computer using standard D i g i t a l Equipment Corp. components. The p h o t o m u l t i p l i e r s i g n a l was a m p l i f i e d and, using one channel of four-channel m u l t i p l e x e r that had been i n t e r f a c e d to the computer, input to a 10-bit A/D converter. The input to the converter was a l s o d i s p l a y e d on a meter. The i n i t i a l program to c o n t r o l the e l l i p s o m e t e r was w r i t t e n i n the PAL8 assembly language. L a t e r , w i t h the a d d i t i o n of a Dectape u n i t , programs were w r i t t e n i n higher l e v e l languages. With the a d d i t i o n of a d i s k and hardware f l o a t i n g point processor, improved algorithms became p o s s i b l e . The new c o n t r o l program w r i t t e n f o r the present work i s discussed i n s e c t i o n 3.3. I t was discovered at the s t a r t of t h i s study that the photomulti-p l i e r o r i g i n a l l y used to measure the l i g h t i n t e n s i t y was s u s c e p t i b l e to microphonic pickup from the stepping motors. Therefore, the p h o t o m u l t i p l i e r was replaced w i t h a s o l i d s t a t e photodetector (EGG model HAV-1000A) and a m p l i f i e r u n i t b u i l t f o r t h i s work (see Appendix D). The detector c o n s i s t s —6 2 of a pn j u n c t i o n photodiode ( a c t i v e area = 5 x 10 m ) and an o p e r a t i o n a l 56 a m p l i f i e r i n t e g r a t e d on the same device. In order to minimize n o i s e , the short c i r c u i t photocurrent was measured , as suggested by the manufacturer, by connecting the diode across the input t e r m i n a l s of the op amp. In t h i s c o n f i g u r a t i o n , the manufacturer had s p e c i f i e d the detector to be l i n e a r over 7 decades of l i g h t i n t e n s i t y . The photocurrent-to-voltage conversion r a t i o of the detector/op amp was f i x e d by the op amp's 10 Mfi feedback r e s i s t o r . The detector s i g n a l was f u r t h e r a m p l i f i e d by the a m p l i f i e r u n i t mounted on the end of the analyzer arm before being input to the 10- b i t A/D converter. One of the eight p o s s i b l e a m p l i f i e r gains could be s e l e c t e d l o c a l l y by the user. The a m p l i f i e r was a l s o i n t e r f a c e d to the PDP8/E computer so that the gain could be s e l e c t e d remotely under computer c o n t r o l . 3.2 P r i n c i p l e of Operation The operation of a n u l l e l l i p s o m e t e r i s based on f i n d i n g a set of azimuths f o r the p o l a r i z e r , compensator, and analyzer which e x t i n g u i s h (the i d e a l ) or minimize (the p r a c t i c a l ) the l i g h t reaching the photodetector. In p r a c t i c a l terms, l i n e a r l y p o l a r i z e d l i g h t of known azimuth at the output s i d e of the p o l a r i z e r i s converted by the compensator t o , i n gene r a l , e l l i p -t i c a l l y p o l a r i z e d l i g h t of known e l l i p t i c i t y . The e l l i p t i c i t y i s adjusted, u s u a l l y by r o t a t i n g the p o l a r i z e r and keeping the compensator azimuth f i x e d , so that l i g h t r e f l e c t e d by the sample i s l i n e a r l y p o l a r i z e d . This c o n d i t i o n i s then determined by e x t i n g u i s h i n g the l i g h t w i t h the analy z e r . The procedure f o r f i n d i n g t h i s set of azimuths i s g e n e r a l l y r e f e r r e d to as balancing or n u l l i n g the e l l i p s o m e t e r . At any s e t t i n g of the compensator azimuth, there are two sets of p o l a r i z e r and analyzer azimuths, other than those r e l a t e d by ±TT, which w i l l 57 balance the e l l i p s o m e t e r . In the i d e a l case, i f the p o l a r i z e r and analyzer azimuths f o r one of the balances are P and A, then the second balance w i l l be found at P + TT/2 and 2ir - A . Under normal operation the compensator azimuth i s set to e i t h e r TT/4 or -TT/4. S e t t i n g the compensator to one of these azimuths minimizes instrument e r r o r s and maximizes the speed of convergence to a n u l l when s t a r t i n g from o f f - n u l l c o n d i t i o n s (Confer, Azzam and Bashara 1976). To d i s t i n g u i s h between the independent balances, McCrackin, P a s s a g l i a , Stromberg and Steinberg (1963) defined 4 zones i n which the e l l i p s o m e t e r can be balanced. The zone of the balance can be determined by i n s p e c t i o n of the azimuths of the analyzer and compensator (see Table 3.1). Unless st a t e d otherwise, measurements i n the present study were made i n zones 1 and 3. Table 3.1 R e l a t i o n between the zones defined by McCrackin et a l (1963) and the azimuths of the compensator and analyzer. ZONE C tan A 1 -TT/4 >0 2 TT/4 >0 3 -TT/4 >0 4 TT/4 >0 T-able 3.2 I d e a l R e l a t i o n s between P and A and the E l l i p s o m e t e r angles A and 4* ZONE A ¥ 1 2P + TT/2 A 2 2P + TT/2 A 3 2P - TT/2 -A 4 2P - TT/2 -A 58 The complete r e l a t i o n between the balance azimuths P, C, and A of the p o l a r i z e r , compensator, and analyzer and the e l l i p s o m e t e r angles A and 4* i s given i n Appendix A. However, f o r the i d e a l case where the compensator constants are T £ = 1 and A^ = TT/2 and C = ±TT/4 , ¥ and A are e a s i l y obtained from P and A. The r e s u l t s are summarized i n Table 3.2, assuming that P and A have been adjusted by ±TT so that they are i n the range -TT/4 ^_~P <_ 3TT/4 and -TT/2 < A < TT/2 . 3.3 The E l l i p s o m e t e r C o n t r o l Program 3.3.1 The Balancing Algorithm Over the years, s e v e r a l e l l i p s o m e t e r c o n t r o l programs have been w r i t t e n f o r the PDP8/E computer. The f e a t u r e common to a l l of these programs i s that the balancing algorithms were based on the s o - c a l l e d method of equal swings. This method makes use of the f a c t that the l i g h t i n t e n s i t y at the photodetector v a r i e s q u a d r a t i c a l l y w i t h the angular d e v i a t i o n of the p o l a r i z e r or a n a l y z e r about t h e i r balance azimuths (see s e c t i o n 3.4). One of these o p t i c a l elements would be r o t a t e d to one side of i t s balance azimuth and the l i g h t i n t e n s i t y recorded. Then, i t would be swung back to the other s i d e of the balance u n t i l l i g h t of equal i n t e n s i t y was measured. The balance azimuth was taken to be the midpoint between these angles of equal i n t e n s i t y . A major disadvantage of t h i s method i s that i t g e n e r a l l y r e l i e s on the accuracy w i t h which one can measure the l i g h t i n t e n s i t y at the endpoints of the swing. This problem was p a r t i a l l y overcome i n other balancing schemes i n v o l v i n g running sums over a small moving window of angles (Cornish 1976). I t would seem, however, that a b e t t e r balancing a l g o r i t h m would take i n t o account' a l l of the data recorded during the swing from one s i d e of the balance to the other. Such an approach was taken i n developing F i g . 3.8 A t y p i c a l p h o t o d e t e c t o r s i g n a l recorded by the computer. 60 the b a l a n c i n g a l g o r i t h m used i n t h i s study. Merkt (1981) has a l s o taken t h i s approach, although the d e t a i l s of h i s bala n c i n g procedure are d i f f e r e n t from those of the present study (see s e c t i o n 3.5). As w i t h the method of equal swings, the new balancing a l g o r i t h m i s a l s o based on the quadratic dependence of the l i g h t i n t e n s i t y on angular d e v i a t i o n about the balance azimuth of the p o l a r i z e r and analyzer. One of these o p t i c a l elements i s f i r s t swung to one s i d e of i t s balance and, as i t i s swung back through the balance, the l i g h t i n t e n s i t y i s measured a f t e r each r o t a t i o n step of 0.01° and the value stored i n the computer memory. The computer then f i t s a quadratic polynomial to the i n t e n s i t y vs angle data, c a l c u l a t e s the balance azimuth from the c o e f f i c i e n t s of the polynomial, and then steps the element to the c a l c u l a t e d balance. D e t a i l s of the a l g o r i t h m are given i n Appendix C. Figure 3.8 shows a t y p i c a l example of the l i g h t i n t e n s i t y recorded by the photodetector f o r small d e v i a t i o n s about the balance azimuth. Table 3.3 P o l a r i z e r and Analyzer Balance Azimuths f o r R e f l e c t i o n from an Ic o n e l S l i d e at an Angl e of Incidence P l A l P 3 A 3 15.727 32.721 105.712 327.100 15.723 32.723 105.711 327.098 15.722 32.720 105.710 327.097 15.721 32.720 105.710 327.102 15.722 32.724 .05.710 327.098 ? 1 = 15. 723 ± 0.003 F 3 = 105, .711 ± 0.001 A~± = 32. 721 ± 0.002 A 3 = 327. .099 ± 0.002 61 The smoothing a c t i o n of the f i t t e d polynomial reduces the e f f e c t s of n oise due p r i m a r i l y to the detector and a m p l i f i e r e l e c t r o n i c s and to v a r i a t i o n s i n the l a s e r i n t e n s i t y . This smoothing i s p a r t i c u l a r l y important i n those p a r t s of the e l l i p s o m e t e r ¥-A curve where the s e n s i t i v i t y of the balance i s reduced and noise becomes a s i g n i f i c a n t part of the photodetector s i g n a l . Table 3 . 3 gives some examples of the c a l c u l a t e d balances and t h e i r estimated standard d e v i a t i o n s f o r r e f l e c t i o n from an I c o n e l s l i d e at an angle of incidence of 70°. The p o l a r i z e r and analyzer were balanced a l t e r n a t e l y , w i t h a t o t a l of 10 balances being made i n each zone. The s l i d e was not r e a l i g n e d between balances. Taking the estimated standard d e v i a t i o n to be a measure of the r e s o l u t i o n of the balance, these r e s u l t s show that the s e n s i t i v i t y i s s i g n i f i -c a n t l y greater than the 0.01° r e s o l u t i o n of the shaft encoders. These r e s u l t s a l s o are t y p i c a l of the r e s u l t s obtained f o r h i g h l y r e f l e c t i v e surfaces. However, a r e d u c t i o n i n the s e n s i t i v i t y of the balance l i k e w i s e reduces the r e s o l u t i o n of the balance. 3.3.2 The C o n t r o l Program The new e l l i p s o m e t e r c o n t r o l program makes use of s e v e r a l p e r i p h e r a l devices t h a t have been added to the PDP8/E computer system s i n c e the l a s t c o n t r o l program was w r i t t e n (Cornish 1975). For example, c a l c u l a t i o n s normally would be done i n software by the PDP8/E processor. However, the recent a d d i t i o n of a hardware f l o a t i n g p o i n t processor to the computer system has reduced by a f a c t o r of about 4 or 5 the execution time of programs i n v o l v i n g a l a r g e number of c a l c u l a t i o n s . This has meant that the speed of the e l l i p s o m e t e r balancing r o u t i n e i n the present a p p l i c a t i o n i s l i m i t e d by the speed of the stepping motors r a t h e r than the speed of the c a l c u l a t i o n s r e q u i r e d f o r f i t t i n g the quadratic polynomial to the balance data. The c o n t r o l 62 program also uses a 5h d i g i t voltmeter (DANA model 5100), a 10"kHz c r y s t a l clock b u i l t for t h i s work (see chapter 5), and a 16-bit D/A converter which have recently been interfaced to the computer. F i n a l l y , the recent additions of extra computer memory, a VT52 video terminal and an RL01 dis k d r i v e to the computer system as well as the use of the FORTRAN IV programming language have permitted the development of a more sophisticated and f l e x i b l e c o n t r o l program than has been possible i n the past. The ellipsometer c o n t r o l program i s i n t e n s i v e l y u s e r - i n t e r a c t i v e . The user s e l e c t s the task to be performed from a "menu" of tasks displayed on the video terminal and i s then prompted for a d d i t i o n a l input to the program when required or. i s reminded of options which may be selected before the task i s executed. For example, the p o l a r i z e r and analyzer may be balanced separately or both repeatedly for a us e r - s p e c i f i e d number of times i n two zones. It was also intended to use the ellipsometer to track a growing anodic oxide f i l m . The user i s therefore given the option of s e l e c t i n g a repeated balance of the p o l a r i z e r and analyzer i n one zone. After each pair of p o l a r i z e r and analyzer balances, the anode voltage i s measured with the DANA voltmeter and the time from the 10 kHz c r y s t a l clock. These data may then be stored, as an option, i n a disk f i l e f o r l a t e r a n a l y s i s . A more de t a i l e d d e s c r i p t i o n of the ellipsometer c o n t r o l program and the user's guide to the c o n t r o l program are given i n Appendix D. 3.A S e n s i t i v i t y of the Ellipsometer Balance The l i g h t i n t e n s i t y at the photodetector of an i d e a l n u l l ellipsometer with i t s quarterwave plate i n c l i n e d at an angle C = ± T T / 4 with respect to the plane of incidence i s given by (Aspnes 1974) 63 + R x h[l - cos 24* cos 2A - s i n 2* s i n 2A s i n (A - 2P)] (7) where I i s the maximum i n t e n s i t y t r a n s m i t t e d i n the s t r a i g h t - t h r o u g h p o s i t i o n , ¥ and A are the e l l i p s o m e t e r angles c h a r a c t e r i s t i c of the r e f l e c t i n g s u r f a c e , and A and P are the azimuths of the analyzer and p o l a r i z e r . The term i n the f i r s t set of brackets i s the maximum i n t e n s i t y reaching the detector w i t h the r e f l e c t i n g surface i n p o s i t i o n . For the n u l l c o n d i t i o n s A = V and P = A/2 - T T / 4 , eqn. 7 gives I = 0 as expected f o r the i d e a l e l l i p s o m e t e r . S u b s t i t u t i n g the n e a r - n u l l c o n d i t i o n s A = ¥ + 6A and P = A/2 - T T / 4 + 6P, w i t h |<5A|, 16p j <<1, i n t o eqn. 7 and expanding i n a Taylor s e r i e s about the n u l l angles gives the i n t e n s i t y at the detector as The s e n s i t i v i t y of I D to small d e v i a t i o n s about the n u l l angles i s given by the r e l a t i v e s e n s i t i v i t i e s where the s u b s c r i p t s A and P denote analyzer and p o l a r i z e r r e s p e c t i v e l y . These r e l a t i o n s show t h a t , as one might expect, the s e n s i t i v i t y of a balance depends on the r e f l e c t i v i t y of the surface. They a l s o show that s i n c e 2 0 < s i n 24" <1, the p o l a r i z e r balance g e n e r a l l y i s l e s s s e n s i t i v e than the analyzer balance. As an example, f i g u r e 3.9 shows the c a l c u l a t e d r e l a t i v e s e n s i t i v i t i e s f o r anodic tantalum oxide f i l m s on tantalum p l o t t e d as a f u n c t i o n of f i l m t h i c k n e s s . To be c o n s i s t e n t w i t h the data given i n chapter 4, the angle of incidence and r e f r a c t i v e index of the ambient medium were taken to be 62.77° and 1.334 r e s p e c t i v e l y . The f i l m and su b s t r a t e (8) 64 THICKNESS / nm Fig. 3.9 The calculated r e l a t i v e s e n s i t i v i t i e s S A and S p for r e f l e c t i o n from Ta 0 Cy/Ta at 6. =62.77°and A= 632.8 nm. The ref r a c t i v e indices were taken as nm=1.334, n^.=2.195, and n =2.46-j2.573. 6 5 F i g . 3 . 1 0 The e r r o r f u n c t i o n s F. and F p d e f i n e d by Merkt ( 1 9 8 1 ) . 66 indices were taken to be = 2.195 and N g = 2.4 6 - j2.57 3, as reported by Cornish (1976). For t h i s p a r t i c u l a r case, the f i g u r e indicates that the ellipsometer balances are most s e n s i t i v e when measuring f i l m thicknesses in the f i r s t c y c l e i n the range 0-40 nm and 90-120 nm. Due to the near-zero value of |Rs| for f i l m s about 65 nm thick, the s e n s i t i v i t y of the p o l a r i z e r f a l l s to almost zero when measuring f i l m s of t h i s thickness. This type of v a r i a t i o n i n the s e n s i t i v i t y of the balance was observed during the present studies of anodic tantalum oxide (chapter 4). Equation 8 also shows that the l i g h t i n t e n s i t y of the detector v a r i e s q u a d r a t i c a l l y with small deviations of the p o l a r i z e r and analyzer from t h e i r n u l l angles. Therefore, we are j u s t i f i e d i n f i t t i n g a quadratic polynomial to the balance data (section 3.3.1), at le a s t for s u f f i c i e n t l y small deviations about the n u l l ( i n pr a c t i c e , about ±1° (Merkt 1981)). 3.5 P r e c i s i o n of Ellipsometer Nu l l i n g Merkt (1981) has recently reported a l e a s t squares method f o r n u l l i n g the ellipsometer. The method i s based on f i t t i n g a parabolic funct ion I(P,A) = aP 2 + bA 2 + CPA + dP + eA + f (9) to the measured i n t e n s i t y vs azimuth data. The c o e f f i c i e n t s a - f are found by minimizing the variance _ 1_ N ~ °i >T2 , z [ I(P. ,A.) - i . . r where N i s the number of data points and I., i s the measured i n t e n s i t y . The balance azimuths P and A are then calculated from the c o e f f i c i e n t s . o o The error a of the calculated p o l a r i z e r balance was shown to be given by P 67 a = C [ f ( a + b)(4b 2 + c2)]h S p 3 / 2 , 2 ^ — (i 2 1 + r-> (10) N |4ab - c | opt where c i s the noise of the detected l i g h t and S = | [ 1 2 f / ( a + b)]H opt N i s the optimal spacing between data points f o r which i s a minimum. The corresponding error a for the calculated analyzer balance i s obtained by replacing b by a and v i c e versa i n eqn. 10. Assuming the step width s has i t s optimal value, i t i s convenient to re-write eqn. 1 0 as CJ A = 2 c T N ~ 3 / 2 y*Fn . P,A I P,A where y = f/b i s the ex t i n c t i o n r a t i o with the sample in place and F and F are found by comparison with eqn. 10. The functions F and F depend, v i a the c o e f f i c i e n t s a, b, and c, on the ellipsometer configuration and the e l l i p s o m e t r i c angles of the sample. For a PCSA ellipsometer with i t s quarter-wave plate set at an azimuth C = ± T T / 4 , comparing eqn. 9 with eqn. 8 of the previous section gives the c o e f f i c i e n t s as 2 a = r l s i n 2V o b = r l o c = 0 2 2 where r = ( i R I + IR I ) / 2 . The fac t that c = 0 for t h i s configuration p s indicates that there i s no coupling between the p o l a r i z e r and analyzer. Substituting these c o e f f i c i e n t s into eqn. 1 0 gives the errors of the calc u l a t e d balances as 68 2 h where F (Y) = [1 + sin 2VJ 2 2 2sin 2V F (¥) = %[1 + sin224']1'5 These relations, which are plotted in figure 3.10 , show that the polarizer balance is undefined as V approaches 0° or ±90° whereas the accuracy of the analyzer balance is almost unchanged over the entire range in 41. By comparison, Merkt reported that the highest precision obtainable with the method of equal swings is given by h -1 a, = c T y , a = a. sin 24' A I p A Although eqn. 11 was derived for a balancing method different from the one used in the present studies, the functions Fp and F A give an indication of the accuracy with which the balance angles can be calculated with the Chebyshev f i t t i n g algorithm. 4. Ellipsometer Alignment and Calibration Alignment and calibration of an ellipsometer are essential i f the ellipsometer measurements are to be meaningful. The alignment and calibration procedures used in the present studies are outlined in the next sections of this chapter. Several different calibration procedures giving basically the same information were used to check the v a l i d i t y of the calibration results. In particular, the calibration procedure of Azzam and Bashara (1971a) was investigated and i t s results compared with those of other methods. 4.1 Response of the Photodetector It was important to establish at the outset of these investigations whether the response of the photodetector was independent of the polarization 69 of the i n c i d e n t l i g h t . Therefore, w i t h the e l l i p s o m e t e r set to the s t r a i g h t - t h r o u g h p o s i t i o n and the p o l a r i z e r and analyzer azimuths constant, photodetector was r o t a t e d about the telescope a x i s and i t s output s i g n a l measured. As f a r as could be determined, the output s i g n a l was independent of the azimuth of the photodetector and, thus, the p o l a r i z a t i o n of the i n c i d e n t beam. 4.2 Telescope Axes The plane of incidence i s defined by the d i r e c t i o n af propagation of the i n c i d e n t and r e f l e c t e d beams and, i n a w e l l a l i g n e d e l l i p s o m e t e r , i s independent of the angle of incidence. T i l t i n g of one of the arms w i t h respect to the other r e s u l t s i n the plane of incidence changing w i t h the angle of incidence. Other secondary e f f e c t s are due to p o s s i b l e t i l t i n g of the p o l a r i z e r prism and compensator i n t h e i r mounts and to l a t e r a l s h i f t i n g of one of the axes w i t h respect to the other. The Rudolph e l l i p s o m e t e r used i n these s t u d i e s does not have adjustments f o r c o r r e c t i n g p o s s i b l e t i l t i n g or a l a t e r a l s h i f t i n e i t h e r the p o l a r i z e r or analyzer arms. These have l i t t l e a f f e c t on the measured angle of incidence or on the alignment of the sample. However, the absolute reference azimuths P ,. and A defined as the azimuths at which the re f r e f p o l a r i z a t i o n v e c t o r s of the p o l a r i z e r and analyzer l i e i n the plane of incidence, would t h e o r e t i c a l l y have to be determined at each angle of incidence i f one of the arms were t i l t e d . As w i l l be shown i n s e c t i o n 4.6.1, the t i l t , i f any, was in. p r a c t i c e too small to s i g n i f i c a n t l y a f f e c t the reference azimuths. A l s o , any e f f e c t due to l a t e r a l s h i f t i n g of one of the arms was too small to measure. 70 4.3 Alignment of the Laser Source With the e l l i p s o m e t e r i n the s t r a i g h t - t h r o u g h p o s i t i o n and the compensator removed, the d i r e c t i o n of the l a s e r beam was adjusted so as to maximize the l i g h t i n t e n s i t y at the photodetector. For t h i s alignment, the a d j u s t a b l e aperture on the p o l a r i z e r was stopped to i t s minimum and two apertures on the analyzer arm, one at the f r o n t end of the telescope and the other between the analyzer prism and the photodetectors, defined i t s telescope a x i s . The photodetector s i g n a l was then recorded as the p o l a r i z e r and analyzer were r o t a t e d together through 360°. As shown i n f i g u r e 3.11(a), the d e t e c t o r s i g n a l v a r i e d s i n u s o i d a l l y w i t h the azimuth of the p o l a r i z e r s . The f o l l o w i n g measurements were then made to determine i f t h i s r e s u l t was due to a p r e f e r e n t i a l l y p o l a r i z e d l a s e r source. The compensator was r e i n s e r t e d i n the e l l i p s o m e t e r and i t s azimuth set to C=315.00°. To o b t a i n c i r c u l a r l y p o l a r i z e d l i g h t at the output of the compensator (which had p r e v i o u s l y been adjusted to act as a quarter-wave p l a t e ) , the azimuth of the p o l a r i z e r was set to P = 0.00°. The detector s i g n a l was then recorded as the analyzer was r o t a t e d through 360°. Since the l i g h t i n c i d e n t on the analyzer prism was c i r c u l a r l y p o l a r i z e d , the detector s i g n a l should have been independent of the a n a l y z e r ' s azimuth. Instead, the s i g n a l v a r i e d s i n u s o i d a l l y w i t h the azimuth, as i n the f i r s t t e s t . I t was n o t i c e d that the r e f l e c t i o n from the f r o n t surface of the r o t a t i n g analyzer prism traced out a c i r c l e , c o n c e n t r i c w i t h the telescope a x i s , about the output i r i s of the compensator. From the r a d i u s of the c i r c l e and the d i s t a n c e between the two components, i t was deduced that the analyzer prism was t i l t e d by about 1.4° w i t h respect to the r o t a t i o n a x i s . As discussed i n the next s e c t i o n , t i l t i n g of the analyzer prism i n i t s P i g . 3 . 1 1 ( a ) The v a r i a t i o n of the p h o t o d e t e c t o r s i g n a l w i t h a n a l y z e r azimuth a f t e r i n i t i a l a l i g n m e n t of t h e l a s e r and w i t h c i r c u l a r l y p o l a r i z e d l i g h t i n c i d e n t on the a n a l y z e r . < o in A = 0 ° A=18CV A=360 F i g . 3 . 1 1 ( h ) The v a r i a t i o n o f the p h o t o d e t e c t o r s i g n a l a f t e r r e a l i g n i n g the l a s e r w i t h a t h i r d a p e r t u r e i n the a n a l y z e r arm. 72 mount eliminates unwanted multiple reflect ions. It seemed that a t i l t angle of only about 0 . 2 ° would have been sufficient for this purpose, part icularly when a small aperture was used at the input to the analyzer arm. Unfortunately, there was no apparent way to reduce the t i l t angle. The l ight beam transmitted by the t i l t e d analyzer prism traces out a c i r c le at the plane of the f i n a l aperture as the analyzer is rotated. The -4 radius of the c i r c l e is estimated to be about 2 x 10 m, compared with the _3 aperture diameter of 10 m. Therefore, the intensity of l ight measured by the detector generally depends on the analyzer azimuth and the i n i t i a l a l i g n -ment of the laser. If the laser is aligned so that the c i r c l e of light at the pinhole is concentric with the pinhole (figure 3.12(a)), the detector signal w i l l be independent of the analyzer azimuth. In the above measurements, however, the laser had been aligned with only one aperture ahead of the analyzer prism. Consequently, the alignment depended on the azimuth at which the analyzer happened to be set. Since the alignment method was based on maximizing the detector signal , the transmitted light beam would them trace out a c i r c le l ike that shown in figure 3.12(b) as the analyzer was rotated. This would explain the observed sinusoidal variation of the detector s ignal . A second small diameter aperture was placed ahead of the analyzer prism in order to minimize or, preferably, eliminate the dependence of the laser alignment on the azimuth of the analyzer prism. The laser was then re-aligned to maximize the detector signal and the above measurements using c i rcular ly polarized l ight were repeated. As shown in figure 3.11(b), the dependence of the detector signal on the analyzer azimuth was eliminated over a large range of angles. The compensator was then removed again and,the detector signal re-corded as the polarizer and analyzer were rotated together through 3 6 0 ° . 73 F i g . 3.12 The path of the centre of the l i g h t beam incident on the photodetector pinhole with (a) a well aligned and' (b) a poorly aligned laser, as the analyzer i s rotated through 360 degrees. > cr < cr m cr < CO or O cr cr UJ 0 1 — 1 1 1 1 i I 89.92 89.96 90.00 ANGLE OF INCIDENCE/ DEGREES F i g . 3.13 Zero correction for the angle of incidence scale. 74 Results s i m i l a r to those shown i n f i g u r e 3.11(b) were obtained, i n d i c a t i n g that, as expected, the output from the l a s e r was randomly p o l a r i z e d . 4.4 T i l t i n g of the O p t i c a l Elements Multiple r e f l e c t i o n s between the o p t i c a l elements of an ellipsometer introduce systematic errors i n the ellipsometer measurement. Winterbottom (1955) pointed out that there are f i v e possible r e f l e c t i o n paths between the p o l a r i z e r , analyzer, and compensator surfaces, ignoring multiple r e f l e c t i o n s within the waveplate i t s e l f . T i l t i n g of the waveplate in i t s holder (usually by about 1°) eliminates a l l but one of the r e f l e c t i o n s (Oldham 1967) . The t i l t introduces a small systematic error (see section 4.8) but a correction i s possible. The remaining r e f l e c t i o n can be eliminated by t i l t i n g e i t h e r the p o l a r i z e r or analyzer prism in i t s mount. Generally, i t i s preferable to have the p o l a r i z e r p e r f e c t l y aligned so that the beam incident on the sample does not sweep out a cone of incidence as the p o l a r i z e r i s rotated. For the ellipsometer used i n the present study, the p o l a r i z e r appears to be t i l t e d by less than 0.1°. However, as mentioned i n the previous section, the analyzer prism as supplied by the manufacturer i s t i l t e d by about 1.4° i n i t s mount. This t i l t angle was considered to be much larger than necessary but there was no apparent way, short of dismantling the analyzer' divided c i r c l e , by which the t i l t could be reduced. On the other hand, by placing rubber washers between the waveplate and i t s divided c i r c l e mount, the t i l t of the waveplate was reduced from ^ 6 ° to ^ 2.5°. In turn, t h i s reduced the dependence of the waveplate constants T^ and A^ on the compensator azimuth. 4.5 The Angle of Incidence The angle of incidence, as read from the divided c i r c l e , corresponding to the ellipsometer being in the straight-through p o s i t i o n was determined by the method used by Cornish (1976). Having a l i g n e d the l a s e r by the method discussed i n s e c t i o n 4.3 and w i t h the p o l a r i z e r and analyzer azimuths both set to 0°, the photodetector s i g n a l was recorded as the analyzer arm was moved i n steps of 0.01° about an angle of incidence of 90°. Figure 3.13 shows the detector s i g n a l p l o t t e d against the angle of incidence. The maximum s i g n a l , . which occurs at 0 = 89.96°, i s taken to correspond to the " r e a l " s t r a i g h t -through p o s i t i o n of the analyzer arm. To check that the l a s e r alignment d i d not a f f e c t t h i s r e s u l t , the angle of incidence was set to 89.96° and the l a s e r r e a l i g n e d . The above c a l i b r a t i o n precedure was then repeated. Within the estimated measurement e r r o r of ±0.01° i n angle of i n c i d e n c e , there was no d i f f e r e n c e between the r e s u l t s . A l l subsequent angles of incidence read from the d i v i d e d c i r c l e were correc t e d f o r the reference angle of 89.96°. 4.6 P o l a r i z e r and Analyzer Aximuths In order to define absolute azimuths of the p o l a r i z a t i o n s referenced to the plane of in c i d e n c e , i t was necessary to f i n d the reference azimuths P r e£ and A r e f , as i n d i c a t e d by the shaft encoders, at which the p o l a r i z a t i o n vectors of the p o l a r i z e r and analyzer were i n the plane of in c i d e n c e . This c a l i b r a t i o n procedure became necessary a f t e r the gear assemblies which drove the shaft encoders were removed f o r c l e a n i n g . When the encoders were r e -connected to the gear assemblies, they were adjusted i n hardware to i n d i c a t e 0° when the p o l a r i z a t i o n v ectors were approximately i n the plane of incidence. The f o l l o w i n g o u t l i n e s the methods which were thus used to determine a c c u r a t e l y the reference azimuths corresponding to the plane of incidence. Several methods were used so as to ob t a i n a check on the r e s u l t s . i Unless otherwise s t a t e d , the compensator was removed from the e l l i p s o m e t e r f o r these c a l i b r a t i o n s . 76 4.6.1 Method 1: R e f l e c t i o n from a Fused Quartz Slab When p or s l i g h t i s i n c i d e n t on an o p t i c a l l y i s o t r o p i c r e f l e c t o r , the r e f l e c t e d l i g h t i s s i m i l a r l y p or s p o l a r i z e d and, t h e r e f o r e , can be extinguished by an analyzer. This f i r s t method, then, i s based on f i n d i n g the azimuths of the p o l a r i z e r and analyzer which e x t i n g u i s h the l i g h t . A fused quartz s l a b was used as the r e f l e c t i n g surface s i n c e i t cancels f i r s t order e l l i p t i c i t y e r r o r s i n the p o l a r i z i n g prism (Azzam and Bashara 1971) . The slab was a l s o t h i c k enough that only the r e f l e c t i o n from the f r o n t surface was measured. Before being used, the slab was cleaned i n methanol and then blown dry with n i t r o g e n . The slab was ali g n e d at an angle of incidence of 70°. With the p o l a r i z e r set to azimuths about P = 90°, the analyzer was balanced near A = 0°. The balance azimuths A were p l o t t e d against P - 90°. The analyzer was then set to values about A = 90° and the p o l a r i z e r balanced about P = 0°. The balance azimuths P were p l o t t e d against A - 90°. Figure 3.14 shows the two sets of data p l o t t e d on the same graph. The i n t e r s e c t i o n of the two l i n e s gives the reference azimuths as P . = 0.062° and A , = 0.005°. The sample re f r e f was r e a l i g n e d and the above c a l i b r a t i o n repeated. The average reference azimuths and t h e i r estimated standard d e v i a t i o n s , c a l c u l a t e d from three sets of measurements, were <P > = 0.040 ± 0.02° and <A > = 0.02 ± 0.02. Unless re f r e f state d otherwise, the measured azimuths of the p o l a r i z e r and analyzer were subsequently c o r r e c t e d by these amounts. The above c a l i b r a t i o n was repeated at d i f f e r e n t angles of incidence to check f o r p o s s i b l e e f f e c t s due to t i l t i n g of one of the arms w i t h respect to the other. The r e s u l t s , c o r r e c t e d by the amounts given above, are given i n Table 3.4 The v a r i a t i o n i n P . and A . seems'to be more random than ref r e f systematic. The average of these r e s u l t s and t h e i r estimated standard I I I I L _ -0.4 0 0.4 POLARIZER ANGLE / DEGREES F i g . 3.14 Zero correction to the polarizer and analyzer scales using r e f l e c t i o n from a fused quartz slab. Fig. 3.15 Zero correction to the polarizer and analyzer scales using r e f l e c t i o n from an 1^ Iconel slide at 0t= 65°. 78 d e v i a t i o n s i s <P > = 0.013 ± 0.022° and <A ,> = 0.007 ± 0.015°. In view r e f r e f of the u n c e r t a i n t y i n these r e s u l t s , the azimuths were taken as c o r r e c t l y referenced to the plane of incidence. TABLE 3.4 Reference Azimuths as a Function of the Angle of Incidence. ANGLE OF P A INCIDENCE/DEG. 70 -0.018 0.007 65 -0.021 -0.005 60 -0.033 -0.001 55 0.017 0.028 4.6.2 Method 2: R e f l e c t i o n from an Icon e l S l i d e The second method, a v a r i a t i o n of the f i r s t , used on Icon e l s l i d e as the r e f l e c t i n g surface instead of the fused quartz s l a b . The h i g h l y r e f l e c t i v e metal f i l m gave a much sharper balance and, thus, increased the accuracy of the balance. The s l i d e was cleaned i n methanol, blown dry w i t h n i t r o g e n , and then a l i g n e d at an angle of incidence near i t s p r i n c i p a l angle. With the analyzer set to azimuths about A = 90°, the p o l a r i z e r was balanced near P = 0°. The measurements were done at angles of incidence of 60°, 65°, and 70°, The data f o r = 65° are p l o t t e d i n f i g u r e 3.15. The i n t e r s e c t i o n of these data w i t h the l i n e P = A - 90° gives an e r r o r c defined by P. . = 0° + e i n t A = 90° + e i n t where P. and A. are the i n t e r s e c t i o n azimuths. A summary of the e r r o r s e i n t i n t as a f u n c t i o n of angle of incidence are given i n Table 3.5. There appears to be a systematic v a r i a t i o n of the e r r o r w i t h the angle of i n c i d e n c e . However, sin c e the c a l i b r a t i o n must be done at the p r i n c i p a l angle of the metal f i l m , i t i s l i k e l y that the apparent v a r i a t i o n i n e i s due to using the incorrect angle of incidence. The p r i n c i p a l angle f or the Iconel s l i d e was estimated to be near 65°. The error at t h i s angle i s e s s e n t i a l l y zero, providing support for the c a l i b r a t i o n r e s u l t s of method 1. TABLE 3 . 5 V a r i a t i o n of the error z with angle of incidence for r e f l e c t i o n from an Iconel s l i d e . Angle of Incidence/degrees e 60 -0.022 65 0.004 70 0.043 4.6.3 Method 3: R e f l e c t i o n at the Brewster Angle When l i g h t of a r b i t r a r y p o l a r i z a t i o n i s incident on an i s o t r o p i c nonabsorbing d i e l e c t r i c at i t s Brewster angle, only l i g h t p olarized perpendi-cular to the plane of incidence (s l i g h t ) i s r e f l e c t e d . This l i g h t can then be extinguished by the analyzer, thereby providing a check on the r e s u l t s of the previous c a l i b r a t i o n s of the analyzer. A fused quartz slab, cleaned as in method 1, was again used as the r e f l e c t i n g surface. At A = 632.8 nm, the r e f r a c t i v e index of the slab i s n^ = 1.457 (Handbook of Chemistry and Physics 1969). The Brewster angle i n a i r i s given by 0 = tan ^ n = 55.54°. At t h i s angle of incidence and for the p o l a r i z e r nominally set to P = 90°, the analyzer was balanced near 0° and 180°. The balance azimuths were 0.00 ± 0.01° and 179.99 0.01°. The uncertainty i n these measurements i s due mainly to the low r e f l e c t i v i t y of the fused quartz. We can conclude, however, that within the error of the balance, the analyzer azimuth i s referenced to the plane of incidence. 80 As a check on the reading of the p o l a r i z e r azimuth, the e l l i p s o m e t e r was set to the s t r a i g h t - t h r o u g h p o s i t i o n and, w i t h the analyzer set to A = 0.00°, the p o l a r i z e r was balanced near 90°. The measured balance was at P = 90.00 ± 0.01°, confirming that the p o l a r i z e r azimuth i s a l s o referenced to the plane of incidence. 4.6.4 Method 4: Two-Zone Balances The f i n a l c a l i b r a t i o n , due to Azzam and Bashara (1971), was done wi t h the compensator r e i n s e r t e d i n the e l l i p s o m e t e r . The compensator, nominally a quarter-wave p l a t e , was set w i t h i t s c o r r e c t e d azimuth at C = 315.00°. (see s e c t i o n 4.7). Using the cleaned fused quartz s l a b as the r e f l e c t i n g s u rface, the e l l i p s o m e t e r was balanced i n two zones to o b t a i n the uncorrected balance azimuths (P',A') and (P",A"). This procedure was repeated at s e v e r a l angles of incidence. D e f i n i n g the two-zone r e s i d u a l ResA (without a f a c t o r of ± m T r /2 , m = integer) as ResA = A' + A" Azzam and Bashara have shown (see s e c t i o n 4.9.8) that f o r an i d e a l l y clean nonabsorbing d i e l e c t r i c ResA = 2sin2Ycos6C-2 6A where \p and A are the e l l i p s o m e t e r angles obtained from the balance azimuths and 6C and 6A the zero o f f s e t s of the compensator and analyzer azimuths r e s p e c t i v e l y . These o f f s e t s are assumed to be s u f f i c i e n t l y s m a l l that second and higher order c o n t r i b u t i o n s can be neglected. Figure 3.16 shows ResA p l o t t e d against 2sin2l<cosA f o r the r e s u l t s obtained at d i f f e r e n t angles of incidence. From the i n t e r c e p t of the graph w i t h the ResA a x i s we get 0.4 J L - 1 0 1 2 2sin2YcosA F i g . 3.16 Zero correction to the polarizer and analyzer scales by the method of Azzam and Bashara. _i i i i i_ 0.3 0.7 1.1 1.5 1.9 COMPENSATOR ANGLE / DEGREES Fig. 3.17 Zero correction to the azimuth scale of the compensator. 82 6A = 0.015 ± 0.02° and from the sl o p e , 6C = 0.29 ± 0.05°. The analyzer o f f s e t i s i n agreement w i t h the e a r l i e r c a l i b r a t i o n s . The zero o f f s e t of the com-pensator w i l l be discussed i n the next s e c t i o n . The main advantage of the graphing method over the other methods i s that p o s i t i o n i n g e r r o r s of the r e f l e c t o r tend to c a n c e l , provided s u f f i c i e n t data are used. Of source, the method al s o assumes that the o f f s e t azimuths are independent of the angle of incidence. 4.6.5 Conclusion The r e s u l t s of the four independent c a l i b r a t i o n s of the p o l a r i z e r and analyzer azimuths are i n agreement w i t h i n an estimated e r r o r of about ± 0.01°. Therefore, we can conclude that the p o l a r i z e r and analyzer azimuths were c o r r e c t l y referenced to the plane of incidence by the method given i n Section 4.6.1. 4.7 The Compensator Azimuth The reference azimuth C r, as read from the d i v i d e d c i r c l e , at re f which the f a s t a x i s of the compensator was p a r a l l e l to the plane of incidence has been determined by Cornish (1976) to be 1.13° f o r the e l l i p s o m e t e r used i n the present s t u d i e s . Since the waveplate was removed temporarily from i t s mount (Section 4.4), the reference azimuth was measured again to check that nothing had changed. Following the method of Cornish, a fused quartz s l a b was a l i g n e d at an angle of incidence of 70° without the compensator i n plac e . With the p o l a r i z e r and analyzer azimuths set to P = 0.00° and A = 90.00° r e s p e c t i v e l y , the photodetector s i g n a l was then measured as the compensator was r o t a t e d about 0°. The idea was that the l i g h t would be extinguished only when the f a s t a x i s of the compensator was p a r a l l e l to the transmission a x i s of the p o l a r i z e r and, thus, to the plane of incidence. The l i g h t i n t e n s i t y 83 measured by the detector i s p l o t t e d i n Figure 3.17 as a f u n c t i o n of the azimuth of the compensator. The minimum s i g n a l corresponds to the f a s t a x i s being p a r a l l e l to the plane of incidence. A quadratic Chebyshev polynomial was f i t t e d to the data shown i n Figure 3.17 using a program l i b r a r y r o u t i n e obtained from the UBC Computing Centre. The minimum s i g n a l was c a l c u l a t e d to be at C „ = 1.13°. In sub-ref sequent s e t t i n g s of the compensator azimuth, t h i s value was subtracted from the measured azimuth to obtain the azimuth referenced to the plane of incidence. In Section 4.6.4, i t was s t a t e d that the compensator was set to an azimuth of C = 315°, c o r r e c t e d f o r the reference azimuth. In the subsequent graphing method, i t was found that the zero o f f s e t of the compensator was 6C = -0.29° and not 0° as expected from the above c a l i b r a t i o n . I t i s not c l e a r why there i s such a l a r g e d i f f e r e n c e . C e r t a i n l y , the d i f f e r e n c e i s too l a r g e to a s c r i b e to measurement e r r o r . One p o s s i b i l i t y i s that the t i l t of the compensator somehow a f f e c t e d the azimuth c a l i b r a t i o n discussed i n the previous s e c t i o n . The r e s u l t s of the present s e c t i o n ' s c a l i b r a t i o n were taken to be c o r r e c t . 4.8 Compensator Transmittance and Retardation E r r o r s The compensator i s c h a r a c t e r i z e d by the constant p = T^exp - A^ where T^ i s the r a t i o of the transmittance along the f a s t and slow axes and A i s the r e l a t i v e phase r e t a r d a t i o n between the two axes. Archer and c Shank (1967) and Oldham (1967) were the f i r s t to recognize that d e v i a t i o n s of T c and A £ from t h e i r i d e a l values of u n i t y and T T/2 can s i g n i f i c a n t l y a f f e c t e l l i p s o m e t r i c measurements. Non-unity values of T^ are due mainly to m u l t i p l e r e f l e c t i o n s i n the compensator and to t i l t i n g of the compensator i n i t s mount. With the B a b i n e t - S o l e i l compensator, the phase r e t a r d a t i o n A^ 84 can be adjusted to be as c l o s e to T T / 2 as the accuracy of the c a l i b r a t i o n procedure a l l o w s . The e f f e c t of these d e v i a t i o n s from i d e a l i t y i s to introduce systematic d i f f e r e n c e s between the values f o r V and A measured i n two zones and c a l c u l a t e d from the i d e a l r e l a t i o n given i n S e c t i o n 3.2. The d i f f e r e n c e s are p a r t i c u l a r l y n o t i c e a b l e w i t h the angle V. Although c o r r e c t i o n s can be made f o r these d e v i a t i o n s (see Appendix A), i t i s convenient to use an i d e a l quarter wave p l a t e . The f o l l o w i n g o u t l i n e s the procedures used to set the compensator contants T and A as c l o s e as p o s s i b l e to t h e i r i d e a l values. c c Following the method of Cornish (1976) , the micrometer s e t t i n g s of the compensator which gave approximately a quarter wave r e t a r d a t i o n were determined by the f o l l o w i n g procedure. With the e l l i p s o m e t e r i n the s t r a i g h t -through p o s i t i o n and p o l a r i z e r , analyzer, and compensator azimuths set to P = 0.00°, A = 90.00°, and C = 315.00° r e s p e c t i v e l y , the micrometer was screwed out s t a r t i n g from zero on i t s s c a l e and the s e t t i n g s corresponding to minimum photodetector s i g n a l s were noted. At these s e t t i n g s , the compensator acts as a f u l l - w a v e p l a t e . To determine the s e t t i n g s which gave a quarter-wave r e t a r d a t i o n , one quarter of the d i f f e r e n c e between e x t i n c t i o n s e t t i n g s was added to the full-wave s e t t i n g s . One of the quarter-wave s e t t i n g s was s e l e c t e d f o r the f i n a l c a l i b r a t i o n . The angle of incidence was set to 70° and a c l e a n I c o n e l s l i d e was c a r e f u l l y a l i g n e d . The e l l i p s o m e t e r was then balanced i n zones 1 and 3 and the data used to c a l c u l a t e T and A f o r that p a r t i c u l a r s e t t i n g of the micro-c c meter (see Appendix B). The measurements were repeated at s l i g h t l y d i f f e r e n t s e t t i n g s of the micrometer u n t i l the d e s i r e d values of T and A were obtained. ° c c The I c o n e l s l i d e was r e a l i g n e d before each set of measurements. The above method assumes the s l i d e to be a l i g n e d p e r f e c t l y . When the f i n a l s e t t i n g of the micrometer i s e s t a b l i s h e d , i t i s e s s e n t i a l to 85 r e a l i g n and measure the s l i d e s e v e r a l times so that average values of T and c A can be obtained, c In general, the compensator cannot be adjusted so that T = 1 and A^ = T T / 2 simultaneously. This i s p a r t i c u l a r l y true i f the waveplate i s t i l t e d i n i t s mount (Oldham 1967), as i s the case w i t h the present e l l i p s o m e t e r . However, i t was p o s s i b l e to minimize the d e v i a t i o n s of T and A from t h e i r c c i d e a l values. The compensator had to be c a l i b r a t e d s e v e r a l times, u s u a l l y a f t e r the micrometer had been moved a c c i d e n t a l l y , and the values T^ = 1.0004 and A £ = 90.023° are t y p i c a l of the waveplate constants thus obtained. I f the compensator i s t i l t e d w i t h respect to the telescope a x i s , as i s u s u a l l y the case, the above c a l i b r a t i o n must be repeated at each azimuth of the compensator. For example, i f 4-zone balances are to be alone, T c and A^ must be determined separately at C = - T T / 4 and C = T F / 4 . In the present i n v e s t i g a t i o n , the compensator was set to C = - T T / 4 , unless otherwise s t a t e d . 4.9 Azimuth E r r o r s and Component Imperfections The e l l i p s o m e t e r equation qiven i n Appendix A i s derived assuming that the azimuths are known e x a c t l y and the o p t i c a l components behave i d e a l l y . However, azimuth e r r o r s and component imperfections introduce zone-dependent d i f f e r e n c e s i n the e l l i p s o m e t e r angle A and ¥. McCrakin et a l . (1963) noted that averaging A and ¥ between zones 1 and 3 or zones 2 and 4 reduces the d i f f e r e n c e s but were unable to e x p l a i n t h i s observation. Several authors (McCrackin 1970, Aspnes 1971) have s i n c e i n v e s t i g a t e d the problem and have proposed c o r r e c t i o n s to the e l l i p s o m e t e r measurements. Azzam and Bashara have been p a r t i c u l a r l y a c t i v e i n t h i s f i e l d . The f o l l o w i n g s e c t i o n b r i e f l y o u t l i n e s t h e i r method of a n a l y s i s (Azzam and Bashara 1971a) and gives some experimental r e s u l t s which are compared w i t h those of the preceding c a l i b r a -t i o n s . I t should be noted that Azzam and Bashara i m p l i c i t l y assume p e r f e c t 86 alignment of the e l l i p s o m e t e r telescopes and of the sample. A Jones-matrix formalism ( S c h u r c l i f f 1962) i s used i n the a n a l y s i s of the component imp e r f e c t i o n s . The Jones matri x T of an o p t i c a l component K. k w i t h small d e v i a t i o n s from i d e a l i t y can be w r i t t e n as T k = T k° + 6T k 6 T k = ( T i j k ) , |6T..k| « 1 where 6T k i s a 2 x 2 complex imper f e c t i o n matrix r e p r e s e n t i n g the d e v i a t i o n s from the i d e a l behaviour described by T k°. To a f i r s t approximation, one can neglect coupling between the p and s components of l i g h t . ^ Therefore, the o f f - d i a g o n a l elements of both T k° and 6T k may be set to zero i . e . T, L l l 0 T, 22 ST, = 11 0 T 0 22 In a s i m i l a r way, the azimuth Z k of the kth component can be w r i t t e n as Z. = Z. ° + 6Z. , 6Z, « 1 k k k k where Z ° i s the azimuth read from the e l l i p s o m e t e r and 6Z i s a small correc-K K t i o n . W r i t i n g the r e f l e c t a n c e r a t i o as p = f ( Z , T..,) and expanding S K. IJ K p i n a Taylor s e r i e s about Z, ° and T. 0 gives S K. XJ K. f ( Z ° T . ° ) + E y, 6Z. + Z y... 6T... k l j k k k k k x J k (12) s s The e f f e c t of both coherent and Incoherent p -«-*- s cross s c a t t e r i n g has been i n v e s t i g a t e d by Azzam and Bashara (1971b). 87 where y^ and a r e coupling c o e f f i c i e n t s and second and higher order terms are excluded. The l i n e a r approximation makes i t p o s s i b l e to analyze, say, the azimuth e r r o r s w h i l e assuming p e r f e c t o p t i c a l components. The c o r r e c t i o n f a c t o r s f o r A and f are determined by d i f f e r e n t i a t i n g the e l l i p s o m e t e r equation SLn(p^) = £n(tan4'expjA) , g i v i n g 6 p s 2 s - z 641 + j6A (13) from which we have p ° sin24' s 64< = sin24' Re(6p /p °) (14a) z s s 6A = Im(6p /p °) (14b) s s Therefore, to de r i v e the c o r r e c t i o n terms 64' and 6A i t i s f i r s t necessary to obtain expressions f o r the cou p l i n g c o e f f i c i e n t s y and y. k I J k 4.9.1 Azimuthal E r r o r s The coupling c o e f f i c i e n t s f o r the azimuthal e r r o r s are obtained by d i f f e r e n t i a t i n g the e l l i p s o m e t e r equation -tanA[tanC + p tan(P-C)] p = 9 s 1 - p tanCtan(P-C) K J c w i t h respect to P, C, or A. In the above expression, P c = T^xp-jA^,. The coupling c o e f f i c i e n t of the an a l y z e r , Ya> f° r example, i s given by -sec 2A[tanC + p tan(P-C)] YA = 1 - P tanCtan(P-C) ( 1 6 ) c When P = - j ( i . e . T c = 1, A £ = 90°) and C = -45°, t h i s expression s i m p l i f i e s to 2 Y A = j s e c Aexp(j2P) Since i n t h i s case P g = j t a n A e x p ( j 2 P ) , then 6 2 , p s . _ j s e c Aexp(j2P)<5A V ° ; jtanAexp(j2P) 88 = 6A/(y sin2A) (17) Therefore, using eqns. 14 and 17, we have 6A = 0 , and = ^sin24'6A/(isin2A) = <5A i n zones 1 and 2 = -<5A i n zones 3 and 4 C o r r e c t i o n terms f o r P and C can be derived i n a s i m i l a r manner. Azzam and Bashara (1971a) i d e n t i f i e d the sources of azimuth e r r o r as (1) r e s i d u a l c a l i b r a t i o n e r r o r s of the d i v i d e d c i r c l e s which are constant from one measurement to the next; (2) random e r r o r s due to e r r o r s i n determining the azimuths of the n u l l ; and (3) specimen misalignment e r r o r s . The l a t t e r two sources of e r r o r can i n p r i n c i p l e be e l i m i n a t e d from a measurement by averaging over s e v e r a l measurements or alignments of the specimen. 4.9.2 Compensator Imperfections The compensator i s represented by the Jones matrices 1 0 0 P e and 6 T k = 0 0 0 6p (18) where P c° = - j f o r an i d e a l quarter-wave p l a t e and 6p^ i s the complex d e v i a t i o n from i d e a l i t y . To f i n d the coupling c o e f f i c i e n t y ' i - n the equation P ° + y ' 6 P , s c c (19) 89 equation 15 i s d i f f e r e n t i a t e d w i t h respect to p g i v i n g , _ -tanAsec zCtan(P-C)  Y c [p °tanCtan(P-C)-l]2 (20) For the s p e c i a l case of C = -45° and P C ° = - j , t h i s expression s i m p l i f i e s to Y ' = -jtanAcos2Pexp(j2P) As before, s o l v i n g f o r ( < S p s / p s ° ) c = (.YC/P°)6P where P° = jtanexp(j2P) gives 6P —I = -cos2P6P P c s c (21) W r i t i n g 6p^ = t ^ + j t 2 and s u b s t i t u t i n g i n t o (19) gives s c = - c o s 2 P ( t l c + j t 2 c ) Therefore, the c o r r e c t i o n terms due to imperfections i n the compensator are <5A = -t„ cos2P , and c 2c 64* = - r t sin2Acos2P i n zones 1 and 2 c 2 l c = r t , sin2Acos2P i n zones 3 and 4 2 l c 4.9.3 P o l a r i z e r and Analyser Imperfections The Jones matrix r e p r e s e n t a t i o n s of the p o l a r i z e r and analyzer prisms are LP,A 1 0 0 6p P.A (22) 90 where 6p . is the deviation from ideality due to transmittance along the extinction axis of the polarizer and analyzer. Substituting the above matrix into equation A-14 of Appendix A and multiplying through gives the coupling coefficients y ' with p = -j and C = -45°, as L j A C Yp « -2tanAexp(j2P) (23) Y A= 0 Writing 6 p ^ = t ^ p + jt^p gives the correction terms due to imperfections in the polarizer prism as 64' = - -jt2pSin2A in zones 1 and 4 = yt2pSin2A in zones 2 and 3 6 A = 2t^p in zones 1 and 3 = -2t in z6nes 2 and 4 IP The analyzer does not contribute a f i r s t order correction to A. 4.9.4 Cell Window Birefringence When in situ measurements are made, generally A and 4" must be corrected for the effects of the c e l l windows on the measurements. The entrance and exit windows can be modelled as birefringent plates with their fast and slow axes in the plane of the windows and at azimuths W and W', respectively, with respect to the plane of incidence. Assuming the windows transmit light equally along both fast and slow axes but introduce a small phase retardation between the two components, the Jones matrix representation for each window has the form 91 T ' = W,W 1 0 0 1 + 6p W,W (24) where Sp^ w , = - J ^ y w i represents the phase r e t a r d a t i o n . S u b s t i t u t i n g (24) i n t o equation A-14, f i r s t f o r the entrance window and then f o r the e x i t window, and s o l v i n g as before f o r p , the window e r r o r s at C = -45° and P C ° = - j are given by 6¥TT = — t„TTsin2Wcos2Psin2A f o r zones 1 and 4 W Z ZV1 t_„sin2Wcos2Psin2A f o r zones 2 and 3 Z ZVI 6Aw = ^ w 0 0 5 ^ 6«F' = 0 w S A ^ = t„I,'cos2W* - 2t„,,'sin2W'cot2A '2W "2W These r e s u l t s show that the e f f e c t of the c e l l windows on A and ¥ can be qu i t e l a r g e , depending on the r e l a t i v e phase r e t a r d a t i o n s and t 9 T J ' of the "2W entrance and e x i t windows. Even f o r la r g e values of t^^ and t 2 w ' , however, the e f f e c t of the windows can be minimized by s e t t i n g t h e i r azimuths to appropriate angles. The c o r r e c t i o n s to ¥ and A due to azimuth e r r o r s and component imperfections are summarized i n Table 3.6. The equations given i n t h i s t a b l e can be i n v e r t e d to give the n u l l i n g azimuths P and A as fu n c t i o n s of the specimen ¥ and A , azimuth e r r o r s and component imperfections (Table 3.7). TABLE 3.6 y and A Corrected to f i r s t order ( a f t e r Azzam and Bashara 1971) ZONE 1 4* = A - t 1 T J s i n 2 A . - ^-t, sin2A 1cos2P 1 + sin2A 1 sin2P.. 6C + 6A + •^t O T Tsin2Wcos2P 1 sin2A n 1 i r 1 Z I c 1 1 1 1 I ZW 1 1 1 1 2 Y = A„ + t O T,sin2A. - -^t sin2A0cos2P„ + sin2A„sin2P06C + 6A - -t n i Tsin2Wcos2P nsin2A n z zF z z Ic z z I I z zw 2 z 3 41 = -A_ + t_„sin2A, + - i t , s i n 2 A 0cos2P 0 - sin2A nsin2P n(SC - 6A - it„t7sin2Wcos2P„sin2A0 3 2 P 3 2 1c 3 3 3 3 2 2W 3 3 4 4* = -A. - t o n s i n 2 A . + -^t. sin2A.cos2P. - sin2A.sin2P.6C - 6A + ^t o t,sin2Wcos2P, sin2A. 4 2P 4 2 l c 4 4 4 4 2 2W 4 4 1 A = 2P + TT/2 + 2 t 1 r i + 26P - t„ cos2P - 26C + t o t Tcos2W + t 'cos2W - t„ 'sin2W cot2A, 1 IP 2c 1 2W 2W 2W 1 2 A = -2P„ - TT/2 - 2 t 1 T , - 26P - t„ cos2P„ + 26C + t O I Icos2W + t 'cos2W' - t 'sin2Wcot2A„ 2 IP 2c 2 2W 2W 2W 2 3 A = 2P_ - TT/2 + 2 t 1 r > + 26P - t„ cos2P„ - 26C + t„,,cos2W + t 'cos2W' - t 'sin2Wcot2A 0 3 IP 2c 3 2W 2W 2W 3 4 A = -2P. + TT/2 - 2 t 1 T , - 26P - t„ cos2P. + 26C + t O I Icos2W + t„ 'cos2W' - t„'sin2W'cot2A. 4 IP 2c 4 2W 2W 2W 4 TABLE 3.7 N u l l i n g azimuthal angles as functions of ¥ and A ( a f t e r Azzam and Bashara 1971) ZONE 1 A, = Y + t„T,sin2'}' + i t , sin2YsinA + sin2*fcosASC - SA - it O I 7 s i n 2 W s i n A s i n 2 1 ; 1 2P 2 l c 2 2W 2 A = H> - t ^ s i ^ V - i t , s i ^ Y s i n A + sin2¥cosA6C - 6A - it_„sin2WsinAsin2Y 2 2P 2 l c 2 2W 3 A- = - V - t O T,sin2Y + i t , sin24'sinA + sin24'cosA6C - SA - it o t,sin2WsinAsin2 ,l' 3 2P 2 l c 2 2W 4 A. = - T + t O T,sin2V - i t , sin2>PsinA + sin2>i'cosA6C - SA - it O I Tsin2WsinAsin24' 4 2P 2 l c 2 2W 1 ?1 = A/2 - T T/4 - t l p - <5P + - | t 2 c s i n A + 6C - |t 2 wcos2W - y t 2 w ' c o s 2 W + yt 2 w'sin2W'cot2 >i' 2 P = - A/2 - T T / 4 - t 1 T J - 6P + sinA + SC + it_„cos2W + i t 'cos2V - | t ' sin2W cot2f z IP Z zc I 2W 2 2W 2 2W 3 P = A/2 + T T/4 - t l p - 6P - i t sinA + 6C - it o t Icos2W - i t 'COS2W - i t ' sin2W'cot24< j I r Z Z C 2 zw Z zw Z zw 4 P = - A/2 + T T/4 - r - SP - i t sinA + SC + it O I tcos2W + i t 'cos2W' + i t •sia2W,cot2¥ 4 l r z Z C z 2W 2 2W 2 2W to 94 4.9.5 Four-Zone Measurements From Table 3.7 , i t can be shown th a t averaging V and A over four zones gives 1/4(A 1 - A 3 + A 2 - A 4) A 1/2 ( P X + P 3 - P 2 - P 4 ) + N, WW where n t 7 T 7 ' = t„.,cos2W + t» 'cos2W' . WW /.VI 2W That i s , averaging over four zones e l i m i n a t e s a l l e r r o r s considered i n 41, and A must be corrected only f o r the b i r e f r i n g e n c e of the c e l l windows i f i n s i t u measurements are made. This r e s u l t holds even i f , as i n the present study, the compensator constants T and A (and, thus, t, and t„ ) vary w i t h the c c l c 2c azimuth of the compensator, provided the d e v i a t i o n from t h e i r i d e a l values i n each zone i s s m a l l . 4.9.6 Two-Zone Measurements Averaging the e l l i p s o m e t e r measurements between zones 1 and 3 or zones 2 and 4 gives the values of 41 and A as • where the ± signs correspond to C = ± 45° and 4*° and A° are the i d e a l averages giv e n , except f o r the a d d i t i o n of some i n t e g r a l m u l t i p l e of 90°, by 4< = y° ± g A = A ° 1 ^P,C + \W r = f ( A 2 - A 4) or I ( A1 " A3> A° = - ( P 2 + P 4 ) or 95 The c o r r e c t i o n f a c t o r s E, and n are given by L ST y L» C P = t 2 p s i n 2 4 ' nP,C = 2 t l P + 2 6 P " 2 6 C Thus, to a f i r s t approximation, a two-zone average of V i s f r e e of compensator, azimuth, and window-birefringence e r r o r s , w i t h the only c o r r e c t i o n being due to the p o l a r i z e r e l l i p t i c i t y t 2 p . For A, however, c o r r e c t i o n s are required f o r the p o l a r i z e r i m p e r f e c t i o n t ^ , azimuth e r r o r s 6P and <5C, and the window b i r e f r i n g e n c e n, ' . WW 4.9.7 One-Zone Measurements Measurements i n one zone u s u a l l y are required when i t i s important to minimize the measurement time i n order to f o l l o w a r e l a t i v e l y f a s t s u r f a c e -phenomenon. In t h i s case, the values of A and ¥ from the one-zone measurements must be corrected f o r the component e r r o r s . I f ¥.° and A.° are the i d e a l one-1 1 zone values of f and A i n the i zone, the co r r e c t e d values according to Azzam and Bashara are given by * = Y.° ± t„„sin2A ± —t^„sin2A. cos2P . ± sin2A.sin2P . 6C l 2P 2 IC l l i i ± 6A ± i-t„..sin2Wcos2P .sin2A. 2 2W i i A " Ai° + ^P.C " ^ C ^ i + ^ - t 2 w ' s i n 2 W c o t 2 A . where the appropriate combination of signs i s taken from Table 3.6. These c o r r e c t i o n s are p a r t i c u l a r l y important w i t h i n s i t u measurements when the c e l l windows are s t r o n g l y b i r e f r i n g e n t . One should a l s o note that the terms t and t„ g e n e r a l l y depend on the azimuth of the compensator and, thus, must be known se p a r a t e l y f o r zones 1 and 3 and zones 2 and 4. 96 4.9.8 Measurement of the E l l i p s o m e t e r E r r o r s Values of the e r r o r terms given i n Tables 3.6 and 3.7 can be deduced from the data f o r 2-zone balances. With the compensator set to C =-45°, the r e s i d u a l s ResA and ResP are defined, w i t h i n an i n t e g r a l m u l t i p l e of 90°, as ResA = A1 + A 3 (25) ResP = P - P 1 (26) From Table 3.7 , these r e s i d u a l s are given by ResA = t, „sin24'sinA + 2sin2YcosA6C - 26A - t„ITsin2Wsin2¥sinA (27) IL ZW ResP = -t_„sinA -. t-'sin2W'cot24< (28) zC zW I f the r e f l e c t i n g surface i s an i d e a l l y clean i s o t r o p i c nonabsorbing d i e l e c t r i c , as approximated by the fused quartz s l a b i n some c a l i b r a t i o n s , then sinA = 0. Therefore, f o r measurements not i n v o l v i n g c e l l windows, one has ResA = 2sin24'cosA6C - 26A This r e l a t i o n was used i n Section 4.6.4 to determine the azimuth e r r o r s 6A and 6C as 0.015° and -0.29°. With the azimuth e r r o r s <5A and 6C known, i t was p o s s i b l e to determine t , i n eqn. 27. A t h i c k zirconium f i l m on a glass substrate was measured i n zones 1 and 3 at an angle of incidence of 70°. Several measurements were made, w i t h the sample r e a l i g n e d a f t e r each s e t , so as to o b t a i n an average r e s i d u a l <ResA>. Rearranging eqn. 27 to solv e f o r t ^ (without windows) gives t = <ResA> - 2sin2ycosA6C + 26A IC s i r ^ Y s i n A 97 (29) S u b s t i t u t i n g the measured values of A and Y i n t o eqn. (29) gives t ^ = 0.030 ± 0.05°. The above measurements were repeated w i t h an I c o n e l s l i d e . From these measurements, t ^ , was c a l c u l a t e d to be 0.043 ± 0.05°. The two values f o r t ^ were i n agreement, w i t h i n the measurement e r r o r . The two-zone data f o r zirconium were a l s o used to c a l c u l a t e t. "2C From eqn. 28, i s given as t 2 = - ResP/sinA (30) assuming there are no windows. From the data, we obtained = 0.15 ± 0.01°. Measurements with the I c o n e l s l i d e confirmed t h i s r e s u l t . The e l l i p s o m e t e r was then set to the s t r a i g h t - t h r o u g h p o s i t i o n and balanced i n two zones. This i s equivalent to a s i t u a t i o n i n which the l i g h t i s i n c i d e n t on a f i c t i t i o u s r e f l e c t i n g surface that does not change i t s s t a t e of p o l a r i z a t i o n . ( i . e . f = 45°, A = 180°). Therefore, from Table 3.7 , we have under these c o n d i t i o n s A i = " 2 t 2 P + 6 C " 6 A ( 3 1 ) p± - - K f C - w ( 3 2 ) where A^ and P^ represent the a c t u a l n u l l i n g angles i n zone 1 or 3, minus t h e i r i d e a l values. From eqns. 31 and 32, we o b t a i n the r e l a t i o n s t 2 p = j ( \ - A 3) (33) Y c - " 2 p i (34) NW 98 where NW means "no windows". S u b s t i t u t i n g the measured data i n t o these equations gave t 2 p = -0.01 ± 0.01° and n p c = -0.23 ± 0.06°. Since n p c = 2 t l p + 2 6P - 2 6C and t ^ p i s expected to be of the same order of magnitude as t 2 p ( i . e . approximately z e r o ) , 6P i s c a l c u l a t e d to be -0.41°. L a s t l y , the c o r r e c t i o n terms due to the c e l l windows were evaluated. From Table 3.7 and eqn. 32, we have the r e l a t i o n s W - ( p i + VNW " ( p i + Vww ( 3 5> t i n 2 w = 2 ( A1NW - A1WW) 2W s ^ f s i n A v ' r • • ( R e S P )NW - (R£SP>WW t 2 w sin2W = (37) where WW means measurements "with the windows" i n place. As an example of the magnitude of these c o r r e c t i o n terms, the f o l l o w i n g r e s u l t s were obtained wit h an Iconel s l i d e and a r b i t r a r y azimuths of the fused quartz windows discussed i n Chapter 4: n, ' = 0.06 ± 0.005° WW t 2 w s i n 2 W = 0.14 ± 0.01° t 2 w' s i n 2 W ' = -0.19 ± 0.01° Of course, r o t a t i n g the c e l l windows i n t h e i r mounts w i l l change these r e s u l t s . Therefore, these c o r r e c t i o n terms must be evaluated each time the windows are r o t a t e d . The measured values of the e l l i p s o m e t e r c o r r e c t i o n f a c t o r s are summarized i n Table 3.7. 99 Table 3.8 Summary of Measured E l l i p s o m e t e r C o r r e c t i o n Factors. t l c 0.03 ± 0.05° t 2 c 0.15 ± 0.01° t 2 p -0.01 ± 0.01° n p c -0.23 ± 0.06° 6A 0.015 ± 0.02° 6C -0.29 ± 0.05° 4.9.9 R e l a t i o n between t^, and TC> AC From the previous s e c t i o n s , we have the r e l a t i o n P = P ° + 6p C C C where 6p c = t l c + j t ^ . I f P c° = - j , then P c = t l c - j ( 1 - t 2 ( J . W r i t i n g i n the form p = |T |exp - j AC, then T c " [ t i c 2 + ( 1 - h c ^ 2 ( 3 8 ) A = - t a n - 1 ! ^ 1 - t 2 c ) ] (39) L t l C S u b s t i t u t i n g the values f o r t and t obtained i n the l a s t s e c t i o n (converted to radians) gives |P | = 0.997 , AC = 90.03° These r e s u l t s are i n c l o s e agreement w i t h those obtained i n Se c t i o n 4.8. 100 Since and t^^ are much l e s s than u n i t y , eqns. 38 and 39 can be approximated by TC * 1 " '2C a n d AC a - (I + t l C ) That i s , t 2 c i s the d e v i a t i o n from u n i t y of the transmittance r a t i o and t ^ i s the d e v i a t i o n from 90° of the compensator's phase r e t a r d a t i o n . 4.9.10 Some Conclusions Regarding the Method of Azzam and Bashara The c a l i b r a t i o n method of Azzam and Bashara gives r e s u l t s f o r the p o l a r i z e r and compensator imperfections and analyzer azimuth e r r o r that agree reasonably w e l l w i t h the r e s u l t s of other c a l i b r a t i o n methods and the manufacturer's s p e c i f i c a t i o n s . A l s o , the method gives a procedure whereby the e r r o r s introduced by c e l l windows can be co r r e c t e d . However, there i s some un c e r t a i n t y regarding the azimuth e r r o r s of the p o l a r i z e r and compensator. The c a l i b r a t i o n procedures discussed i n Sections 4.6 and 4.7 referenced both of these components to the plane of incidence. As t h i s was done p r i o r to the c a l i b r a t i o n procedure of t h i s s e c t i o n , one might have expected both 6P and 6C to be zero. I t i s not c l e a r why the values 6P = -0.41° and 6C = -0.29° were obtained i n s t e a d . These values are too l a r g e to a t t r i b u t e to measurement e r r o r . One p o s s i b i l i t y i s that the t i l t of the compensator somehow a f f e c t e d the azimuth c a l i b r a t i o n s . Therefore, i n the f o l l o w i n g e l l i p s o m e t r y measurements, the r e s u l t s of the previous c a l i b r a t i o n s w i l l be taken as c o r r e c t and the method of Azzam and Bashara w i l l be used only to c o r r e c t f o r the c e l l window e r r o r s . 101 IV. OPTICAL AND DIELECTRIC PROPERTIES OF ANODICALLY OXIDIZED TANTALUM 1. I n t r o d u c t i o n In previous o p t i c a l s t u d i e s of a n o d i c a l l y o x i d i z e d tantalum, i t was found that oxides grown i n d i l u t e s u l f u r i c a c i d (Young and Zobel 1966, Ord 1972, Cornish 1976) and a c e t i c a c i d (Muth 1969) could be tre a t e d as s i n g l e -l a y e r homogeneous nonabsorbing f i l m s . An exception to these r e s u l t s was the observation of a t h i n absorbing l a y e r at the outer surface of the f i l m by measurements of the r e f l e c t i v i t y of p l i g h t at the Brewster angle of the oxide (Masing, Orme, and Young 1961). Although t h i s r e s u l t i s anomalous, i t i s c o n s i s t e n t w i t h observations that f i l m s grown i n concentrated s u l f u r i c a c i d are absorbing and c o n t a i n a high c o n c e n t r a t i o n of s u l f a t e ions i n the outer part of the f i l m . A l s o , a small amount of s u l f a t e i s incorporated i n the outer part of the oxide when i t i s grown i n d i l u t e s u l f u r i c a c i d s o l u t i o n s ( R a n d a l l , Bernard, and Wilkinson 1965). Films grown i n d i l u t e phosphoric a c i d a l s o have a two l a y e r s t r u c t u r e which can be detected o p t i c a l l y (Dell'Oca and Young 1970a) as w e l l as by e l e c t r i c a l measurements (Randall 1975) . In the present work, i t was thought that the new e l l i p s o m e t e r balancing a l g o r i t h m discussed i n the previous chapter might provide s u f f i c i e n t accuracy to detect the double l a y e r s t r u c t u r e of oxide f i l m s grown i n d i l u t e H2SO4. The r e s u l t s of t h i s study are presented i n the f i r s t h a l f of t h i s chapter. In the second h a l f , some r e s u l t s are given f o r the e l e c t r i c a l p r o p e r t i e s of the oxide f i l m s . 2. Experimental Apparatus and Procedure 2.1 Anodization C e l l The a n o d i z a t i o n c e l l ( f i g u r e 4.1) used i n the present s t u d i e s had been designed and b u i l t p r e v i o u s l y f o r i n s i t u e l l i p s o m e t r y (Yee 1974, Cornish 1972). To be r e s i s t a n t to any chemicals used, the c e l l was made 102 WATER CURTAINS WINDOW BASE TEFLON LID & CYLINDER SCREWS F i g . 4.1 S i d e and t o p view o f the a n o d i z a t i o n c e l l used i n the pr e s e n t work. C = c a l o m e l e l e c t r o d e ; C1,C2 = c o o l i n g c o i l ; T = thermometer; TP = t h e r m i s t o r probe; G1,G2 = gas b u b b l i n g t u b e s ; H = 150 W h e a t e r ; S = s t i r r i n g p r o p e l l o r ; E1,E2,E3'= Pt/H, e l e c t r o d e s . ' 103 from a t e f l o n c y l i n d e r w i t h a t e f l o n baseplate glued i n t o one end. Holes were cut i n the side of the c e l l to hold t e f l o n f i x t u r e s on which were mounted 2.54 x 10~ 2 m diameter windows (see Section 2.2). The angle of incidence (62.77°) was f i x e d by the angle between the two windows. The c e l l was mounted on a brass baseplate which was i n turn mounted at the pi v o t a x i s of the e l l i p s o m e t e r . Both the c e l l and sample holder could be r o t a t e d independently about the p i v o t a x i s . The c e l l could a l s o be t i l t e d by a d j u s t i n g screws on the baseplate and then locked i n place once the des i r e d t i l t was obtained. The t i l t of the c e l l was adjusted so that the surface of the input window was almost perpendicular to the l a s e r beam. A s l i g h t misalignment was d e s i r a b l e to e l i m i n a t e a m u l t i p l e r e f l e c t i o n path (see Chapter 3 , Section 4.4). In the present study, a t e f l o n l i d was threaded onto the top of the c e l l . The l i d supported three p l a t i n i z e d platinum e l e c t r o d e s and a calomel reference e l e c t r o d e . One of the platinum e l e c t r o d e s formed a r i n g around the sample, with openings f o r the e l l i p s o m e t e r beam. The other two platinum -4 2 ' ele c t r o d e s were used as the cathode (area x 10 m ) and a P t / ^ reference e l e c t r o d e . The l i d a l s o supported a 150 watt heating element (Watlow model G1J31) and a Yellow Springs Instruments (YSI) model 406 thermistor temperature sensor. Both the heating element and temperature sensor were encased i n _3 s t a i n l e s s s t e e l j a c k e t s . A c o o l i n g c o i l made from 6.4 x 10 m diameter s t a i n l e s s s t e e l tubing, a t e f l o n s t i r r i n g p r o p e l l o r , and a thermometer a l s o were i n s e r t e d through the l i d of the c e l l . The sample was i n s e r t e d through an opening at the center of the l i d . The hole was l a r g e enough to a l l o w r e l a t i v e l y easy alignment of the sample on the e l l i p s o m e t e r . A l l samples were anodized i n 0.1 M ^SO^. I t was planned, i n some experiments, to sa t u r a t e the s o l u t i o n w i t h hydrogen gas. To a l l o w f o r these 104 experiments, the s t i r r i n g p r o p e l l o r and sample were i n s e r t e d i n t o the c e l l through water c u r t a i n s . A l l other components were e i t h e r press f i t t e d or glued i n t o the l i d . The hydrogen gas was bubbled i n t o the c e l l through a narrow g l a s s tube. So that the c e l l could not o v e r - p r e s s u r i z e , an o u t l e t tube was a l s o i n s e r t e d i n t o the l i d . The gas flow through the c e l l was monitored by bubbling both the input and output gas through Drechsel b o t t l e s . Above room temperature, the c e l l was thermostatted to w i t h i n 0.1° using a YSI model 72 p r o p o r t i o n a l temperature c o n t r o l l e r . To ob t a i n temperatures near 0°C, water from an i c e bath was pumped through the c o o l i n g c o i l . At temperatures much d i f f e r e n t from room temperature, the c e l l was wrapped w i t h sponge foam to avoid excessive heating of the i c e bath or heat l o s s from the c e l l . The data reported i n t h i s chapter were obtained at 25°C. 2.2 Measurement of the C e l l Window B i r e f r i n g e n c e As mentioned i n Chapter 3 measurement e r r o r s due to the c e l l windows can, i n p r i n c i p l e , be c o r r e c t e d . The window e r r o r s must be kept s m a l l , however, so that second and higher order e f f e c t s can be neglected. In a d d i t i o n , the window surfaces should be as f l a t and p a r a l l e l as p o s s i b l e i n order to minimize r e f r a c t i v e e f f e c t s . Both requirements were met by using -3 -2 6.4 x 10 m t h i c k fused s i l i c a windows (2.54 x 10 m diameter, 1/20 wave i n f l a t n e s s , and 1 sec. p a r a l l e l i s m ) obtained from O r i e l Corporation. The b i r e f r i n g e n c e of each window was measured before being i n s e r t e d i n the c e l l . The window was mounted i n a d i v i d e d c i r c l e holder and, w i t h the e l l i p s o m e t e r i n the s t r a i g h t - t h r o u g h p o s i t i o n , measured at i t s center. The window was r o t a t e d about i t s surface normal and measured every 15° r o t a t i o n . Before each measurement, the window was a l i g n e d to be perpendicular to the i n c i d e n t beam. 105 Within the measurement e r r o r of ± 0.002°, the window had no a f f e c t on the analyzer balance. The balance angle of the p o l a r i z e r , however, v a r i e d s i n u s o i d a l l y w i t h the azimuth of the window. Figure 4.2 shows AP =P, -P, 6 1 1NW 1WW where NW and WW i n d i c a t e "no window" and "with window" r e s p e c t i v e l y , p l o t t e d against the azimuth angle referenced to an a r b i t r a r y zero. Note that AP goes through two complete c y c l e s as the window i s rot a t e d through 360°. Assuming the window can be t r e a t e d as a b i r e f r i n g e n t p l a t e w i t h i t s f a s t and slow axes p a r a l l e l to the window su r f a c e s , the above r e s u l t s are pr e d i c t e d by the equations given i n Table 3.7. In the present measurements, the window can be considered to be the input window W i n a PCWSW'A e l l i p s o m e t e r c o n f i g u r a t i o n where sample S i s a f i c t i t i o u s r e f l e c t i n g surface w i t h R /R = 1 p s and the e x i t window W' i s i d e a l (eg. t^y, = 0 ) . Therefore, c o n s i d e r i n g only the data f o r zone 1, f o r example, we have M l = A1NW " A1WW = j t 2 W S i n 2 W s i n A s i n 2 ' 1 ' ( 1 )  A P 1 = P1NW " P1WW " I t2W C O s 2 W ( 2 ) In eqn. 2, AP^ i s p r e d i c t e d to vary s i n u s o i d a l l y w i t h W, with peak-to-peak amplitude t O T T . From the data i n Figure 4.2 , t„„ = 0.045 ± 0.005°. AA, a l s o zw zw 1 i s p r e d i c t e d to vary s i n u s o i d a l l y w i t h W but the peak-to-peak amplitude i s damped by the term xsinAsin24'. Since the windows are n e a r l y i d e a l nonabsorbing d i e l e c t r i c s ( i . e . sinA-0) the e f f e c t of the window on the analyzer balance i n these measurements i s too small to measure. The window was clamped more t i g h t l y i n i t s holder and the above measurements repeated. The r e s u l t s f o r the p o l a r i z e r are shown as the dashed l i n e i n Figure 4.2. These data show the e f f e c t of st r e s s - i n d u c e d b i r e f r i n -gence. To minimize t h i s e f f e c t i n subsequent i n s i t u measurements, the windows were tightened i n t h e i r c e l l 0-ring f i x t u r e s only enough to prevent 106 F i g . 4.2 The change i n the p o l a r i z e r balance a n g l e due t o n a t u r a l (0) and s t r e s s - i n d u c e d (+) b i r e f r i n g e n c e of the c e l l 107 the e l e c t r o l y t e l e a k i n g f roin the c e l l . The above r e s u l t s a l s o show that the c o r r e c t i o n term l y ^ t (see Chapter 3 , S e c t i o n 4.9.4) f o r two-zone i n s i t u measurements can be made n e g l i g i b l y small by c a r e f u l l y a d j u s t i n g the azimuth of the input or e x i t window i n the c e l l . 2.3 Sample Preparation _3 The samples were cut as ^ 3 x 10 m t h i c k d i s k s w i t h a diamond saw _2 from 1.2 x 10 m diameter s i n g l e c r y s t a l tantalum obtained from M a t e r i a l s Research Corporation. Both <100> and <111> o r i e n t e d c r y s t a l s were used. The major contaminants, as s p e c i f i e d by the manufacturer, were Nb (25 ppm), C (10 ppm), 0 2 (3.5 ppm), N 2 (2.3 ppm), N i (1.5 ppm) and W (1.2 ppm). The concentrations of the remaining contaminants were g e n e r a l l y l e s s than 0.1 ppm. -3 The d i s k s were cut i n the shape of shovels and a 1.3 x 10 m diameter Fansteel tantalum w i r e , which had p r e v i o u s l y been chemically p o l i s h e d (see Chapter 5, Section 2.1), was spot welded to the narrow neck of each sample. The samples were then mechanically p o l i s h e d on both faces w i t h a sequence of 1/0, 2/0, 3/0 and 4/0 emery paper. P o l i s h i n g w i t h each paper was continued u n t i l the scratches from the previous paper had been e l i m i n a t e d . When changing papers, the samples were thoroughly r i n s e d w i t h d e i o n i z e d water and blown dry w i t h n i t r o g e n . The p o l i s h i n g d i r e c t i o n was a l s o r o t a t e d by 90°. A f t e r u l t r a s o n i c degreasing i n b o i l i n g t r i c h l o r o e t h y l e n e , acetone, and then methanol, the samples were immersed i n hot n i t r i c a c i d to remove any copper residue from the spot welder. The samples were then r i n s e d i n de-i o n i z e d water and e l e c t r o p o l i s h e d i n a s o l u t i o n of 9:1 p a r t s by volume 98% H 2S0^ and 48% HF (Tegart 1959). The heat of r e a c t i o n when mixing the two a c i d s was s u f f i c i e n t to b r i n g the mixture to the temperature required f o r e l e c t r o p o l i s h i n g . The e l e c t r o p o l i s h i n g was done i n a t e f l o n container that had been b u i l t p r e v i o u s l y (Young and Zobel 1966). The s o l u t i o n was v i g o r o u s l y 108 s t i r r e d by a t e f l o n i m p e l l o r b u i l t i n t o the t e f l o n c o n t a i n e r . The samples were e l e c t r o p o l i s h e d f o r about 20-30 minutes at a 3 9 current d e n s i t y i n the range 1 - 2 x 10 A/nr and a voltage of ^ 20 v o l t s . -4 2 A p l a t i n i z e d platinum e l e c t r o d e M.0 m i n area was used as the cathode. _2 The cathode and anode were separated by ^ 3 x 10 m. The q u a l i t y of the surface depended g r e a t l y on the current d e n s i t y and v o l t a g e . A t r i a l and e r r o r procedure was used to f i n d the optimum p o l i s h i n g c o n d i t i o n s f o r each sample. The f i l m l e f t by the p o l i s h i n g s o l u t i o n was leached by immersion i n b o i l i n g water f o r s e v e r a l minutes (Young 1961a). A t h i c k anodic oxide (^  300 nm) was then grown on the sample and supporting tantalum wire. The working area of the sample u s u a l l y was defined by repeated immer-sions of the supporting wire i n t o a heated bath of Pyseal wax obtained from F i s h e r Chemicals. The wire and spot-welded neck were he l d i n the wax f o r about 1 minute to heat the metal and thus ensure a good s e a l . The coat of wax was then b u i l t up by s e v e r a l quick dips i n t o the wax. In some experiments (Chapters 5 and 6), the Pyseal wax was replaced w i t h Canada balsam so that measurements could be made at temperatures higher than the melting p o i n t of the Pyseal wax (^ 40°C). However, s i n c e the balsam d i d not give as good a s e a l as the Pyseal wax, i t was used only f o r measurements r e q u i r i n g i t s higher m e l t i n g point (^ 80°C). Both the Pyseal wax and Canada balsam were r e l a t i v e l y r e s i s t a n t to a t t a c k by concentrated HF. Therefore, i t was p o s s i b l e to etch the working area of the sample w i t h l i t t l e damage to the wax mask, provided only the working area was dipped i n t o the HF. This procedure enabled the same sample to be used i n s e v e r a l experiments. I f there was any s i g n of oxide r e c r y s t a l l i -z a t i o n , however, the sample was r e p o l i s h e d . 109 Several other m a t e r i a l s were i n v e s t i g a t e d as p o s s i b l e masking m a t e r i a l . These included Microstop, v a r i o u s epoxies, and a i r p l a n e glue. A l l were r e j e c t e d , however, because they e i t h e r could not withstand exposure to HF or the ano d i z a t i o n e l e c t r o l y t e , or the an o d i z a t i o n v o l t a g e s . 2.4 Procedure The samples were anodized at constant current using a Northeast S c i e n t i f i c model RI-233 current source. The voltage between the anode and calomel reference e l e c t r o d e was measured w i t h the DANA d i g i t a l voltmeter i n t e r f a c e d to the PDP 8/E computer. The growth of the oxide f i l m s was followed by i n s i t u e l l i p s o m e t r y under computer c o n t r o l (see Chapter 3). As a check on the q u a l i t y of the sample, the computer c a l c u l a t e d AV/AT and output the value to a 16-bit D/A converter (DAC) i n t e r f a c e d to the computer. The s i g n a l from the DAC was dis p l a y e d on a s t r i p chart recorder. A non-constant value of AV/AT, a l l o w i n g f o r a few percent n o i s e , was taken to i n d i c a t e a bad sample. The anodization was i n t e r r u p t e d p e r i o d i c a l l y and the sample measured i n zones 1 and 3. To avoid p o s s i b l e e l e c t r o o p t i c e f f e c t s (Cornish 1972), the anode was shorted to the platinum cathode during these measurements. In some experiments, the small s i g n a l ac capacitance of the oxide f i l m was a l s o measured w i t h respect to the r i n g of p l a t i n i z e d platinum e l e c t r o d e s surrounding the sample. 3. E l l i p s o m e t r y 3.1 Tracking a Growing Oxide F i l m The repeated balance feature of the e l l i p s o m e t e r c o n t r o l program was used to t r a c k the growing oxide during the constant current a n o d i z a t i o n s . The p o l a r i z e r and analyzer were balanced a l t e r n a t e l y and the balance, formation 110 v o l t a g e s , and the time elapsed from the s t a r t of the repeated balancing procedure were stored i n a d i s k f i l e f o r l a t e r a n a l y s i s . In order to improve i t s a b i l i t y to t r a c k the growing f i l m , the c o n t r o l program p r e d i c t e d where the balance would be, based on the past r a t e of change of the balance, and then stepped the o p t i c a l component to t h i s angle before s t a r t i n g the balance procedure. When the balance was completed, the component was set to i t s new balance angle. The balancing procedure was repeated u n t i l e i t h e r a user-s p e c i f i e d formation voltage was reached or stopped by the user. Each balance took about 1-2 seconds, depending on the sweep width of the balance. Since the f i l m was growing during these balances, one would expect the " t r u e " balances to be d i f f e r e n t from the measured balances. This could be c o r r e c t e d f o r . However, the d i f f e r e n c e s can be neglected i f the balance angles change by only a small amount during the time when i n t e n s i t y vs. angle data are being recorded. In a d d i t i o n , because the p o l a r i z e r and analyzer balances are not coupled (see Chapter 3 ), the f a c t that one of these components may be s l i g h t l y o f f i t s balance azimuth does not a f f e c t the balance of the other component. Figure 4.3 shows t y p i c a l e l l i p s o m e t e r angle vs. time data f o r a 2 t f i l m at 0.66 A/m . The (¥,A) data were c o r r e c t e d f o r the c e l l window e r r o r s by the method of Azzam and Bashara and the time was measured at the end of each balance. In the region of the e l l i p s o m e t e r curve f o r these measurements, both the p o l a r i z e r and analyzer balances were r e l a t i v e l y s e n s i t i v e , and, t h e r e f o r e , the sweep width was set to only 1.3°. The i n i t i a l l y r a p i d de-creases i n A i s due to the e l e c t r o - o p t i c and e l e c t r o - s t r i c t i v e e f f e c t s on applying the a n o d i z a t i o n current (discussed below). Once t h i s i n i t i a l ^The computer program to p l o t the e l l i p s o m e t e r data was w r i t t e n by G. Boyd. p 0 O X H* a. cm o *: CR) cm H J H B O P H P Pi (Ti > > e+ B rv> Pi P P o 4 PS! / DEGREES 50 234 236 238 DELTA / DEGREES 240 P 3 £ PSI / DEGREES 3 54 54.5 55 55.5 H -P* i 1 1 1 r — —I l I i i _ 235.6 235.8 236.0 236.2 236.4 DELTA / DEGREES TIT ( 112 t r a n s i e n t phase i s completed, both ¥ and A change continuously as the oxide f i l m grows i n t h i c k n e s s . Because of the greater s e n s i t i v i t y of the analyzer balance the changes i n ¥ are smoother than those i n A. For these r e s u l t s , ¥ and A change on average by about 0.06° and 0.03° per second, r e s p e c t i v e l y , during the measurements. Since the stepping motors are being d r i v e n at about A00 steps/second during the sweeps through the balance angles, the a c t u a l changes i n ¥ and A during these c r i t i c a l measurements are l e s s than 0.02° and 0.01° r e s p e c t i v e l y . As these changes are l e s s than the estimated e r r o r of measure-ments made on an unchanging f i l m of s i m i l a r t h i c k n e s s , we can conclude f o r t h i s example that the measured balances correspond to the " t r u e " balances w i t h i n the measurement e r r o r . This conclusion might not apply to other examples i n which ¥ and A change more r a p i d l y w i t h time. Figure 4.A shows some of the data i n Figure A.3 on an expanded s c a l e . These data show more c l e a r l y the time d i f f e r e n c e s between balances of the p o l a r i z e r and analyzer. In order to p l o t V against A, Lagrangian 3-point i n t e r p o l a t i o n between successive p o l a r i z e r balances was used to c a l c u l a t e the p o l a r i z e r balance corresponding to the time of the bracketed analyzer balance. The r e s u l t s f o r three sets of data obtained at a formation 2 current d e n s i t y of 0.66 A/cm are p l o t t e d i n Figure A.5. The s o l i d l i n e i s the computed e l l i p s o m e t e r curve that was f i t t e d to the two-zone data obtained at zero f i e l d at i n t e r v a l s during the ano d i z a t i o n (see Sect i o n 3.7). The short v e r t i c a l dashes represent the data recorded during the an o d i z a t i o n s . The d i s c o n t i n u i t y i n these data near the peak i n A i s due to the break i n the constant current a n o d i z a t i o n i n order to measure the z e r o - f i e l d values of ¥ and A. The experimental data show q u i t e c l e a r l y , however, the e f f e c t of the a n o d i z a t i o n f i e l d on the o p t i c a l p r o p e r t i e s of the oxide f i l m . The oxide i s i s o t r o p i c under z e r o - f i e l d c o n d i t i o n s but becomes u n i a x i a l l y a n i s o t r o p i c 113 ^ 35 45 55 PSI/DEGREES F i g . 4 .5 The e l l i p s o m e t e r d a t a f o r an ox ide f i l m growing at 0.66 A /m^. The s o l i d l i n e i s the computed e l l i p s o m e t e r curve f i t t e d to the z e r o - f i e l d d a t a . 114 ( b i r e f r i n g e n t ) , w i t h the o p t i c a x i s perpendicular to the oxide s u r f a c e , on applying an e l e c t r i c f i e l d (Cornish 1972, Cornish and Young 1973). Both the o r d i n a r y and e x t r a o r d i n a r y r e f r a c t i v e i n d i c e s of the oxide change w i t h the a p p l i e d f i e l d v i a the e l e c t r o - o p t i c e f f e c t . In a d d i t i o n , i f measurements were made under c o n d i t i o n s such that no i o n i c current flowed, the oxide t h i c k -ness a l s o would be observed to change with the a p p l i e d f i e l d due to the e l e c t r o s t r i c t i v e e f f e c t . The r e s u l t of the oxide becoming b i r e f r i n g e n t i s that the (4*,A) data s p i r a l e i t h e r upwards or downwards i n the (WjA) domain on successive e l l i p s o m e t e r c y c l e s i n s t e a d of f o l l o w i n g a closed loop as expected f o r a nonabsorbing i s o t r o p i c f i l m . The d i r e c t i o n of the s p i r a l depends on whether the ordinary r e f r a c t i v e index i s l a r g e r than the e x t r a -ordinary index. This type of s p i r a l l i n g i s q u i t e d i f f e r e n t from what would be observed with a double l a y e r f i l m as discussed i n Chapter 3. The present r e s u l t s i n d i c a t e the e x t r a o r d i n a r y r e f r a c t i v e index i s l a r g e r than the o r d i n a r y index, i n agreement with Cornish (1972). The e l e c t r o - o p t i c and e l e c t r o - s t r i c t i v e e f f e c t s were not i n v e s t i g a t e d f u r t h e r i n t h i s work. The r e s u l t s i n Figure 4.5 i n d i c a t e , however, that the repeated balancing feature of the e l l i p s o m e t e r c o n t r o l program could be used i n such a study. The data a l s o show that the measurements are most accurate i n the range 35° < 4" < 55°. Although the e l l i p s o m e t e r measurements change more r a p i d l y w i t h o p t i c a l t hickness at l a r g e values of 4*, the s e n s i t i v i t y of the balances and, thus, t h e i r accuracy i s g r e a t l y reduced as 4" approaches 90°. The present e l l i p s o m e t e r c o n t r o l program can t r a c k f a s t e r growing f i l m s than discussed above, provided the balances do not change too r a p i d l y . \n p r a c t i c e t h i s allows measurements w i t h formation currents up to about 2 5 A/m . I t i s l i k e l y that l a r g e r formation currents could be used but 115 allowances would have to be made f o r the balance changing during the measurement, p o s s i b l y by i n c l u d i n g a cubic term i n the balancing a l g o r i t h m . 3.2 Evidence f o r an I s o t r o p i c Nonabsorbing Single-Layer Oxide F i l m . To confirm that the present oxide f i l m s were nonabsorbing (at A = 632.8 nm) s i n g l e - l a y e r s t r u c t u r e s and i s o t r o p i c w i t h zero f i e l d a p p l i e d , 2 an oxide was grown at 0.48 A/m to a maximum formation voltage of 250 v o l t s . The an o d i z a t i o n was i n t e r r u p t e d p e r i o d i c a l l y , u s u a l l y at 5 v o l t s i n t e r v a l s , and the oxide measured i n s i t u i n zones 1 and 3. Smaller voltage i n t e r v a l s were used when the balance angles changed r a p i d l y i n the region of ¥ near 90°. The anode and cathode were shorted together f o r these measurements. The zone-averaged data, c o r r e c t e d f o r the c e l l windows, are shown i n Figure 4.6. The data go through almost three complete e l l i p s o m e t e r c y c l e s and, w i t h i n an estimated measurement e r r o r of about 0.05° and 0.10° i n ¥ and A, are reproduced a f t e r only one c y c l e . Therefore, we can conclude that the present oxide f i l m i s nonabsorbing and i s o t r o p i c at zero f i e l d . The s o l i d l i n e i n the f i g u r e i s the computed e l l i p s o m e t e r curve f i t t e d to the experimental data. The model, discussed i n greater d e t a i l i n Section 3.7, i s f o r a nonabsorbing f i l m w i t h a t h i n i n t e r f a c i a l l a y e r of constant t h i c k n e s s sandwiched between the oxide and subs t r a t e . The data i n Figure 4.6 als o show no evidence of a double l a y e r s t r u c t u r e (discussed i n Chapter 3 ) , i n agreement w i t h Dell'Oca (1969). For such a s t r u c t u r e to be c o n s i s t e n t w i t h the present data, at l e a s t w i t h i n the measurement e r r o r , the d i f f e r e n c e -in r e f r a c t i v e index between the two l a y e r s would have to be l e s s than An = 0.01 as determined by comparing computed curves w i t h the data. Since t h i s model i n v o l v e s e x t r a parameters, i t was r e j e c t e d i n favour of the more simple s i n g l e - l a y e r model which gives an eq u a l l y good, i f not b e t t e r , d e s c r i p t i o n of the data. 116 O CVI O (NI CN) LO UJ LU cr o LU Q o LU Q o C N I o 4 - x o - + " ° - x - "O-i + X X I X + = FIRST CYCLE X = SECOND CYCLE O = THIRD CYCLE X X + --rO-X-20 45 PSI / DEGREES 70 F i g . 4 . 6 Z e r o - f i e l d e l l i p s o m e t e r d a t a o b t a i n e d a t i n t e r v a l s d u r i n g the c o n s t a n t c u r r e n t a n o d i z a t i o n o f Ta. The s o l i d l i n e i s t he computed cu r v e f o r the i n t e r f a c i a l f i l m . m o d e l ( s e c t i o n 3.7) 117 3.3 Dependence on the Surface F i n i s h The i n s i t u ellipsometer data were r e l a t i v e l y s e n s i t i v e to the quality of the single c r y s t a l tantalum surface. Figure 4.7 shows the two-zone data for two d i f f e r e n t samples. Both c r y s t a l s had been mechanically polished and then electropolished by presumably i d e n t i c a l procedures. Therefore, one might reasonably have expected them to give i d e n t i c a l ellipsometer r e s u l t s . Instead, the ellipsometer measurements of the two samples are s i g n i f i c a n t l y d i f f e r e n t . To show more c l e a r l y t h i s d i f f e r e n c e , a computed curve based on the i n t e r f a c i a l f i l m model discussed in Section 3.7 i s plotted for one of the samples. The differences between the data shown in t h i s figure are t y p i c a l of the differences generally observed between data for a l l samples measured in s i t u . It i s proposed in the next section that these differences are due either to varying amounts of surface roughness or to varying thicknesses of i n i t i a l "oxide" l e f t from the preparation process. 3.4 Single Layer Film Model The problem of f i t t i n g computed curves to the in s i t u ellipsometer data was simplifed by the r e s u l t s of the previous sections. In a d d i t i o n , previously reported values for the r e f r a c t i v e indices of anodic Ta^O^ and bulk tantalum (Table 4.1 ) were used as a guide i n s e l e c t i n g the o p t i c a l constants for the present f i l m s . For a l l i n s i t u measurements, the r e f r a c t i v e index of the d i l u t e H SO. e l e c t r o l y t e was taken to be n = 1.334 (Handbook of z 4 m Chemistry and Physics 1969). Since the single c r y s t a l samples used i n t h i s w o r k were cut from the same c r y s t a l as used by Cornish (1976), the o p t i c a l constants which he r e -ported were taken as a s t a r t i n g point i n f i t t i n g computed curves to the present data. Figure 4.8 shows the i n s i t u data for one sample pl o t t e d with the 118 PSI / DEGREES F i g . 4 . 7 Z e r o - f i e l d e l l i p s o m e t e r d a t a f o r two d i f f e r e n t samples. The s o l i d l i n e i s the computed curve f o r the i n t e r f a c i a l f i l m model ( s e c t i o n 3 . 7 ) . 119 computed curve f o r n^ = 2.195 and n^ = 2.46 - J2.573. Oxide f i l m t hicknesses are i n d i c a t e d f o r the f i r s t c y c l e of the computed curve. TABLE 4 .1 P r e v i o u s l y Reported Values f o r the R e f r a c t i v e Indices of Anodic T a 2 ^ and Bulk Tantalum. A/nm uc n r e f . f s 435.8 2.26 a 3.50 ± 0.11 Masing.Orme, and -j2.44 ± 0.1 Young (1961) 546.1 2.26 3.5 - j2.4 Kumagai and Young (1964) 546.1 2.22 3.30 ± 0.02 Young and Zobel -J2.30 ± 0.05 (1966) 546.1 2.21 3.30 - J2.26 Dell'Oca (1969) 546.1 2.22 3.43 - j 3 . 6 6 b Muth (1969) 590.0 2.20 ± 0.02 Young (1958) 632.8 2.185 2.46 ± 0.01 Cornish (1972) -J2.56 ± 0.01 632.8 2.20 3.02 - J2.57 Ord (1972) 632.8 2.195 ± 0.005 2.46 - j2.573 Cornish and Young (1973) 2 a) For f i l m s grown at 250 A/m and 25°C b) Sputtered tantalum ( t e t r a g o n a l ) anodized i n 0.01% c i t r i c a c i d The data are i n r e l a t i v e l y c l o s e agreement w i t h the computed curve f o r oxide thicknesses l e s s than about 50 nm but l i e i n s i d e of the curve f o r thicknesses i n the range 50 - 110 nm. As w i l l be discussed below, the d i f f e r e n c e s at the peak values of A are of p a r t i c u l a r s i g n i f i c a n c e . In p r i n c i p l e , the r e f r a c t i v e i n d i c e s of the oxide and su b s t r a t e could be adjusted to f i t a computed curve to the data i n Figure 4.8. However, t h i s would r e q u i r e values considerably d i f f e r e n t from those given i n Table 4.1, PSI / DEGREES F i g . 4 . 8 Z e r o - f i e l d e l l i p s o m e t e r d a t a f o r a t y p i c a l sample. The s o l i d l i n e was computed f o r t h e r e f r a c t i v e i n d i c e s r e p o r t e d by C o r n i s h ( 1 9 7 2 ) . I 1 1 I 15 40 65 90 PSI / DEGREES F i g . 4 . 9 The e f f e c t o f a 3 nm t h i c k i n t e r -f a c i a l f i l m on the computed e l l i p s o m e t e r curve f o r a s i n g l e l a y e r f i l m . 121 p a r t i c u l a r l y f o r the substrate r e f r a c t i v e index. Moreover, because of the d i f f e r e n c e s between samples, as discussed i n the previous s e c t i o n , t h i s f i t t i n g procedure would have to be repeated f o r each sample. That i s , each sample would have to be assigned i t s own substrate r e f r a c t i v e index. I t does not seem reasonable to expect such a s i t u a t i o n when the samples had been cut from the same c r y s t a l and, i n some cases, had simply been r e p o l i s h e d a f t e r the oxide f i l m had r e c r y s t a l l i z e d . A more l i k e l y s o l u t i o n to these curve f i t t i n g problems seems to be a model i n v o l v i n g an i n t e r f a c i a l f i l m model, as discussed i n Chapter 3. For convenience, Figure 3.4 showing the e f f e c t of a 3 nm t h i c k i n t e r f a c i a l f i l m w i t h index n^ = 1.54 - j l . 5 5 on the computed e l l i p s o m e t e r curve f o r a bulk oxide w i t h index n^ = 2.195 i s reproduced i n Figure 4.9. The important feature to note i n Figure 4.9 i s that the (4*,A) data f o r the i n t e r f a c i a l f i l m model are d i s p l a c e d a s y m e t r i c a l l y from the computed curve f o r the s i n g l e f i l m model. That i s , the curves near the peak values i n A are separated by s e v e r a l degrees i n A but superimpose almost p e r f e c t l y near the minimum values of A . In a d d i t i o n , the curve f o r the i n t e r f a c i a l f i l m model g e n e r a l l y i s on the i n s i d e of the s i n g l e f i l m curve. These e f f e c t s could not be d u p l i c a t e d by simply changing the value of the r e f r a t i v e index of the oxide f i l m . A comparison of Figures 4.7 and 4.9 shows that the main features of the present experimental data are d u p l i c a t e d by the t r a n s i t i o n f i l m model. As w i l l be shown i n the next s e c t i o n , the apparent i n t e r f a c i a l f i l m can be d i r e c t l y r e l a t e d to the surface roughness of the sample. The i n i t i a l "oxide" due to the preparation process a l s o c o n t r i b u t e s to the detected i n t e r f a c i a l f i l m (Maurel, Dieumegard and Amsel 1972). 1 2 2 3.5 Ellipsometry on Rough Surfaces In deriving the ellipsometer equations i n Chapter 3 , we i m p l i c i t l y assumed i d e a l l y smooth surfaces or inter f a c e s when, i n f a c t , a l l r e a l surfaces exhibit some degree of roughness. Several theories of the e f f e c t of a randomly rough surface on V and A have been developed (Ohlidal and Lukes 1 9 7 2 , Church and Zavada 1 9 7 4 ) . The theory of equivalent films (Fenstermaker and McCrackin 1 9 6 9 , Berreman 1 9 7 0 ) i s p a r t i c u l a r l y useful as a f i r s t approximation to modelling the e f f e c t of a rough boundary. The following i s a b r i e f o u t l i n e of t h i s theory. According to the theory of equivalent f i l m s , the roughened surface or i n t e r f a c e region can be represented by a t r a n s i t i o n layer with plane-p a r a l l e l boundaries (Figure 4 . 1 0 ) , provided the dimensions of the roughness are much l e s s than the wavelength of the incident l i g h t . The thickness of the layer i s equal to the rms height of the roughness and i t s e f f e c t i v e r e f r a c t i v e index i s given by the Garnett ( 1 9 0 4 ) theory of a discontinuous f i l m as 2 2 2 2 N - i n N - N A -f = Q(z) -f K ( 3 ) N + 2 N ~ N + 2 N ~ e A s A where, r e f e r r i n g to Figure 4 . 1 0 , N and N are the r e f r a c t i v e indices of the A S ambient and substrate r e s p e c t i v e l y . The function Q(z) i s the volume f r a c t i o n of substrate material i n the f i l m at a distance z from the i d e a l l y smooth substrate surface. In the simplest model Q(z) = constant and the t r a n s i t i o n layer i s treated as a single homogeneous i n t e r f a c i a l f i l m . I f Q(z) changes with z, the t r a n s i t i o n layer can be divided into a series of sublayers each of which i s assigned a constant value of Q and, thus, an e f f e c t i v e layer index N E > In eit h e r case, the e f f e c t of the t r a n s i t i o n layer can be taken into account when computing ¥ and A. \ 123 Imaginary F i g . 4 .11 R p and R s ( f o r n m =1.334, n f =2 .195 , n .=2 .46 - j 2 .573 , 9j=62.77 , and X =632.8 nm) p l o t t e d i n the complex p lane as a f u n c t i o n of f i l m t h i c k n e s s . 124 Fenstermaker and McCrackin (1969) used numerical methods to i n -v e s t i g a t e the e f f e c t s of a d e t e r m i n i s t i c a l l y rough surface on the apparent r e f r a c t i v e index of the s u b s t r a t e . M o d e l l i n g the rough surface by a square r i d g e , t r i a n g u l a r r i d g e , and pyramid topology, the v a r i a t i o n of the e f f e c t i v e r e f r a c t i v e index of the t r a n s i t i o n l a y e r w i t h distance from the i d e a l l y smooth surface was c a l c u l a t e d by equation 3. These r e s u l t s were then used to c a l c u l a t e V and A f o r the rough surface and thus i t s apparent r e f r a c t i v e index. As might be expected, the e f f e c t on the apparent index of the substrate increased w i t h i n c r e a s i n g surface roughness. These numerical r e s u l t s were l a t e r supported by e l l i p s o m e t r i c measurements of d i f f r a c t i o n g r a t i n g s , which are d e t e r m i n i s t i c a l l y rough surfaces. (Assam and Bashara 1972). More r e c e n t l y , Brudzewski (1979) has i n v e s t i g a t e d the problem of a s t a t i s t i c a l l y rough surface. For such a s u r f a c e , the v a r i a t i o n i n Q from the mean l e v e l of roughness i s given by Q(z) = j {1 ± e r f ( z / o / 2 ) } where o i s the standard d e v i a t i o n . Therefore, assuming the ambient-film and f i l m - s u b s t r a t e i n t e r f a c e s to be s t a t i s t i c a l l y independent i n terms of rough-ness, t h i s expression can be used to c h a r a c t e r i z e the t r a n s i t i o n l a y e r at each i n t e r f a c e . Experimental data were given f o r tungsten e l e c t r o d e s which has beed roughened i n t e n t i o n a l l y , anodized, and then measured by e l l i p s o m e t r y . The data were analyzed by d i v i d i n g the rough oxide/substrate and a i r / o x i d e i n t e r f a c e regions i n t o 200 equivalent f i l m s , w i t h Q(z) given by the above expression, and then a d j u s t i n g a to give the best f i t to the data. The optimal value of a was w i t h i n about 5% of the value obtained by Talystep measurements, f o r o up to about 12 nm. 125 3.6 S e n s i t i v i t y to an I n t e r f a c i a l F i l m The e l l i p s o m e t e r 1 s s e n s i t i v i t y to a t h i n i n t e r f a c i a l l a y e r can be determined q u a l i t a t i v e l y from the expression f o r the t o t a l r e f l e c t a n c e of a three phase system v v R r 0 1 + r 1 2 X 1 + r Q 1 r 1 2 X where X = exp - j(4Tm^d^cos0^/A), the s u b s c r i p t s 0, 1, and 2 r e f e r to the ambient, f i l m , and substrate r e s p e c t i v e l y , and v denotes p- or s- p o l a r i z a t i o n . For a nonabsorbing f i l m , when d, or A are v a r i e d , the locus of R i n the 1 v complex plane t r a c e s out a c i r c l e g e n e r a l l y o f f s e t from the o r i g i n along the r e a l a x i s (Theeten, Aspnes, and Chang 1978). The r e f l e c t a n c e c o e f f i c i e n t s R and R c a l c u l a t e d as a f u n c t i o n of f i l m t hickness f o r the case n = 1.334, p s m n f = 2.203 and n g = 2.46 - J2.573 are p l o t t e d i n Figure 4.11. For t h i s example, |R. | remains n e a r l y constant whereas the locus of R g passes q u i t e c l o s e to the o r i g i n . Since the e l l i p s o m e t e r measures the r e f l e c t a n c e r a t i o p = R /R = tan^expjA, the rapi d v a r i a t i o n i n R I near the o r i g i n gives p s 1 s' r i s e to a sharp maximum i n tanY, and thus the sharpness of the maximum increases as the locus of R g approaches c l o s e r to the o r i g i n . The i n t r o d u c t i o n of a t h i n i n t e r f a c i a l f i l m between the oxide and substrate changes the e f f e c t i v e r e f l e c t a n c e of the substrate according to the r e l a t i o n v v r -I- r Y v, ... 13 32 A3 1 + r l 3 r 3 2 X 3 where X^ = exp - j (^-^n^d^cosQ^/A) and n^ and d^ are the r e f r a c t i v e index and thickness of the i n t e r f a c i a l f i l m This expression i s s u b s t i t u t e d i n t o 126 eqn. 4 i n place of r ^ 2 V * ^ e e f f e c t of r ^ ( e f f ) i s to d i s p l a c e the c i r c l e s i n Figure 4.11, I f d^ << A, the c i r c l e s w i l l be only s l i g h t l y d i s p l a c e d . Since the locus of i s approximately centred on the o r i g i n , t h i s displacement has a n e g l i g i b l e e f f e c t on |R |. However, due to the pro x i m i t y of the R c i r c l e p s to the o r i g i n , the small displacement w i l l have a l a r g e e f f e c t on the value of | R gT^ near the o r i g i n and, t h e r e f o r e , the maximum value of tanY. This e f f e c t becomes i n c r e a s i n g l y pronounced as the locus of R g, without the i n t e r -f a c i a l f i l m present, approaches c l o s e r to the o r i g i n . That i s , the s e n s i t i -v i t y of the e l l i p s o m e t e r to an i n t e r f a c i a l f i l m increases as the maximum value of V approaches. This would e x p l a i n the s e n s i t i v i t y of the i n s i t u measurements to an i n t e r f a c i a l f i l m . Theeten et a l . (1978) have used a technique s i m i l a r to the one out-l i n e d above to c h a r a c t e r i z e the i n t e r f a c e between GaAs and i t s plasma-grown oxide. Instead of making measurements as a f u n c t i o n of oxide t h i c k n e s s , however, measurements were made w i t h a f i x e d oxide t h i c k n e s s by va r y i n g the wavelength of the i n c i d e n t l i g h t . 3.7 I n t e r f a c i a l F i l m Model Using the i n t e r f a c i a l f i l m model, computed (¥,A) curves were f i t t e d i n d i v i d u a l l y to the i n s i t u e l l i p s o m e t e r data. A s i n g l e i n t e r f a c i a l f i l m of constant thickness was assumed and the agreement between experimental and computed data was determined v i s u a l l y . The r e f r a c t i v e i n d i c e s of the f i l m and substrate were taken to be those obtained f o r the oxide f i l m s measured i n a i r . This was considered to be a reasonable assumption since the measurements i n a i r were r e l a t i v e l y i n s e n s i t i v e to the roughness of the surface. For each set of data, the r e f r a c t i v e index and th i c k n e s s of the i n t e r f a c i a l f i l m at the o O J o 20 45 PSI / DEGREES 70 F i g . 4.12 The d a t a o f f i g . 4.8 and the computed c u r v e f o r the i n t e r f a c i a l f i l m model. 128 oxide/substrate i n t e r f a c e was then adjusted to give the best f i t to the data . The r e s u l t s of the curve f i t t i n g are summarized i n Table 4.2. The range of f i l m thicknesses and r e f r a c t i v e i n d i c e s f o r the i n t e r f a c i a l i s l i k e l y due to v a r y i n g amounts of surface roughness from one sample to the next. TABLE 4.2 Summary of Results f o r the I n t e r f a c i a l F i l m Model Sample n_^  d^/nm DA071379 1.70 - j l . 5 5 2.0 DA102279 1.90 - j l . 2 0 2.0 AUG2980 2.00 - j l . 2 0 2.0 OCT1980 1.54 - j l . 4 5 2.0 OCT1680 1.54 - j l . 5 5 4.5 n f = 2.203 ± 0.005 n = 2.47 - J2.575 s J Figure 4.12 shows the data given i n Figure 4.8 p l o t t e d w i t h the computed curve f o r the i n t e r f a c i a l f i l m model w i t h index n_^  ,= 1.54 - j l . 4 5 and thickness d^ = 2.0 nm. The almost p e r f e c t agreement between the data and computed curve i s t y p i c a l of the agreement obtained w i t h other samples. The s l i g h t divergence of the computed curve from the data near the peak value of A was a l s o observed w i t h other samples. This s l i g h t disagreement i s probably due to the s i m p l i f i e d nature of the i n t e r f a c i a l f i l m model f o r surface rough-ness. More complicated models were not t r i e d i n the present a n a l y s i s but probably would have improved the agreement near these peak values. An i n t e r f a c i a l f i l m at the e l e c t r o l y t e / o x i d e i n t e r f a c e was t r i e d i n the curve f i t t i n g but had e i t h e r a n e g l i g i b l e or undesirable e f f e c t on the computed curve, depending on the values used f o r i t s r e f r a c t i v e index and t h i c k n e s s . 190 nm 25 30 35 40 PSI / DEGREES F i g . 4.13 Data f o r sample 0GT1980 measured i n a i r at several angles of incidence. The contours of constant f i l m thickness and ref r a c t i v e index were computed using the constants given i n Table 4.2. 130 2 One sample which had been anodized a t 0.64 A/m to 100 v o l t s w i t h respect to calomel was removed from the c e l l , and measured i n a i r at i n t e r v a l s of 5° i n angle of i n c i d e n c e , s t a r t i n g from 35° to a maximum of 80°. The data are p l o t t e d i n Figure 4.13 along with " l i n e s " of constant index and thickness of the oxide f i l m . These curves were computed f o r the i n t e r f a c i a l f i l m model f i t t e d to the i n s i t u data f o r the sample. The e x c e p t i o n a l agreement between the experimental data and the computed curves i s shown most c l e a r l y by the l i n e of constant oxide thickness drawn through the data p o i n t s . I t should be noted that the oxide thickness (177 nm) does not includ e the t h i c k -ness of the i n t e r f a c i a l f i l m . The data given i n Figure 4.13 were a l s o compared w i t h l i n e s of constant index and thickness computed f o r the s i n g l e f i l m model. The r e f r a c -t i v e i n d i c e s of the f i l m and substrate were taken to be the same as used i n Figure 4.13. As mentioned e a r l i e r , the l i n e s of constant index i n the range of 4* p l o t t e d i n the f i g u r e are r e l a t i v e l y i n s e n s i t i v e to the presence of an i n t e r f a c i a l f i l m . However, the l i n e s of constant thickness gave q u i t e d i s t i n c t d i f f e r e n c e s between the two models, w i t h the s i n g l e f i l m model g i v i n g a n o t i c e a b l y poorer f i t to the data than obtained w i t h the i n t e r f a c i a l f i l m model. 3.8 Comparison w i t h the Spectrophotometric Method The oxide thickness of some samples was measured by the s p e c t r o -photometric method reported by Young (1958). The wavelengths of minimum r e f l e c t i v i t y of the samples at 11° angle of incidence were determined using a Carey 14 double-beam recording spectrophotometer. A s t r i p of chemically p o l i s h e d but unanodized tantalum f o i l was used as the reference surface. The thickness of the oxide f i l m was then obtained from a chart (Figure 4.14) 400 300 -E c i 200 [ 2 100 k 0 I 1 1 1 1 300 400 500 600 A / nm F i g . 4.14 Wavelengths o f minimum r e f l e c t i v i t y at 9^ = 11° v e r s u s Ta20 f i l m t h i c k n e s s (Young 1958). F i g . 4.15 The spectrophotometer t r a c e a t h i r d order minimum i n r e f l e c t i v i t y ( i . e . maximum i n apparent a b s o r p t i o n ) . of i—* C O 132 r e l a t i n g the oxide thickness and the wavelengths of minimum r e f l e c t i v i t y . (Young 1958). Figure 4.15 shows a t y p i c a l r e c ording of the r e f l e c t i v i t y p l o t t e d against wavelength near the t h i r d order of minimum r e f l e c t i v i t y . For t h i s example, the wavelength corresponding to the minimum r e f l e c t i v i t y (or maximum absorption) can be determined to w i t h i n about 0.5 nm. Lower order minima are l e s s sharp but have about the same s e n s i t i v i t y to changes i n thickness because the wavelength of the minimum r e f l e c t i v i t y v a r i e s more r a p i d l y w i t h oxide t h i c k n e s s . The absolute accuracy of the method has been estimated to be about 0.5 nm i n oxide thickness (Young 1958). The wavelengths of minimum r e f l e c t i v i t y and the corresponding oxide thicknesses (determined from Figure 4.14 ) are given i n Table 4.3 . The oxide f i l m s were t h i c k enough to give three absorption peaks f o r each sample. The s l i g h t d i f f e r e n c e s between the oxide thicknesses determined f o r each sample are l i k e l y due to small e r r o r s i n the c a l i b r a t i o n curves given i n Figure 4.14. The thicknesses of the oxide f i l m s determined by e l l i p s o m e t r y are a l s o given i n Table 4.3. The f i r s t two samples l i s t e d i n the t a b l e were measured i n a i r and the data were f i t t e d by a s i n g l e f i l m model. The l a t t e r three samples l i s t e d i n the t a b l e were measured i n s i t u and the i n t e r f a c i a l f i l m model was f i t t e d to the data. The oxide thicknesses given i n the t a b l e f o r these three samples inc l u d e s the thickness of the i n t e r f a c i a l f i l m . The e l l i p s o m e t e r r e s u l t s f o r most of the samples are s l i g h t l y l a r g e r than those obtained by the spectrophotometric method. However, the r e s u l t s g e n e r a l l y agree w i t h i n the estimated e r r o r of the spectrophotometric method. . Young (1960) used the spectrophotometric method to measure the thickness of anodic tantalum oxide f i l m s grown over a range of steady s t a t e c o n d i t i o n s . At each formation current d e n s i t y , the oxide was grown to 133 TABLE 4.3 Oxide F i l m Thicknesses Measured by Spectrophotometry and E l l i p s o m e t r y and C a l c u l a t e d from the Steady State Data Reported by Young (1960). Sample DA060779 DAQ61179 AUG2980 OCT1980 SEP 038 0 J A/m2 0.25 2.5 0.48 0.64 0.48 Spectrophotometer Ap nm 291 362 585 283 344 547.5 271 321 493.5 285 352 567.5 287.8 353.8 565 d nm 181.4 182.9 182.9 170.4 170 170 152.3 152.6 153.1 173.7 175.7 176.8 176.8 177 176 E l l i p s o m e t e r d nm 182.5 171.2 152.7 179.5 175.4 Ca l c u l a t e d Thickness nm 181.7 170.5 152.3 177.4 175.3 voltages d i f f e r i n g by 1 v o l t which bracketed the thicknesses g i v i n g r e f l e c t i o n minima at X = 350 nm. The voltage corresponding to r e f l e c t i o n minima at 2 350 nm were found by i n t e r p o l a t i o n . An expression of the form £nJ<= aE - 3E , where the o v e r f i e l d E was determined from the f i l m t hickness and the formation v o l t a g e , was then f i t t e d to the J-E data. Working backwards, the o v e r f i e l d s p r e d i c t e d by t h i s expression were c a l c u l a t e d f o r the present work and the oxide thicknesses determined from the formation vol t a g e . The r e s u l t s are 134 given i n Table 4.3. On average, these c a l c u l a t e d thicknesses are about 0.5% l e s s than the values obtained by e l l i p s o m e t r y . This r e s u l t i s w i t h i n the 1% e r r o r estimated by Young f o r the oxide thicknesses used i n c a l c u l a t i n g the o v e r f i e l d . The apparently systematic nature of the d i f f e r e n c e s may be due, i n p a r t , to the d i f f e r e n t wavelengths used f o r the measurements. 3.9 D i f f e r e n t i a l E l e c t r i c F i e l d F igure 4.16 shows the measured oxide thickness p l o t t e d against 2 the formation voltage f o r a sample anodized at 0.64 A/m . The measurements were made at 5 v o l t i n t e r v a l s during the oxide growth. As w i l l be discussed i n Section 4.3.3, the oxide thickness does not inc l u d e the thickness of the i n t e r f a c i a l f i l m . However, t h i s assumption has no a f f e c t on the slope of the data. The data i n the f i g u r e were f i t t e d by l i n e a r r e g r e s s i o n , which gave i t s slope as 1.73 nm/V. The slope c a l c u l a t e d piecewise by f i r s t d i v i d e d d i f f e r e n c e s from the data i s p l o t t e d against formation voltage i n the top h a l f of the f i g u r e . The s t r a i g h t l i n e drown through these data corresponds to the slope obtained from the l i n e a r f i t to a l l of the data. The piecewise slopes show no evidence of i n c r e a s i n g i n value w i t h i n c r e a s i n g f i l m t h i c k n e s s , as expected according to a theory r e c e n t l y advanced by Dignam (1979), but i n s t e a d vary randomly about the average value. These r e s u l t s are t y p i c a l of the r e s u l t s obtained w i t h three other samples, two of which were anodized at d i f f e r e n t current d e n s i t i e s . The r e s u l t s f o r the d i f f e r e n t i a l f i e l d s (the i n v e r s e of the slope of the data i n Figure 4.16) are summarized i n Table 4.4. For each sample, the measured f i e l d was about 1.5% l a r g e r than the f i e l d expected according to Young's (1960) steady s t a t e J-E data. Part of t h i s d i f f e r e n c e can be a t t r i b u t e d to the 1% 135 VOLTAGE/ VOLTS F i g . 4.16 The f i l m t h i c k n e s s measured by e l l i p s o m e t r y a t i n t e r v a l s d u r i n g o x i d e growth at 0.66 A/m . The f i l m t h i c k n e s s does not i n c l u d e the t h i c k n e s s o f the i n t e r f a c i a l f i l m . 136 e r r o r i n E estimated by Young. The remaining d i f f e r e n c e may be due i n part to an e r r o r i n measuring the working area of the anode. For example, a 5% e r r o r i n the area r e s u l t s i n about a 0.2% e r r o r i n the expected f i e l d . An a d d i t i o n a l f a c t o r to keep i n mind i s the approximate nature of the i n t e r f a c i a l f i l m model used to represent the rough surface. This i s estimated to introduce an e r r o r of about 0.5% i n the f i e l d . The e r r o r would appear to be a systematic 2 ra t h e r than random, since the two samples anodized at 0.64 A/m give the same r e s u l t f o r the d i f f e r e n t i a l f i e l d . The d i f f e r e n c e s i n wavelengths used to measure the thickness of the oxide f i l m might also c o n t r i b u t e to the d i f f e r e n c e s i n E. However, t h i s e f f e c t i s probably q u i t e small i n view of the c l o s e agreement between the measured and c a l c u l a t e d f i l m thicknesses given i n the previous s e c t i o n . F i n a l l y , the surface preparation and the r e s u l t i n g i n i t i a l "oxide" f i l m might have a f f e c t e d the measured f i e l d ( S e i j k a et a l . 1971, Maurel et a l . 1972). TABLE 4.4 C u r r e n t - F i e l d Data and R e l a t i v e P e r m i t t i v i t y of Anodic Ta.,0,. Sample Formation Current A/m2 Expected F i e l d 1 " MV/m Measured F i e l d MV/m e r DA 060779 0.25 555 563 27.2 OCT1680 0.64 570 578 27.7 OCT1980 0.64 570 578 27.6 DA061179 2.5 592 600 27.7 <e > = 27.6 ± 0.2 r C a l c u l a t e d from Young's (1960) steady s t a t e data. 137 4. Small Signal Capacitance 4.1 Theory The small s i g n a l ac impedance of an e l e c t r o l y t i c c e l l can be represented by a p a r a l l e l R C combination of the oxide f i l m i n ser i e s with P P the e l e c t r o l y t e resistance Rg, provided C^ i s much less than the capacitance of the cathode/electrolyte i n t e r f a c e . This representation and i t s equivalent series representation i s shown in Figure 4.17 The two representations are related by the expressions R = R (1 - a" 2) = R a" 2 p s s C = C (1 + a 2) = C p s s where a = wC R = (wC R ) 1 = tan6 i s the loss tangent. In p r a c t i c e , tan6 < 0 s s p p ° *- » . so the above approximation i s v a l i d . 4.2 Method The small signal capacitance of some samples was measured in s i t u . To approximate c y l i n d r i c a l symmetry, a r i n g of platinum electrodes surrounding the anode formed the counter electrode for these measurements. The large capacitance of t h i s electrode also ensured that i t did not contribute s i g n i f i -cantly to the measurement of the c e l l capacitance. Some samples were measured with a General Radio (GR) model 1615-A capacitance bridge. The bridge consists of a transformer r a t i o arm with an a i r capacitor standard and i s accurate to 6 d i g i t s . Since t h i s bridge could not accurately measure the lossy films encountered with oxides grown to l e s s than "V 5 v o l t s , the c e l l capacitance for some samples was measured with an RC Wien bridge, as shown i n Figure 4.18. To minimize noise a l l wiring was done with coaxial cable. In addition, the leads were kept as short as 138 R e R s C s F i g . 4.17 R e p r e s e n t a t i o n of the a n o d i c o x i d e by p a r a l l e l R C and s e r i e s R C c o m b i n a t i o n s . R i s the e l e c t r o l y t e p p s s e J r e s i s t a n c e . F i g . 4.18 A Wien c a p a c i t a n c e b r i d g e . 139 p o s s i b l e to minimize measurement e r r o r . The c e l l was balanced against a s e r i e s RC combination by v a r y i n g R^ and R^ and keeping R^  and f i x e d . The r e s i s t a n c e s (GR decade r e s i s t a n c e boxes) were claimed by the manufacturer to be accurate to ± 0.05%. The p o l y s t y r e n e reference c a p a c i t o r was c a l i b r a t e d as C^ = 0.99998 yF using the GR bridge. Both bridges used a tuned a m p l i f i e r (GR model 1232-A) to detect the n u l l . The ac s i g n a l was about 0.1 v o l t peak-to-peak. The ensure the anode did not become cathodic during part of the ac c y c l e , the anode was u s u a l l y biased to about + 1 v o l t w i t h respect to the counter e l e c t r o d e . The GR bridge gave d i r e c t readings of the e f f e c t i v e s e r i e s capacitance and tan6 of the c e l l . The s e r i e s capacitance and r e s i s t a n c e as determined by the Wien bridge i s given by the r e l a t i o n s s 3 1 2 R = R + R = R.R„/R. s s e 3 2 1 The bridge accuracy, estimated to be about 0.1%, was checked against the GR b r i d g e . 4.3 Results and D i s c u s s i o n 4.3.1 E l e c t r o l y t e Resistance The e l e c t r o l y t e r e s i s t a n c e R^ was determined from the zero i n t e r c e p t of R g p l o t t e d against f ^ as f ^ goes to zero (Young 1961a). At i n f i n i t e l y high frequency ( i . e . f ^ = 0), of the oxide i s short c i r c u i t e d by the oxide capacitance and, t h e r e f o r e , only R^ remains i n the measured value of the r e s i s t a n c e . An example of the data obtained at 25°C i s shown i n Figure 4.19. For the c o n f i g u r a t i o n of e l e c t r o d e s used i n the measurement, R Q = 5.6 £2. The 141 e l e c t r o l y t e r e s i s t a n c e between the anode and other e l e c t r o d e s i n the c e l l should be of comparable magnitude. 4.3.2 Loss Tangent The l o s s tangent was found f o r most samples to be about tan6 = 0.009 and almost independent of frequency over the range of frequencies i n Figure 4.19. 4.3.3 R e v e r s i b l e P o t e n t i a l Several samples were grown at constant current d e n s i t y and, p e r i o d -i c a l l y , the growth was i n t e r r u p t e d and the capacitance measured at a frequency of 1 kHz. The measurements were done on the order of 2-3 minutes a f t e r p l a c i n g the c e l l on open c i r c u i t . Figure 4.20 shows a t y p i c a l example of the inverse of the capacitance p l o t t e d against the formation voltage (measured w i t h respect to calomel). The l i n e a r r e l a t i o n i s expected s i n c e , f o r steady s t a t e growth, the oxide thickness (<*C ^) increase l i n e a r l y , w i t h i n about 1%, w i t h the forma-t i o n v o l tage (Young 1960). Assuming that a l l of the e r r o r was i n measuring the capacitance, a l i n e a r r e l a t i o n was f i t t e d to the data by l i n e a r r e g r e s s i o n . The d i f f e r e n c e s between the experimental values of C 1 and the values c a l c u l a -ted from the l i n e a r r e l a t i o n are p l o t t e d i n the top h a l f of the f i g u r e . The small but apparently systematic d e v i a t i o n i s p o s s i b l y due i n part to e r r o r s i n measuring small changes i n capacitance at l a r g e formation voltages and to an increase i n the working area of the anode w i t h the formation v o l t a g e . An a d d i t i o n a l f a c t o r i s the decay of the capacitance during the p e r i o d at zero f i e l d (Young 1956). The d e v i a t i o n about the f i t t e d l i n e v a r i e d from one sample to the next. E x t r a p o l a t i n g the data i n Figure 4.20 to zero f i l m t hickness ( i . e . C 1 = 0) gives the r e v e r s i b l e p o t e n t i a l of the oxide-calomel reference 142 F i g . 4.20 The i n v e r s e o f the s m a l l s i g n a l c a p a c i t a n c e o f the o x i d e f i l m , measured a t 1 kHz, as a f u n c t i o n o f the f o r m a t i o n v o l t a g e . 143 e l e c t r o d e system as V = -1.17 v o l t s . This value i s w i t h i n 10 mV of the rev r e v e r s i b l e p o t e n t i a l c a l c u l a t e d from thermodynamic data f o r t h i s e l e c t r o d e system (-1.16 v o l t s ) . A n a l y s i s of other capacitance data gave s i m i l a r r e s u l t s f o r the r e v e r s i b l e p o t e n t i a l of the Ta^O^/calomel e l e c t r o d e combination. The c l o s e agreement w i t h the t h e o r e t i c a l r e v e r s i b l e p o t e n t i a l i n d i c a t e s that the s e r i e s capacitance of the o x i d e / e l e c t r o l y t e and c a t h o d e / e l e c t r o l y t e double -4 l a y e r s i s greater than 10 F and, t h e r e f o r e , does not c o n t r i b u t e appreciably to the measured capacitance. When the s o l u t i o n was saturated w i t h hydrogen and the p o t e n t i a l measured w i t h respect to a P t / ^ e l e c t r o d e , the r e v e r s i b l e p o t e n t i a l was found by the above method to be -0.70 ± 0.02 v o l t s averaged over s e v e r a l samples. This value i s i n c l o s e agreement w i t h the value -0.710 v o l t s f o r the standard h a l f - c e l l p o t e n t i a l w i t h respect to a standard hydrogen e l e c t r o d e (Handbook of Chemistry and Physics 1969). In p r i n c i p l e , the r e v e r s i b l e p o t e n t i a l can a l s o be determined by e x t r a p o l a t i n g the p l o t of oxide thickness vs. formation voltage to zero oxide t h i c k n e s s , as shown i n Figure 4.16. As mentioned e a r l i e r , these data do not i n c l u d e the t h i c k n e s s of the i n t e r f a c i a l f i l m . From the slope and i n t e r c e p t of the l i n e f i t t e d to the data i n Figure 4.18 by l i n e a r r e g r e s s i o n , the r e v e r s i b l e p o t e n t i a l i s c a l c u l a t e d to be -1.26 v o l t s w i t h respect to calomel. I f the l i n e a r f i t i s r e s t r i c t e d to the f i r s t 6 data p o i n t s , the r e v e r s i b l e p o t e n t i a l i s c a l c u l a t e d to be -1.16 v o l t s , i n c l o s e r agreement w i t h the r e s u l t s of the capacitance method. The values of the r e v e r s i b l e p o t e n t i a l determined by the above method f o r the other samples were g e n e r a l l y i n the range -2 to -3.5 v o l t s , depending on the reference e l e c t r o d e used. These values are considerably l a r g e r than the r e s u l t s of the capacitance method. I t should be noted that 144 the disagreement would have been greater s t i l l i f the thickness of the i n t e r f a c i a l f i l m had been included i n the oxide thickness p l o t t e d against v o l t a g e . This r e s u l t , combined w i t h the r e s u l t of S e c t i o n 3.8, suggests that the i n t e r f a c i a l f i l m i s d e t e c t a b l e o p t i c a l l y but does not c o n t r i b u t e to the oxide thickness f o r e l e c t r i c a l measurements. This would seem to be a reason-able c o n c l u s i o n since f o r a rough i n t e r f a c e the i n t e r f a c i a l f i l m s probably contains mostly tantalum metal, as suggested by t h e i r r e l a t i v e l y l a r g e absorp-t i o n c o e f f i c i e n t s . A l t e r n a t i v e l y , the i n i t i a l oxide f i l m due to the surface preparation might a l s o be expected to be h i g h l y conducting. As regards the general disagreement between the values of the r e v e r s i b l e p o t e n t i a l determined from the capacitance and thickness measurements, even a l l o w i n g f o r the t h i c k -ness of the i n t e r f a c i a l f i l m , the s i n g l e l a y e r i n t e r f a c i a l f i l m model i s only an approximation to the rough surface. 4.3.4 Oxide P e r m i t t i v i t y Figure 4.21 shows the inverse capacitance data of Figure 4.20 p l o t -ted against the measured oxide thickness d. As expected from the constancy of the o x i d a t i o n f i e l d , C 1 i s l i n e a r l y dependent on the oxide t h i c k n e s s , w i t h the r e l a t i o n given by C - 1 = d/(e e A) o r where A i s the sample area and the p e r m i t t i v i t y of f r e e space. The r e l a t i v e p e r m i t t i v i t y e of the oxide f i l m , deduced from the slope of the l i n e a r r e l a t i o n f i t t e d to the data, has the value = 27.6 f o r t h i s sample. Note that i n c l u d i n g the t h i c k n e s s of the i n t e r f a c i a l l a y e r i n the thickness of the oxide only s h i f t s the p l o t of C 1 vs. d but has no a f f e c t on the slope of the graph and, t h e r e f o r e , the above value of E^. 145 0 50 100 THICKNESS / n m 150 F i g . 4.21 The c a p a c i t a n c e d a t a of f i g . 4.20 p l o t t e d a g a i n s t the o x i d e f i l m t h i c k n e s s measured by e l l i p s o m e t r y 146 A l s o p l o t t e d i n Figure 4.21 are the values of c a l c u l a t e d piecewise from the f i r s t d i v i d e d d i f f e r e n c e s of the data. These r e s u l t s roughly match the d e v i a t i o n of the C data about the l i n e a r r e l a t i o n p l o t t e d i n Figure 4.20, and, t h e r e f o r e , are probably due to systematic e r r o r s i n measuring C. The r e l a t i v e p e r m i t t i v i t y of three other samples was determined by the method o u t l i n e d above. The r e s u l t s are summarized i n Table 4.4. An e r r o r of only about 2% i n measuring the working area of the samples would account f o r the disagreement between the values of e f o r sample DA060779 and the other three samples. In a d d i t i o n , there does not appear to be any depen-dence of e on the formation current d e n s i t y , at l e a s t w i t h i n t h i s e r r o r . r J The average value of the r e s u l t s i n Table 4.4 i s < e r > = 27.6, w i t h an estimated d e v i a t i o n of ± 0.2. This r e s u l t f o r happens to be i n p e r f e c t agreement w i t h the value = 27.6 ± 5% reported by Young (1958) f o r 2 f i l m s grown at 100 A/m i n 0.1 M U^SO^ and measured at 1 kHz. The present r e s u l t i s a l s o i n c l o s e agreement w i t h the values of reported f o r anodic Ta 90^ on deposited Ta f i l m s (Muth 1969, Croset and Velasco 1971). 147 V. THE OPEN CIRCUIT TRANSIENT 1. I n t r o d u c t i o n When an el e c t r o d e of tantalum (or a s i m i l a r oxide f i l m - f o r m i n g metal) i s a n o d i c a l l y p o l a r i z e d at constant c u r r e n t , and i s then placed on open c i r c u i t , the s e l f - d i s c h a r g e of the c a p a c i t o r formed by the m e t a l / o x i d e / e l e c t r o -l y t e i s due, at l e a s t i n i t i a l l y , to i o n i c conduction i n the oxide. I f i t i s assumed that C, the i n t e g r a l capacitance per u n i t area, i s constant during the discharge, then the i o n i c current d e n s i t y J can be obtained as a f u n c t i o n of the o v e r f i e l d E i n the oxide from J = - 4 r [C(V-V ) ] = -CdV/dt = - edE/dt (1) dt rev where V i s the measured p o t e n t i a l , V i s the r e v e r s i b l e p o t e n t i a l of the rev anode/reference e l e c t r o d e system, and e i s the p e r m i t t i v i t y of the oxide. Methods based on t h i s p r i n c i p l e have been widely used by Ord et a l . and o t h e r s T under the name of "the open c i r c u i t t r a n s i e n t " technique. In the s o - c a l l e d f i r s t - o r d e r a n a l y s i s , a l i n e a r l o g J-E dependence and a constant C are assumed and the data are f i t t e d to an int e g r a t e d expression f o r the time dependence of E. The p e r m i t t i v i t y of the oxide i s then deduced from the f i t t i n g parameters. More r e c e n t l y , l i n e a r (Ord, Clayton, and Wang 1977) and non l i n e a r (Ord 1980) dependences of C on E have been assumed i n order to e x p l a i n the apparent n o n - l i n e a r i t y of the lo g J-E r e l a t i o n . These methods of a n a l y s i s , which are discussed i n greater d e t a i l i n a l a t e r s e c t i o n of t h i s chapter are perhaps somewhat misleading i n that they depend on the assumed J(E) and C(E) r e l a t i o n s . An a l t e r n a t i v e method, r e q u i r i n g assumptions only about the C(E) r e l a t i o n , i s to n u m e r i c a l l y d i f f e r e n t i a t e the measured E(t) +See f o r example: Ord and DeSmet 1965; Ord, Hopper and Wang 1972; Ord and Lushiku 1979. 148 data to obtain d i r e c t l y the dependence of J on E. To our knowledge, t h i s method of a n a l y s i s has not been used p r e v i o u s l y . The purpose of the present work i s to i n v e s t i g a t e the open c i r c u i t t r a n s i e n t as a method of studying the k i n e t i c s of i o n i c conduction. To t h i s end, the r e s u l t s of the f i r s t - o r d e r and numerical d i f f e r e n t i a t i o n methods of a n a l y s i s are compared with each other and w i t h p r e v i o u s l y reported data obtained by other methods. 2. Experimental Procedure 2.1 Sample Preparation The samples were cut i n the shape of paddles from 99.999% p u r i t y _3 tantalum sheet (1.3 x 10 m t h i c k ) obtained from F a n s t e e l . A f t e r degreasing by the method o u t l i n e d i n Chapter 4, the samples were c h e m i c a l l y p o l i s h e d i n a s o l u t i o n of 5:2:2 p a r t s of volume ^SO^, 98% HNO^ and 48% HF (Tegart 1959). The heat of r e a c t i o n when mixing these a c i d s was s u f f i c i e n t to b r i n g the mixture to the de s i r e d temperature f o r p o l i s h i n g . The samples were held f o r about 10-15 seconds i n the p o l i s h i n g s o l u t i o n and then r i n s e d i n deionized water. The f i l m l e f t by the p o l i s h i n g s o l u t i o n was leached by immersing the sample i n b o i l i n g water f o r s e v e r a l minutes (Young 1961a). The working area of the samples was defined by the method o u t l i n e d i n the previous chapter. 2.2 Oxide Formation The a n o d i z a t i o n c e l l i s described i n the previous chapter. In some experiments the an o d i z a t i o n s o l u t i o n (0.1 M H SO.) was saturated w i t h 2 4 hydrogen and the anode voltage measured w i t h respect to a p l a t i n i z e d platinum e l e c t r o d e l o c a t e d adjacent to the gas bubbling tube. The s o l u t i o n was taken to be saturated when the vo l t a g e between two platinum e l e c t r o d e s separated 149 by the c e l l diameter was l e s s than 1 mV. In a l l other experiments, the anode vo l t a g e was measured w i t h respect to a saturated calomel reference e l e c t r o d e . The c e l l s o l u t i o n was s t i r r e d c o n t i n u o u s l y during the experiments and the temperature was thermostated to w i t h i n 0.05°C. 2.3 Voltage Measurements As shown sc h e m a t i c a l l y i n Figure 5.1, the anode voltage was measured w i t h a DANA model 5100 d i g i t a l voltmeter which i s a l s o i n t e r f a c e d to the PDP8 computer. This voltmeter has v o l t a g e ranges w i t h f u l l s c a l e v o l t a g e s of 0.16, 1.6, 160, and 1200 v o l t s . On the l a t t e r two ranges, the input r e s i s t a n c e , as s p e c i f i e d by the manufacturer, i s l O ^ f i . The r e s i s t a n c e 9 i s greater than 10 ft on the lower three ranges. Therefore, to minimize the e f f e c t of the voltmeter on the discharge of the anode, the open c i r c u i t t r a n s i e n t s were recorded w i t h the voltmeter set to one of i t s low v o l t a g e -high input r e s i s t a n c e ranges unless stated otherwise. The measurement range of the voltmeter was extended beyond 16 v o l t s by backing o f f the anode vo l t a g e w i t h a b a t t e r y . The voltmeter uses the method of dual slope i n t e g r a t i o n to d i g i t i z e the input v o l t a g e . In normal o p e r a t i o n , the voltmeter has a r e s o l u t i o n of 5^5 d i g i t s w i t h a conversion time of ^50 msec. In a s o - c a l l e d f a s t mode, the conversion time i s reduced to ^8 msec but at the expense of l o s i n g 1 d i g i t of r e s o l u t i o n . For both conversion modes, the v o l t a g e sampling time i s about 1/5 of the t o t a l conversion time. The voltmeter was c a l i b r a t e d on the lower voltage ranges against a standard c e l l . A c a l i b r a t e d dc d i f f e r e n t i a l voltmeter and a high v o l t a g e power supply were used to c a l i b r a t e the two high-voltage ranges. 150 45 V TO PDP8/E COMPUTER F i g . 5.1 Apparatus f o r c o n s t a n t c u r r e n t a n o d i z a t i o n and measurement of the anode v o l t a g e . ^ 6 0 - — 40 — CLOCK PULSE — 8 DATA READY F i g . 5.2 T i m i n g diagram f o r the 10 kHz c r y s t a l c l o c k . The times a r e g i v e n i n mic r o s e c o n d s . 151 The voltmeter provides as one output a 0-16 v o l t s i g n a l p r o p o r t i o n a l to the measured v o l t a g e . This permitted the voltmeter to be used as a high impedance b u f f e r between the anode and the r e l a t i v e l y low impedance 10- b i t A/D converter which was a l s o i n t e r f a c e d to the computer.^ The 10-usec conversion time of t h i s converter made i t p o s s i b l e to record f a s t open c i r c u i t t r a n s i e n t s . The converter's f u l l - s c a l e v o l t a g e i s 10 v o l t s . The analog output from the voltmeter i s referenced to the negative side of the input v o l t a g e whereas the input to the A/D converter i s referenced to ground. This presented no problem when measuring the anode vo l t a g e w i t h respect to a P t / ^ e l e c t r o d e since t h i s reference e l e c t r o d e was w i t h i n a m i l l i v o l t of the cathode, which was at ground p o t e n t i a l . However, the p o t e n t i a l of the calomel reference e l e c t r o d e was g e n e r a l l y about -0.2 v o l t s w i t h respect to the cathode. Therefore, to avoid s h o r t i n g t h i s e l e c t r o d e to the cathode, the output from the voltmeter was f u r t h e r buffered by a d i f f e r e n t i a l a m p l i f i e r . 2.4 Time Measurements A 10 kHz c r y s t a l c l o c k was designed, b u i l t , and i n t e r f a c e d to the PDP8/E computer f o r the present work. The 10 kHz c l o c k r a t e i s derived from a 1 MHz quartz c r y s t a l o s c i l l a t o r . The timing diagram i s shown i n Figure 5.2. The number of c l o c k periods i s counted by a 2 4 - b i t binary r i p p l e - c o u n t e r , p e r m i t t i n g an elapsed time of about 28 minutes before the counter "wraps around" and s t a r t s counting from zero again. The c l o c k r a t e was c a l i b r a t e d w i t h a frequency counter which had p r e v i o u s l y been c a l i b r a t e d to' 1 part i n 10^ w i t h a temperature-controlled ^The A/D converter i s connected to 1 of 4 inputs through a 4 channel m u l t i p l e x e r operated under computer c o n t r o l . 152 time mark generator. The c l o c k frequency was found to be 9.9995 kHz, g i v i n g a c l o c k p e r i o d of 100.005 usee. 2.5 Synchronization Between Time and Voltage Measurements For a l l computer-controlled experiments i n v o l v i n g voltage-time measurements, the c o n t r o l programs were w r i t t e n so that the computer waited f o r the c l o c k word to be updated and then immediately issued a s t a r t - c o n v e r s i o n command to e i t h e r the voltmeter or A/D converter. Therefore, although the clo c k period was 100 usee, the time of the voltage measurements was, under worst case c o n d i t i o n s , accurate to w i t h i n ^10 usee. The u n c e r t a i n t y i s due to the time-width of the "Data Ready" window and to the c y c l e time of the computer. On average, the time i s probably accurate to w i t h i n 5 usee. 2.6 Data A c q u i s i t i o n The PDP8/E'computer c o n t r o l l e d both the steady s t a t e a n o d i z a t i o n and the data a c q u i s i t i o n during the open c i r c u i t t r a n s i e n t . The experiment c o n t r o l program was w r i t t e n l a r g e l y i n the Fortran I I programming language obtained from DEC. Some sec t i o n s o f the program r e q u i r i n g s p e c i a l I/O c o n t r o l or greater speed than p o s s i b l e w i t h the higher l e v e l language, were w r i t t e n i n the PDP8 Assembler language. The computer c o n f i g u r a t i o n i s described i n Appendix D. The computer i n i t i a t e d the ano d i z a t i o n by opening a reed r e l a y across the output of the current source and simultaneously c l o s i n g another reed r e l a y connecting the current source and anode. These r e l a y s are designated R0 and RI r e s p e c t i v e l y i n Figure 5.1. During the constant current growth, the time and voltage (measured by the DANA voltmeter) were c o n t i n u a l l y recorded by the computer and the c a l c u l a t e d value of AV/At was p l o t t e d against time on a Houston DP-10 incremental p l o t t e r . The voltmeter was operated i n 153 i t s h i g h - r e s o l u t i o n mode f o r these measurements. A constant value of AV/At was taken to i n d i c a t e low leakage steady s t a t e growth; otherwise, the sample was r e j e c t e d . When the anode voltage reached a pre-set v a l u e , r e l a y R l was opened, the c l o c k reset to zero, and, during the subsequent s e l f - d i s c h a r g e of the anod between 100 and 300 voltage-time data were recorded. I f the i n i t i a l r a t e of discharge was s u f f i c i e n t l y s m a l l , the voltage was recorded by the voltmeter operating i n i t s f a s t (lower r e s o l u t i o n ) mode. Otherwise, the A/D converter was used. To avoid p o s s i b l e e f f e c t s due to e l e c t r o n i c current at low f i e l d s , only about the f i r s t 1.5 seconds of the t r a n s i e n t was recorded. At the end of the t r a n s i e n t , r e l a y RO was closed i n prep a r a t i o n f o r another set of measurements. A l l time data were correct e d f o r the de-energizing time of r e l a y R l (0.4 msec). In a d d i t i o n , the c e l l voltage at the i n s t a n t the r e l a y con-t a c t s opened was c a l c u l a t e d by e x t r a p o l a t i n g from the f i n a l steady-state voltage recorded, a l l o w i n g f o r the time delays due to A/D conversion i n the voltmeter and de-energizing of r e l a y R l . The voltage-time data were then stored i n f i l e s on DECTAPE or the RL01 d i s k f o r l a t e r a n a l y s i s . 3. Equipment Tests and Experimental Simulation 3.1 Voltmeter Response The a b i l i t y of the voltmeter to measure a c c u r a t e l y a r a p i d l y changing voltage was i n v e s t i g a t e d i n the f o l l o w i n g way. A computer-triggered + T e s t measurements showed that the contacts of the reed r e l a y s "bounced" f o r about 100 ysec a f t e r making i n i t i a l contact on e n e r g i z i n g . On de-energizing, the r e l a y contact was broken c l e a n l y . 154 ramp waveform from a s i g n a l generator was input to both the voltmeter and A/D converter. To simulate worst-case c o n d i t i o n s , the peak-to-peak voltage swing was set to 10 v o l t s . The voltage was then measured a l t e r n a t e l y by the A/D converter and then by the voltmeter. The measured voltages were compared by p l o t t i n g them against the time of measurements. With the voltmeter operating i n i t s f a s t conversion mode, the waveforms recorded by the voltmeter and A/D converter were i n c l o s e agreement f o r slew rates up to about 200 V/sec (as discussed i n the next s e c t i o n , the A/D converter can a c c u r a t e l y measure much f a s t e r changing v o l t a g e s ) . Because of the r e l a t i v e l y long conversion time of the voltmeter (^ 8 msec i n the f a s t mode), an i n s u f f i c i e n t number of data were recorded to draw any conclusions regarding the accuracy at l a r g e r slew r a t e s . 3.2 A/D Converter Response The e f f e c t of the d i f f e r e n t i a l a m p l i f i e r (shown s c h e m a t i c a l l y i n Figure 5.1), 4-channel m u l t i p l e x , and input e l e c t r o n i c s a s s o c i a t e d w i t h the A/D converter was i n v e s t i g a t e d by i n p u t t i n g a s i n g l e - s h o t ramp waveform d i r e c t l y to the d i f f e r e n t i a l a m p l i f i e r and observing on an o s c i l l o s c o p e the s i g n a l at the input to the A/D converter. To simulate worst case c o n d i t i o n s , the peak-to-peak voltage swing was set to 10 v o l t s . The slew r a t e of the ramp waveform was then increased u n t i l the o s c i l l o s c o p e t r a c e was d i s t o r t e d . Under these c o n d i t i o n s , a slew ra t e greater than 10^V/sec was required before the trace was n o t i c e a b l y d i s t o r t e d . The e f f e c t of the voltmeter on the s i g n a l measured by the 10- b i t A/D converter was a l s o i n v e s t i g a t e d . A computer-triggered t r i a n g u l a r wave-form from a s i g n a l generator was input to the voltmeter and i t s analog output measured w i t h the converter. The waveform was al s o input d i r e c t l y to the converter through a 4-channel m u l t i p l e x e r operated under computer c o n t r o l . 155 The voltage swing of the waveform was set to 0-10 v o l t s to simulate worst-case c o n d i t i o n s . Except f o r a time delay of about 0.1 msec between the two s i g n a l s , the voltmeter-buffered and d i r e c t s i g n a l s as measured by the A/D converter were i d e n t i c a l f o r ra t e s of change of voltage up to about 7000 V/sec. This 3 compares w i t h a maximum slew r a t e of ^10 V/sec expected f o r the open c i r c u i t t r a n s i e n t . I t should a l s o be noted that f o r a s i g n a l changing at a r a t e of 3 10 V/sec, the s i g n a l changes by 10 mV during the A/D conversion p e r i o d . Since t h i s value equals the estimated 10 mV e r r o r due to n o n - l i n e a r i t y , the s i g n a l can be t r e a t e d as being constant during the conversion p e r i o d of the 10-b i t A/D converter. The A/D converter was a l s o c a l i b r a t e d against a c a l i b r a t e d dc d i f f e r e n t i a l voltmeter. The voltage as measured by the converter was w i t h i n 10 mV of the value measured by the voltmeter. 3.3 Experimental Simulation To determine that the experimental equipment and computer c o n t r o l program were working p r o p e r l y , the open c i r c u i t t r a n s i e n t was simulated by charging a c a p a c i t o r at constant current and then a l l o w i n g i t to discharge through a r e s i s t o r . The voltage across the c a p a c i t o r was measured w i t h the DANA voltmeter operated on both low and high impedance ranges. A 1.005 yF mylar c a p a c i t o r ( c a l i b r a t e d at 1 kHz w i t h the GR model 1615 capacitance bridge) and a 250 K.Q ± 0.05% GR r e s i s t o r were used i n the s i m u l a t i o n c i r c u i t . These values were chosen so that the discharge time was comparable to that expected f o r the open c i r c u i t t r a n s i e n t . The c a p a c i t o r was charged at 10 ^A. The voltage-time data f o r the discharge were analyzed by two methods. F i r s t , the data were f i t t e d by l i n e a r r e g r e s s i o n to the V(t) r e l a t i o n p r e d i c t e d by c i r c u i t theory 0.001 < -0.001 1.5 b o > g> 1.0 0.5 0 0.2 0.4 0.6 TIME / SECONDS F i g . 5.3 V ( t ) f o r a 1 pF c a p a c i t o r d i s c h a r g i n g t h r o u g h a 250 k A r e s i s t o r . 0.2 0.4 0.6 TIME / SECONDS F i g . 5.4 The d e r i v a t i v e (dV/dt) o f the d a t a p l o t t e d i n f i g . 5.3. 1 5 7 £n[V(t)/volt] = An[V(o)/volt] - t / x ( 2 ) where the time constant T=RC and R i s due to a l l resistances, including the input resistance of the voltmeter, in p a r a l l e l with the capacitor C. As a check on the r e s u l t s of t h i s analysis and also the accuracy of the routine used to numerically d i f f e r e n t i a t e the open c i r c u i t transient data, the V(t) data were also numerically d i f f e r e n t i a t e d and then f i t t e d to the r e l a t i o n £n[-dV/dt/Vsec - 1] = Jtn[V(o)/T/vsec - 1] - t / x (3) Typical r e s u l t s for the l i n e a r r e l a t i o n s predicted by eqns. 2 and 3 are shown in Figures 5.3 and 5.4. Both methods gave the same value of T( = 0 . 2 5 sec) to within 1%. The difference between the data and the r e l a t i o n f i t t e d by l i n e a r regression (plotted at the top of the figures) i s small and random. Due to quantization errors and e l e c t r i c a l noise, the difference increases s l i g h t l y with increasing discharge time. The e f f e c t of the voltmeter's input resistance on the rate of d i s -charge of the capacitor was c l e a r l y indicated by the r e s u l t s of the analyses. Assuming the capacitor to be i d e a l , the t o t a l p a r a l l e l resistance was c a l -culated from the value of x to be R = 2 4 9 . 9 kfi when the voltmeter was P operated on a low voltage-high input resistance range. Further assuming that the discharge r e s i s t o r i s exactly 2 5 0 kfi, the voltmeter's input resistance 9 (in p a r a l l e l with the discharge r e s i s t o r ) was calculated to be ^10 fi, i n agreement with the manufacturer's s p e c i f i c a t i o n s . The e f f e c t of the voltmeter i s much more dramatic on the high-voltage-low impedance ranges. For measurements made with the voltmeter operated on these ranges, the calculated p a r a l l e l resistance was reduced to R = 2 4 4 kfi, P giving, by the above method, the voltmeter input resistance as VLO ft. This value again i s in agreement with the manufacturer's s p e c i f i c a t i o n s . 158 Given t h a t the input r e s i s t a n c e of the voltmeter i s known, i t s e f f e c t on the s e l f - d i s c h a r g i n g i n an o p e n - c i r c u i t t r a n s i e n t g e n e r a l l y can be c o r r e c t e d . However, as w i l l be discussed l a t e r , the f i r s t - o r d e r a n a l y s i s of the data i m p l i c i t l y assumes the voltmeter has no e f f e c t on the measurements. Therefore, most open c i r c u i t t r a n s i e n t s were recorded w i t h the voltmeter on a low v o l t a g e - h i g h input r e s i s t a n c e range. An a d d i t i o n a l advantage to using the high input r e s i s t a n c e ranges i s that the current f l o w i n g through the reference e l e c t r o d e i s kept n e g l i g i b l y s m a l l . Thus, the p o l a r i z a t i o n of the e l e c t r o d e would be expected to remain n e a r l y constant during the t r a n s i e n t , thereby g i v i n g a constant reference p o t e n t i a l . 4. Results and Data A n a l y s i s Open c i r c u i t t r a n s i e n t s were recorded at temperatures of 1, 25, 50, 2 and 75°C f o r i n i t i a l steady s t a t e current d e n s i t i e s i n the range 0.25 - 30 A/m , At each formation current and temperature, t r a n s i e n t s were recorded at s u c c e s s i v e l y higher formation voltages u s u a l l y to a maximum of about 30 v o l t s . The steady s t a t e f i e l d E g at each was c a l c u l a t e d from Young's (1960) steady s t a t e data f o r tantalum. T y p i c a l voltage-time data recorded w i t h the voltmeter f o r the open 2 c i r c u i t discharge are shown i n Figure 5.5. The oxide was grown at 0.25 A/m to 15 v o l t s . The V(t) data have an almost e x p o n e n t i a l dependence, as would be expected f o r a c a p a c i t o r d i s c h a r g i n g through a r e s i s t o r . However, the voltage dependence of the e f f e c t i v e p a r a l l e l r e s i s t a n c e of the oxide r e s u l t s i n a more complex V(t) dependence than a simple e x p o n e n t i a l . The i n i t i a l r a t e of change of the V(t) data increases w i t h i n i t i a l 2 steady s t a t e J . For oxide f i l m s grown at greater than about 0.5 A/m , t h i s 0.2 0.4 0.6 TIME / SECONDS 5.5 5.0 E > O 4.0 o 1 i — 1 T = 25° C A = 2 V + J s = 0.25 Am"2 + = 6 • o = 10 • = 12 O + • + ' -• A O o • 1 1 0 0.3 0.6 0.9 TIME/SECONDS F i g . 5 . 5 T y p i c a l open c i r c u i t t r a n s i e n t F i g . 5 . 6 E ( t ) data f o r oxide f i l m s grown V ( t ) d a t a f o r an o x i d e f i l m grown t o 1 5 V. under the same c o n d i t i o n s as f o r f i g . 5 . 5 160 i n i t i a l r a t e of change was too l a r g e f o r the voltmeter to record a c c u r a t e l y much of the i n i t i a l part of the t r a n s i e n t . These " f a s t " t r a n s i e n t s were recorded w i t h the 1 0 - b i t A/D converter. The increase i n t h i c k n e s s of the oxide due to i o n i c conduction during _2 the s e l f - d i s c h a r g e was c a l c u l a t e d to be l e s s than 10 nm. Therefore, i t was considered reasonable to assume a constant oxide thickness D that could be c a l c u l a t e d from the formation voltage and the i n i t i a l steady s t a t e f i e l d E by the r e l a t i o n s J D = (V - V )/E f rev s The agreement between the c a l c u l a t e d value of oxide thickness and the t h i c k -ness measured by e l l i p s o m e t r y and spectrophotometry i s discussed i n the previous chapter. The c a l c u l a t e d thickness of the oxide was used to convert the V(t) data, as shown i n Figure 5.5, to E ( t ) data. This procedure e f f e c t i v e l y normalized the open c i r c u i t t r a n s i e n t data and thus made p o s s i b l e a d i r e c t comparison of data obtained at d i f f e r e n t formation voltages., Figure 5.6 shows t y p i c a l E ( t ) data obtained at 25°C f o r an i n i t i a l steady s t a t e current 2 d e n s i t y J g = 0.25 A/m and formation voltages V = 2, 6, 10, and 12 v o l t s . For t h i s example, the anode p o t e n t i a l was measured with respect to a Pt/H^ e l e c t r o d e . These r e s u l t s show that the s e l f - d i s c h a r g e i s , w i t h i n an estimated experimental e r r o r of l e s s than 0.1%, independent of oxide thickness f o r oxide f i l m s as t h i n as 5 nm. This would seem to i n d i c a t e that space charge e f f e c t s , which presumably would be g r e a t e s t w i t h very t h i n f i l m s , are n e g l i g i b l y small w i t h the present oxide f i l m s . ) 161 4.1 Oxide P e r m i t t i v i t y The p e r m i t t i v i t y of the oxide was deduced from the i n i t i a l stages of the open c i r c u i t t r a n s i e n t by the r e l a t i o n where i s the steady s t a t e current d e n s i t y and e i s assumed to be i n -:awn dependent of E. The value of ( d E / d t ) t _ 0 was c a l c u l a t e d from the l i n e drs through the f i r s t few data p o i n t s of the t r a n s i e n t , as shown i n Figure 5.7. The values of the r e l a t i v e p e r m i t t i v i t y f o r the s e v e r a l samples measured g e n e r a l l y were i n the range 29-32. These values are about 10-15% greater than the value determined from the small s i g n a l capacitance measured at 1 kHz. This d i f f e r e n c e could be due to e r r o r s i n determining the i n i t i a l slope of the E ( t ) data. 4.2 F i r s t - O r d e r A n a l y s i s The f i r s t - o r d e r a n a l y s i s of open c i r c u i t t r a n s i e n t s i s based on assuming a constant p e r m i t t i v i t y e f o r the oxide and a l i n e a r l o g J-E r e l a t i o n given by J = J e x p ( a E ) + (4) o where J and a are constants. S u b s t i t u t i n g (4) i n t o the d i f f e r e n t i a l o equation which describes the s e l f - d i s c h a r g e J = - edE/dt (5) In most published work on the open c i r c u i t t r a n s i e n t , the i o n i c current d e n s i t y i s w r i t t e n J = J D e x p ( v / v G ) where V i s the o v e r p o t e n t i a l and V Q i s taken to be a constant during the t r a n s i e n t . This i s equivalent to eqn. 4 w i t h V D = D/cx where D i s the oxide t h i c k n e s s . Neither form commits one to a model i n v o l v i n g bulk as opposed to i n t e r f a c e c o n t r o l of the current, 162 0 0.04 0.08 0.12 TIME / S E C O N D S F i g . 5.7 V ( t ) data f o r the i n i t i a l stages of an open c i r c u i t s e l f - d i s c h a r g e . 163 (assuming only i o n i c current) and integrating gives an a n a l y t i c expression for the time depedence of E E(t) = E(t=o) - a " 1 £n(l + t/x) (6) where T = e/aJ and J i s the i n i t i a l steady state current, s s The analysis of the open c i r c u i t transient data proceeds as follows. A value for x i s guessed and the time data are converted to £n(l + t/x). Since the error in t, and thus 2n(l + t/x) i s n e g l i g i b l y small compared to the error in E, l i n e a r regression in E i s used to f i t a l i n e a r r e l a t i o n to the E. vs 1 £n(l + t^/x) data. New values for x are selected and the above procedure repeated u n t i l the penalty function N 2 J-f = 1/N[ Z (E - a - a £n(l + t ^ x ) ) ] 2 (7) 1=1 i s minimized. The values for E(t=o) and a are then calculated from the values of the l i n e a r f i t parameters a and a, . The value of e i s determined by the o 1 . J values of a^ and x. It should be noted that t h i s analysis does not constrain E(t=o) to equal E , the steady state f i e l d . However, the value of E(t=o) obtained from the optimally f i t t e d l i n e a r r e l a t i o n generally agreed with the experimental value to better than 0.5%. It i s also assumed i m p l i c i t l y that the voltmeter used to measure the anode voltage (and thus the f i e l d ) has an i n f i n i t e l y large input resistance; the f i r s t - o r d e r method of analysis does not allow corrections to be made for the current discharge through the voltmeter. A program was written for the PDP8/E computer to analyze the open c i r c u i t data by the above method. The minimum value of the penalty function and the s e n s i t i v i t y of the penalty function to small changes in x increased with the i n i t i a l steady state 164 current d e n s i t y . In a l l cases, the minimum i n the penalty f u n c t i o n was determined to w i t h i n 0.1%. The value of a was much l e s s s e n s i t i v e to small changes i n x . Therefore, the u n c e r t a i n t y i n the optimum value of a. i s l a r g e r than i n the optimum value of x . As mentioned above, the oxide p e r m i t t i v i t y e may be c a l c u l a t e d from a and x but t h e i r combined u n c e r t a i n t y makes the e r r o r i n e at l e a s t 5-10%. Figure 5.8 shows the data of Figure 5.6 f i t t e d to the l i n e a r r e l a t i o n given by (6). The data i n the f i g u r e are approximately e q u a l l y spaced i n f i e l d so that the a n a l y s i s i s weighted e q u a l l y over the e n t i r e range of the data. The d i f f e r e n c e between the f i e l d data and the l i n e a r r e -l a t i o n p r e d i c t e d by eqn. 6 i s p l o t t e d at the top of the f i g u r e . The systema-t i c d e v i a t i o n of the data about the f i t t e d r e l a t i o n i s t y p i c a l of a l l open c i r c u i t t r a n s i e n t data analyzed by the f i r s t - o r d e r method and forms the b a s i s f o r the s o - c a l l e d second-order a n a l y s i s which w i l l be discussed i n the next s e c t i o n . For the example shown i n Figure 5.8, the maximum d e v i a t i o n i s l e s s than 0.1% of the o x i d a t i o n f i e l d . The maximum percent d e v i a t i o n increased w i t h the formation current d e n s i t y . For comparison, Figure 5.9 shows the f i t t e d open c i r c u i t data f o r an oxide f i l m grown at 3 A/m and 25°C. Increas-ing the formation current d e n s i t y by about a f a c t o r of 10 al s o increased the maximum percent d e v i a t i o n (shown at the top of Figure 5.9 ) by a s i m i l a r amount i' The r e s u l t s of the f i r s t - o r d e r a n a l y s i s depend on the extent of the t r a n s i e n t included i n the a n a l y s i s . For example, the data i n Figure 5.6 extend to about 0.9 sec a f t e r p l a c i n g the c e l l on open c i r c u i t . F i t t i n g a l i n e a r r e l a t i o n to the e n t i r e range of data gave the r e l a t i v e p e r m i t t i v i t y of the oxide as 26.4. R e s t r i c t i n g the a n a l y s i s to shorter periods of elapsed time increased the values of x and e and decreased the value of a. F i g . 5 . 8 F i r s t - o r d e r a n a l y s i s of the E ( t ) d a t a p l o t t e d i n f i g . 5 . 6 . F i g . 5 . 9 F i r s t - o r d e r a n a l y s i s of E ( t ) d a t a . F i g . 5.10 V a r i a t i o n of the optimum v a l u e o f T w i t h the time range o f the f i r s t -o r d e r a n a l y s i s . I i i i I 0 0.2 OA 0.6 0.8 Tmax / SECONDS F i g . 5 . 1 1 V a r i a t i o n o f a w i t h the time range of the f i r s t - o r d e r a n a l y s i s . 167 F i g . 5.12 The v a r i a t i o n of € r w i t h the time range of the f i r s t -o r d e r a n a l y s i s . 168 Figures 5.10, 11, and 12 show the v a r i a t i o n of T , a, and e w i t h the time range 2 of the f i r s t - o r d e r a n a l y s i s f o r oxide f i l m s grown at 0.33 A/m and 25°C. The dependence on the time range increased r a p i d l y w i t h the steady s t a t e formation current d e n s i t y . A minimum of 8 data points was used i n each of the f i r s t -order analyses. The r e l a t i v e p e r m i t t i v i t y of the oxide f i l m s g e n e r a l l y e x t r a p o l a t e d to a value i n the range 26-30 as the time range was reduced to zero. The u n c e r t a i n t y i n the r e s u l t s f o r e was too l a r g e to e s t a b l i s h any dependence on temperature or formation current d e n s i t y . A l l o w i n g f o r t h i s u n c e r t a i n t y (probably at l e a s t 10%) , the c a l c u l a t e d value of obtained by e x t r a p o l a t i n g to t=0 agrees with the value obtained by measuring the small s i g n a l capacitance of the oxide at 1 kHz. Figure 5.13 shows the i n i t i a l slope ctkT = kTdlogJ/dE (as determined by e x t r a p o l a t i n g to t=o the r e s u l t s of the f i r s t - o r d e r a n a l y s i s f o r s e v e r a l samples at d i f f e r e n t temperature) p l o t t e d against the i n i t i a l steady s t a t e f i e l d . The l i n e drawn through the data i s c a l c u l a t e d using the data reported by Young (1961b) f o r the stepped f i e l d t r a n s i e n t . The c l o s e agreement between the present r e s u l t s and the c a l c u l a t e d l i n e i n d i c a t e s t h a t , at l e a s t i n the i n i t i a l stages of the open c i r c u i t discharge, the open c i r c u i t t r a n s i e n t c l o s e l y approximates the stepped f i e l d t r a n s i e n t . The r e l a t i o n between these t r a n -s i e n t s w i l l be discussed i n greater d e t a i l i n Chapter 7. A p o s s i b l e explanation f o r the dependence of x , a, and, thus, on the range of the data i n the f i r s t - o r d e r a n a l y s i s i s that the l o g J-E r e l a t i o n f o r the open c i r c u i t discharge i s n o n l i n e a r . Assuming a n o n l i n e a r r e l a t i o n , eqn. 4 i s a good approximation only to the i n i t i a l stages of the t r a n s i e n t . As shown i n a somewhat exaggerated way i n Figure 5.14, the l o g j - E l i n e described by eqn. 4 would be t a n g e n t i a l to the n o n l i n e a r l o g J-E F i g . 5.13 The dependence of the i n i t i a l 3lope d l o g J / d E of the open c i r c u i t t r a n s i e n t on the s t e a d y s t a t e f i e l d . 170 FIELD /10° Vm1 J7.ig. 5.14 An exaggerated r e p r e s e n t a t i o n o f the apparent n o n l i n e a r l o g J - E r e l a t i o n f o r the open c i r c u i t s e l f - d i s c h a r g e . The dashed l i n e s r e p r e s e n t the l i n e a r l o g J - E r e l a t i o n s assumed i n the f i r s t - o r d e r a n a l y s i s and t h e i r dependence on t h e e x t e n t o f the t r a n s i e n t i n c l u d e d i n the a n a l y s i s . 171 curve only at t=o. Including l a t e r stages of the transient i n the f i r s t - o r d e r a n a l y s i s would skew the f i t t e d l i n e a r r e l a t i o n , thereby increasing the slope a and t=o intercept of the f i t t e d log J-E l i n e . This i s exactly the behaviour shown in Figure 5.11. It should also be noted that the systematic deviation of the non-linear log J-E curve about the f i t t e d l i n e , as suggested by Figure 5.14, would be s i m i l a r to the deviation shown in Figure 5.6. Ord and DeSmet (1969), using the simple exp(cxE) dependence of current on f i e l d , where a has d i f f e r e n t values f o r open-circuit transient and steady state, have shown that the pre-exponential factor J q in eqn. 4 for open c i r c u i t transient conditions i s proportional to J D where n i s a constant. This s follows by noting that at t=o, the steady state and transient equations give the same J . 4.3 Second-Order Analysis Recently, Ord et a l proposed a so-called second-order analysis of the open c i r c u i t transient data to account for the systematic deviations of the type shown in Figures 5.8 and 5.9 for the open c i r c u i t t r ansient. None of the present data were analyzed by t h i s second-order method; however, the method i s b r i e f l y reviewed for completeness. It should be noted that i n the following the variables are s l i g h t l y d i f f e r e n t from those used by Ord et a l but the equations are equivalent. The second-order analysis i s based on assuming the current i s co n t r o l l e d by an e f f e c t i v e f i e l d E £ according to the r e l a t i o n J = J exp(aE ) (8) o e 172 where E ^KE^ and K i s a function of the o v e r f i e l d E. The f i e l d dependence e of K can be expressed as a power series in E i n which cubic and higher order terms are usually neglected. For computational purposes, however, Ord and Lushiku (1979) and Ord, Clayton, and Brudzewski (1978) assumed a power law of the form K = K Q ( 1 - Y ( E / E S ) N ) (9) where K i s the z e r o - f i e l d value of K E i s the i n i t i a l steady state f i e l d , o s and y and n are constants. Substituting eqn. 9 into (8) and defining the variable u by the r e l a t i o n u = (K/K 1)E (10) where i s the p e r m i t t i v i t y of the oxide at the oxidation f i e l d E^, eqn. 5 for the self-discharge can be re-written as K l o T = " J 0 e x p ( a " u ) where a" = K^a. Equation 11 can be integrated a n a l y t i c a l l y , as in the f i r s t -order a n a l y s i s , giving u(t) = u(t=o) - i j r An(l + t/x') (12) where T ' = K /a" J . 1 s The second order analysis of the open c i r c u i t data proceeds as follows. F i r s t , values for the parameters y and n are chosen and the f i e l d data are transformed using equations 9 and 10. The analysis then proceeds "'"The e f f e c t i v e f i e l d should be written (K + 2)E/2 i f a c a v i t y model i s assumed. However, the second order analysis does not have s u f f i c i e n t accuracy to d i s t i n g u i s h between the two forms for large values of K. Therefore, the simple form KE i s used for convenience. 173 as in the case of the f i r s t - o r d e r a n a l y s i s , with T ' chosen to minimize the standard deviation i n E when u i s f i t t e d to a l i n e a r dependence on £n(l + t / i ' The procedure i s repeated for other values of y and n u n t i l values are found which minimize the deviation in E. Values for the parameters , a", and J Q can then be cal c u l a t e d . In the above a n a l y s i s , the deviation in E rather than in u i s minimized in order to avoid errors due to the n o n l i n e a r i t y of the transforma-tion from E to u. The deviation i n E i s calculated by multiplying the deviation in u at each data point by the value of dE = 1 - y ( E / E s ) " du 1 - ( n + l ) Y ( E / E s ) " Data have been published for the second-order analysis of open c i r c u i t transients for tantalum, niobium, and tungsten. The r e s u l t s are summarized in Table 5.1. It should be noted that the values for K , the or r e l a t i v e p e r m i t t i v i t y of the oxide at zero f i e l d , are s u r p r i s i n g l y large compared to the published values measured at 1 kHz. These large values are a d i r e c t consequence of the assumption that the p e r m i t t i v i t y depends on the f i e l d . The r e s u l t s of the second order analysis have also been used to r e l a t e by the Clausius-Mossotti r e l a t i o n , the f i e l d dependence of the s t r a i n i n the oxide, assuming the s t r a i n i n the d i r e c t i o n perpendicular to the f i e l d i s zero. 174 TABLE 5.1 Reported Results f o r the Second-Order A n a l y s i s of Open C i r c u i t Transients f o r Ta, Nb, and W. Metal J K @ 1 kHz ' K. y n Reference r or A/m2 (Young 1961a) Ta 0.40 27.6 46.1 0.215 1 Ord, C l a y t o n , and Wang 1977 Nb 1.91 41.4 180 0.275 1.85 Ord and Lushiku 1979 W 2.00 41.7 122 0.290 1.75 Ord and Lushiku 1979 W 2.00 41.7 230 0.44 1.0 Ord, Clayton and Brudzewski 1978 4.4 Dependence of the P e r m i t t i v i t y on F i e l d In t h i s s e c t i o n , we w i l l analyze the s e l f - d i s c h a r g e data assuming a field-dependent p e r m i t t i v i t y of the form K = K Q ( 1 - YE) (13) where, as i n the second-order a n a l y s i s , K q and y are constants. Instead of an e f f e c t i v e f i e l d , however, we w i l l assume that the " i d e a l " open c i r c u i t J(E) r e l a t i o n i s given by J = J exp(a gAE) (14) where a = a - BE i s determined from Figure 5.13 and AE = E - E . The b a s i s s s . s f o r t h i s assumption w i l l be discussed i n g r e a t e r d e t a i l i n Chapter 7. The observed n o n l i n e a r i t y i n the l o g J-E r e l a t i o n , where J i s c a l c u l a t e d from J = - edE/dt, i s then due to the f i e l d dependence of the p e r m i t t i v i t y . S u b s t i t u t i n g eqns. 13 and 14 i n t o eqn. 4 gives the d i f f e r e n t i a l equation d e s c r i b i n g the open c i r c u i t s e l f - d i s c h a r g e as 175 |- (K (1 - YE)E) = - J exp(a AE) (15) at o s s I n t e g r a t i n g eqn. 15 and making the s u b s i t u t i o n s a l " E s a 3 " 2 Y a„ = a a. = x = J a /K I s 4 s s o gives the E( t ) dependence during the discharge as E(t) - a 2 £n(l + a 3 ( E ( t ) + a £ ) ) (16) + - £n(l + a-jCa^ + a,,) + a^t) Using a parameter o p t i m i z a t i o n r o u t i n e obtained from the UBC Computing Centre, a computer program was w r i t t e n to f i n d the values a^, a 2 , a^, and a^ which minimize the penalty f u n c t i o n N dE 2 PF = I [ ( y . - a± + a 2 to(l + a3(a± + a ) + a 4 t ) i = l y i where y_^  = E_^  - a 2 £n(l + ^ ( E . ^ + a 2 ) ) and the summation i s over N data p o i n t s , No c o n s t r a i n t s were placed on any of the parameters. A l s o , the data p o i n t s were approximately e q u a l l y spaced i n f i e l d to ensure equal weighting over the e n t i r e range of data. The optimum value of a^ ( i . e . y) was e s s e n t i a l l y zero i n a l l cases, independent of the i n i t i a l o x i d a t i o n c u r r e n t , f i e l d , or temperature. As i n the f i r s t - o r d e r a n a l y s i s , the values of the other parameters depended on the time range of the data. The values of the parameters obtained by the two methods were g e n e r a l l y equal w i t h i n a few percent. Figure 5.15 shows the 2 experimental r e s u l t s f o r a f i l m grown at 0.26 A/m and 25°C and the l i n e f i t t e d to the data by the above method. Figure 5.16 shows the d e v i a t i o n 5-5 h | 5.0 00 O Q _J UJ 4.5 r 4.0 + + \ + + T = 25 C Je c 0.26 Am' + \ + \ + \ \ V 0.0 0.2 0.4 0.6 TIME / SECONDS 0.8 F i g . 5 - 1 5 E x p e r i m e n t a l E ( t ) d a t a and the r e l a t i o n f i t t e d t o the d a t a by parameter o p t i m i z a t i o n . 0.0 0.2 0.4 0.6 TIME /SECONDS 0.8 F i g . 5 . 1 6 The d e v i a t i o n between the d a t a i n f i g . 5 . 1 5 and the f i t t e d r e l a t i o n . 177 between the data and the f i t t e d E ( t ) r e l a t i o n . Again as w i t h the f i r s t - o r d e r a n a l y s i s , the d e v i a t i o n i s small but systematic, i n d i c a t i n g that eqns. 13 and 14 do not describe the open c i r c u i t discharge data any b e t t e r than do eqns. 4 and 5 of the f i r s t - o r d e r a n a l y s i s . 4.5 Numerical D i f f e r e n t i a t i o n of the Open C i r c u i t Transient E ( t ) Data A computer program was w r i t t e n to n u m e r i c a l l y d i f f e r e n t i a t e the open c i r c u i t t r a n s i e n t E ( t ) data by the method of cubic s p l i n e s . The program was w r i t t e n f o r the Amdahl 470 computer at the UBC Computing Centre and used a s p l i n e r o u t i n e w i t h l e a s t squares smoothing obtained from the Computing Centre's program l i b r a r y . The smoothing feature was p a r t i c u l a r l y u s e f u l i n reducing the e f f e c t s of n o i s e and q u a n t i z a t i o n e r r o r s i n the voltage measurements. The program was tested by using data c a l c u l a t e d from a n a l y t i c a l expressions and the c a p a c i t o r discharge data discussed i n Section 3.3. Figure 5.17 shows the c a l c u l a t e d J-E dependence f o r t r a n s i e n t s recorded at 25°C and steady s t a t e formation current d e n s i t i e s of 0.33, 3.0, 2 and 30 A/m . The r e l a t i v e p e r m i t t i v i t y of the oxide was taken to be = 28. To avoid the graph from becoming overcrowded, only a few r e p r e s e n t a t i v e data p o i n t s are p l o t t e d i n the f i g u r e . The l i n e drawn through these p o i n t s r e -presents the remainder of the data. The c a l c u l a t e d steady s t a t e J(E) r e l a t i o n (Young 1960) i s a l s o p l o t t e d f o r comparison. The data shown i n Figure 5.17 are t y p i c a l of the data obtained at other temperatures. The l o g J-E r e l a t i o n f o r the open c i r c u i t t r a n s i e n t i s almost l i n e a r , at l e a s t i n the i n i t i a l stages of the s e l f - d i s c h a r g e and f o r small formation current d e n s i t i e s . The l o g J-E r e l a t i o n becomes i n c r e a s i n g l y n o n l i n e a r , however, w i t h i n c r e a s i n g steady s t a t e formation current d e n s i t y . To show t h i s more c l e a r l y , a quadratic r e l a t i o n was f i t t e d by l e a s t mean 178 T r J I I ; L 3.7 4.4 5.1 5.8 6.5 7.2 FIELD 108Vm1 Pig. 5.17 Typical J(E) data calculated from the E(t) data by numerical d i f f e r e n t i a t i o n . The steady state l i n e was calculated from Young's (I960) data. 179 squares method to some of the c a l c u l a t e d log J-E data. The r e s u l t s obtained f o r 3 current d e n s i t i e s J are summarized i n Table 5.2, where the current s 5Q 2 during the s e l f discharge i s w r i t t e n as J = J e x p ( — (aE - BE )) . The estimated O K. i. d e v i a t i o n s were determined by averaging over 5 sets of data f o r s u c c e s s i v e l y higher formation voltage at each J . The i n c r e a s i n g magnitude of 8 as J i n -s s creases shows c l e a r l y the dependence of the curvature of the l o g J-E p l o t on the steady s t a t e current d e n s i t y . A p o s s i b l e explanation f o r t h i s n o n l i n e a r i t y may be that r e l a x a t i o n processes tend to skew on otherwise l i n e a r l o g J-E r e l a t i o n toward the steady s t a t e r e l a t i o n . TABLE 5.2 Quadratic C o e f f i c i e n t s i n the F i t t e d R e l a t i o n log( J/J 0)=ijpkaE-BE 2) f o r the Open C i r c u i t T ransient. J s a 8 A/m2 nm nm/10 8Vm - 1 0.25 0.193 ± 0.003 0.0087 ± 0.0002 2.5 0.239 ± 0.004 0.0137 ± 0.0003 25 0.261 ± 0.003 0.0158 ± 0.0005 5. Discussion The determination of the p e r m i t t i v i t y of the oxide and i t s f i e l d dependence during the s e l f - d i s c h a r g e i s c e n t r a l to the open c i r c u i t t r a n s i e n t method. This problem i s common to both the i n t e g r a t i o n and d i f f e r e n t i a t i o n methods of a n a l y s i s discussed i n t h i s chapter. I f the p e r m i t t i v i t y i s tre a t e d as a field-dependent v a r i a b l e , one has an a d d i t i o n a l problem regarding the form of the f i e l d dependence. Assuming a f u n c t i o n a l dependence on f i e l d f o r 180 both the current d e n s i t y and p e r m i t t i v i t y and then minimizing a penalty f u n c t i o n as i n the second order method of a n a l y s i s does not, i n our view, provide s u f f i c i e n t evidence that the assumed dependence i s c o r r e c t . Since the r a p i d decay of f i e l d during the i n i t i a l stages of the discharge occurs on a time s c a l e comparable to about 1 msec, an i n d i c a t i o n of the dependence of the p e r m i t t i v i t y on f i e l d and the h i s t o r y of the oxide can be obtained from the small s i g n a l value measured at 1 kHz. The clos e agreement between the measured p e r m i t t i v i t y and the value obtained from the i n i t i a l c o n d i t i o n s of the t r a n s i e n t (Section A.1) supports t h i s approach, at l e a s t as a f i r s t approximation. In a d d i t i o n , the capacitance increases by only a few percent at lower frequencies (Dell'Oca, P u l f r e y , and Young 1971). The small s i g n a l capacitance of the oxide depends i n a r a t h e r compli-cated way on the f i e l d i n the oxide as w e l l as the h i s t o r y of the oxide. When measured wet a f t e r annealing, presumably to remove mobile defects or c e n t r e s , the capacitance decreases by seve r a l percent as the b i a s f i e l d i s increased toward the o x i d a t i o n f i e l d (Dell'Oca 1969). When the f i e l d i s reduced to zero, the capacitance i s increased by a few percent from i t s s t a r t i n g value. In a s i m i l a r s i t u a t i o n , the capacitance of an annealed oxide decreases by se v e r a l percent during the i n i t i a l stages of the constant f i e l d t r a n s i e n t when (presumably) no i o n i c current i s flo w i n g (Dell'Oca 1969). A s i m i l a r e f f e c t i s a l s o observed when the capacitance i s measured as a f u n c t i o n of time near zero f i e l d a f t e r oxide growth at constant current d e n s i t y . I t i s not e n t i r e l y c l e a r how the above r e s u l t s apply to the open ( c i r c u i t s e l f - d i s c h a r g e . However, they suggest a strong dependence of the p e r m i t t i v i t y on whatever i s re s p o n s i b l e f o r the h i s t o r y e f f e c t s i n the oxide f i l m s . This dependence might be due to changes i n p o l a r i z a t i o n , s t r u c t u r e of the oxide, or i n t e r f a c e charge d e n s i t y depending on the d e t a i l s of the 181 model used to describe the i o n i c conduction process. None of the methods of analysis discussed i n t h i s chapter take these " r e l a x a t i o n " e f f e c t s into consideration. This e n t i r e question of the oxide p e r m i t t i v i t y leaves the open c i r c u i t transient method in a somewhat unsatisfactory p o s i t i o n as regards i t s u t i l i t y in obtaining information about the i o n i c conduction process. 182 VI. THE STEPPED CURRENT TRANSIENT 1. I n t r o d u c t i o n The stepped current (or g a l v a n o s t a t i c ) t r a n s i e n t i s a more general case of the open c i r c u i t t r a n s i e n t , discussed i n the previous chapter, i n which the formation current i s stepped from the i n i t i a l steady s t a t e to some new value J other than zero. On stepping the current w i t h > ^' the f i e l d i n the oxide r i s e s r a p i d l y from the i n i t i a l f i e l d E^ to a peak value Ep before decreasing slowly to the new steady s t a t e f i e l d E^• Analogous t r a n s i e n t s are observed when - ^ ^ l < t n e d i f f e r e n c e being that the f i e l d decreases to a minimum value E before i n c r e a s i n g slowly to the new f i e l d E„. m / In t h i s chapter, the stepped current t r a n s i e n t i s i n v e s t i g a t e d as a fu n c t i o n of the i n i t i a l steady s t a t e , current r a t i o J^/J-^, and temperature and some comparisons are then made w i t h p r e v i o u s l y reported data. The r e s u l t s are a l s o compared i n the next chapter w i t h the p r e d i c t i o n s of phenomenological equations used to describe the t r a n s i e n t . 2. Experimental Procedure 2.1 Sample Preparation The samples were prepared both from s i n g l e c r y s t a l and sheet tantalum by the methods described i n the previous two chapters. The working area of -4 2 the samples was t y p i c a l l y 1. - 1.5 x 10 m . 2.2 Oxide Formation The a n o d i z a t i o n c e l l and constant current oxide formation are described i n the previous two chapters. As before, the s o l u t i o n was s t i r r e d continuously and the temperature thermostatted to w i t h i n 0.05°C during the experiments. 183 2.3 Voltage Measurements The computer-controlled d i g i t a l voltmeter measured the voltage of the anode with respect to the saturated calomel reference e l e c t r o d e . The voltmeter was operated on one of i t s low volt a g e - h i g h input r e s i s t a n c e ranges. The range was e f f e c t i v e l y extended to about 60 v o l t s by backing o f f the anode voltage w i t h a b a t t e r y . Because stepped current t r a n s i e n t s can occur on a time s c a l e much l e s s than the conversion time of the voltm e t e r , they were u s u a l l y recorded by the much f a s t e r 1 0 - b i t A/D converter, b u f f e r e d by the voltmeter as described i n the previous chapter. In some experiments, the t r a n s i e n t s were a l s o recorded on a storage o s c i l l o s c o p e as a check on the r e s u l t s of the A/D converter. Because one side of the input to the o s c i l l o s c o p e was grounded, the voltage was measured w i t h respect to the grounded cathode. The anode voltage w i t h respect to the calomel reference e l e c t r o d e was then deduced from the voltage of the calomel e l e c t r o d e measured w i t h respect to the cathode. -4 2 As before, the cathode i n a l l experiments was a 4 x 10 m p l a t i n -_2 iz e d platinum e l e c t r o d e about 4 x 10 m from the anode. Since the e l e c t r o -l y t e r e s i s t a n c e was about 5 fi, the p o t e n t i a l drop across the s o l u t i o n was neglected. 2.4 Current Source The output from the Northeast S c i e n t i f i c current source i s c o n t r o l l e d by a p a r a l l e l r e s i s t o r network i n s e r i e s w i t h the output, as shown s c h e m a t i c a l l y i n Figure 6.1. By maintaining a constant voltage across the network, i n -dependent of the t o t a l r e s i s t a n c e , the output current i s kept constant. In normal operation the output i s s e l e c t e d by r e s i s t o r s mounted on 4 r o t a r y switches on the f r o n t panel of the current source. The current can be stepped i n increments of 10 ^ A from zero output to a maximum of 0.25 A. F i g . 6.1 Schematic r e p r e s e n t a t i o n of the No r t h e a s t S c i e n t i f i c model RI-233 c u r r e n t s o u r c e . 00 4>-185 To enable remote c o n t r o l of the current source, the f u n c t i o n of the swi t c h mounted c o n t r o l r e s i s t o r s was d u p l i c a t e d by an e x t e r n a l decade r e s i s t a n c e box. As shown i n Figure 6.1 , the input to the d i f f e r e n t i a l a m p l i f i e r i n the c o n t r o l c i r c u i t of the source was connected to the e x t e r n a l r e s i s t o r s through a computer-controlled r e l a y RO and a manually operated switch SI. With e i t h e r the switch or r e l a y c l o s e d , the output from the current source was determined by the p a r a l l e l combination of the switch-mounted r e s i s t o r s and the e x t e r n a l r e s i s t a n c e box. In most experiments, the current was stepped from the i n i t i a l steady s t a t e current to a l a r g e r value. Therefore, p r i o r to each experiment, the i n i t i a l current was f i r s t set with the e x t e r n a l c o n t r o l r e s i s t o r s disconnected from the current source. The e x t e r n a l r e s i s t o r s were then connected and adjusted to set the f i n a l c u rrent. For experiments i n which the current was to be stepped down from the formation c u r r e n t , the f i n a l current was f i r s t set and the e x t e r n a l r e s i s t o r s were then connected and adjusted to set the i n i t i a l c u r r e n t . A mercury-wetted r e l a y ( P o t t e r and Brumfield JMF-1120) was used f o r r e l a y RO. Tests showed t h a t , on e n e r g i z i n g the r e l a y , the r e l a y contacts d i d not "bounce" once i n i t i a l contact was made. I t was a l s o important to minimize s t r a y inductance i n the current c o n t r o l c i r c u i t , p a r t i c u l a r l y s i n c e the current was to be stepped as r a p i d l y as p o s s i b l e . Several decade r e s i s t -ance boxes were t r i e d before d e c i d i n g on one manufactured by J . B i d d l e Co. Co a x i a l cable was used to connect the current source, s w i t c h i n g r e l a y , and e x t e r n a l r e s i s t o r s . 2.5 Experimental C o n t r o l and Data A c q u i s i t i o n The stepped current t r a n s i e n t s were recorded by the PDP8/E computer under the c o n t r o l of the same computer program as described i n the previous 186 chapter f o r the open c i r c u i t t r a n s i e n t . A f t e r constant current formation to a preset anode v o l t a g e , the computer stepped the current to a new value by opening or c l o s i n g r e l a y RO and, during the subsequent t r a n s i e n t , recorded the anode voltage w i t h the 10-bit A/D converter. During the i n i t i a l stages of the t r a n s i e n t , the voltage was recorded as o f t e n as necessary to f o l l o w the r a p i d l y changing vol t a g e . In l a t e r stages, the voltage was recorded only i f i t changed by a minimum amount from the preceding value. As before, between 100 and 300 V(t) data were recorded f o r each t r a n s i e n t . These data were stored on DECTAPE or d i s k f o r l a t e r a n a l y s i s . As w i t h the open c i r c u i t t r a n -s i e n t , a l l time data were corrected f o r the e n e r g i z i n g time of the s w i t c h i n g r e l a y (^2.5 msec), A/D conversion, and program r e l a t e d delays. 3. Experimental Tests and Simulations 3.1 Current Source Response The response of the current source to connecting the e x t e r n a l r e s i s t o r s to i t s c o n t r o l c i r c u i t was i n v e s t i g a t e d i n the f o l l o w i n g way. The output of the source was connected to a GR decade r e s i s t a n c e box set to 1 kfi and the voltage across the r e s i s t o r d i s p l a y e d on a storage o s c i l l o s c o p e . When -4 stepping the current by a f a c t o r of 2 from an i n i t i a l current of 10 A, the output overshot i t s f i n a l value by about 20% but s e t t l e d to w i t h i n a few percent of i t s f i n a l value w i t h i n about 50 usee of i n i t i a t i n g the step. The overshoot decreases w i t h the t e s t r e s i s t a n c e and i n c r e a s i n g current step. The s e t t l i n g time was always about 50 usee. In another t e s t , the current to the a n o d i z a t i o n c e l l was monitored by measuring the voltage across a 1 kfi r e s i s t o r i n s e r i e s w i t h the cathode. The capacitance of the anode d i d not appear to a f f e c t the extent or d u r a t i o n of the overshoot of the voltage across the r e s i s t o r . 187 3.2 Experimental Simulation The a b i l i t y of the 10-bit A/D converter to record a c c u r a t e l y the r a p i d l y changing voltage expected f o r the stepped current t r a n s i e n t was tes t e d i n the f o l l o w i n g way. Under computer-control, a s i n g l e c y c l e square wave from a s i g n a l generator was a p p l i e d to a s e r i e s LCR c i r c u i t and the voltage across the ca p a c i t o r measured w i t h the A/D converter, b u f f e r e d by the voltmeter as before. The values of R, L, and C were s e l e c t e d to simulate as c l o s e l y as p o s s i b l e the r a p i d l y changing voltage of the stepped current t r a n s i e n t . Figure 6.2 shows t y p i c a l data and the V(t) curve c a l c u l a t e d from c i r c u i t theory (the s e r i e s r e s i s t a n c e of the inductor was included i n the c a l c u l a t i o n ) . , Except f o r a small overshoot i n the f i r s t c y c l e , the data agree w i t h the computed curve. The overshoot, which i s probably due to the response of the voltmeter b u f f e r , sets a l i m i t of V3500 V/sec on the maximum slew r a t e of the input v o l t a g e . This i s i n agreement w i t h the t e s t r e s u l t s described i n the previous chapter. 4. Results and A n a l y s i s Stepped current t r a n s i e n t s were recorded at temperatures of 1, 25, 50, and 75°C and oxide formation current d e n s i t i e s of 0.08, 0.3, 1.0, and 2 3.0 A/m . Several t r a n s i e n t s at s u c c e s s i v e l y higher formation voltage (to a maximum of about 45 v o l t s ) were recorded at each temperature and cu r r e n t . Figure 6.3 shows t y p i c a l v o l tage data f o r a stepped current 2 t r a n s i e n t w i t h i n i t i a l steady s t a t e current = 0.08 A/m and current r a t i o J ^ / J ^ = 2. Following Dewald (1957), the data are p l o t t e d against the charge passed at J . On stepping the current from to J ^ , the voltage drop across the oxide increases r a p i d l y to a maximum, then decreases to a minimum before 6 188 LU O _ l o 0 0 R = 1.95 k A C= 0.02 uF L = 5.3 H A A 0.01 TIME / SECONDS 0.02 Pig. 6.2 V(t) data for a series RLC c i r c u i t subjected to a voltage step. The dashed l i n e was calculated using the values of R, L, and G given i n the figure. T = 25 C J,= 0 .08Am 5 J 2 / J , = 2 J L 0.0 0.2 04 0.6 0.8 CHARGE PASSED / 10Cm"2 F i g . 6.3 T y p i c a l V(Q) d a t a f o r a stepped c u r r e n t t r a n s i e n t . T= 25°C 3 k 2 h 1 h J,= 0.08 Am J 2 / j , = 64 0.0 0.2 0.4 0.6 CHARGE P A S S E D / l O C m -2 0.8 F i g . 6.4 T y p i c a l c h a r g i n g c u r r e n t . 00 190 f i n a l l y approaching the steady s t a t e c o n d i t i o n of l i n e a r increase w i t h time c h a r a c t e r i s t i c of . Analogous t r a n s i e n t s are observed when < J-^> the d i f f e r e n c e being that the voltage decreases r a p i d l y to a minimum and then increases toward the new steady s t a t e c o n d i t i o n of l i n e a r increase w i t h time. As mentioned i n Section 2.3, the voltage t r a n s i e n t s were al s o r e c o r -ded with a storage o s c i l l o s c o p e to check the accuracy of the A/D converter. The values of the peak f i e l d s f o r a t y p i c a l s e r i e s of t r a n s i e n t s are given i n Table 6.1 (the voltages were converted to f i e l d s by the method o u t l i n e d i n the next s e c t i o n ) . For t h i s example, the peak f i e l d s as measured w i t h the con-v e r t e r and o s c i l l o s c o p e were i n very c l o s e agreement f o r current r a t i o s up to about ^ 2 ^ 1 = " ^ e o v e r s n c , o t of the converter measurement, as d i s -cussed i n the previous s e c t i o n , r e s u l t e d i n a l a r g e disagreement at l a r g e r current r a t i o s . At these large current r a t i o s , the o s c i l l o s c o p e measurement was taken to be c o r r e c t . The maximum current r a t i o f o r which the t r a n s i e n t could be recorded a c c u r a t e l y w i t h the converter increased w i t h decreasing formation current . TABLE 6.1 Values of the Peak F i e l d s as Measured w i t h the A/D Converter and Storage O s c i l l o s c o p e . T=25°C J 2 / J l J ^ l . O A/m2 E (A/D) P 108V/m E^(scope) 108V/m 2 4 8 16 32 64 6.08 6.39 6.69 7.00 7.31 7.90 6.09 6.40 6.70 6.99 7.29 7.59 191 4.1 V o l t a g e - t o - f i e l d Conversions As w i t h the open c i r c u i t t r a n s i e n t data, the stepped current V(t) data were converted to E ( t ) data, thus a l l o w i n g a d i r e c t comparison of data obtained at d i f f e r e n t formation v o l t a g e s . I t should a l s o be noted that s i n c e the time s c a l e of the t r a n s i e n t decreases w i t h i n c r e a s i n g and current r a t i o p l o t t i n g the data against charge passed makes p o s s i b l e a comparison of data obtained f o r d i f f e r e n t i n i t i a l c o n d i t i o n s . Converting the elapsed time t to charge passed i s a simple m u l t i -p l i c a t i o n Q(t)=J2 t . Converting the voltages V to f i e l d strengths E, however, i s a more complicated problem s i n c e , i n the expression E ( t ) = ( V ( t ) ~ v r e v ) / D ( t ) , the oxide thickness D increases continuously during the t r a n s i e n t . Assuming s t o i c h i o m e t r i c Ta 20^ with molecular weight M and d e n s i t y p , the thickness D(t) i s given by the i n t e g r a l equation D(t) = D(0) + T - ^ — / j . ( t ) d t (1) 10 Fp g l where D(0) i s the oxide thickness at t=0, F i s Faraday's constant, and n i s the current e f f i c i e n c y (assumed to be u n i t y ) . J ( t ) , the i o n i c current component of the t o t a l c u r r e n t , i s given by J . ( t ) = J . - J (2) I 2 c where = d(CV)/dt i s the displacement current due to charging the metal/ o x i d e / e l e c t r o l y t e c a p a c i t o r . A computer program was w r i t t e n to evaluate eqn. 1 f o r the stepped current data, given the i n i t i a l boundary c o n d i t i o n s J^(0) = and D(0) = (V(0) - V )/E,. The value of E, was determined from the steady s t a t e rev 1 1 J(E) data reported by Young (1960). The r e v e r s i b l e p o t e n t i a l f o r measurements w i t h respect to the calomel e l e c t r o d e was taken to be -1.16 v o l t s , as discussed i n Chapter 4. 192 Since the data were spaced so c l o s e l y i n time and, towards the end of the t r a n s i e n t , i n v o l t a g e , the i n t e g r a l i n eqn. 1 was approximated by summing over the N data p o i n t s t N / J . ( t ) d t - I (J„ - J )At (3) 0 1 k=2 2 c where At = t ^ - t ^ _ ^ and J c a ( C k \ " C k - l \ - l ) / A t ^ Assuming a constant p e r m i t t i v i t y e, the capacitance i n eqn. 4 was c a l c u l a t e d by the r e l a t i o n C = e/D. As a f i r s t approximation, was c a l c u l a t e d assuming that C^ = The thickness was then c a l c u l a t e d and, using t h i s value, a new value c a l c u l a t e d f o r C^. The procedure was repeated u n t i l the c a l c u l a t e d value of changed by l e s s than 0.01%. U s u a l l y , only one or two i t e r a t i o n s were re q u i r e d f o r convergence. Figure 6.4 shows a t y p i c a l example of the charging current J c 2 c a l c u l a t e d by the above method f o r i n i t i a l c o n d i t i o n s = 0.08 A/m and T = 25°C and current r a t i o J ^ ^ l = 64. The r e l a t i v e p e r m i t t i v i t y was taken to be = 28, although i t s value had a n e g l i g i b l e a f f e c t on the i n t e g r a t e d thickness and f i e l d . Comparing t h i s f i g u r e w i t h the E(Q) data f o r the t r a n s i e n t (discussed below) shows that J can be taken as zero i n a l l but the c i n i t i a l stages of the t r a n s i e n t . That i s , the i o n i c current i s e s s e n t i a l l y constant at i t s f i n a l steady s t a t e value once the f i e l d reaches i t s maximum. Figure 6.5 shows the E(Q) data f o r 3 stepped current t r a n s i e n t s 2 recorded at 25°C f o r an i n i t i a l current d e n s i t y = 0.08 A/m and current r a t i o ^ 2 ^ 1 = ^ • The oxide f i l m s were formed to 7.2, 16.6, and 27.9 v o l t s . The E(Q) data f o r the 3 t r a n s i e n t s superimpose almost e x a c t l y , i n d i c a t i n g that the t r a n s i e n t s are independent of the formation v o l t a g e , at l e a s t over 193 7.0 6.5 > CO O Q _ J LU 6 . 0 5.5 1 X 1 1 1 | T = 2 5 ° C O + ^ = 0 . 0 8 A m2 X J 2 / J i = 6 4 -r 0 + X — o % X 4-I o X +o -° + X ° + o x x + o + x o + = 7.2 V o l t s x =16.6 0 + o = 27.9 i i i i 0 2 4 6 C H A R G E P A S S E D / C m " 2 8 F i g . 6.5 E(Q) d a t a f o r 3 stepped c u r r e n t t r a n s i e n t s r e c o r d e d f o r the same and c u r r e n t r a t i o Z^/Z^. 1 1 o F i g . 6.6 E(Q) d a t a f o r the i n i t i a l s t a g e s of a stepped c u r r e n t t r a n s i e n t . 195 the range of v o l t a g e s of the data. These data are a l s o t y p i c a l of the data obtained w i t h other i n i t i a l c u rrent d e n s i t i e s , current r a t i o s , and tempera-t u r e s . In a l l , the stepped f i e l d t r a n s i e n t was found to be independent of oxide thickness over the voltage range M..5 to ^45 v o l t s (the maximum voltage used). 4.2 P e r m i t t i v i t y of the Oxide The p e r m i t t i v i t y of the oxide was deduced from the i n i t i a l slope dE/dt of the t r a n s i e n t . Assuming that the oxide thickness and i o n i c current = are constant over a small i n t e r v a l At near t=0, the p e r m i t t i v i t y of the oxide i s given by e = ( J 2 - J ]_)/(dE/dt) t = ( ) Figure 6.6 shows t y p i c a l E(Q) data f o r the i n i t i a l stages of a t r a n s i e n t f o r 2 i n i t i a l c o n d i t i o n s = 0.3 A/m , T = 25°C, and J ^ ^ l = 64. A value of = 27.2 i s obtained from the slope of the l i n e drawn through the f i r s t few data p o i n t s . Several stepped current t r a n s i e n t s were recorded under d i f f e r e n t c o n d i t i o n s and analyzed i n the above way. The average value of = 28.6 ± 1.2 agrees, w i t h i n the estimated u n c e r t a i n t y , w i t h the value obtained from measurements of the oxide's small s i g n a l capacitance at 1 kHz. 4.3 Peak F i e l d Data 4.3.1 F i r s t - O r d e r A n a l y s i s As mentioned i n Chapter 2, a u s e f u l (and simple) i n t e r p r e t a t i o n of the stepped current t r a n s i e n t i s that the J" 2 - E^ data approximate the data obtained under stepped f i e l d c o n d i t i o n s . That i s , as a " f i r s t - o r d e r " approximation, J 0 may be taken as the i o n i c current that would flow on 196 applying the peak f i e l d E^ instantaneously, s t a r t i n g from an i n i t i a l steady state J ^ ( E ^ ) . The r e l a t i o n between these d i f f e r e n t data w i l l be discussed in greater d e t a i l i n the next chapter. Figure 6.7 shows the - E^ data for a s e r i e s of stepped current transients recorded at 25°C and formation current d e n s i t i e s of 0.08, 0.3, 2 1.0, and 3.0 A/m . Similar data were obtained at other temperatures. The peak voltages, and thus the peak f i e l d s , were usually determined to within about 20 mV with the 10-bit A/D converter. When the slew rate of the voltage transient was too large for the converter to record accurately, the peak voltage was determined with a storage o s c i l l o s c o p e . Since the duration of the transient was r e l a t i v e l y short, p a r t i c u l a r l y with large values of J^, l o c a l heating of the anode during the transient was not thought to be s i g n i f -i c a nt. The - data plotted in Figure 6.7 appear to be remarkably l i n e a r over quite a large range of current r a t i o s . This i s shown perhaps more c l e a r l y by Figure 6.8 i n which the slopes AlogJ/AE calculated piece-wise by f i r s t divided differencesare plotted against E for data set P 2 D(J^ = 0.08 A/m ) in Figure 6.7. The slope seems to be r e l a t i v e l y constant over the e n t i r e range of data. The very s l i g h t decrease i n slope at large values of E i s probably due to small errors i n determining the peak voltage. Using a routine obtained from the program l i b r a r y of the UBC Computing Centre, a computer program was written to f i t both l i n e a r and quadratic r e l a t i o n s to the logJ„ - E data. T y p i c a l l y , the uncertainty i n 1 P the c o e f f i c i e n t s was 2 - 4 times greater for the quadratic f i t than for the l i n e a r f i t . This r e s u l t was taken to indicate that the dependence of E i s best described by a l i n e a r r e l a t i o n of the form P l o g J 2 = a Q + a ; L ( E p - E x) on 197 F i g . 6.7 J 2-E data for a series of stepped current transients with J±(k) = 3 A/m2, J^B) = 1 A/m2, J - J C ) = 0.3 A/m2, and J,(D) = 0.08 A/m2. 0.36 E c LU 0.28 o 0 .20 T = 25 C J 1 = 0.08 Am -2 o o o ° o o 6 FIELD / 10 8 Vm" 1 8 F i g . 6.8 The s l o p e o f d a t a s e t D i n f i g . 6.7 c a l c u l a t e d by f i r s t d i v i d e d d i f f e r e n c e s 199 I t should be noted that the value of a^ gives the slope AlogJ/AE of the data p l o t t e d i n Figure 6.7 . Although no c o n s t r a i n t s were placed on the value of a Q , i t was g e n e r a l l y found to equal l o g J ^ (as expected) w i t h i n about 0.5%. The value of a^ depended on both the temperature and formation c u r r e n t . For 2 2 example, f i t t i n g data sets D ( J 1 = 0.08 A/m ) and A ( J = 3.0 A/m ) i n Figure 6.7 to the above l i n e a r r e l a t i o n gave the values of a^ as 1.086 ± 0.004 cm/MV and 0.924 ± 0.003 cm/MV r e s p e c t i v e l y . The u n c e r t a i n t y i n these r e s u l t s i s t y p i c a l of that f o r data obtained at other formation currents and tempera-tures . 4.3.2 Temperature Dependence According to the elementary theory discussed i n Chapter 2 , the stepped f i e l d t r a n s i e n t data have a temperature dependence of the form AE/AlogJ = kT/a (5) where a i s the a c t i v a t i o n d i p o l e . I d e n t i f y i n g a^ f o r the present stepped current experiments w i t h AlogJ/AE i n eqn. 5, Figure 6.9 shows the values of 1/a-^  f o r data sets A and B i n Figure 6.7 p l o t t e d against temperature of formation. The estimated standard d e v i a t i o n s i n the r e s u l t s are a l s o p l o t t e d . For a given formation c u r r e n t , the t r a n s i e n t slope AE/AlogJ decreased s l i g h t l y w i t h i n c r e a s i n g temperature r a t h e r than i n c r e a s i n g as p r e d i c t e d by eqn. 5 . For comparison, Figure 6.9 a l s o shows the expected temperature dependence c a l c u l a t e d from eqn. 5 assuming on a c t i v a t i o n d i p o l e a = 0.5 e nm. The d i f f e r e n c e between theory and data cannot be a t t r i b u t e d to instrument e r r o r s s i n c e these should be independent of the s o l u t i o n temperature. Following Young (1961b) the slope a^ = AlogJ/AE, m u l t i p l i e d by kT/e, i s p l o t t e d i n Figure 6.10 against the i n i t i a l steady s t a t e o x i d a t i o n 200 F i g . 6 . 9 The temperature dependence of the s l o p e of the l o g ^ - E d a t a . The d i a g o n a l l i n e o f p o s i t i v e s l o p e shows the temperature dependence t h a t might be expected on elementary t h e o r i e s . 0.35 4.5 5.0 5.5 6.0 FORMATION FIELD / 108Vm"1 F i g . 6.10 The dependence of the s l o p e of the l o g ^ - E data on the i n i t i a l s t e a d y s t a t e f i e l d . 202 fi e l d E 1 ' The data can be approximated within experimental significance by the linear relation (kT/e)AlogJ/AE = a' -where a' = 0.62 ± 0.025 nm and g' = 0.0624 ±0.0045 nm/10 Vm Converting these results to Young's (1961b) notation for the stepped f i e l d transient, we have where a = 0.28 6 ± 0.012 nm and 3 = 0.0144 ± 0.001 nm/10°Vm These results 8 —1 are clo se to the values ot — 0.223 nm and 3nT, — 0.0106 nm/10 Vm reported SF SF by Young for the stepped f i e l d transient. 4.3.3 Comparison with the Open Circuit Transient In one series of stepped current experiments, the current was decreased rather than increased. The results for l o g ^ plotted against the' minimum f i e l d E^ are shown in Figure 6.11. As before a straight line can be drawn through the data. For comparison, the open circuit transient data recorded for the same i n i t i a l steady state conditions and calculated by numerical differentiation (see the previous chapter) are also plotted in the figure. The results indicate that the J„ - E relation of the stepped current 2 m transient is more closely related to the J(E) relation of the open circuit transient than the steady state. Since both transients occur on a similar time scale, however, i t is somewhat surprising that the J 0 - E m data so 8„ -1 m poorly approximate the open circuit transient data. 203 F i g . 6.U ^ ( E p ) d a t a f o r stepped c u r r e n t t r a n s i e n t s w i t h ^2^1^ 1 * ° P e n c i r c u i t t r a n s i e n t J ( E ) l i n e r e p r e s e n t s the e x p e r i m e n t a l d a t a o b t a i n e d f o r J = 3 A/m^. 204 4.3.4 Comparison w i t h P r e v i o u s l y Published Data Figure 6.12 shows the inverse slope AE/AlogJ of the J - E data I p at 25°C p l o t t e d against the formation current d e n s i t y . To be c o n s i s t e n t w i t h published data (discussed below), the slope was c a l c u l a t e d by d i v i d e d d i f f e r e n c e s of the J„ - E data obtained f o r J„/J, = 2. The data reported 2 p 2 1 r by Young (1961b) f o r both the stepped current and stepped f i e l d t r a n s i e n t s are a l s o reproduced i n the f i g u r e . The present r e s u l t s f o r the stepped current t r a n s i e n t are i n c l o s e agreement w i t h the data reported by Young. As i n d i c a t e d i n the f i g u r e , the values of AE/AlogJ f o r the stepped current t r a n s i e n t are s i g n i f i c a n t l y smaller than those obtained f o r the stepped f i e l d t r a n s i e n t . This would seem to i n d i c a t e that whatever i s r e s p o n s i b l e f o r the h i s t o r y e f f e c t s changes s i g n i f i c a n t l y from i t s i n i t i a l value c h a r a c t e r i s t i c of J. i n the time required f o r the f i e l d to reach E . That i s , the J„ - E 1 n p 2 p data are not as good an approximation to stepped f i e l d data as might have been expected. This conclusion i s supported by the comparison w i t h the open c i r c u i t t r a n s i e n t data, as discussed i n the previous s e c t i o n . However, the stepped cu r r e n t t r a n s i e n t data show the same trend w i t h J ^ as do the stepped f i e l d data, suggesting a c l o s e r e l a t i o n between these t r a n s i e n t s . 4.3.5 The Excess F i e l d Dewald (1957) defined the excess f i e l d of the stepped cu r r e n t t r a n s i e n t as the d i f f e r e n c e AE = E - E where E i s the steady s t a t e f i e l d p I 1 at J ^ . For experiments i n v o l v i n g Ta^O^ i n d i l u t e H^SO^, the excess f i e l d was found f o r a given current r a t i o to be independent of f i l m t h i c k n e s s , at l e a s t f o r f i l m s t h i c k e r than about 10 nm, and i n i t i a l current d e n s i t y J ^ . These r e s u l t s agree w i t h the present - E^ data w i t h i n an estimated e r r o r of about 2% i n determining the peak f i e l d . i 1 1 1 r I i i i i i 1 - 3 - 2 - 1 0 1 2 log ( J / A m " 2 ) F i g . 6.12 The dependence of the s l o p e of the J ? ^ ^ d a t a (•) o n 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 . The d a t a r e p o r t e d by Young (1961b) f o r the stepped f i e l d ( J — V ) and stepped c u r r e n t (J,-"-Jp) t r a n s i e n t s a r e reproduced f o r comparison. 206 1.25 0 4 16 6 4 2 5 6 J2/J1 F i g . 6.13 The dependence of the excess f i e l d on and the c u r r e n t r a t i o J0/«7i • 207 Figure 6.13 shows how the excess f i e l d f o r the present stepped current data depend on and the current r a t i o ^ 2 ^ 1 * P r e s e n t data show a dependence on J ^ , in disagreement with Dewald. Tn ad d i t i o n , as indicat in the f i g u r e , the excess f i e l d for a given has a nonlinear dependence on lo g J ^ / J ^ . Since the l o g j ^ - data are l i n e a r l y r e l a t e d , the n o n l i n e a r i t y indicated by the figu r e i s due e n t i r e l y to the curvature of the steady state logJ-E l i n e . Dewald reported a s i m i l a r n o n l i n e a r i t y i n the excess f i e l d data but a t t r i b u t e d t h i s behaviour to changes in concentration of mobile c a r r i e r s during the transient. 208 VII. COMPARISON WITH THEORY 1. Introduction In t h i s chapter, the mathematical basis for numerically c a l c u l a t i n g J(E) data for ioni c current transients using the phenomenological equations of Dignam's d i e l e c t r i c p o l a r i z a t i o n theory i s described. Some of the values of the parameters used i n the equations are f i r s t deduced from comparisons with previously published J(E) data for the steady state and stepped f i e l d t r a nsient. The J(E) r e l a t i o n s predicted by the equations for the stepped current and open c i r c u i t transients are then compared with the experimental r e s u l t s present i n the previous two chapters. 2. Mathematical Framework 2.1 The J(E) Relation According to Dignam's d i e l e c t r i c p o l a r i z a t i o n theory (discussed in Chapter 2), the ionic current J i s given by J = J exp[-(W - aE + BE 2)/kT] (1) o e e where the e f f e c t i v e f i e l d E i s related to the o v e r f i e l d E by e J E o = (e E + P)/e K_ (2) e o o D and i s the p e r m i t t i v i t y of the double layer. Since the values of a, 6, and are generally unknown, the problem i s to rewrite eqn. 1 i n terms of measurable parameters. Dividing the p o l a r i z a t i o n P into a " f a s t " component P^ ='e x^E and one or more "slow".components P., eqn. 2 may be rewritten as 209 E E = ( E ^ E + E P . ) ^ (3) where = 1 + X-^  i s the dynamic p e r m i t t i v i t y (taken to be the small s i g n a l value at 1 kHz). S u b s t i t u t i n g (3) i n t o (1) and rearranging gives J = J exp[-{W - a(K /K_)(E + Z P./E K n) o I D ^ l o 1 + 6 ( K 1 / K D ) 2 ( E + I P ± / E o K 1 ) 2 } / k T ] (4) Since i n the steady s t a t e P. = E Y - e » the steady s t a t e J (E ) r e l a t i o n i s 1 o l s s s given by J = J exp[-(W - a(K /K )E + 6(K / 0 2 E 2 ) / k T ] (5) s o s D s s D s where K g = + Z i s the s t a t i c p e r m i t t i v i t y . From Young's (1960) steady state data f o r anodic Ta20^, we have the e m p i r i c a l r e l a t i o n J = J exp[-(W - 5qa E + 5qB E 2 ) / k T ] (6) s o s s s s s s The values f o r J , w, a , and 8 are given i n Table 7.1. Comparing o s s s s c o e f f i c i e n t s between (5) and (6) gives 5 ^ s s = o(W 7(a) 5^ss • 6(VV2 7(b) Using these r e s u l t s , eqn. 4 can be r e w r i t t e n as J ='J exp[-{w- 5qa (K./K ) ( E + E P./E K.) o ss 1 s - J ^ l O l + 5qB (K./K ) 2 ( E + E P./E K n ) 2 } k T ] s s l s £ 1 O 1 210 This expression can now be used to c a l c u l a t e J(E) data, given the values of X. and the time dependence of P.. 1 1 TABLE 7.1 Values of Parameters Used i n C a l c u l a t i n g J(E) Data f o r T a ^ Parameter Value Reference o W x ss 3 ss XSF BSF K, 1 n12.24 . 2 10 A/m 2.185 eV 0.6995 nm 0.0335 nm/10"Vm 0.223 nm -1 0.0106 nm/108Vm 1 27.6 Young (1960) Young (1961b) 2.2 The I d e a l Stepped F i e l d Transient Following Vermilyea (1957a), the stepped f i e l d t r a n s i e n t can be taken as representing a l i m i t i n g c o n d i t i o n f o r i o n i c conduction inasmuch as whatever i s re s p o n s i b l e f o r the h i s t o r y e f f e c t s does not change a p p r e c i a b l y from i t s i n i t i a l value when the f i e l d i s stepped from the i n i t i a l steady s t a t e f i e l d E g to a new value E. In the d i e l e c t r i c p o l a r i z a t i o n theory, these h i s t o r y e f f e c t s are i d e n t i f i e d w i t h the slow components of p o l a r i z a t i o n . Therefore, assuming that AP. = 0 ( i . e . P. = e x.E )> when the f i e l d i s ' B 1 1 o 1 s stepped from E g to E, the e f f e c t i v e f i e l d i s given by Ee = (VV ( E +V^V (9) 211 where K g - = ^X^- S u b s t i t u t i n g (9) i n t o (1) and rearranging gives the current of the i d e a l stepped f i e l d t r a n s i e n t as K - K J t = J oexpf-{W - a C K ^ M E + i . E g) (10) s 1 + e ( K 1 / K D ) ' ( E + E s)"}/kT] Eqn. 10 p r e d i c t s a no n l i n e a r l o g J-E r e l a t i o n w i t h a slope that v a r i e s as K — K kT d £ n J t = aCKj/Kjj) - 2 6 ( K 1 / K D ) 2 ( E + i ( 1 1 ) dE 1 De f i n i n g AE = E - E , eqn. 11 can be r e w r i t t e n as dtoJ kT = a(K 1/K D) - 26(1^/1^) ( K g / K D ) E s - 23(K^/K^) AE (12) The term i n v o l v i n g AE i n eqn. 12 determines the magnitude of the n o n l i n e a r i t y i n the stepped f i e l d l o g J-E curve. R e s t r i c t i n g the a n a l y s i s to small values of AE gives the dependence of the t r a n s i e n t slope on the i n i t i a l steady s t a t e f i e l d as d£nJ kT- 1 dE AE+0 - a(W " 2 e ( K l / K D ) ( K s / K D ) E s ( 1 3 ) The values of the constants i n eqn. 13 may be deduced by comparison w i t h the e m p i r i c a l r e l a t i o n A&nJ  R i AE AE+0 " 5< aSF " 1 0 « B S F E s ( U ) reported by Young (1961b) f o r the stepped f i e l d t r a n s i e n t w i t h anodic Ta20,-. The values of a and 8 are given i n Table 7.1. Equating c o e f f i c i e n t s SF SF between (13) and (14) gives 212 5qcx S F = aC^/I^) (15a) 5 q B S F = BC^/KpX^/Kp) (15b) By i n s p e c t i o n , the f i n a l term i n Eqn. 11 i s given by B C W 2 = 5 ^ S F e S F / a s s ( 1 5 c ) S u b s t i t u t i n g these r e s u l t s i n t o Eqn. 10 gives the i o n i c current of the i d e a l stepped f i e l d t r a n s i e n t i n terms of experimentally measured parameters: o o J = J exp{5qa C T ?AE[l - — - AE - 2 E )]/kT} (16) t s ^ SF a a„_ s ss SF 2.3 Comments on the Stepped F i e l d Transient There appears to be some u n c e r t a i n t y as to the i n t e r p r e t a t i o n of Eqn. 14 reported by Young (1961b) to describe the i n i t i a l slope of the stepped f i e l d t r a n s i e n t l o g J-E curve. Following the approach used to i n t e r p r e t the steady s t a t e data, Young t r e a t e d Eqn. 14 as approximating a d i f f e r e n t i a l equation and, i m p l i c i t l y t a k i n g E^ as a v a r i a b l e , i n t e g r a t e d the expression to o b t a i n J = J exp{[a (E - E ) - B„„(E2 - E 2 ) ] 5 q / k T } (17) t S o r s o r S The quadratic l o g J-E dependence given by (17) seemed to be very appealing s i n c e i t explained the anomalous temperature dependence of dE/dlogJ ( s i m i l a r to the temperature dependence of the stepped current t r a n s i e n t ) and was co n s i s t e n t with the quadratic l o g J-E dependence reported f o r the steady s t a t e . However, to our knowledge, i t has never been e s t a b l i s h e d that the l o g J-E dependence f o r the stepped f i e l d t r a n s i e n t i s described by anything other than a l i n e a r r e l a t i o n . The change of slope w i t h i n i t i a l steady s t a t e f i e l d 213 described by Eqn. 13 i s not s u f f i c i e n t evidence f o r a quadratic log J-E r e l a t i o n . In a d d i t i o n , the l o g J-E data o r i g i n a l l y reported by Young a c t u a l l y appear to be l i n e a r but t h i s may be due to the small step s i z e i n f i e l d used i n the experiments (over a small range, a quadratic r e l a t i o n can be approxima-ted as l i n e a r ) . The stepped f i e l d l o g J-E data reported by Dignam and Ryan (1968a) f o r aluminum a l s o appear to be l i n e a r but, again, the step s i z e i n the f i e l d was too small to draw any c o n c l u s i o n s . However, the r e s u l t s of the stepped current t r a n s i e n t s discussed i n the previous chapter c l e a r l y show a l i n e a r l o g J^-E^ r e l a t i o n over a l a r g e range of f i e l d steps^ Although these stepped current data were shown to be not as good an approximation to stepped f i e l d c o n d i t i o n s as expected, the data suggest that the stepped f i e l d l o g J-E data are a l s o l i n e a r . This c o n c l u s i o n c o n t r a d i c t s the p r e d i c t i o n s of the d i e l e c t r i c p o l a r i z a t i o n theory. However, Dignam and Ryan asserted that any d i f f e r e n c e s between the i d e a l and r e a l stepped f i e l d behaviour i s due to r e l a x a t i o n of the p o l a r i z a t i o n toward the steady s t a t e s u f f i c i e n t to mask the p r e d i c t e d n o n l i n e a r log J-E dependence. I t seems improbable that the non-l i n e a r i t y i s compensated so e x a c t l y to give the observed l i n e a r dependence. Assuming a l i n e a r l o g J-E r e l a t i o n f o r the stepped f i e l d t r a n s i e n t , i n t e g r a t i o n of eqn. 13 gives the t r a n s i e n t i o n i c current as J = J exp{(a c_ - 28 E )5qAE/kT} (18) t s SF SF s where AE = E - E . The term a - 28 E might be considered an a c t i v a t i o n s t t s d i s t a n c e which depends on the steady s t a t e f i e l d and changes s l u g g i s h l y w i t h changes i n the f i e l d i . e . the term i s constant during the t r a n s i e n t . Figure 7,1 shows the stepped f i e l d l o g J-E r e l a t i o n s c a l c u l a t e d by the 3 methods discussed above: (1) l i n e a r l o g J-E, (2) quadratic l o g J-E, and (3) i d e a l l o g J-E p r e d i c t e d by the d i e l e c t r i c p o l a r i z a t i o n theory. The values 214 0 h -1 CM I < o -2 -3 T = 2 5 ° C S T E P P E D FIELD LINEAR IDEAL S TEADY STATE FIELD 10 8Vm 1 F i g . 7.1 C a l c u l a t e d stepped f i e l d l o g J - E r e l a t i o n s . 215 of the parameters used i n the c a l c u l a t i o n s are given i n Table 7.1. For small values of AE, there i s no s i g n i f i c a n t d i f f e r e n c e between the quadratic and " i d e a l " curves. Both curves are q u i t e d i f f e r e n t from the l i n e a r l o g J-E r e l a t i o n , p a r t i c u l a r l y i n view of the l o g a r i t h m i c current s c a l e . The two curves separate at l a r g e values of AE, w i t h the " i d e a l " curve l e s s n o n l i n e a r than the quadratic r e l a t i o n . 2.4 Comparison of Steady State and Stepped F i e l d Parameters Since the J(E) r e l a t i o n s d e s c r i b i n g the steady s t a t e and i d e a l stepped f i e l d t r a n s i e n t are both based on eqn. 1, the experimental r e s u l t s obtained by the two methods are presumably l i n k e d by the common parameters a and 3 i n eqn. 1. By i n s p e c t i o n of these parameters i n eqns. 7 and 15, we have the r e l a t i o n ( B / a ) ( K s / K D ) = 3 s s / « s s = 3 ^ / a ^ Using the values given i n Table 7.1, the r a t i o s are c a l c u l a t e d as 3 /a 0.479 nm/V ss ss 3 S F / a s p = 0.475 nm/V As p r e d i c t e d by the d i e l e c t r i c p o l a r i z a t i o n theory, the r a t i o s are equal, w e l l w i t h i n the experimental e r r o r given by Young (1960, 1961b) f o r the values of the above parameters. Dignam and Ryan (1968a) found a s i m i l a r agreement f o r aluminum. In comparison, the present r e s u l t s f o r the stepped current t r a n s i e n t given i n Section 4.3.2 of the previous chapter give the r a t i o as a /3 = 0.504 nm/V. sc sc From eqns. 7(a) and 15(a), the theory a l s o p r e d i c t s that a /a = K /K, ss SF s 1 216 S u b s t i t u t i n g the values from Table 7.1 gives the r a t i o as K /K., = 3.1 or s i K g % 87 i f i s taken to be the r e l a t i v e p e r m i t t i v i t y measured at 1 kHz. Although K g has not been measured d i r e c t l y , Taylor and Dignam (1973b) estimated a lower bound of K^/K^ > 2, from the charging current observed i n i t i a l l y i n constant f i e l d t r a n s i e n t s . However, Dell'Oca et a l (1971) esimated the value of K to be no more than 3.5% l a r g e r than based on s 1 small s i g n a l capacitance measurements. 2.5 Numerical C a l c u l a t i o n of Stepped Current and Open C i r c u i t Transients Dignam's phenomenological equations were used to nu m e r i c a l l y c a l -c u l a t e J(E) data f o r the stepped current and open c i r c u i t t r a n s i e n t s . The f o l l o w i n g b r i e f l y o u t l i n e s the met.hod used. Assuming (somewhat a r b i t r a r i l y ) two components of slow p o l a r i z a t i o n , the current J f l o w i n g i n t o the anode from the e x t e r n a l c i r c u i t i s given by dP dP J = J + e K. §^ + — - + — - (19) ex o l dt dt dt where J i s the i o n i c current and the l a s t three terms c o n s t i t u t e the d i s -placement c u r r e n t . Taking the time dependence of a n ^ as given by Eqn. 22 i n Chapter 2,^ the time e v o l u t i o n of E, ?^, and P^ during a current t r a n s i e n t i s obtained by simultaneously s o l v i n g the d i f f e r e n t i a l equations dP 1 = B.J (e X o E - P.) (20a) dt 2 v o A2 2' d P 3 J = B 0 J (E X,E " P,) ( 2 0 B ) dt "3 v o A3 J J A 3 ; The thermally a c t i v a t e d process described by Eqn. 21 i n Chapter 2 does not apply at the r e l a t i v e l y l a r g e current u s u a l l y involved i n these experiments. 217 dT= ( J e x " J - - d t 7 - - l T ) / e o K l < 2 0 c> subject to the i n i t i a l c o n d i t i o n s J (0) = J P (0) = e XE s 2 o s E (0) = E P (0) = e X E s 3 o s where J g and E g are the i n i t i a l steady s t a t e current d e n s i t y and f i e l d r e s p e c t i v e l y . The i o n i c current J i s given by eqn. 8 f o r a l l E, P^, and P^. A computer program was w r i t t e n to n u m e r i c a l l y i n t e g r a t e eqns. 20(a)-(c) using a 4th order Runge-Kutta r o u t i n e w i t h e r r o r c o n t r o l obtained from the UBC Computing Centre program l i b r a r y . The program was run on both the Amdahl 470 computer at the Computing Centre and on the PDP8/E minicomputer i n the l a b o r a t o r y . The accuracy of the i n t e g r a t i o n was checked by comparing the r e s u l t s of s i n g l e and double p r e c i s i o n v e r s i o n s of the program. Both stepped current and open c i r c u i t t r a n s i e n t J(E) data were c a l c u l a t e d by the above method and compared with the experimental data d i s -cussed i n the previous two chapters. For the stepped current t r a n s i e n t , J = £,J where £ i s the current r a t i o J„/J, . For the open c i r c u i t t r a n s i e n t , ex s 2 1 J g x = 0. Since experimentally i t was not p o s s i b l e to separate the i o n i c current and a presumed displacement current due to and P^, the measured i o n i c current was compared with J ' defined by dP dP J ' = J + " d f + "dF ( 2 1 ) 3. Stepped Current Transient 3.1 Comparison with Experimental Data 2 Figure 7.2 shows E(Q) data recorded at 1°C w i t h J = 0.08 A/m and current r a t i o s J^/J = ^» ^~^> a n c^ ^ ' ^ n e E(Q) data p r e d i c t e d by the 218 7.5 1 1 1 1 1 T=1°C J , = 0 .08 A m 2 7.0 - -> OO O \ Q _i 6 .5 J 2 / J 1 = 6 A LU \ * — — * Li_ =16 \ ^ ^ M — + _ -+—i—i H 6 .0 i i ^ K—+ + + 1 1 1 1 i J I I I 0 1 2 3 C H A R G E P A S S E D / Cm F i g . 7.2 E x p e r i m e n t a l (+) and computed ( ) E(Q) d a t a f o r stepped c u r r e n t t r a n s i e n t s w i t h the same but d i f f e r e n t c u r r e n t r a t i o s J 5 / J \ . F i g . 7.3 E x p e r i m e n t a l and computed J0-E d a t a . E 220 phenomenological equations are a l s o p l o t t e d i n the f i g u r e . The values of and (given i n Table 7.2 ) were taken as the values reported by Goad and Dignam (1972) from an a n a l y s i s of Dewald's (1957) data f o r stepped current t r a n s i e n t s with tantalum. I t should be noted that Goad and Dignam obtained t h e i r values of x^  and by a method of a n a l y s i s d i f f e r e n t from the one used here. The experimental and c a l c u l a t e d data are i n c l o s e agreement everywhere except near the peak f i e l d . S i m i l a r agreement between experiment and theory was obtained f o r stepped current t r a n s i e n t s recorded at other values of and temperature. The temperature independence of x^  and i s c o n s i s t e n t w i t h previous data obtained w i t h the constant f i e l d t r a n s i e n t (Cornish 1972) . TABLE 7.2 Parameter Values Used i n F i t t i n g Stepped Current Transient J(E) Data. Parameter Value X 2 36 X 3 • .16.4 2 B 2 0.657 m /coul B 3 4.34 m 2/coul Other values of x. and B. were t r i e d but the f i t to the data l l (determined v i s u a l l y ) could not be improved s i g n i f i c a n t l y . In p a r t i c u l a r , i f only one p o l a r i z a t i o n term was used, no combination of \^ and B^ could be found which gave as good a f i t to the data as when two terms were used. Figure 7.3 shows 2 sets of experimental - E data f o r stepped 2 2 curr e n t t r a n s i e n t s recorded at 25°C with = 0.08 A/m and 3.0 A/m . The . J 2 - Ep r e l a t i o n s p r e d i c t e d by the p o l a r i z a t i o n model (using the parameter values i n Table 7.2) are a l s o p l o t t e d f o r these formation c u r r e n t s . The computed l o g - E^ r e l a t i o n s are c l e a r l y n o n l i n e a r whereas the experimental 0.20 6 FIELD 8 / 10' 8 V m 1 F i g . 7.4 The s l o p e of the e x p e r i m e n t a l J 2 ~ E p d a t a a n d t h e s l o P e p r e d i c t e d by the phenomenological e q u a t i o n s . 222 data, as discussed i n Chapter 6 , are best represented by a l i n e a r r e l a t i o n . 2 This i s most apparent w i t h the data f o r = 0.08 A/m since the data extend over a very l a r g e range of f i e l d s . As w i l l be shown i n the next s e c t i o n , the p o l a r i z a t i o n changes s i g n i f i c a n t l y during the increase i n f i e l d to E . However, P as w i t h the i d e a l stepped f i e l d t r a n s i e n t , i t seems improbable that a r e l a x a -t i o n process would so e x a c t l y compensate the p r e d i c t e d n o n l i n e a r i t y . Some of the d i f f e r e n c e may a l s o be due to experimental e r r o r s . The curvature of the experimental and computed l o g J - E data f o r 2 = 0.08 A/m i n Figure 7.3 are compared i n Figure 7.4. The slope dlogj/dE of the experimental data was approximated by t a k i n g f i r s t d i v i d e d d i f f e r e n c e s between adjacent data p o i n t s . A l l o w i n g f o r s c a t t e r i n the data, the slope of the experimental data appears to be constant. On the other hand, the slope of the computed curve decreases at a constant r a t e . The r a t e of change of t h i s slope was almost independent of and temperature. Introducing a t h i r d slow p o l a r i z a t i o n term d i d not e l i m i n a t e the curvature of the computed curve. 3.2 P r e d i c t e d Change i n P o l a r i z a t i o n Figure 7.5 shows the amount by which the t o t a l p o l a r i z a t i o n i s pr e d i c t e d to increase during the increase i n f i e l d to E . The change i n P p o l a r i z a t i o n increases both w i t h the current r a t i o and i n i t i a l current d e n s i t y . The r e l a t i v e l y small increase i n p o l a r i z a t i o n (about 1-2%) r e s u l t s i n about a 2-4% re d u c t i o n i n E (about 0.1 - 0.3 MV/cm) from the value expected f o r the i d e a l stepped f i e l d t r a n s i e n t of the same value of J^. Therefore, although the change i n p o l a r i z a t i o n i s r e l a t i v e l y s m a l l , i t s a f f e c t on the f i e l d introduces a s i g n i f i c a n t d i f f e r e n c e between the ^ ( E ^ ) data and the J(E) data of the stepped f i e l d t r a n s i e n t . 223 O l o g ( J 2 / J 1 ) F i g . 7 . 5 Dependence o f the change i n t o t a l p o l a r i z a t i o n on J and c u r r e n t r a t i o J^/J-, . 224 4. Open C i r c u i t Transient 4.1 Comparison w i t h Experimental Data Figure 7.6 shows the E ( t ) data f o r an open c i r c u i t t r a n s i e n t 2 recorded at 25°C with an i n i t i a l steady s t a t e current = 0.33 A/m . To avoid c l u t t e r i n g the graph, only every t h i r d data point i s p l o t t e d . The E ( t ) r e l a t i o n c a l c u l a t e d w i t h the same parameter values as f o r the stepped current t r a n s i e n t (Table 7.2 ) i s p l o t t e d as curve A i n the f i g u r e . The experimental and computed data are i n r e l a t i v e l y c l o s e agreement during the i n i t i a l stages of the t r a n s i e n t but diverge with i n c r e a s i n g time, the experimental data f a l l i n g o f f more r a p i d l y w i t h time than the computed data. The f a s t e r r a t e of decrease of the experimental data i n d i c a t e s a l a r g e r i o n i c current than p r e d i c t e d w i t h the values of y. and B. i n Table 7.2. With the values of B„ and B„ taken as i i 2 3 c o r r e c t , the values of and x^  w e r e "adjusted" to o b t a i n a b e t t e r f i t ( v i s u a l l y ) to the data. Curve B i n Figure 7.6 was obtained w i t h x2 = 45 and X^ = 30. For t h i s combination of x^ and B^, the experimental and computed data are i n c l o s e agreement over the e n t i r e range of data p l o t t e d i n the f i g u r e . TABLE 7.3 Values of Parameters X i and B^ Used i n F i t t i n g the Open C i r c u i t Transient Data. Parameter (a) X 2 45 X 3 30 B 2 0.657 B 3 4.34 Value (b) 62.5 0 0.657 m 2/coul 2 0 m /coul 225 5.5 h 5.0 E > oo O y A.5 lx. 4.0 0.0 0.2 0.4 0.6 TIME / SECONDS 0.8 F i g . 7.6 E x p e r i m e n t a l (+) and computed ( ) E ( t ) d a t a f o r an open c i r c u i t s e l f - d i s c h a r g e . 226 The E ( t ) were a l s o f i t t e d u s i n g only one slow p o l a r i z a t i o n term. 2 Taking = 0.657 m /coul (as above), the computed curve w i t h X2 = 62.5 appeared to give as good a f i t v i s u a l l y to the data as with two terms. I t i s not c l e a r which of these r e s u l t s provided the b e t t e r d e s c r i p t i o n of the open c i r c u i t t r a n s i e n t . However, a model having the fewest parameters would seem to be p r e f e r a b l e . As shown i n Figure 7.7 } the open c i r c u i t t r a n s i e n t E ( t ) recorded at other i n i t i a l steady s t a t e current d e n s i t i e s and temperatures were i n cl o s e agreement with the computed data. The computed curves were obtained using the values of x i and B^ given i n Table 7.3(b). The temperature independence of B^ and B^ i s c o n s i s t e n t with the a n a l y s i s of the stepped current t r a n s i e n t data i n the previous s e c t i o n . I t should be noted that the open c i r c u i t t r a n s i e n t was recorded to a f i e l d strength of about 1/2 the i n i t i a l o x i d a t i o n f i e l d . The r e l a t i o n between the experimental and computed data at l a t e r times during the s e l f -discharge ( i . e . at lower f i e l d s ) was not i n v e s t i g a t e d . T y p i c a l open c i r c u i t t r a n s i e n t J(E) data c a l c u l a t e d by numerical d i f f e r e n t i a t i o n of the experimental E ( t ) data recorded at 25°C (see Chapter 5) are p l o t t e d i n Figure 7.8. The i n i t i a l current d e n s i t i e s are 0.33, 3.0, and 2 30 A/m . Only some of the experimental data are p l o t t e d so as to avoid c l u t t e r i n g the graph. The s o l i d l i n e drawn through the data p o i n t s f o r each t r a n s i e n t i s the l o g J-E r e l a t i o n p r e d i c t e d by the equation f o r the i d e a l stepped f i e l d t r a n s i e n t (eqn. 16) . Assuming that the stepped f i e l d t r a n s i e n t i s a c c u r a t e l y described by eqn. 16, the c l o s e agreement between experimental and computed data suggests that the open c i r c u i t t r a n s i e n t i s an e x c e l l e n t approximation to the stepped f i e l d t r a n s i e n t . The J(E) data c a l c u l a t e d f o r the open c i r c u i t t r a n s i e n t s w i t h the parameter values given i n Table 7.3 F i g . 7.7 E x p e r i m e n t a l ( + ) and computed ( 3 open c i r c u i t t r a n s i e n t s . ) E ( t ) d a t a f o r 228 FIELD / 10 Vm F i g . 7.8 J ( E ) d a t a c a l c u l a t e d by n u m e r i c a l d i f f e r e n t i a t i o n of the e x p e r i m e n t a l E ( t ) d a t a . The l i n e drawn through the d a t a p o i n t s i s the J ( E ) r e l a t i o n c a l c u l a t e d f o r the i d e a l stepped f i e l d t r a n s i e n t . 229 a l s o agreed w i t h the experimental data, as expected from the agreement w i t h the E ( t ) data i n Figures 7.6 and 7. A.2 P r e d i c t e d Change i n P o l a r i z a t i o n The p r e d i c t e d change i n p o l a r i z a t i o n during the open c i r c u i t s e l f -discharge i s shown i n Figure 7.9. The computed J(E) data were obtained w i t h the parameter values given i n Table 7.3(b) i . e . only one slow p o l a r i z a -t i o n term was used. On decreasing the f i e l d to 1/2 the o x i d a t i o n f i e l d , the p o l a r i z a t i o n decreases by about 1%. The amount by which the p o l a r i z a t i o n decreases i s almost independent of the i n i t i a l o x i d a t i o n current d e n s i t y ; i n c r e a s i n g the i n i t i a l current by a f a c t o r of 100 increases only s l i g h t l y the ra t e of decrease of the p o l a r i z a t i o n . Considering the agreement between the experimental J(E) data and the J(E) r e l a t i o n f o r the i d e a l stepped f i e l d t r a n s i e n t , i t i s somewhat s u r p r i s i n g that the p o l a r i z a t i o n decreases by as much as i t does. That i s , one might have expected the p o l a r i z a t i o n to change by no more than a few tenths of 1%. In comparison, the p o l a r i z a t i o n a s s o c i a t e d with the J - E^ data of the stepped current t r a n s i e n t would have changed by more than 2.5% at the f i e l d E s t a r t i n g from the same steady s t a t e c u r r e n t . m 5. D i s c u s s i o n The r e s u l t s of the previous two s e c t i o n s show that Dignam's phenom-e n o l o g i c a l equations provide a remarkably good but not p e r f e c t d e s c r i p t i o n both of the stepped current and open c i r c u i t t r a n s i e n t s . That i s , f o r a set of parameter v a l u e s , the experimental J(E) data are reproduced reasonably w e l l by the computed data over a l a r g e range of i n i t i a l c u rrent d e n s i t i e s and temperatures. In a d d i t i o n , the c l o s e agreement between the r a t i o s of the e m p i r i c a l l y determined parameters of the steady s t a t e and stepped f i e l d t r a n s i e n t , as p r e d i c t e d by the equations (see Section 2.4), f u r t h e r supports the v a l i d i t y of the equations. 230 0.0 0.2 0.4 0.6 1 - E ( t ) / E ( 0 ) F i g . 7 . 9 Change i n t o t a l p o l a r i z a t i o n d u r i n g the s e l f - d i s c h a r g e . 231 Notwithstanding the c l o s e agreement between theory and experiment, however, there are a few apparent anomalies which need to be explained. I f the equations provide a general d e s c r i p t i o n of i o n i c conduction, i t would seem reasonable to expect the values of x^  and B_^  to be constant, independent of the type of current t r a n s i e n t being s t u d i e d . This appears to be the case wit h B 2 and B^ (although B^ was taken as zero i n one a n a l y s i s of the open c i r c u i t t r a n s i e n t data). The values of x^» however, c l e a r l y depend on whether the t r a n s i e n t i s a stepped current or open c i r c u i t t r a n s i e n t . There does not appear to be any obvious r e l a t i o n between the values of x2 and X3 obtained f o r these t r a n s i e n t s . A l s o , there does not appear to be an obvious r e l a t i o n between those values of x. and B. and those reported f o r anodic aluminum oxide, 1 1 except to say that B_^  increases as x^  decreases. The f a c t that the a n a l y s i s i s l i m i t e d to two slow p o l a r i z a t i o n terms would a l s o seem to be somewhat a r b i t r a r y . One might expect a large number of terms corresponding to a range of " r e l a x a t i o n " times as suggested by measurements of the capacitance (Young 1956) and " l a t e n t " i o n i c c o n d u c t i v i t y (Young 1964). Taylor and Dignam (1973b) r e j e c t e d the i n c l u s i o n of more than two terms, arguing that a t h i r d term would correspond to an unreasonably l a r g e i n t e r a c t i o n radius f o r the mobile d e f e c t s . This c o n c l u s i o n depends on the i n t e r p r e t a t i o n of the B^ parameter. Another anomaly i s the f a c t that the l o g j ^ - E^ data f o r the stepped current t r a n s i e n t are remarkably l i n e a r over a l a r g e range of f i e l d s whereas a n o n l i n e a r r e l a t i o n i s p r e d i c t e d by Dignam's equations. A r e l a x a t i o n process a c t i n g to p u l l the stepped current data toward the steady s t a t e l i n e would p a r t l y e x p l a i n the discrepancy between the experimental and computed data. However, as mentioned before, i t seems improbable that the p r e d i c t e d n o n l i n e a r i t y would be so e x a c t l y compensated as to give a l i n e a r logJ-E r e l a t i o n . 232 A fact not taken into consideration by Dignam's equations i s the e f f e c t of e l e c t r o l y t e incorporation on i o n i c conductivity. For example, the steady state logJ-E curve for tantalum oxide films grown in d i l u t e H^PO^ i s more nonlinear than f o r films grown i n d i l u t e H^SO^. In addition, the curvature of the steady state l i n e increases dramatically as the concentration of ILjSO^ increases (Young 1960). Dell'Oca (1969) suggested a model i n which the curvature i s due to e l e c t r o l y t e incorporation i n the oxide f i l m , with the r e s u l t i n g two layers having d i f f e r e n t p e r m i t t i v i t i e s . Assuming that the i o n i c conduction i n each layer i s characterized by a l i n e a r logJ-E dependence ( i . e . logJ^cc^E^) and that the r a t i o of the outer layer thickness to t o t a l oxide thickness G^logJ, the n o n l i n e a r i t y of the steady state logJ-E curve (where E i s the average f i e l d across the oxide) i s due to the continuity of the e l e c t r i c displacement across the oxide. Such a model might explain the present r e s u l t s for the logJ - E data. 233 V I I I OPTICAL AND ELECTRICAL PROPERTIES OF THERMAL TANTALUM OXIDE FILMS ON  SILICON 1. I n t r o d u c t i o n The o p t i c a l p r o p e r t i e s of thermally o x i d i z e d tantalum f i l m s de-po s i t e d on s i l i c o n by r f s p u t t e r i n g were i n v e s t i g a t e d by e l l i p s o m e t r y . The e l e c t r i c a l p r o p e r t i e s of the oxide f i l m s were also studied to t e s t t h e i r s u i t a b i l i t y f o r device a p p l i c a t i o n s (Smith and Young 1981). 2. Some Recent A p p l i c a t i o n s of Ta^O^ Films Tantalum oxide f i l m s have been used f o r s e v e r a l years as a d i e l e c t r i c m a t e r i a l i n t h i n f i l m RC c i r c u i t s (Westwood et a l 1975) and e l e c t r o l y t i c c a p a c i t o r s (Harrop and Campbell 1968). More r e c e n t l y , tantalum oxide f i l m s have a l s o been used i n s p e c i a l i z e d a p p l i c a t i o n s i n s i l i c o n devices. For example, because of i t s r e l a t i v e l y high p e r m i t t i v i t y , anodic la^O^ has been used as the storage c a p a c i t o r d i e l e c t r i c i n high d e n s i t y dynamic random access memories (Tarui 1980). The anodic oxide has als o been used as a gate i n s u l a t o r i n a t h i n f i l m t r a n s i s t o r ( K a l l f o s s and Lueder 1979) and as a nonl i n e a r r e s i s t o r (Baraff et a l 1981) i n devices intended f o r addressing d i s p l a y s . Anodic Ta^O^ has a l s o been i n v e s t i g a t e d as an a l t e r n a t i v e to the n a t i v e oxide on GaAs de-v i c e s ( N i s h i and Revesz 1979). Thermally grown Ta20,- i s used as an a n t i r e f l e c t i o n (AR) coating on s i l i c o n s o l a r c e l l s . The oxide i s n e a r l y i d e a l l y s u i t e d to t h i s a p p l i c a t i o n since i t s r e f r a c t i v e index i s near the optimum value which, f o r the appropriate f i l m t h i c k n e s s , reduces to ne a r l y zero the r e f l e c t i v i t y of the s i l i c o n s u r f a ce. (Revesz, A l l i s o n , and Reynolds 1976). Combined w i t h i t s l a r g e bandgap (^  4.1 eV) and r e p o r t e d l y low Ta^O^/Si i n t e r f a c e recombination v e l o c i t y (Revesz and A l l i s o n 1976), the thermal oxide has been reported to be e f f e c t i v e 234 i n i n c r e a s i n g the e f f i c i e n c y of pn j u n c t i o n c e l l s i n the short wavelength (0.31 - 0.45 um) r e g i o n of the spectrum. Revesz et a l reported f i l l f a c t o r s i n the range 0.78 - 0.80 and power output of 180 W/m f o r s o - c a l l e d v i o l e t c e l l s having a Ta^O^ AR c o a t i n g . These c e l l s were al s o found to be l e s s s u s c e p t i b l e to damage by exposure to UV r a d i a t i o n than conventional c e l l s using SiO AR c o a t i n g , x Ta^O^ f i l m s have a l s o been used on s i l i c o n MIS s o l a r c e l l s (Thomas et a l 1980). In a d d i t i o n to p r o v i d i n g an AR coat, the oxide was used as a source of p o s i t i v e charge w i t h which the s i l i c o n surface was i n v e r t e d thereby c r e a t i n g a shallow pn j u n c t i o n . Although thermally grown and evaporated Ta 0^ f i l m s were more e f f i c i e n t AR c o a t i n g s , Ta^O^ f i l m s spun onto the c e l l from an alcohol-based s o l u t i o n gave c e l l s w i t h the highest conversion e f f i c i e n c y (17% under simulated AMI c o n d i t i o n s ) . Tantalum pertoxide f i l m s may a l s o be u s e f u l i n i n t e g r a t e d o p t i c s . Low l o s s o p t i c a l waveguides on fused quartz or g l a s s substrates have been made by r e a c t i v e s p u t t e r i n g (Paulson et a l 1979) , s p u t t e r i n g from a T^*^ " composite t a r g e t ( T e r u i and Kobayashi 1978) and thermal o x i d a t i o n of a de-p o s i t e d Ta f i l m (Hensler et a l 1971). 3. Experimental Procedure 3.1 Sample Preparation The r e s i s t i v i t y of the s i l i c o n s u b s t r a t e s (n and p-type doping, <100> and <111> o r i e n t a t i o n s ) was measured w i t h a f o u r - p o i n t probe and the impurity c o n c e n t r a t i o n determined from I r v i n ' s chart of r e s i s t i v i t y p l o t t e d against doping d e n s i t y (Sze 1969, page 43). A f t e r u l t r a s o n i c c l e a n i n g i n hot methanol f o r 10 minutes, the samples were chemi c a l l y cleaned by the RCA a c i d -peroxide process (Kern and Puotinen 1970) as f o l l o w s : 235 (1) 10 minutes i n a s o l u t i o n of 5:1:1 pa r t s by volume H^ O ( d e i o n i z e d ) , 30% H 20 2 and 58% NH^OH heated to a temperature i n the range 75-85°C. (2) 10 minute r i n s e i n deioniz e d water. (3) 30 second dip i n 10% HF. (4) 10 minute r i n s e i n deionized water (5) 10 minutes i n a s o l u t i o n of 5:1:1 pa r t s by volume H^ O ( d e i o n i z e d ) , 38% HC1, and 30% HO heated to a temperature i n the range 75-85°C. (6) 10 minute r i n s e i n deioniz e d water. A l l glassware had p r e v i o u s l y been cleaned i n a s o l u t i o n of 1:1 pa r t s by volume 98% ll^SO^ and 30% ^2^2 w a s u s e c * o n l y f ° r t n e RCA c l e a n i n g procedure. The samples were then blown dry with n i t r o g e n and examined under a microscope f o r dust or other l a r g e contaminants. 3.2 Sputte r i n g Immediately a f t e r being cleaned, tantalum f i l m s were deposited on the s i l i c o n substrates by r f s p u t t e r i n g i n argon using a Perkin-Elmer 3140 s i n g l e - t a r g e t system. The 6 inch diameter tantalum target (99.95% p a r i t y ) was obtained from V a r i a n . The substrate t a b l e and target were separated by about 5 cm. Before a d m i t t i n g the argon, the s p u t t e r i n g chamber was evacuated to a pressure l e s s than 1.3 x 10 Pascals (10 t o r r ) as i n d i c a t e d by an i o n i z a t i o n gauge attached to the baseplate. The b u t t e r f l y valve i n the throat of the s p u t t e r i n g system was then close d and the argon pressure i n the _2 chamber adjusted to 3.3 Pascals (2.5 x 10 t o r r ) , as i n d i c a t e d by a pressure gauge lo c a t e d above the b u t t e r f l y v a l v e . The spu t t e r d e p o s i t i o n r a t e was determined by d e p o s i t i n g tantalum f i l m s on g l a s s s l i d e s f o r f i x e d times, w i t h part of the s l i d e covered by a 236 s i l i c o n wafer (To ensure a r e l a t i v e l y uniform d e p o s i t i o n across the s l i d e , the s l i d e was placed at the center of the s p u t t e r i n g t a b l e . ) - The mask was then removed and a tantalum f i l m deposited over the e n t i r e s l i d e . The step height at the mask edge was then measured w i t h a Sloon Angstrometer. Figure 7.1 shows the measured f i l m t h i c k n e s s p l o t t e d against time f o r s p u t t e r i n g at 3 forward power l e v e l s . The r e f l e c t e d power was always kept l e s s than 10 watts. Figure 7.2 shows the d e p o s i t i o n r a t e (deduced from the data i n Figure 7.1 ) p l o t t e d as a f u n c t i o n of the forward r f power. In the f o l l o w i n g work, d e p o s i t i o n s were done at a s p u t t e r i n g power of 200 watts forward power, g i v i n g a d e p o s i t i o n r a t e of about 15 nm/min. F i l m t h i c k n e s s e s , estimated from the s p u t t e r i n g time, were i n the range 10-100 nm. The deposited and o x i d i z e d tantalum f i l m s e x h i b i t e d e x c e l l e n t adhesion to the s i l i c o n sub-s t r a t e s . 3.3 Oxidation The tantalum f i l m s were o x i d i z e d i n a 6.A cm diameter r e s i s t a n c e heated quartz-tube furnace (Mini-Brute) at 500°C with an oxygen flow r a t e of 1 1/min At t h i s temperature, s i l i c i d e formation was not expected ( C h r i s t o u and Day 1976). Previous s t u d i e s have a l s o shown the oxide f i l m s to be non-c r y s t a l l i n e (Revesz, A l l i s o n , K i r k e n d a l l and Reynolds 1974) and s t o i c h i o m e t r i c Ta^O^ (Revesz, Reynolds, and A l l i s o n 1976). 3.4 E l l i p s o m e t r y A l l e l l i p s o m e t r y measurements were made at an angle of incidence of 70°, A = 632.8 nm and i n zones 1 and 3 as disucssed i n Chapter 3. The e l l i p s o m e t e r angles A and ty were obtained by averaging between the zones. F i g . 8.1 Dependence of the Ta f i l m t h i c k n e s s on the s p u t t e r i n g t i m e . F i g . 8.2 The d e p o s i t i o n r a t e as a f u n c t i o n of the s p u t t e r i n g power. 238 4. Results 4.1 Time of Oxidation To determine how long the tantalum f i l m s needed to be o x i d i z e d i n order to get complete conversion of metal to oxide, samples of various thickne were p e r i o d i c a l l y removed from the furnace and measured w i t h the e l l i p s o m e t e r . T y p i c a l data i n Figure 8.3 show that a f t e r a time, which v a r i e d from 15 minutes f o r 10 nm of Ta to ^ 2 . 5 hours f o r 80 nm of Ta, only a very slow r a t e of change of the e l l i p s o m e t e r angles ty and A was found, so that any f u r t h e r slow change could be ascribed to s t r u c t u r a l annealing e f f e c t s and slow changes i n s t o i c h i o m e t r y . The r e f r a c t i v e index of the oxide has been reported to change continuously w i t h continued heat treatment (Revesz et a l 1974) but t h i s was not i n v e s t i g a t e d . 4.2 Model f o r O p t i c a l P r o p e r t i e s Figure 8.4 shows ty and A values f o r a s e r i e s of tantalum f i l m s which had been deposited on p-type, 0.03 fi-m (111) o r i e n t e d s i l i c o n and f u l l y o x i d i z e d i n the above sense. The data f o r the second c y c l e of the (ty,&)-curve l i e c l o s e to those f o r the f i r s t c y c l e . This shows that l i g h t absorp-t i o n by the f i l m and any b i r e f r i n g e n c e i n the f i l m due to thermal s t r e s s e s was not s i g n i f i c a n t e l l i p s o m e t r i c a l l y . The (<JJ,A) data were found to be not very f a r from c a l c u l a t e d curves f o r a homogeneous oxide of constant r e f r a c t i v e index. However, the d i f f e r e n c e was s i g n i f i c a n t . Figures 8.5(a) and(b)show a comparison w i t h curves which were computed f o r homogeneous oxides of i n d i c e s 2.20, 2.22 and 2.24 on a substrate of index n g = 3.87 - j0.025. The data f o r A > 180° agree reasonably w e l l w i t h the computed curve f o r n = 2.24 w h i l e the data f o r A < 180° agree b e t t e r w i t h the computed curve f o r n = 2.20. A reasonable f i t to a l l of these 239 0 15 30 4 5 6 0 MINUTES F i g . 8.3 Y and A measured at intervals during thermal oxidation of a 40 nm thick Ta f i l m on s i l i c o n . 240 80 Y / D E G F i g . 8.4 Ellipsometry data for several completely oxidized films. 0 = f i r s t cycle, • = second cycle. The s o l i d l i n e was computed for the i n t e r f a c i a l f i l m model with n^ = 2.22 and n = 3.87 - jO.025. Bulk oxide thicknesses (nm) are indicated s for the f i r s t cycle. 100 O l i i i i 1 1 15 20 25 30 Y/DEG F i g . 8.5 The d a t a o f f i g . 8.4 w i t h computed and n = 3.87 - jO.025. ip/DEG curves f o r s i n g l e l a y e r homogeneous f i l m s 242 data can be obtained w i t h n = 2.22 i f a substrate index of n = 3.76 - i0.585 s J i s used but t h i s s u b s t r a t e index i s unacceptably f a r from w e l l - e s t a b l i s h e d published data (Hopper, Clarke and Young 1975, Chang and B o u l i n 1977). The d i f f e r e n c e between the data and computed curves was too l a r g e to be due to measurement e r r o r . A l s o , the e l l i p s o m e t e r was r e c a l i b r a t e d and some samples measured again w i t h no s i g n i f i c a n t d i f f e r e n c e i n the r e s u l t s . In previous work (Revesz et a l 1974; Revesz, Reynolds, and A l l i s o n 1976), e l l i p s o m e t r y i n d i c a t e d a graded r e f r a c t i v e index through the whole thickness of the oxide. By c o n t r a s t , i t was found that the |present data, which are f o r f i l m s made by s l i g h t l y d i f f e r e n t procedures, could be f i t t e d by a model with a r e l a t i v e l y narrow t r a n s i t i o n l a y e r at the s i l i c o n i n t e r f a c e of constant thickness independent of the t o t a l f i l m t h i c k n e s s , and the r e s t of the f i l m homogeneous. This r e s u l t i s not s u r p r i s i n g since one would expect a n a t i v e SiO^ l a y e r on the s i l i c o n s u b s t r a t e s . E l l i p s o m e t e r measurements of some s i l i c o n samples immediately p r i o r to d e p o s i t i n g the tantalum f i l m i n d i c a t e d the presence of a f i l m between 2 and 2.5 nm t h i c k . The t r a n s i t i o n l a y e r w i t h tapered index was represented by f i v e l a y e r s each of 1.5 nm thickness and w i t h i n d i c e s as shown i n Figure 8.6(a). The r e f r a c t i v e i n d i c e s of the l a y e r s were chosen to range approximately from the index of S i 0 2 (n = 1.46) to that of the bulk of the f i l m . The bulk of the oxide f i l m had an index of 2.22 which i s w i t h i n 1% of the value f o r anodic oxide f i l m s grown i n d i l u t e H^SO^ (see Chapter 4 ). The substrate was taken as having a r e f r a c t i v e index n g = 3.87 - jO.025, i n agreement w i t h published values (Hopper et a l 1975, Chang et a l 1977). A comparison of the (ty,A) data w i t h the model i s shown i n Figures 8.7(a) and (b) . C l e a r l y the data are i n c l o s e agreement w i t h the model. 243 n 2.2 1.8 1.4 n— BULK OF THE OXIDE 4 5-d/i SURFACE nm F i g . 8.6(a) V a r i a t i o n of r e f r a c t i v e index w i t h d i s t a n c e from the S i - o x i d e i n t e r f a c e . x S i 0 2 .5 d / n m F i g . 8.6(b) The c a l c u l a t e d mole f r a c t i o n of 3 i 0 ? i n the i n t e r f a c e r e g i o n . 245 To check the c o r r e c t n e s s of the model, etchback experiments were performed. Two samples w i t h oxide thicknesses of 130 nm and 220 nm were etched back i n stages and measured w i t h the e l l i p s o m e t e r at each stage. The etching s o l u t i o n (80g NH^F i n 200 ml 48% HF) gave an etch r a t e of ^ 6 nm/min, as reported by P r i n g l e (1972), when f r e s h l y prepared. Except f o r p r e f e r e n t i a l etching of apparent weak spots i n very t h i n oxide f i l m s , the oxide was etched unifo r m l y over the sample surface. A f t e r each e t c h , the samples were r i n s e d w i t h deionized water f o r at l e a s t 5 minutes and then blown dry w i t h n i t r o g e n . The e l l i p s o m e t e r data are shown i n Figures 8.8(a) and (b) together w i t h computed (i^,A) data f o r the t r a n s i t i o n l a y e r model discussed above. The f i t i s good, thus showing that the outer part of the f i l m i s homogeneous. 4.3 Disc u s s i o n of the O p t i c a l Model Revesz et a l a t t r i b u t e d the gradient of r e f r a c t i v e index which they found through t h e i r f i l m s to i n c o r p o r a t i o n of s i l i c o n , p r i m a r i l y i n i t s o x i d i z e d form, i n the tantalum oxide. Their secondary ion mass spectroscopy r e s u l t s (Revesz and K i r k e n d a l l 1976) i n d i c a t e d the presence of s i l i c o n but Rutherford b a c k s c a t t e r i n g s t u d i e s d i d not (Hiroven, Revesz and K i r k e n d a l l 1976). Following these authors we may use the Lorentz-Lorenz equation 2 2 (n - l ) / ( n + 2) = (1/3) £ N i a i ' w n e r e and are the c o n c e n t r a t i o n and molecular p o l a r i z a b i l i t y of the i t h molecular species, to estimate the SiO^ content needed to g i v e the tapered index i n the model f i t t e d to our f i l m s . The r e s u l t i s shown i n Figure 8.6(b) where x i s the c a l c u l a t e d mole f r a c -t i o n of Si02. The d e n s i t i e s and molecular p o l a r i z a b i l i t i e s used were 7.9 x 10 3 kg/m3 and 1.58 x 1 0 ~ 2 3 m3 f o r T a ^ , and 2.2 x 10 3 kg/m3 and —2 9 3 3.72 x 10 m f o r S i 0 2 (Revesz, Reynolds, and A l l i s o n 197 6). The explana-t i o n of the tapered index as due to i n c o r p o r a t i o n of Si02 seems very reasonable. t / D E G Y / D E G F i g . 8.8 Etchback d a t a f o r i n i t i a l o x i d e t h i c k n e s s e s of 130 nm (0) and 220 nm ( t ) . B u l k £ ON oxide t h i c k n e s s e s (nm) a r e i n d i c a t e d f o r the f i r s t c y c l e ( i n t e r f a c i a l f i l m model). 247 As regards the d i f f e r e n c e with previous work, we have already noted that the preparative methods are somewhat d i f f e r e n t . In p a r t i c u l a r , Revesz et a l used electron beam evaporated tantalum f i l m s and t y p i c a l l y oxidized a 28 nm t h i c k f i l m for 15 minutes at 530°C (Revesz and A l l i s o n 197 6). In addition, t h e i r ellipsometer measurements were made at a d i f f e r e n t wave-length (A = 546.1 nm). Incorporation of s i l i c o n into the Ta^O^ could involve absorption of the native oxide (^ 2 nm) o r i g i n a l l y present on the substrate or the transfer of s i l i c o n from the substrate to the Ta^O^. Both processes would c l e a r l y depend on the time and temperature used f o r the oxidation. They may also be affected by s t r u c t u r a l and compositional differences between sputtered and electron-beam-evaporated f i l m s . A d i f f u s i o n - a i d e d implantation process during sputtering (Murty, L a l e v i c , Lee, Suga, and Weissmann 1978) also may have contributed to the formation of a t r a n s i t i o n layer. 4.4 E l e c t r i c a l Properties Al/Ta^O^/Si MOS capacitors were made by thermal evaporation of aluminum through a metal s t e n c i l mask to a thickness of about 500 nm as indicated by a f i l m thickness monitor (Inficon model 321). The monitor had previously been c a l i b r a t e d against f i l m thicknesses measured with a Sloan Angstrometer. The area of the c i r c u l a r aluminum electrodes was determined by measuring the electrode diameter i n 2 perpendicular d i r e c t i o n s with a -7 2 t r a v e l l i n g microscope. T y p i c a l l y , the electrode area was about 3.5 x 10 m . 4.4.1 Current-Voltage Dependence The MOS sample was placed inside a grounded metal box and current-voltage measurements made at room temperature (^  22°C) point by point with a Keithley 602 electrometer. To ensure good e l e c t r i c a l contact to the substrate, 248 small pieces of gold f o i l were placed between the substrate and brass block on which the sample was placed. The measurements were recorded about 1 minute a f t e r applying voltage. Figure 8.9 i l l u s t r a t e s the agreement between two d i f f e r e n t MOS capacitors on the same substrate (p-type, <111> o r i e n t a t i o n , 0.06 ft-m). The deposited tantalum f i l m had been oxidized for a period determined from the o p t i c a l studies discussed in Section 3.1. The good agreement between the data was t y p i c a l and seems to s i g n i f y that pinholes were not too s i g n i f i c a n t . (With anodic film s , weak spots often contribute much of the current (Vermilyea 1965)). Similar r e s u l t s were also obtained with n-type, <100> oriented sub-strates . r Figure 8.10 shows data which indicate a decrease in current with increasing time of oxidation (beyond the time giving constant (ij;,A))up to a c e r t a i n time beyond which no further improvement was obtained. Such an e f f e c t could be due to the completion of conversion to oxide, to better stoichiometry, to s t r u c t u r a l changes, or to incorporation of SiOj^ * ° n e s P e c : i - m e n w a s annealed in hydrogen at 350°C for 15 minutes with the electrodes in place. This gave a s i g n i f i c a n t d e t e r i o r a t i o n in r e s i s t i v i t y . The data of Kaplan, Balog and Frohman-Bentchowsky (1976) for CVD la.^)^ are also reproduced in Figure 8.10. Evidently, the present films are generally comparable to the CVD f i l m s , or even s l i g h t l y better, given s u f f i c i e n t oxidation time. The conclusion of Kaplan et a l that t h e i r CVD films would be suitable for a second but not a f i r s t d i e l e c t r i c layers seems equally a p p l i c a b l e . However, the present r e -s u l t s may be a f f e c t e d by use of aluminum as the gate electrode. The current-voltage dependence with other gate metals was not investigated. The current tended to decrease with time during the measurement. This e f f e c t was more important f o r MOS capacitors giving lower currents. 249 c s i I E < o o CJ) O -2 <f NEGATIVE GATE o o o o o • • o* o • o o o • o • o POSITIVE GATE i i i i i I 0 10 20 E V (10 3 V , /2m' /2) F i g . 8.9 J(E) data for two MIS capacitors on the same p-type substrate ( ( i l l ) orientation, 0.06 ohm-m). The Ta f i l m (70 nm thick) was oxidized for 2.5 h giving an oxide thickness of 213 nm. 250 F i g . 8.10 The e f f e c t of the l e n g t h of o x i d a t i o n on the c o n d u c t i v i t y o f the o x i d e f i l m s . F o r an i n i t i a l Ta f i l m t h i c k n e s s of 40 nm on p-type s u b s t r a t e s ( ( i l l ) o r i e n t a t i o n , 0.06 ohm-m), the o x i d a t i o n times were 1 h ( + ), 1.5 h (0) and 3 h (•). The d a t a r e p o r t e d by K a p l a n et a l f o r CVD T a 2 0 5 on p-type s i l i c o n a r e reproduced f o r comparison. 251 This i s b e l i e v e d to be due to the increased importance of d i e l e c t r i c p o l a r i z a -t i o n current (associated w i t h Debye type processes of long r e l a x a t i o n times) i n r e l a t i o n to the conduction c u r r e n t . However, space change e f f e c t s could a l s o , i n p r i n c i p l e , give such an e f f e c t . In the forward conduction p l o t s ( n e g a t i v e l y biased gate f o r p-type substrates) of l o g j vs E £ given i n Figure 8.9, two l i n e a r segments were found, followed at higher f i e l d s by a region i n which the current increased more r a p i d l y w i t h f i e l d . Excursions i n t o t h i s region gave i r r e v e r s i b l y i n -creased current at lower f i e l d s . This breakdown e f f e c t set i n at about g 2 x 10 V/m. Under reverse b i a s , a s i m i l a r e f f e c t occurred but at higher 9 f i e l d s (y 10 V/m), comparable to the breakdown f i e l d s t r e n g t h g e n e r a l l y observed w i t h thermally grown Si02- T ^ £ r e v e r s e b i - a s current f o r a l l samples (both p- and n-type substrates and i n c l u d i n g those w i t h longer 2 o x i d a t i o n times), saturated at about 0.1 A/m . h The existence of two l i n e a r segments i n the l o g J vs E p l o t s agrees w i t h the r e s u l t s of Kaplan et a l (197 6) f o r CVD la.^^ o n p-type s i l i c o n but c o n t r a s t s w i t h the r e s u l t s of Angle and T a l l e y (1978) f o r thermally o x i d i z e d tantalum f i l m s . S i m i l a r behaviour has been reported by Mackus et a l (1977) f o r V^O^ on n-type s i l i c o n and by Murty et a l (1978) f o r thermally grown Si02 ^^^ m s w i t n sputtered Ta gate e l e c t r o d e s . The r e s u l t s of the l a t t e r reference suggests an i n t e r a c t i o n i n the present f i l m s of the deposited Ta and the n a t i v e oxide p r i o r to o x i d a t i o n . I f the slope d l o g J / d E 5 f o r the higher f i e l d segment i n Figure 8.9 i s i n t e r p r e t e d as equal to q/qMire^e^ / (2.303 kT) , where q = e l e c t r o n i c charge, E q = p e r m i t t i v i t y of f r e e space, and = o p t i c a l r e l a t i v e p e r m i t t i v i t y , the reasonable value of /e % 2 i s found (which compares w i t h the observed r e f r a c t i v e index of 2.2 at A = 632.8 nm). The lower f i e l d segment gave an i 252 unreasonable value f o r the r e f r a c t i v e index ( l e s s than u n i t y ) . The explana-t i o n proposed by Mackus et a l f o r t h e i r ^2®$ f i l m s may be e q u a l l y a p p l i c a b l e . I t i n v o l v e d a higher l o c a l f i e l d near the i n t e r f a c e due to space charge e f f e c t s . With the present f i l m s , the e f f e c t may a l s o r e l a t e to the t r a n s i t i o n l a y e r discussed i n Section 3.2. The above r e s u l t f o r the high f i e l d segment suggests an e l e c t r o d e -l i m i t e d conduction mechanism ( i . e . Schottky conduction) f o r the present f i l m s . To o b t a i n s u f f i c i e n t evidence, however, measurements would have to be made over a range of temperatures. The e f f e c t of the metal used f o r the gate electrode would a l s o have to be i n v e s t i g a t e d . F i n a l l y , the r e s u l t s i n Figure 8.10 i n d i c a t e a dependence of the conduction mechanism on the annealing h i s t o r y of the oxide f i l m , i n agreement w i t h recent data f o r anodic Ta^O^ (Matsumoto, Suzuki, and Yabumoto 1980). 4.4.2 Capacitance-Voltage Dependence The capacitance-voltage dependence of the MOS c a p a c i t o r s was measured point by point at s e v e r a l frequencies w i t h a Boonton 75C capacitance bridge. The ac s i g n a l was l e s s than 20 mV peak-to-peak and the measurements were recorded about one minute a f t e r a p p lying v o l t a g e . Figure 8.11 shows the C(V) data f o r the MOS c a p a c i t o r s used to obtain curve A i n Figure 8.10. The data were obtained immediately a f t e r making the c a p a c i t o r s and showed n e g l i g i b l e h y s t e r e s i s e f f e c t s . The strong frequency dependence of the capacitance at negative b i a s i s t y p i c a l of that observed f o r f i l m s that had been o x i d i z e d f o r the minimum time r e q u i r e d to give constant (I|J,A) data. The extent of the frequency dependence v a r i e d from one sample to the next. In a l l cases, however, the capacitance tended to a constant value of high frequency. 253 The frequency dependence might be r e l a t e d to the t r a n s i t i o n l a y e r of the oxide. A d i f f e r e n c e i n c o n d u c t i v i t y between the t r a n s i t i o n region and the bulk of the oxide would lead to an apparent frequency dependence (the Maxwell-Wagner e f f e c t ) . Microscopic i n s p e c t i o n of the gate e l e c t r o d e d i s -counted a frequency dependence due to taper i n g of the e l e c t r o d e around i t s perimeter (Zaininger 1964). Figure 8.12 shows the C(V) data obtained a t 5 kHz and 500 kHz f o r a sample w i t h 40 nm of Ta o x i d i z e d f o r 3 hours (curve B i n Figure 8.10). The capacitance i n accumulation (negative gate voltage) was almost independent of frequency, changing by l e s s than 2% over the above frequency range. I d e n t i f y i n g t h i s capacitance as being due to the oxide gave the r e l a t i v e p e r m i t t i v i t y of the oxide as 26 ± 2, based on measurements of s e v e r a l samples on the same sub s t r a t e . This r e s u l t i s i n the range f o r anodic Ta20,- and al s o f o r CVD f i l m s (Kaplan et a l 1976; Wang, Zaininger, and Duffy 1970) but i s more than twice the value reported by Revesz and A l l i s o n (1976). Taking the work f u n c t i o n of aluminum as 4.1 eV (Sze 1969), the gate voltage g i v i n g the c a l c u l a -ted f l a t - b a n d capacitance of samples on t h i s s ubstrate i n d i c a t e d a l a r g e 16 2 negative charge on the oxide at f l a t - b a n d , equivalent to 5 ± 2 x 10 q/m . This i s the opposite s i g n of charge to that normally observed on Si02 ^ u t ^ S s i m i l a r to the value found by Revesz and A l l i s o n (1976) . When the sample was annealed i n hydrogen at 350°C f o r 15 minutes w i t h the electr o d e s i n place, the MOS c a p a c i t o r s gave a strong frequency dependence i n accummulation. I d e n t i f y i n g the accummulation capacitance of the annealed sample, measured at 500 kHz, w i t h the oxide capacitance gave a p o s i t i v e oxide charge about equal i n magnitude to the above negative charge. 3 c^cocu 0-o-o-o-QI 1 1 1 1 1 - 8 0 8 B I A S V O L T A G E / VOLTS F i g . 8.11 C a p a c i t a n c e as a f u n c t i o n of b i a s v o l t a g e f o r the MIS c a p a c i t o r used t o o b t a i n curve A i n f i g . 8.10. Lu o 0 8 BIAS VOLTAGE (VOLTS) F i g . 8.12 C(V) d a t a f o r a 40 nm t h i c k Ta f i l m on p-type s i l i c o n ((111) o r i e n t a t i o n , 0.06 ohm-m) o x i d i z e d f o r 3 h. Measurement f r e q u e n c i e s were 5 kHz (•) and 500 kHz ( 0 ) . 255 5. Summary Sputtered tantalum f i l m s on s i l i c o n t hermally o x i d i z e d at 500°C were found by e l l i p s o m e t r y to be of uniform r e f r a c t i v e index (2.22 at 632.8 nm) except f o r a t h i n t r a n s i t i o n l a y e r of tapered lower index l o c a t e d at the s i l i c o n i n t e r f a c e . The r e l a t i v e p e r m i t t i v i t y was 26 ± 2, s i m i l a r to that of anodic and CVD tantalum p e r t o x i d e f i l m s . Al/Ta^O^/Si MOS c a p a c i t o r s showed leakage c u r r e n t s which depended on o x i d a t i o n time but comparable to or s l i g h t l y b e t t e r than devices made w i t h CVD la^O^ (and worse by s e v e r a l decades than given by devices using thermal SiO^). Reasonable C(V) p l o t s were obtained but these e x h i b i t e d l a r g e h y s t e r e s i s and t y p i c a l l y i n d i c a t e d a l a r g e negative oxide change at f l a t band. These r e s u l t s i n d i c a t e that the present oxide f i l m s would be s u i t a b l e f o r a second but not a f i r s t d i e l e c t r i c l a y e r i n a s i l i c o n device. IX. CONCLUSIONS Anodic oxide films are of increasing interest i n semiconductor as well as i n other device applications. An understanding of high f i e l d i o n i c conduc-t i v i t y i n these oxide films i s of technical as well as theoretical i n t e r e s t . In the present work, the op t i c a l and e l e c t r i c a l properties of anodic oxide films on tantalum were investigated by computer-controlled methods. The topics covered i n this work included (1) i n s i t u ellipsometry of anodic Ta 20 5 films grown i n dil u t e I^SO^, (2) an analysis of data obtained for transient conditions and comparison of these data with the predictions of a model for the ionic conduc-tion process, and (3) an Investigation of the op t i c a l and e l e c t r i c a l properties of Ta 20 5 films on s i l i c o n . The results of this work can be summarized as follows: (1) Using a new computer control program, anodic Ta,,05 films grown i n dilut e H^ SO^  were found by ellipsometry at 632.8 nm wavelength to be nonabsorbing, and isotropic at zero f i e l d . The ellipsometer data also (a) indicated a thin i n t e r f a c i a l layer between the homogeneous oxide and Ta substrate, due to surface preparation, (b) gave oxide thicknesses i n agreement with previous data obtained by spectrophotometry, (c) indicated a constant elec-t r i c f i e l d i n oxide films grown at constant current density, and, (d) com-bined with measurements of the oxide capacitance, gave the oxide permit-t i v i t y as e r = 27.6, i n agreement with a previously reported value. As regards the high f i e l d ionic conduction, several contributions were made. (2) I t i s argued that the observed history dependence of the J(E) re l a t i o n i s more probably due to changes i n the structure of the oxide f i l m than to current-driven polarization as claimed by Dignam. Based on such a struc-t u r a l model, i t was shown that equations can be derived which are equiva-lent to those of Dignam. Thus, the success of these equations does not 257 prove the v a l i d i t y of the o r i g i n a l model. The double layer nature of the films should also probably be taken into account. Analysis of the open c i r c u i t transient data confirmed the previously reported r e s u l t s that the deviation of the E(t) data about a f i t t e d l i n e a r l o g j vs. E r e l a t i o n with constant p e r m i t t i v i t y are not random. However, the re s u l t s strongly depend on the extent of the transient included i n the an a l y s i s . The problem with this method i s some assumptions must be made regarding the f i e l d dependence of e i t h e r or both the oxide p e r m i t t i v i t y and i o n i c current. The i n i t i a l slope kT d(logJ)/dE obtained from the open c i r c u i t transient data and i t s dependence on the steady state formation f i e l d i s close to previously reported data for the stepped f i e l d transient. The p e r m i t t i v i t y of the oxide f i l m , obtained by extrapolating the results of the analysis to t = 0 + i s close to the value measured at 1 kHz. Assuming a constant p e r m i t t i v i t y for the open c i r c u i t self-discharge, numerical d i f f e r e n t i a t i o n of the E(t) data indicated a nonlinear l o g j vs. E r e l a t i o n . The n o n l i n e a r i t y increased with increasing formation current. The l o g J 2 vs. Ep r e l a t i o n from the stepped current transient data was l i n e a r over a large range of current. As with the open c i r c u i t t ransient, the slopes kT k ( l o g J 2 ) / d E p were also l i n e a r l y dependent on the i n i t i a l steady state f i e l d . Both the open c i r c u i t and stepped current transient J(E) data are described quite well by the phenomenological equations of Dignam's d i e l e c t r i c p o l a r i z a t i o n theory. However, the equations do not predict the l i n e a r i t y of the logJ 2~Ep data for the stepped current transient. Moreover, d i f -ferent sets of parameters must be used to describe the two t r a n s i e n t s . 258 (8) Thermally oxidized sputtered Ta films on s i l i c o n were shown to be homo-geneous i s o t r o p i c films except for a narrow t r a n s i t i o n layer at the Si i n t e r f a c e . The p e r m i t t i v i t y of the oxide i s comparable to the value for the anodic oxide. The r e l a t i v e l y large leakage current observed for these films indicate that as presently prepared they would be suitable as the second but not the f i r s t d i e l e c t r i c layer i n a Si MIS device. The r e l a -t i v e l y large oxide charge and i t s dependence on annealing history might also be useful i n c e r t a i n solar c e l l a p p l i c a t i o n s . 9.1 Suggestions for Future Work Accurate experimental data are needed to determine the f i e l d depend-ence of the stepped f i e l d transient over a large range of f i e l d s to check the l i n e a r i t y of the logJ-E r e l a t i o n for these conditions. Information i s needed on the e f f e c t of e l e c t r o l y t e incorporation on the constant f i e l d , stepped f i e l d , stepped current, and open c i r c u i t transients. More data for i o n i c current transients are required for other metals such as niobium and aluminum to further test the phenomenological equations of the d i e l e c t r i c p o l a r i z a t i o n theory. 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