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Microwave dielectric measurements on MEM(TCNQ)₂ and TTF-TCNQ Morrow, Michael Robert 1979

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MICROWAVE DIELECTRIC MEASUREMENTS ON MEM (TCNQ)4 AND TTF-TCNQ by MICHAEL ROBERT MORROW B.Sc. McMaster U n i v e r s i t y , 1977 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November, 1979 ©Michael Robert Morrow, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f 7 The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 D a t e ^ — • 6 i i ABSTRACT The microwave d i e l e c t r i c c o n s t a n t and c o n d u c t i v i t y o f MEM(TCNQ)a were s t u d i e d i n the neighbourhood o f the monomer to dimer t r a n s i t i o n a t 61°C u s i n g c a v i t y - p e r t u r b a t i o n t e c h n i q u e s a t 9 GHz. The c o n d u c t i v i t y was found to be i n g e n e r a l agreement w i t h f o u r - p r o b e d.c. r e s u l t s . Thus, doubts about the r e l i a b i l i t y o f the d.c. measurements above the d e s t r u c t i v e t r a n s i t i o n have been removed. The complex d i e l e c t r i c c o n s t a n t o f TTF-TCNQ a t l i q u i d h e l i u m t e m p e r a t u r e s was s t u d i e d u s i n g d i e l e c t r i c resonance t e c h n i q u e s . Some anomalies r e g a r d i n g the i n t e r p r e t a t i o n o f the d i e l e c t r i c resonance mode p l o t s were r e s o l v e d . V a l u e s f o r €^ o f (3.0-0.4)X10 3 and f o r o f g r e a t e r than 9 were i m p l i e d by the r e s u l t s . F i n a l l y , p r e l i m i n a r y r e s u l t s and proposed d i r e c t i o n s f o r a b o l o m e t r i c a b s o r p t i o n measurement i n TTF-TCNQ i n the microwave bands a r e p r e s e n t e d . T h i s t e c h n i q u e may prove u s e f u l f o r the d i r e c t o b s e r v a t i o n o f the pinned charge d e n s i t y wave i n TTF-TCNQ. i i i TABLE OF CONTENTS Page A b s t r a c t T a b l e o f Co n t e n t s L i s t o f T a b l e s L i s t o f F i g u r e s Acknowledgements CHAPTER I - INTRODUCTION 1 1.1 Quasi-One-Dimensional TCNQ S a l t s 2 1.2 MEM (TCNQ)^ and the E f f e c t o f I n t e r a c t i o n s i n TCNQ S t a c k s 7 1.2a MEM(TCNQ^ 1 1.2b Use o f the Hubbard Model to D e s c r i b e TCNQ S t a c k s 14 1.2c Purpose o f the C a v i t y P e r t u r b a t i o n Measurements on MEM(TCNQ)^ 20 1.3 TTF-TCNQ and the Pinned Charge D e n s i t y Wave 21 1.3a TTF-TCNQ 21 1.3b D i e l e c t r i c C o n s t a n t f o r a Pinned Charge D e n s i t y Wave 25 1.3c Purpose o f D i e l e c t r i c Resonance s t u d i e s on TTF-TCNQ 29 CHAPTER I I - THE MEASUREMENT OF ELECTRICAL PROPERTIES OF MEM(TCNQ) ABOVE ROOM TEMPERATURE 31 2.1 C a v i t y p e r t u r b a t i o n 31 i i i i i v i i v i i i x i i i iv 2.2 The Apparatus 37 2.3 The Measurements 42 CHAPTER I I I - PERTURBATION RESULTS 49 3.1 Regime o f v a l i d i t y o f p e r t u r b a t i o n r e s u l t s 49 3.2 D i e l e c t r i c C o n s t a n t s and C o n d u c t i v i t y 56 3.3 D i s c u s s i o n 63 CHAPTER IV - DIELECTRIC RESONATOR STUDIES OF TTF-TCNQ 67 4.1 D i e l e c t r i c r e s o n a t o r a p p a r a t u s 67 4.2 The Spectrometer 73 4.3 C r y s t a l c u t t i n g 75 4.4 C r y s t a l Mounting 79 4.5 Measuring the C r y s t a l 82 4.6 Computer C o n t r o l 83 4.7 F i t t i n g of the S p e c t r a 87 CHAPTER V - THEORY OF ANISOTROPIC DIELECTRIC RESONATORS 91 5.1 I n t r o d u c t i o n 91 5.2 Open C i r c u i t Boundary C o n d i t i o n s 93 5.3 A Review o f R e s u l t s f o r I n f i n i t e D i e l e c t r i c R e s o n a t o r s 101 5.4 Other Models o f F i n i t e R e s o nators 108 5.4a "Magnetic Tube" P a r a l l e l to X 109 5.4b "Magnetic Tube" P a r a l l e l t o Z 116 5.4c "Magnetic W a l l s " P e r p e n d i c u l a r to X 118 V 5.5 I n f i n i t e Waveguides and T r a n s m i s s i o n L i n e s 120 5.5a n=0 A z i m u t h a l Dependence 121 5.5b n=l A z i m u t h a l Dependence 126 CHAPTER VI - RESULTS AND DISCUSSION OF DIELECTIRC MEASUREMENTS ON TTF TCNQ 131 6.1 The Mode P l o t s 131 6.1a The A Mode 133 6.1b The D i e l e c t r i c Mode P l o t s 137 6.2 The Imaginary P a r t o f the D i e l e c t r i c C o n s tant 148 CHAPTER V I I - MICROWAVE ABSORPTION STUDIES OF TTF TCNQ 157 7.1 The Experiment 159 7.1a The Spectrometer 159 7.1b The C r y s t a l Mount 160 7.1c D e t e c t i o n 162 7.2 R e s u l t s 165 7.3 F u t u r e D i r e c t i o n s 171 CHAPTER V I I I - SUMMARY 172 APPENDIX A - TIGHT BINDING CALCULATIONS FOR A LINEAR CHAIN WITH A BASIS 174 APPENDIX B - ANISOTROPIC DIELECTRIC WAVE GUIDES AND TRANSMISSION LINES 176 V I APPENDIX C - DIELECTRIC RESONANCE MODE PLOTS FOR SEVERAL TTF TCNQ CRYSTALS 184 REFERENCES 1 9 1 \ v i i LIST OF TABLES Page 1 Physical Data for MEMfTCNQ)^ Crystals Used 43 2 Calculated Results for MEM(TCNQ)a Crystals Studied 61 3 Dimensions of Copper Replicas Used for Coaxial Mode Studies 134 4 Values of Obtained from D i e l e c t r i c Resonance Studies of TTF-TCNQ 139 v i i i LIST OF FIGURES Page 1 The organic acceptor molecule TCNQ. 3 2 The organic donor molecule MEM. 8 3 The c r y s t a l structure of MEM(TCNQ)^ at 93 K. 8 4 a. The intra-dimer TCNQ-TCNQ overlap in the dimerized phase of MEM(TCNQ)^ . b. The inter-dimer overlap in the dimerized phase of MEM (TCNQ)^ . 10 5 The TCNQ-TCNQ spacing perpendicular to the TCNQ planes in the dimerized phase of MEM(TCNQ) . 11 6 The two types of TCNQ-TCNQ overlap in the undimerized phase of MEM (TCNQ)^ . 12 7 The organic donor molecule TTF. 22 8 The c r y s t a l structure of TTF-TCNQ. 22 9 The 8.5 GHz resonant cavity. 38 10 The cavity perturbation apparatus. 41 11 Photograph of MEM(TCNQ)^ cr y s t a l before (A) and after (B) having passed through the t r a n s i t i o n . 44 12 Typical cavity resonant frequency versus temperature with MEM (TCNQ)^ c r y s t a l inserted. 46 i x 13 T y p i c a l w i d t h o f c a v i t y resonance v e r s u s temperature w i t h MEM(TCNQ)^ c r y s t a l i n s e r t e d . 47 14 T y p i c a l t r a c e s o f resonance d e r i v a t i v e w i t h and w i t h o u t c r y s t a l i n s e r t e d . 49 15 a. Showing the r e l a t i o n between w i d t h and <r u s i n g Eqs. 2.10, 2.13, and 2.17 f o r e x p e r i m e n t a l c o n d i t i o n s as found f o r MEM(TCNQ) c r y s t a l s 1 and 4. 52 b. <J- v e r s u s l/a. c f o r assumption o f l a r g e and s m a l l s k i n d e pth f o r c r y s t a l s 1 and 4. 53 16 Microwave c o n d u c t i v i t i e s (8.5GHz) f o r MEM(TCNQ)^ c r y s t a l s s t u d i e d . 57 17 Comparison o f microwave c o n d u c t i v i t y o f c r y s t a l 4 and d.c. c o n d u c t i v i t y o f a second MEM(TCNQ)i c r y s t a l p l o t t e d a g a i n s t 1/T. 58 18 The r e a l p a r t o f the d i e l e c t r i c c o n s t a n t v e r s u s t e m p e r a t u r e f o r some MEM(TCNQ)^ c r y s t a l s . 59 19 C r y s t a l m a n i p u l a t i n g rod and rod h o l d e r f o r the d i e l e c t r i c resonance e x p e r i m e n t . 68 20 The dewar assembly f o r the d i e l e c t r i c resonance s t u d i e s o f TTF-TCNQ. 71 21 Schematic drawing o f the microwave a p p a r a t u s used f o r d i e l e c t r i c resonance measurements. 74 22 The c r y s t a l c u t t i n g a p p a r a t u s mounted below m i c r o s c o p e . 76 23 C r y s t a l and b l a d e b e f o r e and a f t e r c u t t i n g o p e r a t i o n . A n v i l has been withdrawn f o r c l a r i t y . 78 24 C r y s t a l mounting a p p a r a t u s . 81 25 Examples o f observed resonances ( s o l i d l i n e s ) and bes t f i t o b t a i n e d u s i n g "AFIT" program (dashed l i n e s ) . 90 26 The l a b e l l i n g o f the d i m e n s i o n s o f TTF-TCNQ c r y s t a l s . 97 27 The f i e l d l i n e s f o r some o f the low l y i n g r e s o n a n c e s . 100 28 S u r f a c e s on which boundary c o n d i t i o n s a r e not s a t i s f i e d i n t h e o r y used f o r a n a l y s i s o f d i e l e c t r i c resonance d a t a . 105 29 Magnetic w a l l s as a p p l i e d i n a d a p t a t i o n o f Yee,s t h e o r y t o r e c t a n g u l a r c r y s t a l s . 110 30 a. n=0 d i e l e c t r i c wave g u i d e mode p l o t f o r Rj =0.015 cm, €A =5, € 2=3000. (no o u t e r conductor) b. n=0 d i e l e c t r i c wave g u i d e mode p l o t f o r R-, =0.015 cm, Rj_=0. 500 cm, ^=5, € z=3000. ( o u t e r c o n d u c t o r p r e s e n t ) 122 31 a. n=0 d i e l e c t r i c wave g u i d e mode p l o t f o r Ri =0.015 cm, Rx =0. 500 cm, €A=5, £^=600. ( o u t e r c o n d u c t o r p r e s e n t ) b. n=0 d i e l e c t r i c wave g u i d e mode p l o t f o r R.,=0.015 cm, R z=0.05 cm, €A=5. € z=3000. ( o u t e r c o n d u c t o r p r e s e n t ) 123 32 Q u a l i t a t i v e d e p i c t i o n o f the v a r i a t i o n o f E x a c r o s s the d i e l e c t r i c rod di a m e t e r f o r s e l e c t e d p o i n t s on mode p l o t o f F i g u r e 30b. 125 x i 33 a. n=l d i e l e c t r i c wave R, =0.015 cm, ^=5, 6Z=3000. b. n=l d i e l e c t r i c wave R-t =0.015 cm, R A =0.500 cm, co n d u c t o r p r e s e n t ) g u i d e mode p l o t f o r (no o u t e r conductor) g u i d e mode p l o t f o r €i.=5, € z=3000. ( o u t e r 127 34 Q u a l i t a t i v e d e p i c t i o n o f the v a r i a t i o n i n magnitude of Ez. a c r o s s the d i a m e t e r o f the d i e l e c t r i c rod f o r the denoted p o i n t on F i g . 33b. 128 35 a. n=l R7 =0.015 co n d u c t o r b. n=l R, =0. 015 co n d u c t o r d i e l e c t r i c cm, R A=0.05 pr e s e n t ) d i e l e c t r i c wave g u i d e mode p l o t f o r cm, <L=5, ^=3000. ( o u t e r wave g u i d e mode cm R5 p r e s e n t ) =0. 500 cm, <j=5, ^=60 0. p l o t f o r ( o u t e r 129 36 Mode p l o t s modes. f o r copper r e p l i c a s i n s t u d y o f c o a x i a l 135 37 Cb p l o t t e d a g a i n s t f„ f o r s e v e r a l TTF-TCNQ c r y s t a l s s t u d i e d 141 38 C o u p l i n g o f the doub l e d i e l e c t r i c resonance i n c r y s t a l 19 at s e v e r a l d i s t a n c e s o f the c r y s t a l c e n t r e from the s h o r t . 145 39 Diagram showing the r e l a t i o n s h i p o f the wave g u i d e f i e l d s to the proposed d i e l e c t r i c resonance f i e l d s f o r d o u b l e resonance. 146 40 P l o t o f c o n d u c t i v i t y v e r s u s f r e q u e n c y f o r s e v e r a l TTF-TCNQ c r y s t a l s s t u d i e d by d i e l e c t r i c resonance. 151 41 P l o t o f ln(er) v e r s u s 1/T f o r c r y s t a l 19 at s e v e r a l f requenc i e s . 153 42 High f r e q u e n c y mount f o r microwave a b s o r p t i o n s t u d i e s i n TTF-TCNQ. 161 x i i 43 Spectrum showing the ratio of absorbed to incident power for the TTF-TCNQ c r y s t a l studied in the 75 to 110 GHz band. 167 44 D i e l e c t r i c resonance mode plot for TTF-TCNQ c r y s t a l 4. 185 45 D i e l e c t r i c resonance mode plot for TTF-TCNQ cr y s t a l 9. 186 46 D i e l e c t r i c resonance mode plot for TTF-TCNQ cr y s t a l 12. 187 47 D i e l e c t r i c resonance mode plot for TTF-TCNQ cr y s t a l 17. 188 48 D i e l e c t r i c resonance mode plot for TTF-TCNQ cr y s t a l 18. 189 49 D i e l e c t r i c c r y s t a l 19. resonance mode plot for TTF-TCNQ 190 x i i i ACKNOWLEDGEMENTS I would l i k e to acknowledge the s u p p o r t and d i r e c t i o n o f Dr. W.N. Hardy i n h i s s u p e r v i s i o n o f t h i s p r o j e c t . I have a l s o been a i d e d by numerous d i s c u s s i o n s w i t h Dr. A . J . B e r l i n s k y . Dr. L a r r y W e i l e r and h i s group p r o v i d e d many o f the c r y s t a l s s t u d i e d i n these e x p e r i m e n t s and I would a l s o l i k e to thank them. The h e l p f u l s u g g e s t i o n s and t e c h n i c a l a s s i s t a n c e o f Dr. J . C a r o l a n and Dr. J . E l d r i d g e a r e a l s o g r a t e f u l l y acknowledged. I a l s o wish t o thank a l l o f the members o f the microwave group f o r t h e i r s u p p o r t and a s s i s t a n c e . B. S t a t t i s to be thanked p a r t i c u l a r l y f o r h i s i n v a l u a b l e a s s i s t a n c e w i t h the computer c o n t r o l l e d s p e c t r o m e t e r . I would a l s o l i k e t o thank the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l f o r t h e i r s u p p o r t i n the form o f a S c i e n c e S c h o l a r s h i p . 1 CHAPTER I I n t r o d u c t i o n In t h i s s e c t i o n we w i l l b e g i n by commenting on q u a s i - o n e -d i m e n s i o n a l TCNQ s a l t s i n g e n e r a l w i t h some d i s c u s s i o n o f the P e i e r l ' s t r a n s i t i o n , charge d e n s i t y wave f o r m a t i o n , and F r o h l i c h s u p e r c o n d u c t i v i t y . We w i l l then d e s c r i b e MEM(TCNQ).j, and, i n c o n n e c t i o n w i t h i t , models o f o n e - d i m e n s i o n a l s t a c k s i n which Coulomb i n t e r a c t i o n s a r e i m p o r t a n t . T h i s w i l l s e r v e as i n t r o d u c t i o n f o r the f i r s t p a r t o f t h i s work. F i n a l l y we w i l l c o n s i d e r TTF-TCNQ , the s u b j e c t o f the second p a r t o f t h i s work, and, i n p a r t i c u l a r , the d i e l e c t r i c c o n s t a n t a p p r o p r i a t e to the pinned charge d e n s i t y wave model o f TTF-TCNQ. f 2 1.1 Quasi-One-Dimensional TCNQ Salts There are now a large number of charge transfer s a l t s based on the organic electron acceptor, tetracyanoquinodimethane, (TCNQ). These are predominantly 1:1 s a l t s , such as TTF-TCNQ, or 1:2 s a l t s , l i k e MEM(TCNQ)i . A common feature of most of these materials i s that the planar TCNQ molecules, seen in Fig. 1, form stacks in which the overlap of the TT molecular o r b i t a l s i s s i g n i f i c a n t . Some planar donor molecules, l i k e t etrathiofulvalene, (TTF) , can also form stacks. The interactions of stacks in directions perpendicular to the stacking axis, while strong enough to determine a three-dimensional structure, are usually considerably weaker than those along the stacks. These highly anisotropic materials thus offer the p o s s i b i l i t y of studying the effects of interactions in one dimension and are often described by the term quasi-one-dimensional . This name r e f l e c t s the fact that real systems cannot be purely one-dimensional. In a system with short range interactions and only one dimension, fluctuations w i l l prevent the system from existing in a phase with long range order at any non-zero temperature. This comes about because the contact between regions of d i f f e r i n g order parameter i s a point and the surface energy involved in creation of such a boundary i s independent of the size of the d i f f e r i n g region. The entropy 3 N H H N \ \ / /// c C = C c \ / \ / c = c c = c / \ / \ / / C = C N \ \ N H H N T C N Q Figure 1. The organic acceptor molecule TCNQ. 4 contribution to the change in the free energy for creating a boundary somewhere along a chain of M s i t e s goes as TlnM. For large M, t h i s entropy change w i l l always dominate the surface energy increase and the system w i l l be able to lower i t s free energy by creating boundaries.(Emery,1977). The fact that phase transitions w i l l not occur in a one-dimensional system does not invalidate the interest in one-dimensional phenomena. In a real quasi-one-dimensional system, the coupling of fluctuations on neighbouring one-dimensional chains can lead to ordering in three dimensions. We go on, then, to discuss b r i e f l y some of the properties of purely one-dimensional chains with the i m p l i c i t understanding that, in any real system, the three-dimensional nature of the l a t t i c e would play a part in any phase t r a n s i t i o n observed. Perhaps the best known feature of one-dimensional models i s the Peierls i n s t a b i l i t y (Peierls ,1955). In a purely one-dimensional system with no interactions and less than 2 electrons per s i t e , the lowest band w i l l be f i l l e d up to k=-k° , where the Fermi vector i s given by (1.1) k°=NG l (/4N 0. Here, N i s the number of electrons, Na the number of s i t e s , each of which can accomodate two electrons, and G,, =2Tr/dH is the reciprocal l a t t i c e vector for a chain with s i t e s separated by d1( . This s i t u a t i o n describes a p a r t i a l l y f i l l e d free electron band and thus a one dimensional metal. The kinetic energy 5 a s s o c i a t e d w i t h the e l e c t r o n s near k=k° can be lowered i f a gap i s opened a t t h a t p o i n t . Such a gap i s opened by 2k° d i s t o r t i o n s o f the e l e c t r o n i c system. One thus e x p e c t s a d e n s i t y f l u c t u a t i o n o f wave v e c t o r 2kl to form f o r a v a n i s h i n g l y s m a l l component o f the a p p l i e d p o t e n t i a l . T h i s i s e x p r e s s e d i n terms o f the e l e c t r o n i c d e n s i t y response f u n c t i o n a t wave v e c t o r q, X(q ) r which measures the r a t i o o f the d e n s i t y f l u c t u a t i o n s a t q to the component o f the a p p l i e d p o t e n t i a l a t t h a t wave v e c t o r . Toombs (1978) and B e r l i n s k y (1979) d i s c u s s the form o f X(2kp). They show t h a t , a t low t e m p e r a t u r e s , i t i s n e g a t i v e and goes as l n ( T ) . The f o r c e on the l a t t i c e i n c l u d e s a term i n v o l v i n g ~X.(q) . Toombs g i v e s the f r e q u e n c y f o r phonons o f wave v e c t o r q, Cl(q) , as (1.2) H 1(q)= 0 0 , * " + (2gauA, /Nfc )"X (q) where g i s the e l e c t r o n - p h o n o n c o u p l i n g c o n s t a n t , «^ i s the unperturbed phonon f r e q u e n c y , and N i s the number o f s i t e s . In t h i s c a s e , X(q) i s d e s c r i b i n g the response o f the e l e c t r o n s t o the p e r t u r b a t i o n due to phonons o f wavevector q. The f i r s t term can be thought o f as a r e s t o r i n g f o r c e due to the l a t t i c e i t s e l f and the second as a f o r c e due to the c o n d u c t i o n e l e c t r o n s . Because X(2k°) i s n e g a t i v e and i n c r e a s i n g as T i s l o w e r e d , t h e r e w i l l be a t e m p e r a t u r e , Tc , a t which PL~(2k° ) goes t o z e r o . B e r l i n s k y (1979) shows t h a t the form o f n. a(2k?) above T^  i s g i v e n by 6 lY(2k* ) = (g^ i k„/£ F ) l n ( T / T 0 ) • He f u r t h e r s t a t e s t h a t , f o r T<T0 , the 2k° d i s t o r t i o n i s s t a t i c and a gap opens i n the e l e c t r o n i c spectrum. T h i s s o f t e n i n g o f the 2k° phonon i s r e f e r r e d to as the Kohn anomaly. Other mechanisms f o r d i s t o r t i n g the l a t t i c e w i l l be d i s c u s s e d when we d e s c r i b e the use o f the Hubbard model to d e a l w i t h Coulomb i n t e r a c t i o n s i n r e f e r e n c e to MEM(TCNQ)2. F r o h l i c h (1954) has shown t h a t , f o r a j e l l i u m model, the t r a n s l a t i o n a l i n v a r i a n c e o f the p o s i t i v e background g i v e s no p r e f e r r e d p o s i t i o n f o r the d i s t o r t i o n and a l l o w s i t to propagate as a s l i d i n g charge d e n s i t y wave w i t h o n l y weak a t t e n u a t i o n due to s c a t t e r i n g from phonons. T h i s s i t u a t i o n a l s o a p p l i e s i f the charge t r a n s f e r does not r e s u l t i n the number o f s i t e s b e i n g an i n t e g r a l m u l t i p l e o f the number o f e l e c t r o n s . The new u n i t c e l l brought about by the d i s t o r t i o n w i l l be i n f i n i t e i n l e n g t h and t h e r e w i l l be no p r e f e r r e d p o s i t i o n f o r the d i s t o r t i o n . The d i s t o r t i o n and l a t t i c e a re s a i d to be incommensurate. In r e a l i t y , t h e charge d e n s i t y wave, i f not commensurate w i t h the l a t t i c e , i s s u s c e p t i b l e to p i n n i n g by i m p u r i t i e s , d e f e c t s , or t h r e e - d i m e n s i o n a l o r d e r i n g . ( Lee, R i c e , and Anderson , 1974). The e f f e c t o f such a pinned mode on the low f r e q u e n c y e l e c t r i c a l p r o p e r t i e s o f a m a t e r i a l w i l l be d i s c u s s e d below i n r e l a t i o n to TTF-TCNQ. 7 1.2 MEM(TCNQ) 2 and the E f f e c t o f I n t e r a c t i o n s on TCNQ S t a c k s 1.2a MEM(TCNQ) M e t h y l e t h y l m o r p h o l i n i u m T e t r a c y a n o q u i n o d i m e t h a n e , MEM(TCNQ) i , i s an o r g a n i c semi-conductor w i t h a room temperature c o n d u c t i v i t y o f about 10" 3 (a-cm) ' and an a c t i v a t i o n energy o f about 0.4eV. I t has been a s o u r c e o f some i n t e r e s t l a t e l y because o f i t s unambiguous charge t r a n s f e r , i t s q u a s i -o n e - d i m e n s i o n a l b e h a v i o u r , and i t s two phase t r a n s i t i o n s i n v o l v i n g d i m e r i z a t i o n and t e t r a m e r i z a t i o n o f the TCNQ m o l e c u l e s a l o n g the s t a c k i n g a x i s . The MEM m o l e c u l e i s shown i n F i g 2. I t i s a v e r y good e l e c t r o n donor and hence i s assumed to donate a f u l l e l e c t r o n per MEM m o l e c u l e to the TCNQ s t a c k . The c r y s t a l s t r u c t u r e c o n s i s t s , g e n e r a l l y , o f s t a c k s o f TCNQ m o l e c u l e s w i t h MEM m o l e c u l e s s e p a r a t i n g p l a n e s c o n t a i n i n g the s t a c k s . ( B o s c h and van Bodegom,1977). See F i g . 3 f o r the c r y s t a l s t r u c t u r e a t 93K. There a re two known phase t r a n s i t i o n s i n MEM(TCNQ) A. Below 19K, the TCNQ m o l e c u l e s a r e t e t r a m e r i z e d a l o n g the s t a c k i n g a x i s w i t h one e l e c t r o n p a i r per t e t r a m e r . T h i s i s thought to be an example o f a S p i n P e i e r l s r t r a n s i t i o n which opens a gap a t the Fermi w a v e v e c t o r . The ground s t a t e i s then non-magnetic w i t h e l e m e n t a r y e x c i t a t i o n s which a r e spin-wave e x c i t a t i o n s . ( H u i z i n g a e t a l . , 1979). Between 20K and 340K, X-ray r e s u l t s i n d i c a t e t h a t the TCNQ m o l e c u l e s form dimers w i t h the i n t r a -Figure 3. The c r y s t a l structure of MEM(TCNQ)t at 93 K. From Bosch and van Bodegom (1977) 9 dimer and i n t e r - d i m e r o v e r l a p s b e i n g e a s i l y d i s t i n g u i s h a b l e . F i g . 4a shows the i n t r a - d i m e r o v e r l a p o f two TCNQ m o l e c u l e s and F i g . 4b shows the i n t e r - d i m e r o v e r l a p . F i g . 5 shows the v a r i a t i o n i n the TCNQ-TCNQ s e p a r a t i o n p e r p e n d i c u l a r to the plan e o f the m o l e c u l e s . Here a=3.15A and b=3.27A. In t h i s phase, the c r y s t a l i s a semiconductor w i t h a gap o f about 0.8.eV. (Morrow e t a l . , 1979). I t has been r e p o r t e d ( C h a i k i n , 1979) t h a t the thermopower i s about -60 /<V/K. Al t h o u g h the low temp e r a t u r e v a l u e s v a r y , t h i s v a l u e f o r the thermopower seems to be common among the 1:2 TCNQ s a l t s i n c l u d i n g Q u i n o l i n i u m (TCNQ) ; and Triethylammonium (TCNQ)^ . (Conwell,1978 and r e f e r e n c e s t h e r e i n ) . Much work has been done on t r y i n g to understand t h i s v a l u e o f the thermopower by u s i n g a on e - d i m e n s i o n a l Hubbard model to d e s c r i b e the TCNQ s t a c k s . I t i s g e n e r a l l y agreed t h a t a v a l u e o f -60 MV/K i m p l i e s t h a t , f o r a Hubbard model, the o n - s i t e r e p u l s i o n U, which i s the i n t e r a c t i o n f o r two e l e c t r o n s on a s i n g l e TCNQ m o l e c u l e , must be g r e a t e r than kT.(Conwell,1978; Kwak and Beni,1976; C h a i k i n and Be n i , 1 9 7 6 ) . The use o f the Hubbard model to d e s c r i b e TCNQ s t a c k s w i l l be d i s c u s s e d below. At 340 K t h e r e i s a v i o l e n t f i r s t o r d e r t r a n s i t i o n to a phase i n which the d i s t i n c t i o n between i n t r a - d i m e r and i n t e r -dimer o v e r l a p almost d i s a p p e a r s . F i g . 6 shows the s e o v e r l a p s i n the u n d i m e r i z e d phase. In t h i s phase the c o n d u c t i v i t y i s between 15 and 30 (ft-cm)-' and does not show an e x p o n e n t i a l 10 11 Figure 5. The TCNQ-TCNQ TCNQ planes in of MEM(TCNQ) spacing perpendicular the dimerized phase to the 12 a Figure 6. The two types of TCNQ-TCNQ overlap in the phase of MEM (TCNQ), . und imerized 13 temperature dependence. As stated above, the MEM molecule should lose one electron to the TCNQ chain. Defining p as N/N0 , where N i s the number of electrons transferred and N D i s the number of TCNQ s i t e s , t h i s corresponds tof>=\/2. Such a band f i l l i n g would be expected to exhibit metallic conduction. The conduction in the high temperature phase, while s t i l l low, has a temperature dependence more l i k e that of a metal than a semiconductor. 14 1.2b Use o f the Hubbard Model to D e s c r i b e TCNQ S t a c k s Many workers have found i t n e c e s s a r y t o i n c l u d e the e f f e c t s o f Coulomb i n t e r a c t i o n s and o v e r l a p i n d e s c r i b i n g the p r o p e r t i e s o f TCNQ s t a c k s i n v a r i o u s TCNQ charge t r a n s f e r s a l t s . These i n c l u d e the s c a t t e r i n g o f d i f f u s e X-rays from TTF TCNQ a t q=4k° (Torranee,1978) and the thermopowers o f the 1:2 s a l t s . These e f f e c t s a r e u s u a l l y i n c l u d e d by t r e a t i n g the s t a c k s u s i n g the Hubbard model or e x t e n s i o n s o f the Hubbard model to i n c l u d e the i n t e r a c t i o n between e l e c t r o n s on n e i g h b o u r i n g TCNQ m o l e c u l e s . T h i s model p r o v i d e s the s i m p l e s t way to d e a l w i t h c o r r e l a t i o n s brought about by Coulomb r e p u l s i o n and by o v e r l a p o f a d j a c e n t s i t e s . I t i s c l e a r t h a t these c o r r e l a t i o n s can be a sour c e o f p e r i o d i c i t y q u i t e d i s t i n c t from t h a t a s s o c i a t e d w i t h the u s u a l P e i e r l s , d i s t o r t i o n o f wavevector 2k°. Hubbard was i n t e r e s t e d i n d e s c r i b i n g c o r r e l a t i o n e f f e c t s i n narrow d - e l e c t r o n bands o f t r a n s i t i o n m e t a l s . For a narrow band, the Wannier f u n c t i o n s , o b t a i n e d by summing the B l o c h f u n c t i o n s over the B r i l l o u i n zone, were w e l l l o c a l i z e d on the l a t t i c e s i t e s . T h i s a l l o w e d him to n e g l e c t a l l Coulomb i n t e r a c t i o n s e x cept f o r o n - s i t e r e p u l s i o n . T h i s H a m i l t o n i a n then had the form,(Hubbard , 1963) , (1.3) H = - t I ( c ^ ciMa +clti, c;„ ) + ( U / 2 ) Z n ( y n,.. Here t i s the o v e r l a p o r hopping i n t e g r a l between a d j a c e n t 15 s i t e s , c£. and c,v are c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r an e l e c t r o n o f s p i n cr on s i t e « , and U i s the Coulomb i n t e r a c t i o n f o r 2 e l e c t r o n s on the same s i t e . The extended Hubbard H a m i l t o n i a n i n c l u d e s terms i n V; f o r the i n t e r a c t i o n o f e l e c t r o n s s e p a r a t e d by i l a t t i c e s p a c i n g s (Hubbard, 1978). We w i l l not be concerned w i t h the g e n e r a l model f o r a l l v a l u e s o f U. R a t h e r , we w i l l l o o k a t the e f f e c t s o f hav i n g a v e r y l a r g e U. F o l l o w i n g Torrance(1977) we note t h a t as U goes to i n f i n i t y , the problem t r a n s f o r m s e x a c t l y t o the case f o r s p i n l e s s f e r m i o n s . T h i s i s because f o r v e r y l a r g e U, doub l e occupancy o f a TCNQ m o l e c u l e i s not a l l o w e d and t h i s i s the s i t u a t i o n f o r s p i n l e s s f e r m i o n s as w e l l . T h i s does not change the t i g h t b i n d i n g problem except t h a t each s t a t e can now accomodate o n l y one e l e c t r o n . The d i s p e r s i o n r e l a t i o n f o r the lo w e s t band i s then i d e n t i c a l to t h a t f o r the U=0 t i g h t b i n d i n g problem, E(k) = - 2 t cos(kb) where b i s the l a t t i c e c o n s t a n t . Now, however, f o r N D l a t t i c e s i t e s , the lower band c o n t a i n s o n l y N Q s t a t e s w i t h i n the B r i l l o u i n zone i n s t e a d of 2N 0 s t a t e s as f o r U = 0. As a r e s u l t , the v a l u e o f the wave v e c t o r f o r the h i g h e s t f i l l e d l e v e l i n the ground s t a t e i s doubled to k F = T t N / ( b N 0 ) r a t h e r than k° = TT N/(2bN„) where k% i s the Fermi wavevector f o r U = 0. There i s now a gap o f U - 4t (Ovchinnikov,1970) above which d o u b l e occupancy o f TCNQ s i t e s o c c u r s . 16 W i t h r e g a r d to t h i s p o i n t , t h e r e i s a m i s l e a d i n g f i g u r e which appears i n a t l e a s t two r e f e r e n c e s . ( C h a i k i n e t a l . , 1973; E p s t e i n e t a l . , 1972). I t shows a gap o f U - 4t w i t h the minimum of the lower band and maximum o f the upper band s e p a r a t e d by U. T h i s i m p l i e s a band w i d t h o f 2t i n the upper and lower band and t h i s i s c l e a r l y i n c o n f l i c t w i t h the r e s u l t f o r the s p i n l e s s f e r m i o n c a s e . That U s h o u l d , i n f a c t , be the s e p a r a t i o n between the c e n t r e s o f the bands i s suggested by two simple-minded arguments. The f i r s t i s t h a t , f o r t=0, we expect 2 l e v e l s s e p a r a t e d by U. As we t u r n on the o v e r l a p , these l e v e l s w i l l spread i n t o bands j u s t as a s i n g l e l e v e l would and t h e r e w i l l c l e a r l y be s t a t e s s e p a r a t e d by more than U. T h i s i s c o n s i s t e n t w i t h the r e s u l t o b t a i n e d by u s i n g the Hubbard model to d e s c r i b e the hydrogen m o l e c u l e . ( A s h c r o f t and Mermin, 1976). For t h i s case the two e l e c t r o n l e v e l s a r e g i v e n by (1.4) E = (l/2 ) U i / 4 t * - + U V 4 * For U>>4t the s p l i t t i n g i s j u s t U. As t i s i n c r e a s e d , the s e p a r a t i o n i n c r e a s e s t o something g r e a t e r than U. T h i s s u g g e s t s some i n c o n s i s t e n c y i n a Hubbard model c a l c u l a t i o n i n w h i c h , f o r U>>4t, the maximum s e p a r a t i o n o f any two l e v e l s i s U. We w i l l now q u a l i t a t i v e l y c o n s i d e r the e f f e c t on a c h a i n , d e s c r i b e d by the extended Hubbard model, o f the dominance o f s p e c i f i c terms i n the H a m i l t o n i a n . We foc u s a t t e n t i o n on the lower band. For l a r g e U t h e r e are N c s t a t e s i n the B r i l l o u i n 17 zone. For fi<l, the wavevector f o r the h i g h e s t f i l l e d l e v e l i n the ground s t a t e i s l e s s than Tr/b and we have a p a r t i a l l y f i l l e d band which can behave l i k e a o n e - d i m e n s i o n a l m e t a l . T h i s f i l l i n g o f the band out to k=i2k° w i l l have a number o f p o s s i b l e consequences. The e l e c t r o n - h o l e e x c i t a t i o n s might have z e r o energy a t q = 4k°.(Coll, 1974). A l t e r n a t i v e l y a gap might be opened i n the band by a l a t t i c e d i s t o r t i o n o f wavevector 4k£. One way i n which t h i s might occur would be a P e i e r l ' s d i s t o r t i o n a r i s i n g because o f the s o f t e n i n g of the 4k£ phonon (Kohn anomaly). A gap might a l s o a r i s e because o f a Wigner c r y s t a l l i z a t i o n . For t h i s s i m p l e model i n which U i s the o n l y Coulomb i n t e r a c t i o n , Wigner c r y s t a l l i z a t i o n can o n l y come about f o r p =1. I f l o n g e r range Coulomb i n t e r a c t i o n s a r e i n c l u d e d , Wigner c r y s t a l l i z a t i o n can r e s u l t f o r p<l. T o r r a n c e a t t r i b u t e s the 4k°F d i f f u s e X-ray s c a t t e r i n g from TTF TCNQ to such a c r y s t a l l i z a t i o n . ( T o r r a n c e , 1978, 1977; T o r r a n c e and S i l v e r m a n , 1977; Klimenko e t a l . , 1976). The s i m p l e s t model d e a l i n g w i t h t h i s s i t u a t i o n i s then the Extended Hubbard model (Hubbard, 1978) f o r which the n e a r e s t neighbour Coulomb i n t e r a c t i o n , V, , i s i n c l u d e d . For l a r g e V, , Hubbard f i n d s t h a t f o r />=l/n the e l e c t r o n s a r e s e p a r a t e d by n s p a c i n g s . For 1/(n+1)</Kl/n,the e l e c t r o n s w i l l be s e p a r a t e d by n or n+1 spaces i n some k i n d o f p e r i o d i c manner. T h i s non-uniform s p a c i n g of e l e c t r o n s might be expected to l e a d to d i s t o r t i o n i n the l a t t i c e due to Coulomb 18 i n t e r a c t i o n s w i t h the l a t t i c e . For the case o f p = l/2, the l a r g e U and V, case would l e a d to e l e c t r o n s on e v e r y second s i t e . T h i s would not d i s t o r t the l a t t i c e but the p e r i o d i c i t y o f the e l e c t r o n s would open a gap i n the e l e c t r o n i c spectrum. T h i s can be seen e a s i l y because o f the f a c t t h a t , f o r the Wigner c r y s t a l ground s t a t e , any e x c i t a t i o n w i l l put two e l e c t r o n s on n e a r e s t neighbour s i t e s and t h i s s t a t e w i l l be r a i s e d by . I f V, i s l a r g e r than the bandwidth, a gap i s formed. We c a n n o t , however, i m m e d i a t e l y a p p l y a l a r g e V, model to MEM(TCNQ)^ w i t h o u t c a u t i o n s i n c e C o n w e l l (1978) notes t h a t the r e s u l t s o f C h a i k i n and Beni (1976) i n d i c a t e t h a t a thermopower o f -60><V/K i s i n c o n s i s t e n t w i t h V) >>kT f o r the Hubbard model. We come, f i n a l l y , t o c o n s i d e r the r o l e t h a t o v e r l a p p l a y s i n d e t e r m i n i n g the b e h a v i o u r o f a Hubbard c h a i n . The s i z e o f the o v e r l a p w i l l have no e f f e c t on the s t a b i l i t y o f a c h a i n u n l e s s we a l l o w the c h a i n to d i s t o r t i n such a way t h a t the i n c r e a s e d o v e r l a p between some o f the s i t e s o f f s e t s the i n c r e a s e d energy o f the d i s t o r t e d l a t t i c e . I f we c o n s i d e r the s p e c i f i c case o f MEM (TCNQ)^ , with/> = l / 2 , we f i n d t h a t i n the presence o f a s m a l l d i s t o r t i o n which d i s t i n g u i s h e s between two o v e r l a p s , t , and t ^ , the i n t r a - d i m e r and i n t e r - d i m e r o v e r l a p s r e s p e c t i v e l y , the bandwidth changes from 4t to 2 ( t 1 + t a ) and a gap o f 2 | t j - t ^ | opens i n the m i d d l e . For a h a l f f i l l e d band, such as f o r l a r g e U and y 0 = l / 2 , t h i s might be an i m p o r t a n t 19 mechanism for driving the system to dimerization. The t i g h t -binding problem for a chain with alternating overlaps and separations i s discussed in Appendix A. 20 1.2c Purpose o f the C a v i t y P e r t u r b a t i o n Measurements on MEM(TCNQ) Z T h i s p a r t o f the experiment was c a r r i e d out to measure the e l e c t r i c a l p r o p e r t i e s above and below the d i m e r i z a t i o n t r a n s i t i o n and, i n p a r t i c u l a r , t o c o n f i r m the d.c. measurements. The d.c. r e s u l t s were, i n some sen s e , s u s p e c t due to the v i o l e n t c r a c k i n g e x p e r i e n c e d by the c r y s t a l on bei n g heated through the t r a n s i t i o n . I t was f e l t t h a t the c o n t a c t l e s s microwave methods, f o r which the c r a c k s would be c a p a c i t a t i v e l y s h o r t e d , would c i r c u m v e n t some o f the problems a s s o c i a t e d w i t h the u n c e r t a i n t y i n the c u r r e n t paths i n a c r a c k e d a n i s o t r o p i c c r y s t a l . Some o f the p r e c e e d i n g d i s c u s s i o n on the r o l e o f d i f f e r e n t i n t e r a c t i o n s i n d e t e r m i n i n g the p r o p e r t i e s o f a m a t e r i a l d e s c r i b e d by the Hubbard model w i l l be a p p l i e d to MEM(TCNQ)Z i n Chapter I I I . There we w i l l f i n d t h a t w h i l e the c o n d u c t i v i t y and d i e l e c t r i c c o n s t a n t do not h e l p us to i d e n t i f y the v a l i d regime i n terms o f r e l a t i v e magnitudes o f i n t e r a c t i o n s , t h e y do c o n f i r m the d.c. r e s u l t s w h i c h , i n t u r n , a l l o w us to use the measured v a l u e f o r the gap to e s t i m a t e some o f the i n t e r a c t i o n s f o r d i f f e r e n t a ssumptions about r e l a t i v e magnitudes o f i n t e r a c t i o n s . 21 1.3 TTF-TCNQ and the Pinned Charge D e n s i t y Wave 1.3a TTF-TCNQ There has been much w r i t t e n about the p r o p e r t i e s and s t r u c t u r e o f TTF TCNQ . A b r i e f summary o f some o f the r e l e v a n t p o i n t s w i l l be p r e s e n t e d h e r e . T h i s m a t e r i a l , l i k e most TCNQ charge t r a n s f e r s a l t s , i n v o l v e s s t a c k s o f TCNQ m o l e c u l e s . U n l i k e some d o n o r s , however, TTF i s a l s o p l a n a r and s t a c k s as w e l l . The TTF m o l e c u l e i s shown i n F i g . 7. The c r y s t a l s t r u c t u r e i s shown i n F i g . 8. The s l i p p e d geometry o f the TCNQ m o l e c u l e s i s found to maximize the o v e r l a p o f the TT o r b i t a l s on n e i g h b o u r i n g m o l e c u l e s . ( B e r l i n s k y e t a l . , 1 9 7 4 ) . The s t r u c t u r e s u g g e s t s h i g h l y a n i s t o t r o p i c b e h a v i o u r and t h i s i s c o n f i r m e d by c o n d u c t i v i t y measurements ( T i e d j e , 1 9 7 5 ) and d i e l e c t r i c c o n s t a n t measurements (Khanna e t al.,1974) . TTF-TCNQ i s one o f the most h i g h l y c o n d u c t i n g o f the one-d i m e n s i o n a l o r g a n i c c o n d u c t o r s found to d a t e . W h i l e the temperature dependence o f the c o n d u c t i v i t y , T r a t h e r than T , i s not e x a c t l y c h a r a c t e r i s t i c o f a m e t a l , i t does e x h i b i t some o f the p r o p e r t i e s expected f o r a m a t e r i a l w i t h a one-d i m e n s i o n a l , p a r t l y f i l l e d band such as the t r a n s i t i o n from h i g h to low c o n d u c t i v i t y and the d i f f u s e s c a t t e r i n g o f X-rays a t q=2k . F r i e n d e t a l . (1978) show t h a t the c o n s t a n t volume temperature dependence, as opposed to the c o n s t a n t p r e s s u r e r e s u l t s u s u a l l y quoted, goes as T . The temperature dependence 23 i n e xcess o f t h i s i s then due to the e f f e c t o f t h e r m a l c o n t r a c t i o n on the c o n d u c t i o n band. As T approaches 54K, c o n d u c t i v i t y r i s e s to a maximum. i t has been suggested t h a t t h i s i s due to a c o l l e c t i v e mechanism i n v o l v i n g charge d e n s i t y waves. (Bardeen,1973; Heegar,1977; A n d r i e u x et a l . , 1978). For the u s u a l P e i e r l s .. t r a n s i t i o n , t hese would have c h a r a c t e r i s t i c wavevector 2k°. T o r r a n c e (1977), however, has argued t h a t on the b a s i s o f the 4kp d i f f u s e X-ray s c a t t e r i n g p r e s e n t a t a l l t e m p e r a t u r e s , TTF-TCNQ s h o u l d be c o n s i d e r e d as a l a r g e U system. T h i s l e a d s to a c h a r a c t e r i s t i c w a v e v e c t o r , f o r the charge system, o f 4kp. For a n t i f e r r o m a g n e t i c c o u p l i n g , the c h a r a c t e r i s t i c wavevector f o r the s p i n system i s s t i l l 2k°. T o r r a n c e s t i l l , however, a c c e p t s the 2kp d i s t o r t i o n as b e i n g dominant a t low t e m p e r a t u r e . The dominant mechanism f o r the h i g h c o n d u c t i v i t y above 54K appears to remain u n s e t t l e d . Below 54K, the c o n d u c t i v i t y drops s h a r p l y . X-ray and n e u t r o n s c a t t e r i n g show e x t r a r e c i p r o c a l l a t t i c e p o i n t s which i n d i c a t e the presence o f a t h r e e - d i m e n s i o n a l s u p e r - l a t t i c e . The p e r i o d i c i t y o f the e x t r a s p o t s i n the s t a c k i n g d i r e c t i o n has been i n t e r p r e t t e d as b e i n g 2k° .(Comes, 1977; Heegar, 1977). The drop i n c o n d u c t i v i t y i s b e l i e v e d to i n d i c a t e the opening of a gap i n the e l e c t r o n i c spectrum due to the 2kp d i s t o r t i o n . T h i s d i s t o r t i o n i s incommensurate w i t h the l a t t i c e and s h o u l d t h u s , i n the absence o f p i n n i n g , have no p r e f e r r e d phase 24 r e l a t i v e to the l a t t i c e . In the apparent absence o f a c o l l e c t i v e mode c o n t r i b u t i o n t o the d.c. c o n d u c t i v i t y , i t i s assumed t h a t the charge d e n s i t y wave i s pinned to the l a t t i c e by i m p u r i t i e s or t h r e e - d i m e n s i o n a l o r d e r i n g e f f e c t s . A pinned c o l l e c t i v e mode would s t r o n g l y enhance the low f r e q u e n c y d i e l e c t r i c c o n s t a n t . Such a mechanism has been advanced (Lee, R i c e , and Anderson, 1974) to account f o r the h i g h d i e l e c t r i c c o n s t a n t o f about 3000 observed below 40 GHz. 25 1.3b D i e l e c t r i c C o n s t a n t f o r a Pinned Charge D e n s i t y Wave As has been d i s c u s s e d , a charge d e n s i t y wave which was incommensurate w i t h the l a t t i c e c o u l d be expected to propagate w i t h o u t a t t e n u a t i o n . Lee, R i c e , and Anderson (1974) p o i n t o u t , however, t h a t both t h r e e - d i m e n s i o n a l o r d e r i n g and i m p u r i t i e s w i l l p i n the charge d e n s i t y waves on d i f f e r e n t c h a i n s to each o t h e r and to the c r y s t a l l a t t i c e . T h r e e - d i m e n s i o n a l o r d e r i n g , t h e y f e l t , s h o u l d r e s u l t i n a sharp t r a n s t i o n to a low tempe r a t u r e i n s u l a t i n g phase w h i l e i m p u r i t y p i n n i n g s h o u l d g i v e r i s e to a more g r a d u a l t r a n s i t i o n . The way i n which such a pinned mode c o u l d c o n t r i b u t e to the low f r e q u e n c y d i e l e c t r i c c o n s t a n t i s d e s c r i b e d below. The d i s t o r t i o n opens a gap o f w i d t h Vij a t the f e r m i w a v e v e c t o r . A s s o c i a t e d w i t h t r a n s i t i o n s a c r o s s t h i s gap w i l l be a p o l a r i z a b i l i t y and a c o n t r i b u t i o n t o the d i e l e c t r i c c o n s t a n t . These are l a b e l l e d as the s i n g l e p a r t i c l e c o n t r i b u t i o n s , °<.%'' and € s f r e s p e c t i v e l y , t o d i s t i n g u i s h them from those due to the c o l l e c t i v e mode i t s e l f . W i t h the c o l l e c t i v e mode we can a s s o c i a t e an e f f e c t i v e c h a r g e , e e , and a reduced mass, M. The e q u a t i o n o f motion f o r the pinned wave i n an a.c. f i e l d , Ee , i s t h e n , ( B a l k a n s k i , 1972) (1.5) M u + M r u + M « 7 * u = e f Ee , M / t where C i s a damping c o n s t a n t and w i l l be the f u l l w i d t h a t h a l f maximum o f the resona n c e , and u i s the c o o r d i n a t e d e s c r i b i n g the 26 motion o f the charge d e n s i t y wave r e l a t i v e to the l a t t i c e . The s o l u t i o n t o (1.5) i s (1.6) u=(e eE/M)e / (&J-UJ-iVw) The a s s o c i a t e d d i p o l e moment i s then ue e so t h a t the p o l a r i z a b i l i t y i s g i v e n by (1.7) * = ° t p + ( e * / M ) - i ? « > ) where the s i n g l e p a r t i c l e c o n t r i b u t i o n has been i n c l u d e d . A s h c r o f t and Mermin (1976) show how the d i e l e c t r i c c o n s t a n t can be o b t a i n e d from a p o l a r i z a b i l i t y o f t h i s form. T h e i r r e s u l t i s o b t a i n e d u s i n g the C l a u s i u s - M o s s o t t i r e l a t i o n and i s thus r e s t r i c t e d to c r y s t a l s w i t h c u b i c symmetry i n which the L o r e n t z l o c a l f i e l d i s a p p r o p r i a t e . TTF-TCNQ c l e a r l y does not p r e s e n t a s i t u a t i o n o f c u b i c symmetry. I f we are i n t e r e s t e d i n the d i e l e c t r i c c o n s t a n t f o r a p r i n c i p a l a x i s , however, i t w i l l s t i l l be t r u e t h a t the m a c r o s c o p i c f i e l d , the p o l a r i z a t i o n , and thus the l o c a l f i e l d are p a r a l l e l . In l i g h t o f t h i s , we assume E*1""1 (r)=K E ( r ) where K i s some c o n s t a n t depending on €<w>. For c u b i c symmetry, K=l+(£ - l ) / 3 . Using the r e l a t i o n s h i p s ( A s h c r o f t and Mermin, 1976) P ( r ) = ((£ - D / 4 T T ) f ( r ) and P ( r ) = (* /v)E,e'°' (r) where v i s the volume a p p r o p r i a t e to <*. , we g e t ( £ -1)/K = 4TT =< /v . 27 Replacing << using Eq. (1.7) g i v e s ( 1 . 8 ) i i S L l l - i S f * " t « / V M ) Here, £ (cv) i s the complex d i e l e c t r i c constant at frequency oO . eE, M, and v can be e l i m i n a t e d by d e f i n i n g the low frequency d i e l e c t r i c c o n s t a n t , £ ( 0 ) , and the high frequency v a l u e , £ (°°) . £ (°0) i s the d i e l e c t r i c constant f o r to much g r e a t e r than CO but sma l l e r than 2TT^ . We thus have (1.9a) ( € ( 0 ) - l ) / K ( 0 ) = (4Tr/v) ( <*,p +e//(M *7*) ) (1.9b) ( £ («o)-l ) /K («o ) = 4TToc J ' / v £(oo) c o r r e c t l y depends o n l y on t r a n s i t i o n s across the gap and we w i l l i d e n t i f y i t with € s p . If we can make the assumption that K(co) = l+(£ («J ) - l ) / ( B + l ) , then i t can be shown that (1.10) £{w) = + (£(0)- £i?) U)*/{ cuf-tv'-irco) where t*J*= (£{<=o) +B) £^/(£(0)+B). T h i s i s the e x p r e s s i o n f o r £ {to) as used by E l d r i d g e and Bates (1979) . While i t i s not c l e a r how we can connect u)o to the m i c r o s c o p i c p i c t u r e o f a pinned charge d e n s i t y wave, t h i s model does pr o v i d e the b a s i s f o r a phenomenolog i c a l f i t to £.{u>) . In Eq. (1.10), f o r U) c l o s e to UJ0 , the imaginary p a r t of £{oa) i s a L o r e n t z i a n of f u l l width at h a l f maximum giv e n by P . £ s p i s given by Lee, R i c e , and Anderson (1974) as ^ = 1+2^/3^ 28 This i s the d i e l e c t r i c constant at frequencies where the pinned mode cannot respond and only t r a n s i t i o n s across the gap can contr i b u t e . E l d r i d g e f i t s h i s i n f r a - r e d measurements to t h i s model with £=1068, £(o)=3600, the pinning frequency equal to 102 GHz, and |=1.5GHz. 29 1.3c Purpose o f D i e l e c t r i c Resonance S t u d i e s on TTF-TCNQ The purpose o f t h i s s e r i e s o f ex p e r i m e n t s was to extend B a r r y ' s (1977) d i e l e c t r i c resonance measurements on TTF-TCNQ a t low t e m p e r a t u r e s i n an attempt to c l a r i f y the f r e q u e n c y dependence o f the r e a l and i m a g i n a r y p a r t s o f the d i e l e c t r i c c o n s t a n t . These are o f p a r t i c u l a r i n t e r e s t i n t h a t the low f r e q u e n c y d i e l e c t r i c c o n s t a n t i s i m p o r t a n t i n f i t t i n g the i n f r a - r e d b o l o m e t r i c measurements o f E l d r i d g e and Bates (1979) to eq u a t i o n ( 1 . 1 0 ) f o r the pinned mode d i e l e c t r i c c o n s t a n t . I t i s a l s o i n t e r e s t i n g to c o n s i d e r the f r e q u e n c y dependence o f the d i e l e c t r i c c o n s t a n t i n l i g h t o f t h i s model. U n f o r t u n a t e l y , the fre q u e n c y range c o n v e n i e n t l y a v a i l a b l e i s too s m a l l to g i v e any c o n c l u s i v e i n d i c a t i o n o f the v a l i d i t y o f the model. There have been two p r e v i o u s measurements o f the d i e l e c t r i c c o n s t a n t p a r a l l e l to the s t a c k i n g a x i s , €'b . Khanna e t a l . (1974) o b t a i n e d £'t= (3 . 2±0. 6) x l O 3 u s i n g c a v i t y p e r t u r b a t i o n a t 10.4 GHz. They a l s o c l a i m e d to have measured a c r y s t a l w i t h the lo n g d i r e c t i o n a l o n g the a a x i s and o b t a i n e d £« = 6 i 2 . B a r r y (1977) attempted to measure €'b a t h i g h e r f r e q u e n c i e s u s i n g the d i e l e c t r i c r e s o n a t o r t e c h n i q u e and o b t a i n e d s i m i l a r v a l u e s . The same measurements, however, gave a v a l u e f o r o f about 2. T h i s seemed too s m a l l f o r such a m a t e r i a l . In a d d i t i o n t o t h i s , t he shape o f the mode p l o t s , showing the res o n a n t f r e q u e n c y squared 30 versus the inverse square length of the c r y s t a l , were not f u l l y understood. One mode was i d e n t i f i e d as c o a x i a l and thus expected to have a slope corresponding to propagation along the b a x i s at about the speed of l i g h t . The d i e l e c t r i c modes were expected to giv e a slope p r o p o r t i o n a l to l/£'a. In f a c t a l l modes gave s t r a i g h t l i n e s with a slope intermediate between the two expected extremes. This was a t t r i b u t e d to end e f f e c t s but the lack of understanding of the slope d i d cast some doubts on the r e l i a b i l i t y of the e x t r a p o l a t i o n to the ordinate a x i s which was used to allow a n a l y s i s i n the i n f i n i t e length l i m i t . The present work was c a r r i e d out i n the hope of extending the measurements to s i t u a t i o n s i n which some of the paradoxes might be re s o l v e d . 31 CHAPTER I I The Measurement o f E l e c t r i c a l P r o p e r t i e s o f MEM(TCNQ)^ Above Room Temperature 2.1 C a v i t y P e r t u r b a t i o n The e l e c t r i c a l p r o p e r t i e s o f s m a l l c r y s t a l s can be s t u d i e d u s i n g c a v i t y p e r t u r b a t i o n methods s i m i l a r to those which have been d e s c r i b e d by a number o f a u t h o r s '• (Buravov and Schegolev,1971; Khanna et a l . ,1975; Waldron, 1969). In t h i s t e c h n i q u e , the s h i f t and change i n w i d t h , due to i n s e r t i o n o f a s m a l l s o l i d sample, o f the resonance o f a c a v i t y o f known c h a r a c t e r i s t i c s can be r e l a t e d to the r e a l and i m a g i n a r y p a r t s o f the d i e l e c t r i c c o n s t a n t o f the sample i f the f i e l d s i n the sample are known. The g e n e r a l f o r m u l a f o r c a v i t y p e r t u r b a t i o n i s d e r i v e d i n Waldron: (M.K.S. u n i t s ) Here, E„ and H e a r e the u n p e r t u r b e d e l e c t r i c and magnetic f i e l d s and D 0 and B 0 a r e the unperturbed e l e c t r i c and magnetic d i s p l a c e m e n t s . Sf/f i s the f r a c t i o n a l s h i f t i n the r e sonant f r e q u e n c y o f the c a v i t y due to the p e r t u r b a t i o n . The s u b s c r i p t 1 l a b e l s f i e l d s which a r e the d i f f e r e n c e between the f i e l d s w i t h the sample p r e s e n t and the u n perturbed f i e l d s . The i n t e g r a l i s over the volume o f the c a v i t y . T h i s e q u a t i o n n e g l e c t s the 32 d i f f e r e n c e between D0 and Bo and the t o t a l TT and B , r e s p e c t i v e l y , i n the i n t e g r a l i n the denominator. I t i s thus c o r r e c t o n l y f o r s m a l l p e r t u r b a t i o n s . The e x p e r i m e n t s were c a r r i e d out w i t h the needle-shaped c r y s t a l s o r i e n t e d p a r r a l l e l to and a t the maximum o f the E f i e l d o f the TM01<? mode i n the c y l i n d r i c a l c a v i t y . For t h i s mode E z= E„z J0 (kr) e w t (2.2) H„= - ( j ^ / k ) E o z J 1 (kr) e!wt E r = E ^ =Hr =H Z, =0 where Eoz i s the a m p l i t u d e o f E z and, f o r a c a v i t y o f r a d i u s a, k a = A, w i t h X b e i n g the f i r s t zero o f J Q ("X) . The denominator o f Eq. (2.1) i s then e a s i l y e v a l u a t e d t o g i v e (2.3) fff^(E0 • D0 -H 0 * B 0 ) dV =2V„ J * (ka) E* Using the f a c t t h a t H=0 along the c a v i t y a x i s and t h a t the f i e l d s w i t h s u b s c r i p t 1 a r e assumed n e g l i g i b l e o u t s i d e o f the sample, we can w r i t e Eq. (2.1) as ,2. 4, If*= ///* (E,-R.-E,-5)ay where the i n t e g r a l i s now over the volume o f the sample. F o l l o w i n g Buravov and Schegolev (1971) one can assume t h a t f o r a sample w i t h a d i m e n s i o n s m a l l e r than the s k i n depth and w i t h d e p o l a r i z i n g f a c t o r n and d i e l e c t r i c c o n s t a n t £ , the f i e l d 33 i n s i d e the sample i n the presence o f e x t e r n a l f i e l d Ez i s g i v e n by (2.5) E £ = E»/(l+n(€-l) ) . n i s d e f i n e d f o r a g e n e r a l e l l i p s o i d o f semi-major a x i s a" and semi-minor axes b" and c" as (Osborne, 1945) (2.6) n = ( b " c " / a " * ) ( I n ( 4 a " / ( b " + c " ) ) - l ) , We then have (2.7a) E 1 =-E7 n ( € - l ) / ( l + n ( € - l ) ) (2.7b) D, =E Z ((£ -l)-n(£-l))/(l+n(e-l)) Ta k i n g E x to be c o n t i n u o u s a c r o s s the sample and equal to the maximum v a l u e , E o z. , a l l o w s us to d i s p o s e o f the i n t e g r a l g i v ing Jf/f=-V, ( € - l ) / ( 2 V e J | (ka) ( l + n ( e - l ) ) ) F o l l o w i n g Waldron, we can i n c l u d e the l o s s e s i n the sample by r e p l a c i n g e by <f'-j t " where £"=2°Vf and cf i s the c o n d u c t i v i t y i n sec."' . St/1 i s then r e p l a c e d by (Af0 +j A<>/2) where A f 0 = Sf/f and Ao/2 i s the change, due to the p e r t u r b a t i o n , i n the f u l l w i d t h a t h a l f maximum o f the resonan c e , d i v i d e d by the unperturbed f r e q u e n c y . Ao/2 i s g i v e n by (1/Q C-1/Q )/2. Buravov and Schegolev d e f i n e (2.9) * =V, E,J /2 \EZ\ dV w h i c h , f o r our mode, becomes V, /2V 0 J * (ka) =1.855Vf /V0 so t h a t we a r r i v e a t t h e i r f o r m u l a e : 34 (2.10) (€-D(i+n(£-l)j + ne X - « 3. and (2.11) 2. » _ « a . These may then be i n v e r t e d to g i v e ; (2.12) £-1 = -1 n and (2.13) f These formulae w i l l , i n g e n e r a l , be a p p l i c a b l e o n l y f o r s m a l l c o n d u c t i v i t i e s and t h i n c r y s t a l s f o r which the s k i n depth i s l a r g e compared to the s m a l l e s t d i m e n s i o n . . The s k i n depth i s g i v e n by £ $ =cL /(2ir<ttJ)i where c L i s the speed o f l i g h t , cr' i s the c o n d u c t i v i t y i n sec. -' , and UJ i s the f r e q u e n c y i n r a d i a n s / s e c . (1 sec.""' = 9X10 " ( J l cm)" ) . In the h i g h c o n d u c t i v i t y regime, o n l y the r e a l p a r t o f the c o n d u c t i v i t y can be d e t e r m i n e d . I f the minimum sample d i m e n s i o n p e r p e n d i c u l a r to the f i e l d i s much l a r g e r than the s k i n d e p t h , 35 as was the case f o r some o f the c r y s t a l s s t u d i e d , one must use the e x p r e s s i o n o f Cohen e t a l . (1975) f o r a p r o l a t e s p h e r o i d , ( 2 * 1 5 ) 2s CL A C where f i s the o p e r a t i n g f r e q u e n c y and r i s the semi-minor a x i s . The d e r i v a t i o n o f t h i s e x p r e s s i o n c o n t a i n s the assumption t h a t Af 0^«/n i n the s k i n depth l i m i t e d regime. °< i s p r o p o r t i o n a l to the s p h e r o i d volume and e n t e r s the e x p r e s s i o n i n a way which s u g g e s t s t h a t one may extend the f o r m u l a to the case o f a g e n e r a l e l l i p s o i d by u s i n g Eq. (2.15) f o r a p r o l a t e s p h e r o i d o f the same volume. In terms o f the e l l i p s o i d semi-axes, a">b">c", the e x p r e s s i o n becomes The q u e s t i o n then a r i s e s as to what g e n e r a l e l l i p s o i d s h o u l d be used to approximate a r e c t a n g u l a r c r y s t a l o f l e n g t h , w i d t h , and t h i c k n e s s , a>b>c. The appearance, i n c<. , o f the sample volume, and, i n n, o f the r a t i o o f the d i m e n s i o n s , s u g g e s t s t h a t the a p p r o p r i a t e e l l i p s o i d has the same volume and d i m e n s i o n a l r a t i o s . In terms o f the c r y s t a l o f l e n g t h , w i d t h , and t h i c k n e s s , a>b>c, the e x p r e s s i o n becomes 36 (2.17) 0 ~ a • + c 2 s * C A 0 I f the s k i n depth i s not much l a r g e r than the sample t h i c k n e s s , then the assumption o f u n i f o r m p e n e t r a t i o n i s a p p r o p r i a t e and one can use Eq. (2.13) to f i n d €" and thus <r . As l o n g as the s k i n depth i s l a r g e , t h i s i s t r u e even f o r the regime where one has ne">>l g i v i n g Afa^<</n, and i n which cannot be determined (Cohen e t a l . , 1 9 7 5 ) . In the regime where the s k i n depth has become s m a l l r e l a t i v e to the minimum sample d i m e n s i o n s , Eq. (2.13) i s no l o n g e r v a l i d and Eq. (2.17) must be used to c a l c u l a t e cr. A d i s c u s s i o n as to which model i s v a l i d i n p a r t i c u l a r c a s e s appears i n more d e t a i l i n Chapter I I I . 37 2.2 The Apparatus Measurements on f o u r MEM(TCNQ)a c r y s t a l s were c a r r i e d out i n a c y l i n d r i c a l copper c a v i t y u s i n g the TM0W mode i n which the E n f i e l d has an a n t i n o d e a l o n g the c a v i t y a x i s . In t h i s mode, the r e s o n a n t f r e q u e n c y i s p r o p o r t i o n a l to the d i a m e t e r o f the c a v i t y and independent o f the h e i g h t . (Hidy e t a l . ,1972). The resonance was s t u d i e d by o b s e r v i n g the t r a n s m i s s i o n o f the c a v i t y which r e s o n a t e d empty a t about 8.5 GHz. The c a v i t y i s shown s c h e m a t i c a l l y i n F i g . 9. The c a v i t y was heated by means o f a c i r c u l a t i n g water b a t h . Heat exchangers were p l a c e d on both f a c e s between the c a v i t y body and the incoming and o u t g o i n g b r a s s waveguides. A thermocouple was clamped to the c a v i t y body midway between the exchangers. T h i s assembly was then p l a c e d i n a c l o s e f i t t i n g s t y r o f o a m e n c l o s u r e w i t h w a l l s o f about 5 cm. t h i c k n e s s . Temperature e q u i l i b r i u m was m o n i t o r e d by a l t e r i n g the heat b a t h t e m p e r a t u r e and then o b s e r v i n g the e q u i l i b r a t i o n o f the temperature by f o l l o w i n g the f r e q u e n c y o f the resonance. The c r y s t a l volume and shape were determined from the weight o f the c r y s t a l , u s i n g the known c r y s t a l l o g r a p h i c d e n s i t y o f 1.261 g/cc, (Bosch and van Bodegom, 1977), t o g e t h e r w i t h m i c r o s c o p i c photography. C r y s t a l s were mounted on the end o f a t h i n q u a r t z f i b r e u s i n g epoxy so t h a t the l o n g a x i s was p a r a l l e l to the E f i e l d s 38 Figure 9. The 8.5 GHz resonant c a v i t y 39 a l o n g the c a v i t y a x i s . The b a s i c experiment i n v o l v e d measuring the f r e q u e n c y s h i f t and change i n w i d t h o f the c a v i t y passband due to i n s e r t i o n o f the c r y s t a l . I t was thus n e c e s s a r y t o use the f r e q u e n c y and w i d t h o f the resonance, w i t h the q u a r t z f i b r e i n s e r t e d , as the unperturbed v a l u e s . For these measurements , a s m a l l drop o f epoxy was p l a c e d on the end o f the f i b r e so t h a t the s m a l l a d d i t i o n a l amount r e q u i r e d to suspend the c r y s t a l c o u l d be n e g l e c t e d . The f r e q u e n c y and w i d t h o f the resonance w i t h o n l y the q u a r t z f i b r e i n p l a c e , was then measured as a f u n c t i o n o f t e m p e r a t u r e to a l l o w f o r c a v i t y e x p a n s i o n . Resonance d a t a a t a g i v e n temperature f o r t h a t f i b r e was then o b t a i n e d by i n t e r p o l a t i o n . Sample h a n d l i n g t e c h n i q u e s were such t h a t the f i b r e c o u l d be a c c u r a t e l y r e p o s i t i o n e d a f t e r mounting of the c r y s t a l . The resonance was observed by f r e q u e n c y m o d u l a t i n g the R.F. s o u r c e a t 5 KHz and demodulating the d e t e c t e d s i g n a l w i t h a l o c k - i n a m p l i f i e r . The r e s u l t i n g s i g n a l c o u l d be used i n two ways. I t c o u l d be i n t e g r a t e d and added to the 5KHz m o d u l a t i n g s i g n a l to g i v e an e r r o r s i g n a l which l o c k e d the R.F. s o u r c e to the c e n t r e f r e q u e n c y o f the resonance. T h i s mode o f o p e r a t i o n was u s e f u l f o r m o n i t o r i n g o f the resonance but i n c o n v e n i e n t f o r a b s o l u t e measurements o f the c e n t r e f r e q u e n c y due to d r i f t i n the i n t e g r a t o r . A l t e r n a t i v e l y , the o u t p u t o f the l o c k - i n c o u l d be r e c o r d e d on an X-Y r e c o r d e r w i t h the X i n p u t b e i n g d r i v e n , t h r o u g h a D/A c o n v e r t e r , d i r e c t l y from a f r e q u e n c y c o u n t e r 40 during a slow sweep. The width and centre frequency of the resonance could then be taken d i r e c t l y from the d e r i v a t i v e t r a c e . F i g . 10 shows the general arrangement of the apparatus. LOCK-IN REF. INPUT OUTPUT h X-Y REC. Y_ X i f — f t D/A CONV. OUTf? ^IN DETECTOR CIRC. COUNTER (± THERMO-COUPLE BATH =*= R.F. AMP. HEATER a n i DIGITAL OUT INSULATION SWEEP OSCILLATOR DIR. COUPLER R.F. SOURCE F.M. ^ INPUT ATTENUATOR+ISOLATOR L CAVITY J ISOLATOR Figure 10. The c a v i t y p e r t u r b a t i o n apparatus . 42 2.3 The Measurements Four MEM(TCNQ)A c r y s t a l s were s t u d i e d i n d e t a i l u s i n g c a v i t y p e r t u r b a t i o n . The p h y s i c a l p r o p e r t i e s , i n c l u d i n g the d e p o l a r i z i n g f a c t o r s and the dimensions a r e summarized i n Tab l e 1. I t has been mentioned t h a t the v i o l e n c e o f the f i r s t o r d e r t r a n s i t i o n a t 60°C p r o v i d e d some i n c e n t i v e f o r the use o f c o n t a c t l e s s microwave measurements. F i g . 11 shows c r y s t a l 2 b e f o r e and a f t e r h a v i n g passed through the 60°C t r a n s i t i o n . The c r y s t a l has been broken o f f o f the q u a r t z f i b r e and thus appears s h o r t e r i n B. The r e a l e f f e c t o f the t r a n s i t i o n i s the appearance o f f l a k i n g on the f a c e which i s b e i n g viewed o b l i q u e l y . The c r y s t a l a l s o appears s l i g h t l y b e n t . The damage to the c r y s t a l w i l l a f f e c t o n l y the d e p o l a r i z i n g f a c t o r s i n c e i t i s the o n l y shape dependent parameter i n the a n a l y s i s . T h i s e f f e c t h a s, however, been assumed n e g l i g i b l e i n comparison to the u n c e r t a i n t y i n the d e p o l a r i z i n g f a c t o r s used to d e s c r i b e r e c t a n g u l a r samples. I t has been assumed t h a t the d e p o l a r i z i n g f i e l d s due to p o l a r i z a t i o n charges a t the s u r f a c e o f the c r a c k s are not a s e r i o u s problem. T h i s assumption i s v a l i d i f e i s not too l a r g e and the gap i s s m a l l so t h a t the c a p a c i t a n c e o f the c r a c k i s l a r g e r e l a t i v e to t h a t o f the whole c r y s t a l . In s p i t e o f the damage, i t was found t h a t the c r y s t a l c o u l d be r e t u r n e d t h rough the t r a n s i t i o n and t h a t the low temperature 43 TABLE 1 P h y s i c a l Data f o r MEM(TCNQ) c r y s t a l s used C r y s t a l Source U.B.C. U.B.C. U.B.C. Groningen Length a (cm) 0.259 0.519 0. 465 0. 690 Width b (cm) 0.028 0.038 0.037 0.062 T h i c k n e s s c (cm) 0.0185 0.032 0.037 0.0267 0.0162 0.0108 0.0141 0.00843 44 F i g u r e 11. Photograph o f MEM (TCNQ)^ c r y s t a l b e f o r e (A) and a f t e r (B) h a v i n g passed through the t r a n s i t i o n . 45 f r e q u e n c y s h i f t and w i d t h were r e p r o d u c e d . There was, however, c o n s i d e r a b l e h y s t e r e s i s i n the t r a n s i t i o n , o c c u r r i n g t y p i c a l l y , on h e a t i n g , a t about 65°C and, on c o o l i n g , a t 60°C. T h i s b e h a v i o u r has a l s o been observed i n d.c. and s i m i l a r a.c. measurements (Morrow e t a l . 1979). The h y s t e r e s i s does not depend on the time taken to e q u i l i b r a t e the c r y s t a l a t t e m p e r a t u r e s c l o s e to the t r a n s i t i o n t e m p e r a t u r e . I t was found t h a t the h y s t e r e s i s was s t i l l p r e s e n t when the c r y s t a l was c y c l e d t h r o u g h the t r a n s i t i o n f o r a second t i m e . Graphs o f the f r e q u e n c y and w i d t h o f the p e r t u r b e d resonance as a f u n c t i o n o f t e m p e r a t u r e , w i t h c r y s t a l 4 i n s e r t e d , appear i n F i g . 12 and F i g . 13. T h i s shows c l e a r l y the h y s t e r e s i s i n the t r a n s i t i o n t e m p e r a t u r e and the r e p r o d u c i b i l i t y o f the low t e m p e r a t u r e p r o p e r t i e s a f t e r the t r a n s i t i o n . The downward and upward p o i n t i n g arrows d e p i c t the t r a n s i t i o n on h e a t i n g and on c o o l i n g r e s p e c t i v e l y . T y p i c a l t r a c e s w i t h j u s t the q u a r t z f i b r e and w i t h the f i b r e and a c r y s t a l a t about 61°C, on the h e a t i n g c y c l e , appear i n F i g . 14. The t r a c e shown i s the d e r i v a t i v e o f the power t r a n s m i t t e d by the c a v i t y . Each d e r i v a t i v e t r a c e was o b t a i n e d w i t h a 10MHz wide s c a n . 8.5 C D o warming • cooling o o C D N X c CD = 3 C J <D 8 . 4 irjiio o 8.3 20 4 0 6 0 temperature (°C) 80 gure 12. Typical cavity resonant frequency versus temperature with MEM(TCNQ)a c r y s t a l inserted. 12, 10 o warming • cooling 8 8 j o • o o I 6 5-TJ £ 4 cP 0 20 4 0 60 ' 80 temperature (°C) Figure 13. Typical width of cavity resonance versus temperature with MEM (TCNQ)a c r y s t a l inserted 4 9 CHAPTER I I I R e s u l t s 3.1 Regime o f V a l i d i t y o f P e r t u r b a t i o n R e s u l t s In o r d e r to i n t e r p r e t the f r e q u e n c y s h i f t and change i n w i d t h i n terms o f the d i e l e c t r i c c o n s t a n t s , i t i s n e c e s s a r y to d e t e r m i n e what assumptions can be made about the f i e l d p e n e t r a t i o n i n the a n a l y s i s . The p o s s i b l e s i t u a t i o n s a r e t h a t 1) the f i e l d s a r e f u l l y p e n e t r a t i n g and n £'' i s s m a l l so t h a t b o th €' and e" can be o b t a i n e d from eqs. 2.12 and 2.13, 2) t h a t the f i e l d s a r e not s k i n depth l i m i t e d but ne" i s l a r g e so t h a t o n l y e" can be o b t a i n e d and Eq. 2.13 i s a p p r o p r i a t e o r 3) t h a t the f i e l d s a r e s k i n depth l i m i t e d and ne" i s l a r g e so t h a t o n l y cr o r , e q u i v a l e n t l y , £•" can be o b t a i n e d from Eq. 2.17. Eqs. 2.12 and 2.13 can be used to c a l c u l a t e e' and e" as l o n g as ne" i s s m a l l and S5 i s l a r g e . We see, i n T a b l e 2, t h a t t h i s i s the case below the t r a n s i t i o n . Above the t r a n s i t i o n , however, the s i t u a t i o n i s somewhat more c o m p l i c a t e d . I f we compare the measured v a l u e o f A f 0 to <*/n, where n i s c a l c u l a t e d f o r a "comparable" e l i p s o i d a l sample, the approximate e q u a l i t y s u g g e s t s t h a t t h i s i s the regime i n which ne" i s l a r g e . In t h i s regime, the observed p e r t u r b a t i o n i s i n s e n s i t i v e to €.' . To i n v e s t i g a t e the e f f e c t o f n o n - e l l i p s o i d a l shape, two copper p i e c e s , o f l e n g t h 5 and 3 mm, and s i m i l a r i n shape to the MEM (TCNQ) a c r y s t a l s , were s t u d i e d . For these samples, (°(/A£0)/n 50 was found to be 1.033 and 1.027 where n was calculated from Eq. 2.6. These deviations from one are s u r p r i s i n g l y small. The somewhat larger discrepances observed in the MEM(TCNQ)^ cr y s t a l s may be due to the e f f e c t of anisotropy on the transverse currents which flow in a non-ellipsoidal sample. These deviations from unity are, at any rate, s u f f i c i e n t l y small that we can assume that, above the t r a n s i t i o n , the c r y s t a l is in the regime where the frequency s h i f t i s insensitive to e'. Having determined that ne' i s large above the t r a n s i t i o n , we are s t i l l faced with the decision as to whether or should be calculated assuming skin depth limited penetration, and thus using Eq. 2.17, or assuming f u l l penetration and thus using Eq. 2.13. Clearly one has to base this decision on whether or not the obtained cr gives a skin depth that i s consistent with the size of skin depth assumed in the c a l c u l a t i o n of a* . We w i l l generally take Eq. 2.17 to be v a l i d for a skin depth smaller than 1/2 of the smallest dimension and use Eq. 2.13 otherwi se. If we analyse the results for c r y s t a l s 2,3, and 4, assuming a small skin depth, and thus using Eq. 2.17, the values of <f obtained imply a skin depth that i s s l i g h t l y smaller than 1/2 of the smallest dimension. The assumptions i m p l i c i t in Eq. 2.17 are thus approximately true. In the analysis of the results for c r y s t a l 1, assuming a small skin depth and thus using Eq. 2.17, y i e l d s a value for (f 51 which i s an order of magnitude smaller that that found for c r y s t a l 4 and hence a skin depth which i s comparable to the smallest sample dimension. Since 6~s is larger than half of the thickness, i t i s thus inappropriate to use the assumption of skin depth limited penetration for Crystal 1. I f , on the other hand, we assume that the skin depth i s large for c r y s t a l 1, we can use Eq. 2.11 to relate €•" to A o/2. Because we are in the regime where n e" i s large z><>/2 becomes independent of (e'-l) and we can invert Eq. 2.11 d i r e c t l y to get (3.1) £"= * / ( n * - U./2)) . Eq. 3.1 i s , in fact, just Eq. 2.13 in which Af a has been set equal to <*/n. The conductivity obtained for c r y s t a l 1 using Eq. 3.1 implies a skin depth which i s only s l i g h t l y smaller than the minimum cr y s t a l dimension. This j u s t i f i e s the assumption of penetrating f i e l d s made in the use of Eq. 3.1. The relationship between the two ways of calcu l a t i n g & in the high conductivity regime i s i l l u s t r a t e d in F i g . 15a. Assuming, for this regime, that e' i s unimportant and setting i t a r b i t r a r i l y to 1, and using the values of f, n, and <*-appropriate to c r y s t a l s 1 and 4 just above the t r a n s i t i o n , we have plotted log l o A f a and log l o versus logio o" using Eqs. 2.12 and 2.13. Using the same values of f, n, and «*. plus the value of A f 0 appropriate to each of the two c r y s t a l s , we have also plotted log Ao versus log & using Eq. 2.17. Eq. 2.12 ceases to be useful for the determination of £' in the region 52 LOG «r F i g u r e 15. Showing the r e l a t i o n between w i d t h and m u s i n g Eqs. 2.10 ,2.13, and 2.17 f o r e x p e r i m e n t a l c o n d i t i o n s as found f o r MEM (TCNQ),, c r y s t a l s 1 and 4. a) l o g | Q ( A 0 ) v e r s u s l o g , , (cr) u s i n g Eq. 2.17 b) l o g ,e (A 0) v e r s u s l o g ,»(c) u s i n g Eq. 2.13 c) l o g ( 0 (Af.) v e r s u s l o g ,0 (**) u s i n g Eq. 2.10 d) observed l o g , 0(^&) j u s t above t r a n s i t i o n e) observed l o g l 0 ( A© ) j u s t above t r a n s i t i o n f) l o g ,0 (tf) c o r r e s p o n d i n g S$ =1/2 o f the s m a l l e s t d i m e n s i o n . 0 ' 100 10 -1 200 100 1/A.X10' ure 15b. c versus 1/A© for assumption of large and small skin depth fo c r y s t a l s 1 and 4. 54 where l o g ) 0 Af0 i s c o n s t a n t . Eq. 2.17 i s taken to be v a l i d to the r i g h t o f t h a t c o n d u c t i v i t y f o r which the s k i n depth i s about h a l f o f the s m a l l e s t d i m e n s i o n . In the i n t e r m e d i a t e r e g i o n , Eq. 2.13 p r o b a b l y g i v e s a b e t t e r measure o f cr than Eq. 2.17. The a c t u a l v a l u e o f A0 observed i s denoted by the arrow l a b e l l e d ( e ) . For c r y s t a l 4, we see t h a t the observed change i n w i d t h puts us onto the c u r v e s i n a r e g i o n where the two formulae g i v e s i m i l a r r e s u l t s . More i m p o r t a n t l y , t h i s i s a r e g i o n where Eq. 2.17 i s expected to be a p p l i c a b l e s i n c e the s k i n - d e p t h i s l e s s than h a l f o f the s m a l l e s t d i m e n s i o n . For c r y s t a l 1, on the o t h e r hand, the observed l o s s i s i n a r e g i o n where Eq. 2.17 i m p l i e s a s k i n depth l a r g e r than the sample and i s thus i n c o n s i s t e n t . Eq. 2.13, however, does appear t o be v a l i d f o r c r y s t a l 1 i n t h i s regime. In F i g . 15b, we have p l o t t e d cr v e r s u s 1/A 0 f o r these two c r y s t a l s . For c r y s t a l 4, the observed A Q y i e l d s a s i m i l a r o" f o r both o f the assumptions about the s k i n d e p t h . The s k i n depth l i m i t e d c a l c u l a t i o n i s p r e f e r r e d , i n t h i s c a s e , as i t a v o i d s some o f the u n c e r t a i n t y a s s o c i a t e d w i t h the ( (<* /n) - At0 ) term i n t h i s regime. The t r u e c u r v e r e l a t i n g cr to A0 p r o b a b l y l i e s somewhat above the two c u r v e s i n the r e g i o n o f the i n t e r s e c t i o n . As a r e s u l t , c o n d u c t i v i t i e s o b t a i n e d c l o s e to the c r o s s over between the s k i n depth l i m i t e d and p e n e t r a t i n g regimes a r e p r o b a b l y u n d e r e s t i m a t e d . For c r y s t a l 1, i t i s c l e a r t h a t s k i n depth l i m i t e d p e n e t r a t i o n i s i n a p p r o p r i a t e and Eq. 2.13 or 3.1 55 i s t o be p r e f e r r e d f o r the c a l c u l a t i o n o f <y . As a r e s u l t o f these c o n s i d e r a t i o n s , the c o n d u c t i v i t i e s f o r c r y s t a l 1 above the t r a n s i t i o n were c a l c u l a t e d u s i n g Eq. 2.13 w i t h A.f0=<*/n, and those f o r c r y s t a l 4 were c a l c u l a t e d u s i n g Eq. 2.17. The o t h e r two c r y s t a l s were i n t e r m e d i a t e c a s e s and t h e i r c o n d u c t i v i t i e s were c a l c u l a t e d u s i n g Eq. 2.17 above the t r a n s i t i o n . We note here t h a t t h e r e i s some d i f f i c u l t y a s s o c i a t e d w i t h the i n t e r c h a n g e a b l e use o f «/n and Af0 i n the c a l c u l a t i o n o f cr i n the l a r g e ne' regime. E q u a t i o n 2.17 r e p l a c e s <*/n by A f 0 whereas i n e q u a t i o n 3.1 i t i s e f f e c t i v e l y Af0 t h a t has been r e p l a c e d by <*/n. T h i s problem r e p r e s e n t s an u n c e r t a i n t y t h a t i s i n t r i n s i c to these measurements i n t h i s regime. 56 3.2 D i e l e c t r i c C o n s t a n t s and C o n d u c t i v i t y The microwave c o n d u c t i v i t i e s f o r the c r y s t a l s measured appear i n F i g . 16. The d.c. c o n d u c t i v i t y f o r one o f the Groningen c r y s t a l s appears a l o n g w i t h the microwave c o n d u c t i v i t y f o r c r y s t a l 4, a l s o from G r o n i n g e n , i n F i g . 17. The r e a l p a r t s o f the d i e l e c t r i c c o n s t a n t below the t r a n s i t i o n temperature a re shown i n F i g . 18. The MEM(TCNQ)^ c r y s t a l s were found to have a t r a n s i t i o n t e m p e r a t u r e o f about 65°C on h e a t i n g and 60°C on c o o l i n g . A l t h o u g h r e s u l t s o b t a i n e d i n another l a b o r a t o r y and r e p o r t e d i n Morrow e t a l . (1979) have found these t e m p e r a t u r e s to be about 57.4°C and 51.5"c r e s p e c t i v e l y t h e r e a re a l s o s i g n i f i c a n t d i f f e r e n c e s i n the c o n d u c t i v i t y and the d i e l e c t r i c c o n s t a n t r e s u l t s below the t r a n s i t i o n which a r e not c o m p l e t e l y u n d e r s t o o d . The h y s t e r e s i s o f between 5 and 6 degrees i s , however, common to both s e t s o f r e s u l t s and i s taken as e v i d e n c e f o r the phase t r a n s i t i o n b e i n g d i s c o n t i n u o u s . The r e a l p a r t o f the d i e l e c t r i c c o n s t a n t , , c o u l d o n l y be o b t a i n e d below the phase t r a n s i t i o n as e x p l a i n e d above, and was measured from j u s t below the t r a n s i t i o n down to room te m p e r a t u r e . An i n c r e a s e w i t h t emperature was observed w i t h g o i n g from between 10 and 12 a t room temperature to between 12 and 16 a t the t r a n s i t i o n t e m p e r a t u r e . As a check, the ap p a r a t u s was used to measure the d i e l e c t r i c c o n s t a n t o f a diamond sawed F i g u r e 16. Microwave c o n d u c t i v i t i e s (8.5 GHz) f o r MEM(TCNQ) c r y s t a l s s t u d i e d 58 I E o g o - 2 - 3 - 4 2.8 T 1 1 r o Crystal #6 D.C. Data + Crystal # 4 M.W. Data Crt" o+ o J 1 I I I I 3.0 3.2 3.4 3.6 3.8 4.0 I0 3 / T ( o K ) Figure 17. Comparison of microwave conductivity of c r y s t a l 4 and d.c. conductivity of a second MEM(TCNQ) cr y s t a l plotted against 1/T. ,4f" 6 ° *° =« 0 o 0 • o D I2h ° * A * * 10' * A A Crystal # I 0 Crystal # 2 • Crystal # 3 o Crystal # 4 L 1 1 1 I . i 30 40 50 60 T CC) Figure 18. The real part of the d i e l e c t r i c constant versus temperature for some MEM(TCNQ) c r y s t a l s . 60 s i l i c o n c h i p o f d i m e n s i o n s 0.820 cm by 0.099 cm by 0.065 cm. T h i s c h i p , t h i c k e r than the MEM(TCNQ)^ samples s t u d i e d and not r e a l l y o p t i m a l l y shaped f o r t h i s t e c h n i q u e , was found to have £ / =10.5 as compared w i t h the accep t e d v a l u e o f 11.7. The microwave c o n d u c t i v i t y i n the low c o n d u c t i v i t y range was found to be between 0.014 and 0.017 (A-cm) . The c o n d u c t i v i t y above the t r a n s i t i o n was found to be between 14 and 32 (fl-cm) ' . These r e s u l t s a r e i n c l u d e d i n the summary g i v e n i n T a b l e 2. The c o n d u c t i v i t y below the t r a n s i t i o n can be a n a l y s e d by assuming an a c t i v a t e d b e h a v i o u r so t h a t 0-(T) = cr0 exp(-A 3/2kT) where A 3 i s the energy gap. In these measurements, A<j was found to be (0.7*0.1)eV i n the range from 310K to 335K. The d.c. measurements had p r e v i o u s l y g i v e n (Morrow e t a l . , 1979) A 3=0.78eV between 250K and 290K and 0.69eV between 310K and 335K. The microwave measurements covered o n l y the upper range o f t e m p e r a t u r e s but were g e n e r a l l y found to c o n f i r m the d.c. r e s u l t s . E r r o r s were e s t i m a t e d f o r a t y p i c a l p o i n t above and below the t r a n s i t i o n . Below the t r a n s i t i o n , the e r r o r i n €•' was found to be about 30% and t h a t i n &" and <r to be about 35%. These were l a r g e l y a s s o c i a t e d w i t h the u n c e r t a i n t y i n the c r y s t a l TABLE 2 C a l c u l a t e d R e s u l t s f o r MEM(TCNQ) C r y s t a l s S t u d i e d C r y s t a l 1 2 CT, (Sl-cm)' (T<65°C) 0.0165 0.0142 5^ (/2-cm)"' (T>65*C) 22. 292 14. 53 0.0149 0.0161 13.62 31.39 €' (T=50°C) 1351 11 1023 15 914 13 8. (cm) (T<65°C) 0.424 0. 457 0. 447 1950 13 0.430 8i (cm) (T>65°C) 0. 0115 0.0144 0.0148 0. 0098 («/Af»)/n (T>65°C) 1. 249 1. 267 1. 264 1. 142 n^" 1(T<65"C) n 6" (T>65°C) 0.0564 0.0324 0.0444 0.0287 76.425 33.42 40.81 57.18 62 dimensions and do not include the small systematic e r r o r i n n due to the n o n - e l l i p s o i d a l shape. Above the t r a n s i t i o n , the experimental u n c e r t a i n t y i n c r i s estimated to be about 20 %. 63 3.3 D i s c u s s i o n MEM(TCNQ) a i s found to be a semiconductor w i t h a gap o f about 0.8eV i n the d i m e r i z e d phase and a poor metal i n the h i g h t e m p e r a t u r e phase. There i s some s u g g e s t i o n t h a t the MEM i o n s p l a y a r o l e i n s t a b i l i z i n g the dimer and t h a t the t r a n s i t i o n i s r e l a t e d to the l o s s o f t h i s s t a b i l i z a t i o n w i t h the i n c r e a s e d m o tion o f the MEM i o n s a t h i g h e r t e m p e r a t u r e . (Sawatzky, 1979) For the purpose o f t h i s d i s c u s s i o n , however, we w i l l j u s t c o n s i d e r the e f f e c t , on the extended Hubbard model f o r the J>=l/2 TCNQ s t a c k , o f the t h r e e k i n d s o f terms i n the extended Hubbard H a m i l t o n i a n . We w i l l be i n t e r e s t e d i n the r e l a t i o n s h i p s between the o v e r l a p i n t e g r a l s and the o n - s i t e r e p u l s i o n , U, and, i n p a r t i c u l a r , on whether or not the r e s u l t s p r e s e n t e d here can d i s t i n g u i s h between the l a r g e U and s m a l l U c a s e s . I t i s c l e a r , from the f a c t t h a t the t r a n s i t i o n i s not d i r e c t l y from the monomer to the t e t r a m e r phase, w i t h c h a r a c t e r i s t i c wavevector 2k° , t h a t Coulomb r e p u l s i o n must be s i g n i f i c a n t . In the monomer phase, we can t h i n k o f t h e r e b e i n g j u s t one t r a n s f e r i n t e g r a l , t . Then, f o r U<<4t, we w i l l have a t i g h t b i n d i n g model w i t h a q u a r t e r f i l l e d band o f w i d t h 4 t . For U>>4t, we w i l l have the s p i n l e s s f e r m i o n model w i t h a gap o f U-4t and a lower band o f w i d t h 4 t . Both o f thes e w i l l g i v e a m e t a l - l i k e b e h a v i o u r . I f , however, the n e a r e s t neighbour 64 r e p u l s i o n were l a r g e , then one would expect, f o r the J>=l/2 and l a r g e U case, a Wigner c r y s t a l s i t u a t i o n with a gap above the high e s t f i l l e d l e v e l s i n c e any e x c i t a t i o n would i n v o l v e occupancy of neighbouring TCNQ molecules. That the c o n d u c t i v i t y i s not extremely h i g h suggests t h a t V( might be important. The c o n d u c t i v i t y does not, however, appear to be a c t i v a t e d so t h i s q u e s t i o n remains unre s o l v e d . The high temperature behaviour, then, does not d i s t i n g u i s h between any of the extreme regimes of the Hubbard model. If we now look a t the semi-conducting dimer phase, we can co n s i d e r two p o s s i b i l i t i e s i n terms of t, being l a r g e or s m a l l . We a l r e a d y suspect that U i s l a r g e because the t e t r a m e r i z a t i o n does not occur immediately. The ground s t a t e of the dimerized phase does not i n c l u d e double occupancy of TCNQ molecules. We w i l l f i r s t c o n s i d e r the s i t u a t i o n where t, i s the dominant i n t e r a c t i o n . I f we ignore the Coulomb r e p u l s i o n f o r the moment, we can co n s i d e r the problem to be a t i g h t b i n d i n g problem i n a l a t t i c e of c e l l dimension c with a b a s i s b. We w i l l take the o v e r l a p o f s i t e s separated by b to be - t , and those separated by c-b to be - t A . The B r i l l o u i n zone then extends to 1TT/C. We f i n d the d i s p e r s i o n r e l a t i o n to be (Appendix A) (3.3) € k=-/t, z +t* +2t, t^cos(kc) . 65 T h i s e x p r e s s i o n d e s c r i b e s a band w i t h l i m i t s a t t (t, + t x ) w i t h a gap between i 11,-tj. I . We can now s t u d y the e f f e c t o f i n c l u d i n g the o n - s i t e r e p u l s i o n . We assume t h a t t 1 i s much l a r g e r than t ^ . T h i s l e a v e s t h r e e c a s e s t o c o n s i d e r . These a r e t 1>>t a>>U f t 1>>U>>t ; L, and U » t 1 > > t 3 L . For the f i r s t c a s e , we ex p e c t the r e s u l t t o be s i m i l a r to t h a t d i s c u s s e d above so t h a t we have a lower band o f w i d t h 2 t 3 L . Double occupancy o f the dimers i s a l l o w e d so t h i s lower band would be h a l f - f i l l e d and thus m e t a l l i c . T h i s i s i n c o n s i s t e n t w i t h the c o n d u c t i v i t y measurements and w i t h the f a c t t h a t the magnetic s u s c e p t i b i l i t y measurements ( H u i z i n g a e t a l . , 1979) i n d i c a t e t h a t MEM(TCNQ)^ i s a c h a i n o f l o c a l i z e d s p i n s w i t h a n t i f e r r o m a g n e t i c c o u p l i n g r a t h e r than a m e t a l l i c c h a i n f o r which the much s m a l l e r P a u l i s u s c e p t i b i l i t y would be appropr i a t e . For the second c a s e , t-, >>U>>tL , the h a l f f i l l e d lower band i s s p l i t by the Coulomb energy a s s o c i a t e d w i t h d o u b l e occupancy o f a di m e r . T h i s w i l l be U/2 s i n c e the p r o b a b i l i t y o f double o c c u p a t i o n o f a g i v e n TCNQ m o l e c u l e i n a d o u b l y o c c u p i e d dimer i s 1/4. The gap i s thus U/2-2t^ below which i s a f i l l e d band o f w i d t h 2 t a . In the t h i r d case,U dominates the problem and we can handle t h i s by r e c o g n i z i n g t h a t , as i n the case f o r a s i n g l e type o f o v e r l a p , t h i s i s j u s t the U=0 case f o r s p i n l e s s f e r m i o n s . The 66 r e s u l t i s then two bands o f w i d t h 2 t ^ w i t h a gap o f 2 ( t 7 - t A ) . Now, however, the lower band i s f i l l e d . We s e e , t h e n , t h a t b oth o f these l a s t two models g i v e a s e m i - c o n d u c t i n g phase and, as f o r the m e t a l l i c s t a t e , we cannot d i s t i n g u i s h between U d o m i n a t i n g and t-j d o m i n a t i n g on the b a s i s o f c o n d u c t i v i t y measurements. For both c a s e s , n e a r e s t neighbour r e p u l s i o n , V , , would a l s o c o n t r i b u t e t o the gap i f i t was non-n e g l i g l i b l e . We thus f i n d t h a t , f o r both U>>t1 and U<<t1 , we would expect to f i n d a gap f o r c o n d u c t i v i t y and a gap o f about 9000 degrees i s o b s e r v e d . The thermopower e v i d e n c e o f C h a i k i n (1979) su g g e s t s t h a t U i s i n f a c t l a r g e and t h i s i s c o n s i s t e n t w i t h the c o n c l u s i o n s drawn by H u i z i n g a e t a l . (1979). I f the gap i s tak e n t o be 2 t 7 - 2 t a , as f o r the l a r g e U c a s e , the i m p l i c a t i o n i s t h a t t 1 i s o f the o r d e r o f 0.4 eV. Based on Hubbard's (1978) e s t i m a t e f o r a TCNQ c h a i n o f o f U=4.5 eV f o r the unscreened case and 2.4 eV i f the i n t e r a c t i o n i s screened by c o n d u c t i o n on n e i g h b o u r i n g c h a i n s and p o l a r i z a t i o n o f n e i g h b o u r i n g m o l e c u l e s , t h i s i s indeed s m a l l compared to U. 67 CHAPTER IV D i e l e c t r i c Resonator S t u d i e s o f TTF-TCNQ The second p a r t o f t h i s work i n v o l v e d the s t u d y o f the complex d i e l e c t r i c c o n s t a n t o f TTF-TCNQ f o r the d i r e c t i o n p a r a l l e l to the c o n d u c t i n g a x i s by o b s e r v i n g d i e l e c t r i c resonances o f the c r y s t a l s . T h i s a x i s i s l a b e l l e d b i n what f o l l o w s . 4.1 D i e l e c t r i c Resonator Apparatus The a p p a r a t u s used f o r t h i s p a r t o f the work was e s s e n t i a l l y the same as t h a t used by B a r r y f o r the o r i g i n a l d i e l e c t r i c resonance s t u d i e s o f TTF-TCNQ ( B a r r y , 1977). The n a t u r e o f the experiment r e q u i r e d t h a t i t be p o s s i b l e to suspend the c r y s t a l i n the wave g u i d e a t low temperature and m a n i p u l a t e i t from room t e m p e r a t u r e . I t was a l s o n e c e s s a r y t h a t i t be p o s s i b l e to e x t r a c t the c r y s t a l f o r c u t t i n g w h i l e l e a v i n g the wave g u i d e a t l i q u i d h e l i u m t e m p e r a t u r e s . T h i s was a c c o m p l i s h e d by suspending the c r y s t a l from a q u a r t z f i b r e on the end o f a l o n g s t a i n l e s s s t e e l rod which c o u l d be m a n i p u l a t e d from the top o f the c r y o s t a t . A d e t a i l e d diagram o f the c r y s t a l h a n d l i n g rod and rod h o l d e r appears i n F i g . 19. The c r y s t a l , w i t h i t s l o n g a x i s p a r a l l e l t o the rod assembly, was b u t t e d a g a i n s t the end o f the q u a r t z f i b r e and h e l d w i t h a type o f f a s t - s e t t i n g epoxy (5 min.) manufactured by 68 stainless steel rod O-ring brass rod set screw hypodermic needle brass rod holder quartz fibre crystal Figure 19. C r y s t a l manipulating rod and rod holder f o r the d i e l e c t r i c resonance experiment. 69 Devcon. T h i s q u a r t z f i b r e was g l u e d i n t o a 26 gatage hypodermic needle w i t h a p p r o x i m a t e l y h a l f an i n c h o f q u a r t z f i b r e p r o j e c t i n g from the n e e d l e . The n e e d l e , i n t u r n , was f i t t e d i n t o a c o n c e n t r i c h o l e i n a b r a s s t i p o f the same di a m e t e r a s , and f i t t e d i n t o the end o f , the s t a i n l e s s s t e e l r o d . T h i s s t a i n l e s s s t e e l r o d , o f l e n g t h 62 cm, was i n s e r t e d i n t o a b r a s s h o l d e r w i t h an O-ring s e a l to s e c u r e the r o d . The b r a s s h o l d e r was threaded so t h a t i t c o u l d be a t t a c h e d to the c r y o s t a t f l a n g e . A s p r i n g was c a p t u r e d between the b r a s s cap o f the CD-r i n g s e a l and a cap on the s t a i n l e s s s t e e l r o d . The c r y s t a l c o u l d be withdrawn i n t o the b r a s s h o l d e r , a f t e r mounting, to a l l o w i t to be moved to the c r y o s t a t . The b r a s s h o l d e r c o u l d then be screwed i n t o a threaded h o l e on the top c r y o s t a t f l a n g e . A s t a i n l e s s s t e e l tube was used to g u i d e the rod and c r y s t a l down to and through a copper b l o c k and i n t o the wave g u i d e . The c r y s t a l was o r i e n t e d w i t h the l o n g a x i s p e r p e n d i c u l a r to the E f i e l d i n the r e c t a n g u l a r wave g u i d e . The copper b l o c k , i n which was i n s e r t e d a 390-fl- carbon r e s i s t o r thermometer, s u p p o r t e d a 120il r e s i s t a n c e w i r e h e a t e r and a l s o formed a s h o r t a t the end o f the wave g u i d e . The wave g u i d e formed a "U" shape w i t h one arm s h o r t e d by the copper b l o c k and the o t h e r e x t e n d i n g through the top f l a n g e o f the c r y o s t a t . A p o l y e t h y l e n e window was used to s e a l the wave g u i d e o u t s i d e o f the dewar. There was a h o l e d r i l l e d i n the waveguide j u s t above the o p e r a t i n g l i q u i d Helium l e v e l to a l l o w He gas to e n t e r and 70 c o o l the c r y s t a l . Two wave g u i d e a s s e m b l i e s , t o cover 18 to 26.5 GHz and 26.5 to 40 GHz r a n g e s , c o u l d be i n t e r c h a n g e d . S i n c e a g i v e n c r y s t a l had to be used f o r a number o f measurements, i n o r d e r to c o n s t r u c t a mode p l o t , i t was d e s i r a b l e to m i n i m i z e the o p p o r t u n i t i e s f o r breakage. One o f the most l i k e l y t i m e s f o r the c r y s t a l to f r a c t u r e was d u r i n g the i n s e r t i o n a f t e r a l i q u i d He t r a n s f e r . To m i n i m i z e the number o f t r a n s f e r s n e c e s s a r y to c o l l e c t d a t a from a g i v e n c r y s t a l , as w e l l as to c o n s e r v e He, the r e l a t i v e l y heavy s t a i n l e s s s t e e l s k i r t o f B a r r y ' s a p p a r a t u s was r e p l a c e d by a f l a n g e s u p p o r t i n g t h r e e t h i n - w a l l e d t u b e s . The s m a l l e s t tube c o n t a i n e d the thermometer and h e a t e r w i r e s as w e l l as the tube i n which the c r y s t a l m a n i p u l a t i n g rod was g u i d e d . The l a r g e s t tube c o n t a i n e d the wave g u i d e assembly and the r e m a i n i n g tube guided the t r a n s f e r s i p h o n to the dewar. These tubes passed through 0 - r i n g s e a l s on the top o f a second f l a n g e which was s e a l e d to the dewar. The tubes and upper f l a n g e s s u p p o r t i n g the waveguide and c r y s t a l h a n d l i n g a s s e m b l i e s were f i x e d i n p l a c e w h i l e the dewar c o u l d be r a i s e d and lowered on the tubes by means o f a s i m p l e e l e v a t o r mechanism. The dewar assembly i s shown i n F i g . 20. I t was found t h a t the v i b r a t i o n s a s s o c i a t e d w i t h the He t r a n s f e r were o f t e n enough to break the c r y s t a l so t h a t the c r y s t a l had to be i n s e r t e d a f t e r the t r a n s f e r . T h i s r e q u i r e d c a r e f u l use o f t e c h n i q u e s to i n s u r e t h a t t h e r e was no c o n d e n s a t i o n o f i c e i n the dewar and p a r t i c u l a r l y i n the c r y s t a l Figure 20. The dewar assembly for the d i e l e c t r i c resonance studies of TTF-TCNQ. 7 2 i n s e r t i o n tube. While the f a i l u r e rate was s t i l l high, i t was found that some success could be achieved by lea v i n g a dummy rod in the c r y s t a l i n s e r t i o n tube during t r a n s f e r s along with maintaining a He overpressure at a l l times. The dummy rod prevented the blockage of the tube by condensable vapours during t r a n s f e r s and could be used to c l e a r blockages from the tube p r i o r to i n s e r t i o n . 73 4.2 The Spectrometer The microwave sweep o s c i l l a t o r employed i n these measurements was a W e i n s c h e l 221 w i t h 18 to 26 GHz and 26 to 40 GHz p l u g - i n s . The microwave power was sampled a t the p l u g - i n w i t h a 10 dB d i r e c t i o n a l c o u p l e r . I t passed through a wave meter and a v a r i a b l e a t t e n u a t o r to a Hewlett Packard model 11517A harmonic mixer which c o u l d be used f o r p h a s e - l o c k i n g purposes as d e s c r i b e d below. The wave meter was used to c o n v e r t the e x t e r n a l sweep v o l t a g e to f r e q u e n c y . The power l e v e l was d e t e c t e d , a t a second d i r e c t i o n a l c o u p l e r , by an HP model R422A c r y s t a l d e t e c t o r and used to l e v e l the backward wave o s c i l l a t o r o u t p u t . A f t e r p a s s i n g through an i s o l a t o r , i t was s e n t to the microwave assembly t h r o u g h a t h i r d d i r e c t i o n a l c o u p l e r . The power r e f l e c t e d from the s h o r t , l e s s any absorbed by the c r y s t a l r e s o n a n c e , r e t u r n e d up the wave g u i d e assembly and t h r o u g h the d i r e c t i o n a l c o u p l e r , a v a r i a b l e a t t e n u a t o r , and an i s o l a t o r , to a second HP harmonic mixer used as a s i m p l e d e t e c t o r . The o u t p u t was a m p l i f i e d i n two s t a g e s by i n v e r t i n g o p e r a t i o n a l a m p l i f i e r s b e f o r e b e i n g fed t o the computer t h r o u g h a d i g i t a l to analog c o n v e r t e r . The d a t a h a n d l i n g from t h i s p o i n t w i l l be d e s c r i b e d i n s e c t i o n 4.6. F i g . 21 i l l u s t r a t e s the microwave assembly s c h e m a t i c a l l y . .MIXER BACKWARD WAVE OSCILLATOR ,FM INPUT r-H EXT SWEEP LEVELLER *» ARABLE ATTENUATOR *UXER Ifl M A Z E M E T E R 9? 'ETEC TOR OR. COUPLER 0 = ? BOLATOR H16BIT D/A COMPUTER **— X10 OP AMP SYNTHESIZER X8 MULTPLER 200 MHz AMPLFIER SAMPLER BANDPASS FILTER & FILTER ERROR SIGNAL * 20 MHz SYNCHRONIZER PRESCALER VARIABLE (ATTENUATOR BOLATOR F i g u r e 21. Schemat ic appara tus resonance drawing o f the microwave used f o r d i e l e c t r i c measurements. 75 4 . 3 C r y s t a l C u t t i n g I t was f e l t t h a t i n o r d e r to b e t t e r understand the d i e l e c t r i c resonance mode p l o t s , i t would be n e c e s s a r y to have a means o f trimming s m a l l l e n g t h s from the c r y s t a l q u i c k l y and w i t h a m i n i m a l r i s k o f c r y s t a l breakage. To t h i s end an assembly was c o n s t r u c t e d which c o u l d be used to mount the c r y s t a l under a mi c r o s c o p e and c o u l d then a l l o w the c r y s t a l to be m a n i p u l a t e d i n two d i m e n s i o n s and c u t w h i l e under o b s e r v a t i o n . The ap p a r a t u s used i s shown i n F i g . 22 as i t appears on the mi c r o s c o p e mount. The mount assembly (A) a t t a c h e s to the mi c r o s c o p e arm u s i n g d o v e t a i l B. The r e s t o f the assembly i s s u p p o r t e d by r a i l s which a r e i n s e r t e d through b l o c k C. T h i s b l o c k can be t r a n s l a t e d p e r p e n d i c u l a r to the d i r e c t i o n o f the r a i l s by means o f an a d j u s t i n g knob. The rod on which the c r y s t a l i s mounted can be t r a n s l a t e d p a r a l l e l to the r a i l s by means o f the f o u r a s s e m b l i e s D, E, F, and G. Support D c o n t a i n s a threaded h o l e i n t o which the b r a s s c r y s t a l s u p p o r t can be screwed. Support E, w i t h the cap i n p l a c e , s t e a d i e s the c r y s t a l mount rod and p r o v i d e s a base f o r the s p r i n g which remains on the rod a t a l l t i m e s . C o l l a r F can be p l a c e d around the rod and clamped to i t . The s p r i n g p r e s s e s a g a i n s t i t t o p r o v i d e the f o r c e t o r e t u r n the c r y s t a l t o the b l o c k . P a r t G c o n t a i n s a f o r k t h r o u g h which the rod i s passed and which p r e s s e s on c o l l a r F to move the c r y s t a l toward the The c r y s t a l c u t t i n g apparatus mounted below microscope. For a d e s c r i p t i o n , see t e x t . 77 c e n t r e o f the a p p a r a t u s . At the o t h e r end o f the r a i l s i s the c u t t e r assembly (H) which can be t r a n s l a t e d a l o n g the r a i l s t o p o s i t i o n the b l a d e below the m i c r o s c o p e . The a c t u a l c u t t e r c o n s i s t s o f an a n v i l ( I ) to s u p p o r t the c r y s t a l and a b l a d e (J) to c u t i t . Both are moved by d i f f e r e n t i a l screws mounted i n removable b r a s s arms. A p i e c e o f r a z o r b l a d e s t e e l epoxied to the a n v i l forms the c u t t i n g t a b l e . T h i s t a b l e must be hard so t h a t g r o o v e s , i n t o which the c r y s t a l can be pushed, a r e not formed. The b l a d e was a s e c t i o n o f r a z o r b l a d e . Best c u t t i n g r e s u l t s were o b t a i n e d when the b l a d e was changed f o r each c r y s t a l . I t was found t h a t , i n the c u t t i n g p r o c e s s , i t was p o s s i b l e f o r the b l a d e to e x e r t a f o r c e on the c r y s t a l t h a t was p a r a l l e l to the c r y s t a l ' s l o n g a x i s . S i n c e the c r y s t a l s a r e e x t r e m e l y weak i n c o m p r e s s i o n , t h i s f o r c e had to be compensated f o r . T h i s was done by b a c k i n g o f f the f o r k on p a r t G once the b l a d e had pinned the c r y s t a l to the a n v i l . T h i s a l l o w e d the c u t to proceed w i t h a s l i g h t t e n s i o n , s u p p l i e d by the s p r i n g , a p p l i e d to the c r y s t a l . Photographs showing the c u t t i n g o p e r a t i o n and the end o f the c r y s t a l appear i n F i g . 23. W h i l e the c u t t i n g o p e r a t i o n i s not a c l e a v i n g of the c r y s t a l , i t was found t h a t r e a s o n a b l y c l e a n c u t s c o u l d be made w i t h no apparent d e g r a d a t i o n o f the d i e l e c t r i c resonance. The major purpose f o r the c u t t e r was to a l l o w the c r y s t a l l e n g t h t o be changed by v e r y s m a l l amounts. I t was found t h a t as l i t t l e as about 0.3 mm c o u l d be trimmed from the c r y s t a l F i g u r e 23. C r y s t a l and b l a d e b e f o r e and a f t e r c u t t i n g o p e r a t i o n . A n v i l has been withdrawn f o r c l a r i t y . 7 9 q u i t e e a s i l y . For cuts of l e s s than 0.1 mm, there was some d i f f i c u l t y with the blade crushing the end of the c r y s t a l r e s u l t i n g i n a s l i g h t l y ragged cut. 80 4.4 Mounting T h i s same assembly was a l s o used to mount the c r y s t a l . For t h i s o p e r a t i o n , i t was mounted v e r t i c a l l y on a stand as shown i n F i g . 24. The b l a d e and a n v i l were r e t r a c t e d to a l l o w the mount rod to pass between them. The r o d , w i t h the n e e d l e and q u a r t z f i b r e mounted i n the end as d e s c r i b e d e a r l i e r , was h e l d i n the assembly i n the same way as f o r the c u t t i n g o p e r a t i o n . The c r y s t a l was st o o d u p r i g h t i n a s m a l l h o l e d r i l l e d i n the b r a s s b l o c k ( K ) . The b l o c k r e s t e d on a t a b l e f i x e d to the c u t t i n g assembly as shown ( L ) . The rod was then lowered a g a i n s t the s p r i n g u s i n g the f o r k and c o l l a r arrangement. W i t h a d r o p o f epoxy on the q u a r t z f i b r e , the c r y s t a l and f i b r e were b u t t e d t o g e t h e r and then r a i s e d s l i g h t l y . The b r a s s b l o c k was p o s i t i o n e d so as to h o l d the c r y s t a l s t r a i g h t w h i l e the epoxy hardened. The rod and c r y s t a l were then withdrawn i n t o the b a r r e l o f t h e b r a s s c r y s t a l s u p p o r t i n p r e p a r a t i o n f o r t r a n s f e r to the dewar. Figure 24. C r y s t a l mounting apparatus. see t e x t for d e s c r i p t i o n . 82 4.5 Measuring o f the C r y s t a l The c r y s t a l c o u l d be measured from photographs taken t h r o u g h the m i c r o s c o p e . T h i s method was a c c u r a t e to about 0.01 mm. L a t e r c r y s t a l s were measured by a t t a c h i n g the c r y s t a l h o l d e r to a s t a g e which c o u l d be t r a n s l a t e d i n two d i m e n s i o n s u s i n g m i c r o m e t e r s . These measurements a r e thought to be a c c u r a t e to 0.005 mm or l e s s . 83 4.6 Computer C o n t r o l The microwave sou r c e was c o n t r o l l e d by a Nova 2 minicomputer as d e s c r i b e d by S t a t t (1979). I t s use i n the c o n t e x t o f t h i s e xperiment w i l l be d e s c r i b e d b r i e f l y . Any o p e r a t i o n i n v o l v i n g sweeping o f the backward wave o s c i l l a t o r (BWO) under computer c o n t r o l r e q u i r e d t h a t a s u i t a b l e v o l t a g e ramp be fed to the e x t e r n a l sweep i n p u t o f the BWO. The f i r s t s t e p i n g e n e r a t i n g t h i s ramp i n v o l v e d c a l i b r a t i o n . T h i s was done u s i n g the BASIC program, "CALIBRATION." A d i g i t a l ramp was o u t p u t t o the 16 b i t D/A c o n v e r t e r . A wavemeter (TRG K551 f o r 18 to 26.5 GHz and TRG A551 f o r 26.5 to 40 GHz) c o u l d be used to i d e n t i f y p r e s e l e c t e d f r e q u e n c y p o i n t s i n the sweep. The computer was s i g n a l l e d m a n u a l l y a t each p r e s e l e c t e d p o i n t and the d i g i t a l o u tput g i v i n g r i s e to the a p p r o p r i a t e v o l t a g e r e c o r d e d on d i s k . In o r d e r t o c o l l e c t d a t a , t h e program "SWCONTROL" was used. Two modes o f sweeping were used f o r t h i s e x p e r i m e n t . Because o f the presence o f s t r o n g s t a n d i n g waves between the s h o r t and the p o l y e t h y l e n e vacuum s e a l , i t was u s u a l l y n e c e s s a r y t o r a t i o s p e c t r a t a k e n w i t h the c r y s t a l i n and out i n o r d e r to l o c a t e a l l but the most s t r o n g l y c o u p l e d r e s o n a n c e s . T h i s i n i t i a l s e a r c h f o r each resonance was performed u s i n g the "FASTSWEEP" r o u t i n e w i t h a r e a l time r a t i o o p t i o n . In t h i s mode o f o p e r a t i o n , the computer used the f r e q u e n c y c a l i b r a t i o n p r e v i o u s l y o b t a i n e d to 8 4 c a l c u l a t e a d i g i t a l ramp c o v e r i n g the fr e q u e n c y range r e q u e s t e d w i t h the number o f p o i n t s r e q u e s t e d . T h i s ramp was then o u t p u t to the 16 b i t D/A a t a s e l e c t e d r a t e . At the same time r e f l e c t e d power was b e i n g sampled and r a t i o e d , p o i n t by p o i n t , w i t h the p r e v i o u s l y s t o r e d background. Each r a t i o e d p o i n t was d i s p l a y e d on a scope m o n i t o r b e f o r e the next p o i n t was t a k e n . T h i s r o u t i n e was t y p i c a l l y used f o r r e p e t i t i v e s c a n n i n g o f 1001 p o i n t s over about 2 GHz a t a r a t e o f 100/csec. per p o i n t . For both t h i s r o u t i n e and the s y n t h e s i z e r phase l o c k i n g r o u t i n e , the s p e c t r a c o u l d be r e t a k e n w i t h o n l y the s t o r a g e l o c a t i o n s a l t e r e d by u s i n g the response "R". T h i s a l l o w e d q u i c k r e c o r d i n g o f background and c r y s t a l s p e c t r a w i t h o u t r e - e n t e r i n g a l l o f the sweep p a r a m e t e r s . F e a t u r e s i n the r a t i o e d spectrum c o u l d be i d e n t i f i e d as c r y s t a l resonances by moving the c r y s t a l , and thus changing the c o u p l i n g , w h i l e o b s e r v i n g the r a t i o e d spectrum i n r e a l t i m e . The g a i n o f the d i s p l a y e d r a t i o c o u l d be i n c r e a s e d by up to a f a c t o r o f 16. Using t h i s o p t i o n , resonances c o u p l e d by l e s s than 5% c o u l d be l o c a t e d . The c o u p l i n g was u s u a l l y a d j u s t e d to a maximum so t h a t f o r s t r o n g l y c o u p l e d resonances t h e r e was no q u e s t i o n as to whether the resonance was o v e r -c o u p l e d o r u n d e r - c o u p l e d . The d a t a was n o r m a l l y c o l l e c t e d w i t h the sour c e phase-l o c k e d to the Rhode and Schwarz 0.01 to 500 MHz f r e q u e n c y s y n t h e s i z e r . These c o u l d both be c o n t r o l l e d by the "SYNSWEEP" r o u t i n e i n "SWCONTROL". These scans t y p i c a l l y c overed 0.5 GHz 85 w i t h 1001 p o i n t s a t 100 m s e c . / p o i n t . A g a i n a 1001 p o i n t d i g i t a l ramp would be c a l c u l a t e d and f e d , p o i n t by p o i n t , t o the BWO. At the same t i m e , the computer would output a s i g n a l to s e t the s y n t h e s i z e r to a f r e q u e n c y such t h a t a chosen harmonic o f 8 t i m e s the s y n t h e s i z e r f r e q u e n c y would be 200 MHz g r e a t e r than the d e s i r e d BWO f r e q u e n c y . ( the s y n t h e s i z e r o utput i s fed t h r ough a t i m e s 8 m u l t i p l i e r c h a i n b e f o r e e n t e r i n g the HP harmonic mixer which was a l s o sampling the BWO o u t p u t . ) The r e s u l t o f m i x i n g the BWO o u t p u t and the chosen harmonic (between 6 a t 18 GHz and 11 a t 40 GHz) of the m u l t i p l i e d s y n t h e s i z e r o u t p u t would i n c l u d e a s i g n a l c l o s e to 200 MHz. A f t e r b e i n g p i c k e d out by a bandpass f i l t e r and d i v i d e d by 10, the s i g n a l was fed to a 20 MHz s y n c h r o n i z e r . T h i s g e n e r a t e d an e r r o r v o l t a g e which was then fed to the F.M. i n p u t o f the BWO to c o r r e c t i t s o u t p u t f r e q u e n c y . The power r e f l e c t e d was sampled a t each f r e q u e n c y s t e p and the r e s u l t s t o r e d . The c o n t e n t s o f the b u f f e r c o n t a i n i n g the spectrum were d i s p l a y e d on the m o n i t o r scope a f t e r each p o i n t was c o l l e c t e d . The u s u a l procedure was to s t o r e the spectrum w i t h the c r y s t a l i n s e r t e d and then to r e c o r d a background s i g n a l . The r a t i o was then c a l c u l a t e d and s t o r e d on d i s k u s i n g the "ANALYSIS" program. T h i s program was a l s o used to p l o t the s p e c t r a on an X-Y r e c o r d e r . B e f o r e b e i n g s t o r e d on d i s k by an assembly language s u b r o u t i n e , the r a t i o e d s p e c t r a were m u l t i p l i e d by a l a r g e number i n o r d e r to reduce the d i g i t a l n o i s e a s s o c i a t e d w i t h f l o a t i n g p o i n t to f i x e d p o i n t 86 conversion of small numbers. Consideration of the monitor scope range led to the choice of 2,if as a convenient m u l t i p l i e r . 87 4.7 F i t t i n g o f the S p e c t r a The resonance was a n a l y s e d by assuming a L o r e n t z i a n o s c i l l a t o r l i n e shape f o r the a b s o r p t i o n due t o the d i e l e c t r i c r esonance. I d e a l r a t i o e d s p e c t r a a r e assumed to have the form of 1 minus a L o r e n t z i a n c u r v e . The s p e c t r a can be f i t t e d to such a l i n e shape u s i n g the "AFIT" or "MFIT" programs. These programs can be c a l l e d from the "ANALYSIS" program by re s p o n d i n g to the prompt w i t h "Z". Both r e q u e s t a d i r e c t o r y i n which they may f i n d r e q u e s t e d f i l e s . They can perform d i s k o p e r a t i o n s and scope m o n i t o r p l o t s by res p o n d i n g to the prompt "%" w i t h the l e t t e r s "D" or "P" r e s p e c t i v e l y . These commands a r e used i n the same way as i n "SWCONTROL" ( S t a t t , 1979). The response "L" r e s u l t s i n the computer r e q u e s t i n g i n f o r m a t i o n on the spectrum to be f i t t e d and then c a r r y i n g out the f i t . The f i r s t query r e q u e s t s the b a s e l i n e f a c t o r , as d e s c r i b e d above, to be e n t e r e d . Next i n f o r m a t i o n i s r e q u e s t e d about the spectrum to be f i t i n c l u d i n g the i n i t i a l f r e q u e n c y , the f r e q u e n c y i n t e r v a l , the number o f p o i n t s , and the l o c a t i o n , i n the BASIC b u f f e r , o f t h e f i r s t p o i n t . I t then asks f o r a guess as to the c e n t r e f r e q u e n c y and w i d t h o f a L o r e n t z i a n w i t h the same h e i g h t as w e l l as the f i r s t l o c a t i o n i n t o which t o s t o r e t h i s L o r e n t z i a n . In the manual program, "MFIT" , t h i s L o r e n t z i a n i s c a l c u l a t e d and s t o r e d . The response "C" b r i n g s a r e q u e s t f o r the l o c a t i o n s and l e n g t h s o f two b l o c k s o f b u f f e r space 88 c o n t a i n i n g the o r i g i n a l and c a l c u l a t e d s p e c t r a and then p l o t s them s i m u l t a n e o u s l y f o r v i s u a l i n s p e c t i o n o f the f i t . "CV" w i l l do t h i s as w e l l but w i l l respond to the escape key by summing the square o f the d i f f e r e n c e s o f the two s p e c t r a . The computer w i l l then respond to " L I " by r e q u e s t i n g a new s e t o f parameters f o r the L o r e n t z i a n but r e t a i n i n g the o r i g i n a l spectrum. " C I " w i l l p l o t the c o n t e n t s o f the b l o c k s s p e c i f i e d i n the most r e c e n t "C" command and, as i s a l s o t r u e f o r "CV", w i l l not work u n l e s s a "C" command has been used p r e v i o u s l y . In manual f i t t i n g , t h e n , one would s i m p l y v a r y the L o r e n t z i a n parameters u n t i l the sum o f the squared d i f f e r e n c e s was m i n i m i z e d . In "AFIT", the re q u e s t e d c e n t r e f r e q u e n c y o f the r e q u i r e d L o r e n t z i a n , i s a dummy v a r i a b l e . The program l o c a t e s the lo w e s t p o i n t i n the spectrum, c o r r e s p o n d i n g to the c e n t r e f o r an i d e a l L o r e n t z i a n a b s o r p t i o n , and c a l c u l a t e s the L o r e n t z i a n a b s o r p t i o n w i t h t h a t h e i g h t and c e n t r e f r e q u e n c y and w i t h the w i d t h s u p p l i e d . The sum o f the squared d i f f e r e n c e s i s then c a l c u l a t e d . The w i d t h i s then changed by t w i c e the f r e q u e n c y i n t e r v a l s u p p l i e d and the proce d u r e r e p e a t e d . The w i d t h w i l l change i n the a p p r o p r i a t e d i r e c t i o n u n t i l the f i t parameter i s m i n i m i z e d a t which p o i n t the procedure w i l l be repea t e d f o r the c e n t r e f r e q u e n c y . The parameters o f the L o r e n t z i a n g i v i n g the b e s t f i t , p l u s the p e r - c e n t c o u p l i n g of the resona n c e , w i l l then be o u t p u t . The f i t can be checked m a n u a l l y by u s i n g the "C" command as d e s c r i b e d above. F i g . 25 shows observed s p e c t r a and 8 9 the best f i t Lorentzian for t y p i c a l cases of a w e l l f i t resonance and for a poorly f i t resonance. Figure 25. Examples of observed resonances (solid lines) and best f i t obtained "AFIT" program (dashed l i n e ) . using 91 CHAPTER V Theory o f A n i s o t r o p i c D i e l e c t r i c R e s o n a t o r s 5.1 I n t r o d u c t i o n Because o f the p o s s i b i l i t y o f t o t a l i n t e r n a l r e f l e c t i o n a t the s u r f a c e o f a d i e l e c t r i c and vacuum, a d i e l e c t r i c s o l i d can be made t o a c t as a microwave c a v i t y . T h i s was f i r s t c o n s i d e r e d by Richtmeyer (1939) and a p p l i e d by Okaya and Barash (1962) who used a r e s o n a t o r made o f r u t i l e . D i e l e c t r i c r e s o n a t o r s have s i n c e been used both i n e n g i n e e r i n g a p p l i c a t i o n s and as a t e c h n i q u e f o r measuring d i e l e c t r i c p r o p e r t i e s o f s m a l l c r y s t a l s . D i e l e c t r i c resonance has been a p p l i e d to s t u d i e s o f K^Pt (CN) + Br 3H^0 ( J a k e l i c and S a i l l a n t , 1974) and TTF-TCNQ (Khanna e t a l . , 1 9 7 5 ; B a r r y , 1977). Most o f the u s e f u l t h e o r y r e g a r d i n g a n i s o t r o p i c d i e l e c t r i c r e s o n a t o r s has been d e a l t w i t h by B a r r y . We w i l l r e v i e w some o f i t here and then d e s c r i b e some at t e m p t s t o account f o r the observe d l e n g t h dependence o f the f r e q u e n c y f o r a f i n i t e r e s o n a t o r u s i n g s p e c i a l i z e d and somewhat u n p h y s i c a l boundary c o n d i t i o n s . We w i l l then d e s c r i b e s o l u t i o n s f o r i n f i n i t e d i e l e c t r i c wave g u i d e s w i t h and w i t h o u t an o u t e r c o n d u c t o r f o r the case o f no a z i m u t h a l dependence and f o r a c o s i n e a z i m u t h a l dependence o f the f i e l d s . We w i l l f i n d t h a t the f i e l d p a t t e r n s f o r these models bear some r e l a t i o n to those f o r the two l o w e s t observed modes i n the d i e l e c t r i c resonance e x p e r i m e n t . The mode 92 p l o t s f o r the wave g u i d e s o l u t i o n s do not p r o v i d e an e x p l a n a t i o n f o r the observed s i m i l a r i t y o f the mode p l o t s f o r the l o w e s t c o a x i a l and d i e l e c t r i c modes i n the TTF-TCNQ c r y s t a l s . 93 5.2 Open C i r c u i t Boundary C o n d i t i o n s The problem o f the d i e l e c t r i c r e s o n a t o r i s more c o m p l i c a t e d than t h a t o f a c a v i t y w i t h p e r f e c t l y c o n d u c t i n g w a l l s because o f the presence o f evanescent f i e l d s o u t s i d e o f the d i e l e c t r i c r e s o n a t o r . The s i m p l e s t way to d e a l w i t h t h i s problem i s to i g n o r e the f i e l d s o u t s i d e . T h i s t r e a t m e n t i s j u s t i f i e d by r e c o g n i z i n g t h a t the component o f the e l e c t r i c d i s p l a c e m e n t p e r p e n d i c u l a r to a boundary, D x, must be c o n t i n u o u s a c r o s s t h a t boundary. The r a t i o o f the p e r p e n d i c u l a r e l e c t r i c f i e l d s i n s i d e and o u t s i d e i s then i n v e r s e l y p r o p o r t i o n a l to the r a t i o o f the d i e l e c t r i c c o n s t a n t . I f the sample d i e l e c t r i c c o n s t a n t i s l a r g e , the p e r p e n d i c u l a r component o f E o u t s i d e o f the boundary w i l l be s m a l l . T h i s s i t u a t i o n i s summarized by the s o - c a l l e d magnetic w a l l boundary c o n d i t i o n s (5.1) nxH=0 (5.2) n-E=0 which l i m i t the f i e l d s a t a boundary to E f i e l d s p a r a l l e l to the boundary and H f i e l d s p e r p e n d i c u l a r to the boundary. As J a w o r s k i (1978) has p o i n t e d o u t , however, such boundary c o n d i t i o n a re r e a l l y o n l y a p p r o p r i a t e f o r h i g h e r modes f o r which the e x a c t b e h a v i o u r a t the boundary i s o f l e s s i m p o r t a n c e . U n f o r t u n a t e l y , we s h a l l see t h a t f o r a n i s o t r o p i c r e c t a n g u l a r 94 c r y s t a l s , open c i r c u i t boundary c o n d i t i o n s p r o v i d e the o n l y s i m p l e means o f d e a l i n g w i t h c r y s t a l s o f f i n i t e l e n g t h . B e f o r e a p p l y i n g the boundary c o n d i t i o n s , we must f i n d a s o l u t i o n f o r the f i e l d s i n a c r y s t a l . M axwell's e q u a t i o n s f o r a m a g n e t i c a l l y i s o t r o p i c d i e l e c t r i c , (5.3) VxE=(i«Wc)H (5.4) VxH=- ( i ^ / c)6 E (5.5) V*-£E=0 (5.6) ^-H=0 can be combined to give a wave equation in E, (5.7) VE-7(V-E)+ (wVc*) £ E=(5 We take our o r i g i n to be the c e n t r e o f the c r y s t a l and r e c o g n i z e t h a t we w i l l be d e a l i n g w i t h c r y s t a l s i n which the X d i m e n s i o n , c*, w i l l be v e r y s m a l l so t h a t we want a s o l u t i o n w i t h o u t a no d a l p l a n e p e r p e n d i c u l a r to X i n Ey and E z . A c o n v e n i e n t s o l u t i o n to t h i s wave e q u a t i o n i s then 95 E y =AX sin(k„X)sin(k yY)sin(k zZ) (5.8a) E y =Ay c o s ( k x X) cos ( k y Y) s i n ( k z Z ) E 2 =A2 cos (k x X ) s i n ( k y Y ) c o s ( k z Z ) The c o r r e s p o n d i n g magnetic f i e l d components a r e , H x = ( ~ i / k 0 ) ( A z k y -A y k 2 ) cos ( k x X ) cos (k y Y) cos ( k z Z) (5.8b) H y = ( - i / k 0 ) ( A x k 2 + A z k y ) s i n ( k y X ) s i n ( k y Y ) c o s ( k z Z ) H z = ( i / k c ) ( Ayk x+A yk y ) s i n ( k„ X) cos ( ky Y) s i n ( k^. Z) where ko = u//c* and we have taken M=1. We have chosen the s o l u t i o n f o r which E z i s an odd f u n c t i o n o f Y i n a n t i c i p a t i o n o f t h i s b e i n g the symmetry o f the l o w e s t mode. The open c i r c u i t boundary (OCB) c o n d i t i o n s r e q u i r e t h a t , f o r a r e s o n a t o r o f X, Y, and Z di m e n s i o n s g i v e n by c*, a, and b, we have (see F i g . 26) k x = l T r / c k/=mfr/a k z = n i r / b where 1 i s an even i n t e g e r , and m and n a r e odd i n t e g e r s . S u b s t i t u t i n g f o r E i n the wave e q u a t i o n , ( 5 . 7 ) , g i v e s r i s e t o 96 t h r e e homogeneous e q u a t i o n s f o r A*, ar, and az. For the case o f a t e t r a g o n a l d i e l e c t r i c t e n s o r , w i t h £ z and £,= £ y=£ x, t h e r e a re n o n - t r i v i a l s o l u t i o n s f o r (5.9) <^c* = ( (k*+k* )U2) + (k */eJ and (5.10) " / ^ = ( k ^ - k * + k £ ) / £ x I f we use Eq. (5.9) to s o l v e f o r A* and A y we f i n d (5.11) A x/A z = € zk 1k x/(k x z+ky )6j A y / A z = -e 2k zk y/(k ; f+k y a-)£ x so t h a t A x k y + A y k x = 0 and = 0. We w i l l r e f e r to t h i s as the t r a n s v e r s e magnetic (TM) mode. For t h i s s o l u t i o n , k z and k y must be non-zero s i n c e E y c o n t a i n s s i n k r Z and E z c o n t a i n s s i n k y Y . ky may be 0. I f we use Eq. 5.10, we f i n d t h a t we can s o l v e f o r hz and obta i n (5.12) A 2(k*+k*+k* ) ( ( £*/0-l) = 0 so t h a t i f £2*£j. , we must have E 2=0. T h i s w i l l be r e f e r r e d to 97 F i g u r e 26. The l a b e l l i n g o f the d i m e n s i o n s o f TTF-TCNQ c r y s t a l s . 98 as TE. For n o n - t r i v i a l s o l u t i o n s f o r t h i s type o f mode, we must have k^O. S i n c e c * i s v e r y s m a l l , i n c a s e s o f i n t e r e s t to us, the s m a l l e s t non-zero k* w i l l be v e r y l a r g e . I t i s thus unnecessary to c o n s i d e r t h i s s o l u t i o n f u r t h e r . We n o t e , a t t h i s p o i n t , t h a t f o r an i s o t r o p i c r e s o n a t o r ( i . e . £ x = £ 2 ) , Eqs. 5.9 and 5.10 would be e q u i v a l e n t . We would then be f r e e to choose an a x i s a l o n g which t o s e t the e l e c t r i c or magnetic f i e l d s t o z e r o . In the p r e s e n t c a s e , however, the a n i s o t r o p y has s p l i t the T.E. and T.M. modes and we are not f r e e to choose any a x i s . The a n i s o t r o p y o f the problem has d e f i n e d the Z a x i s as the one to which the l a b e l s T.E. and T.M. a r e r e f e r r e d . From Eq. 5.9, the f r e q u e n c i e s f o r the magnetic w a l l s o l u t i o n a re g i v e n by where 1 can be z e r o . The a n a l y s i s f o r the even modes i s i d e n t i c a l so t h a t Eq. 5.13 i s a p p r o p r i a t e f o r a l l i n t e g e r v a l u e s o f m and n. F i e l d l i n e s f o r some o f the lower l y i n g modes appear i n F i g . 27. In a model w i t h more r e a l i s t i c boundary c o n d i t i o n s , as we s h a l l d i s c u s s below, the s e p a r a t i o n i n t o T.E. and T.M. modes becomes l e s s d i s t i n c t . B a r r y (1977) used the n o t a t i o n E?* to denote d i e l e c t r i c resonances w h i c h , i n the l i m i t o f l a r g e a s p e c t r a t i o (b>>a>>c* ) would have e l e c t r i c f i e l d s (5.13) 99 p r i m a r i l y a l o n g the Z a x i s . We w i l l use t h i s n o t a t i o n i n a s i m i l a r way w i t h 1 and m g i v i n g the number o f e l e c t r i c f i e l d minima i n the Y and Z d i r e c t i o n s r e s p e c t i v e l y . Each mode i n F i g . 27, t h e n , i s l a b e l l e d by the E ^ mode which would reduce to t h a t mode i n the l i m i t o f magnetic w a l l boundary c o n d i t i o n s . f  ' ' • * • i, • 4 • • X ' X ' X ^ ^ ^ 2 t 2 2 •31 r •? X • • • X x " X I— - J "32 a nor" J u u z X * — • Figure 27. The f i e l d l i n e s f o r some of the low l y i n g d i e l e c t r i c resonances. 101 5.3 A Review o f R e s u l t s f o r I n f i n i t e D i e l e c t r i c R e s o n a t o r s F o l l o w i n g B a r r y (1977) , we go on to c o n s i d e r models i n which the sample i s i n f i n i t e i n the Z d i r e c t i o n (b=«?,kz = rr/b=0) and f o r which we can t a k e some account o f f i e l d s o u t s i d e o f the c r y s t a l . I n s i d e o f the c r y s t a l , we w i l l a g a i n use the plan e wave s o l u t i o n s o f Eqs. 5.8a and 5.8b. Because o f the i n f i n i t e l e n g t h k z i s zero and the o n l y s p a t i a l dependence w i l l be i n the X and Y d i r e c t i o n s . We w i l l take t h i s Y dependence to be e y f o r now and thus i n c l u d e both even and odd s o l u t i o n s . For these assumptions we o b t a i n , f o r the f i e l d s , E„=Ey =H2=0 (5.14) H y=-k yE 2/k c Hy = ( - i / k 0 ) (2>Ez/ax) where E z s a t i s f i e s the wave e q u a t i o n (7 aE 2+k o a£ 2E z=0 and k„ = <*J /c . The l o w e s t mode w i l l then have 102 E z =Az;Cos (k xiX) e (5.16) = (-ky/k0)Az,cos(k„X)e' r ikyY H / = ( i / k 0 ) A 2;k,;sin(k y,X)e 2. a a. w i t h Eq. 5.9 g i v i n g (k„,- +ky ) /£z=ke> . We can then go on to d e a l w i t h the f i e l d s o u t s i d e o f the c r y s t a l . We f i r s t c o n s i d e r the boundary p e r p e n d i c u l a r to X s i n c e c* i s the s m a l l e s t d i m e n s i o n and OCB c o n d i t i o n s a r e most d e f i c i e n t i n d e s c r i b i n g the f i e l d s a t t h i s boundary. The f i e l d s o u t s i d e o f the X f a c e boundary a r e assumed to decay e x p o n e n t i a l l y g i v i n g , (5.18) HX =- ( k y / k 0 )A z oe e By = ( i k x p /k Q ) A 2 0e e The wave e q u a t i o n a p p l i e d o u t s i d e o f the c r y s t a l g i v e s (5.19) k j =ky - k * Boundary c o n d i t i o n s a t |X|=c/2 r e q u i r e the c o n t i n u i t y o f 103 a l l three f i e l d s . E l i m i n a t i n g A Z O and Az; from the c o n t i n u i t y equations leaves (5.20) kXo = kxi tan(k„,- c^2) We now consider the Y dependence. I f we j u s t take k/=mir/a, we have appl i e d magnetic w a l l boundaries at Y= a/2. This i s equivalent to an a n i s o t r o p i c s l a b for which the Y dimension i s i n f i n i t e and a wave length of 2a i n the Y d i r e c t i o n has been imposed. Barry (1977) discusses t h i s s o l u t i o n i n d e t a i l . He shows * a. that f o r £2>>1, so that k e i s small r e l a t i v e to k*, and ky/ , one obtains k*a» k* tan ( (k y; c* )/2 ) which, f o r k ) f ic*<<l, becomes k y * k* c*/2 I n s e r t i n g t h i s i n t o Eq. 5.9 gives an e x p l i c i t expression for the resonant frequency. (5.24) £^{ky +2k y/c* } 104 or p u t t i n g k y=mir/a, (5.24a) f = QZ { (m/a) +2m/ac* } ft* A l t e r n a t i v e l y , one c o u l d a p p l y e x a c t boundary c o n d i t i o n s on the f a c e p e r p e n d i c u l a r to Y. In d o i n g s o , one i s f a i l i n g to s a t i s f y t he boundary c o n d i t i o n s f o r the s u r f a c e s o f t h e r e g i o n s bounded by \X\>c/2, |Y|>a/2 as i l l u s t r a t e d i n F i g . 28. We d i s c u s s such a model below. We note here t h a t the s o l u t i o n t o be d i s c u s s e d below i s the one which has been used to a n a l y s e the d i e l e c t r i c resonance modeplots o b t a i n e d e x p e r i m e n t a l l y . We must a l s o remember t h a t we now have both even and odd dependence o f the f i e l d s on Y. For even modes we have (5.25a) E Z (=A Z lcos(k ) f ( X ) c o s ( k y ( Y) i n s i d e the r e s o n a t o r and i n the r e g i o n bounded by |Y|>a/2; |X|<c/2. The wave e q u a t i o n o u t s i d e the c r y s t a l g i v e s (5.25b) E Z = A Y O cos(k y,X) e -* y o (IYI-<ya) 105 Figure 28. Surfaces on which boundary c o n d i t i o n s are not s a t i s f i e d i n theory used f o r a n a l y s i s of d i e l e c t r i c resonance data. 106 (5.26) k£ = ky* -ky0 and matching f i e l d s a t |Y|=a/2 g i v e s (5.27) ky„ =kyi t a n ( k y t - a / 2 ) One can then use Eqs. 5.26 and 5.19 to e l i m i n a t e k y o and k ^ g i v ing (5.28a) ky- -k* =C tan(k„ c*/2) (5.28b) k£ - C = ky* tan A(k y (-a/2) For the odd dependence on Y we o b t a i n (5.29) E 2=A z;Cos(k x- X ) s i n ( k y < - Y) i n s i d e and (5.30) E z=A y o c o s ( k v i X) e i n the r e g i o n bounded by |Y|>a/2; |X|<c/2. These l e a d to Eq. 5.28b b e i n g r e p l a c e d by 107 (5.28c) x a. ky- cot (k y, a/2) . In the a n a l y s i s of the experimental d i e l e c t r i c resonance data, k y j and k y; have been obtained numerically from the c r y s t a l dimensions using the appropriate p a i r o f J E q s . 5.28a and 5.28b or 5.28c. These values have then been used, with the observed resonant frequency contained i n k f , i n (5.17) a. a. a ( k w +kfi )/k 0 to y i e l d a value for €• •2. • 108 5.4 Other Models o f F i n i t e R e s o n a t o r s As mentioned, the o n l y model p r e s e n t e d so f a r which was c a p a b l e o f d e a l i n g w i t h f i n i t e r e s o n a t o r s was the magnetic w a l l or OCB model. For t h i s model, the l e n g t h dependence o f the squared f r e q u e n c y e n t e r e d as kz/£u where k z=mr/b. The s l o p e s o f B a r r y ' s mode p l o t s , however, gave a v a l u e f o r ex o f about 2. T h i s was c o n s i d e r e d to be too s m a l l . In a d d i t i o n , i t was found t h a t the d i e l e c t r i c modes had mode p l o t s l o p e s almost i d e n t i c a l to those o f the l o w e s t ( c o a x i a l ) mode. The models to be d i s c u s s e d below were s t u d i e d i n the hope o f a c c o u n t i n g f o r these a n o m a l i e s . We w i l l b e g i n by c o n s i d e r i n g , i n s e c t i o n 5.4a, a model i n which magnetic w a l l s a r e a p p l i e d to the X a x i s and extended beyond the c r y s t a l b o u n d a r i e s . In s e c t i o n 5.4b, we w i l l c o n s i d e r a s i m i l a r model w i t h the "magnetic tube" p a r a l l e l to the Z a x i s . In s e c t i o n 5.4c, we l e a v e o n l y the magnetic w a l l s p e r p e n d i c u l a r to the X a x i s . 109 5.4a "Magnetic Tube" P a r a l l e l to X Because o f the f i n i t e l e n g t h o f the c r y s t a l , the Y component o f the E f i e l d might be i m p o r t a n t i n the r e g i o n |Z|>b/2 and might be thought to be sampling a d i e l e c t r i c c o n s t a n t o f 1 r a t h e r than £j_ thus l e a d i n g to an i n c r e a s e i n the s l o p e o f the mode p l o t (towards t h a t o f the c o a x i a l mode). The d e p a r t u r e o f the s l o p e from the expected s l o p e f o r a c o a x i a l mode might then be understood i n terms o f t h e f i n i t e s i z e as w e l l . The f i r s t s t e p i n i n v e s t i g a t i n g f i n i t e r e s o n a t o r models was to attempt to extend the OCB model to i n c l u d e matching o f f i e l d s on some b o u n d a r i e s i n o r d e r to see i f t h i s r e s u l t e d i n a d e p a r t u r e from the l/£j_ s l o p e . J a w o r s k i (1978) suggested t h a t i t might be p o s s i b l e to a p p l y a model o r i g i n a l l y proposed by Yee (1965) f o r c i r c u l a r c y l i n d r i c a l r e s o n a t o r s . T h i s model i n v o l v e s magnetic w a l l s i n the form o f a c y l i n d r i c a l tube c o n c e n t r i c w i t h , and o f the same r a d i u s a s , the r e s o n a t o r . The m o d i f i e d model had magnetic w a l l b o u n d a r i e s on the f a c e s p e r p e n d i c u l a r to Y and Z. These w a l l s c o n t i n u e d i n t o the r e g i o n |X|>c*/2 as shown i n F i g . 29. On the f a c e p e r p e n d i c u l a r t o the X a x i s , t he f i e l d s i n s i d e a r e matched to e x p o n e n t i a l l y d e c a y i n g f i e l d s o u t s i d e . The X dependence o u t s i d e o f the f a c e p e r p e n d i c u l a r to the X a x i s goes as Figure 29. Magnetic w a l l s as a p p l i e d i n adaptation of Yee's theory to rectangular c r y s t a l s . I l l e A p p l i c a t i o n o f Yee's r e s u l t f o r an i s o t r o p i c d i e l e c t r i c y i e l d s (5.29a) kxe, = k x t a n ( k x c / 2 ) w i t h (5.29b) k* = k *+k* -kx* In f a c t , as we s h a l l s e e , t h i s r e s u l t i s not a p p r o p r i a t e f o r a n i s o t r o p i c d i e l e c t r i c s . We now a n a l y s e the problem i n d e t a i l . We b e g i n by w r i t i n g the f i e l d s i n s i d e and o u t s i d e . I n s i d e , we have, E,(. =AXl- s i n ( k x < X ) s i n ( k y Y ) s i n ( k i Z ) (5.30) Ey;=Ay, c o s ( k x < X) cos (k y Y) s i n ( k z Z) E2i=Ax; c o s ( k x , X) s i n ( k y Y) cos ( k z Z ) and H X l=(-i/k 0) (Az,- ky-Ay,- k 2 ) cos (kx,X) cos ( k y Y) cos ( k z Z ) 112 (5. 31) H/( = ( - i / k 0 ) (Aw kz+AZ(- k> ) s i n (k X l X) s i n ( k y Y) cos ( k z Z ) H2, = ( i / k 0 ) (A y; k x +AX<- ky ) s i n ( k w X ) c o s ( k y Y ) s i n ( k z Z ) which a r e the u s u a l p l a n e wave s o l u t i o n s . As b e f o r e , the c o n d i t i o n f o r a n o n - t r i v i a l s o l u t i o n w i t h Hz=0 i s Eq. 5.9. B e f o r e g o i n g on to c o n s i d e r the f i e l d s o u t s i d e the c r y s t a l , i t i s i m p o r t a n t to ask whether, f o r our c h o i c e o f the TM mode, one can have a s i t u a t i o n i n which can be zero and k y be non-z e r o . In f a c t , i f one chooses A*,- =0, one can o b t a i n , from the wave e q u a t i o n , t h r e e e q u a t i o n s i n Ay/ and A z ! . Ay,. k y k y +AZ(- k y k z =0 (5.32) A y. ( ( 4 , 7 c * ) £ x - k ^ - k j )+A z r k zky=0 Ay/ k y k 2 +A Z, ( ( ^ Vc*)£z -k^-ky 1) =0 The c o n d i t i o n f o r a n o n - t r i v i a l s o l u t i o n f o r Ay<- and AZ(- i s found to be kx,=0 or ex=.£.z. The second c o n d i t i o n can be understood s i n c e , i f ^ . = £ z, our s e p a r a t i o n i n t o TE and TM modes becomes a r b i t r a r y and one has the u s u a l case f o r a m e t a l l i c c a v i t y i n which one can a s s i g n a TE or TM mode. S i n c e we are i n t e r e s t e d i n £j.*€-z , we are f o r c e d to d e a l w i t h a case i n which E y cannot be zero u n l e s s k x=0. S i n c e we are i n t e r e s t e d i n the 113 s p a t i a l dependence a l o n g the X a x i s , we must a c c e p t E x as be i n g non-zero. We can now w r i t e down the f i e l d s f o r |X|>c/2. They a r e g i v e n by, -kxe(iy\-Exo = A x o e s i n ( k y Y ) s i n ( k z Z ) (5.33) E y o = A y o e cos (k VY) s i n ( k z Z ) EZp=Azoe s i n ( k y Y ) cos ( k z Z ) and - k (M - c\) Hxo = ( - i / k 0 ) ( A 2 0 k y - A y o k z ) e *" cos ( k y Y) cos (k^, Z) (5. 34) H y o = (-i/k») ( A ^ k 2 + A Z o kxo) e *° s i n ( k y Y) cos ( k 2 Z ) H z o = ( i / k 0 ) ( A y ok X o+A X o k y ) e *° c o s ( k y Y ) s i n ( k z Z ) We now have 6 c o n d i t i o n s a t the c r y s t a l boundary |X|=c72. These are E y, =E y o ; Ez; =E z o ; €.xEyi =^Xo and Hy, =tix0 ; Hyi- =Hyo ; Hz,=0 114 g i v i n g A y j c o s ( k x t c/2)=Ayo A Z ( cos ( k v < c*/2)=Azc ( t A / i s i n (ky, c/^ 2 ) =A X 0 (5.35) ( A z < k^  -Ayk^,. ) c o s ( k w c72)=Azo k y-A y < ) k z (A^, k z+A zk^- Jsinfk^-c72)=(A V o k z + A Z o k X e ) (A y<> k>a +A x o ky ) =0 Because o f the r e l a t i o n s between AX(- and A y i and A z ; , as g i v e n by Eq. 5.11, some o f thes e c o n d i t i o n s a r e not independent. One u l t i m a t e l y a r r i v e s a t two e x p r e s s i o n s f o r kxo . (5.36) k w = ( (k* -k z ) / (eM K? -kj) )^ky ; t a n (k w c/2 ) (5.37) k x 0 =£j.k^ t a n ( k x i c*/2) These a re o b v i o u s l y i n c o m p a t i b l e u n l e s s £1 = 1, which i s not an i n t e r e s t i n g s o l u t i o n , or kx-=kya=0, or tan ( k w c/2) =0 which r e t u r n s us to OCB c o n d i t i o n s . We thus f i n d t h a t t r y i n g to a p p l y 115 more r e a l i s t i c boundary c o n d i t i o n s to the face perpendicular to the X a x i s cannot be done for the a n i s o t r o p i c s i t u a t i o n . I t can be e a s i l y shown that for the i s o t r o p i c case, where Ax, can be zero while kx, i s non-zero, that Yee's r e s u l t i s obtained. 116 5.4b "Magnetic Tube" P a r a l l e l to Z The s i t u a t i o n i s s l i g h t l y d i f f e r e n t i f we c o n s i d e r matching to d e c a y i n g f i e l d s on the f a c e s p e r p e n d i c u l a r to the Z and Y axes . The s i m p l e s t way to proceed i s to a p p l y magnetic w a l l s to a l l but the f a c e s p e r p e n d i c u l a r to the Z a x i s . O u t s i d e o f t h i s f a c e , we w i l l c o n s i d e r the f i e l d s t o decay as e . W e can a n t i c i p a t e the be h a v i o u r o f such a model. The fr e q u e n c y w i l l s t i l l be g i v e n by Eq. 5.9 and k y and k y w i l l s t i l l be d e f i n e d by the OCB c o n d i t i o n s a t the a p p r o p r i a t e f a c e s . By a l l o w i n g the f i e l d s t o extend i n t o the r e g i o n |Z|>b/2, we can o n l y o b t a i n a k z t h a t i s s m a l l e r than i t s OCB v a l u e o f nrr/b. T h i s , t h e n , would be the k i n d o f end e f f e c t which would g i v e r i s e to an e f f e c t i v e v a l u e f o r £j_ t h a t was too l a r g e . When the boundary c o n d i t i o n s a r e a p p l i e d , k^0 i s found to be g i v e n by (5.38) k z o=(k z,/£ x) t a n ( k a ; b / 2 ) The c h a r a c t e r i s t i c e q u a t i o n o u t s i d e o f the c r y s t a l i s (5.39) C=C+k* "C Using i t t o e l i m i n a t e k* from Eq. 5.9 g i v e s a second e q u a t i o n i n k Z o and k z i 117 (5.40) (k* +k* ) (£-2-l)/£z =k^+k^/^ For a t y p i c a l c r y s t a l , k* w i l l be much l a r g e r than kz; /£ x with the r e s u l t that k z o w i l l be l a r g e and k2( w i l l be s l i g h t l y smaller than tr/b. I f one thinks of the end e f f e c t as an e f f e c t i v e l e n g t h , b eg =b+^b, then for kx=0 and k; = n/a, numerical s o l u t i o n s using t h i s model f o r a c r y s t a l of dimension a=0.05 cm, give Ab ranging from 0.05 cm, f o r b=lcm, to 0.06 cm for b=0.2cm. A small constant 4b would appear i n the observed €± as £ e i 5 = Q(l+(Ab/b) ) . 118 5.4c Magnetic Walls Perpendicular to X One can extend this model to one in which the magnetic walls are placed only on faces perpendicular to the X axis. When we assume exponential decay beyond both the Z and Y faces, we obtain, as for the previous model, Eq. 5.38 for the re l a t i o n between k Z o and k z as well as a similar condition (5.41) k y e =(ky,/£z) tan(ky«-a/2) The c h a r a c t e r i s t i c equations outside of the c r y s t a l are Eq. 5.39 and Eliminating ke , k y o , and k z o , and taking kx=0, leads to two equations in kzi and ky,-. (5.42) (5.43) (kz;/ky/ )*£*( ( 6X -1) /S.) - £z = tan*( k y; a/2 ) (5.44) (kY,/k2,f £*( (e z-!)/£, )-<^=tan*(kzi b/2) This model, unfortunately, just gives an eff e c t similar to that found previously. For a cr y s t a l with a=0.05 cm and taking £".2 = 3000 and .^= 5, t h i s model would y i e l d a mode plot from which 119 an inverse slope of about 5.14 would be obtained. This i s greater than £x . I t was g e n e r a l l y found, then, that using simple models, which t r y to match plane wave s o l u t i o n s i n s i d e the c r y s t a l to decaying f i e l d s at the faces without matching the f i e l d s i n the corner regions, to analyse mode p l o t slopes would give values of €j_ which were lower than the inverse slope. They are thus unable to account f o r the low observed values of the inverse slope. 120 5.5 I n f i n i t e Waveguides With and Without Outer Conductor The observed s i m i l a r i t y of coaxial and d i e l e c t r i c mode plot slopes suggests that one should consider the ef f e c t of an outer conductor, i . e . the wave guide, on the d i e l e c t r i c resonances. In this section we compare d i e l e c t r i c wave guides with and without a metallic outer conductor for the two lowest azimuthal dependences of the modes. 121 5.5a n=0 A z i m u t h a l Dependence B a r r y (1977) d i s s c u s s e s the case f o r an i n f i n i t e c y l i n d r i c a l rod o f r a d i u s R1 i n a c i r c u l a r m e t a l l i c waveguide o f r a d i u s R^ f o r ^=£Y=€s.. H i s s o l u t i o n s a r e reviewed i n Appendix 2. The s o l u t i o n s can be c l a s s i f i e d by the a z i m u t h a l dependence o f the f i e l d s where t h i s i s g i v e n by e""^ . B a r r y has c a l c u l a t e d the s o l u t i o n s f o r the n=0 c a s e s . Some examples o f c a l c u l a t e d n=0 mode p l o t s appear i n F i g s . 30 and 31. In F i g . 30a, we have shown, f o r a wave g u i d e w i t h and w i t h o u t an o u t e r c o n d u c t o r , the s o l u t i o n f o r £j. = 5, £z=3000 and an R^  g i v i n g a c r o s s - s e c t i o n a l area t y p i c a l o f the c r y s t a l s s t u d i e d . In F i g . 30b, we have shown the mode f o r an i n n e r c o r e i d e n t i c a l to the wave g u i d e o f F i g . 30a and an o u t e r c o n d u c t o r w i t h a c r o s s -s e c t i o n a l area s i m i l a r to t h a t o f wave g u i d e s used i n the ex p e r i m e n t s . A l s o shown are the l i n e s a l o n g which k* = 0 and k a=0. For k A<0, which i s the r e g i o n to the r i g h t o f the k A=0 l i n e , the f i e l d s d i e o f f o u t s i d e o f the d i e l e c t r i c . The s o l u t i o n s a r e seen to be s i m i l a r f o r both the d i e l e c t r i c waveguide w i t h and w i t h o u t an o u t e r c o n d u c t o r . In F i g . 30b, the s e c t i o n o f the l o w e s t mode l a b e l l e d l a to l b i s one i n which much o f the f i e l d i s o u t s i d e o f the d i e l e c t r i c . As we go to h i g h e r f r e q u e n c i e s , the f i e l d s a r e i n c r e a s i n g l y c o n t a i n e d i n s i d e a. the d i e l e c t r i c . At l b , where k^=0, t h e r e i s a node i n a t the s u r f a c e o f the d i e l e c t r i c . The v a r i a t i o n o f the a m p l i t u d e o f E x 122 Figure 30a. n=0 d i e l e c t r i c wave guide mode plot for R., =0.015 cm, €x=5, €*=3000. (no outer conductor) Figure 3 0 b . n=0 d i e l e c t r i c wave guide mode plot for R,= 0 . 0 1 5 cm, RA = 0 . 5 0 0 cm, € x = 5 , € , = 3 0 0 0 . (outer conductor present) 123 Figure 31a. n=0 d i e l e c t r i c wave guide mode plot for PM =0.015 cm, Rx=0.5 cm, €^ =5, £r=600. (outer conductor present) Figure 31b. n=0 d i e l e c t r i c wave guide mode plot for R, =0.015 cm, Rx=0.05 cm, C=5, £z=3000. (outer conductor present) 124 a c r o s s a d i a m e t e r , f o r each o f the l a b e l l e d p o i n t s , i s shown q u a l i t a t i v e l y i n F i g . 32. At 2, the f i e l d s have drawn i n t o the d i e l e c t r i c so t h a t t h e r e i s a c y l i n d r i c a l nodal s u r f a c e a t r^O.Ol cm. k 1 i s a p p r o x i m a t e l y c o n s t a n t a l o n g 4, 2, 3 and the mode p l o t has a s l o p e o f about l/e±. . T h i s i s p r e d o m i n a n t l y a d i e l e c t r i c mode but i s not o f e x p e r i m e n t a l i n t e r e s t because the •A s m a l l c d i m e n s i o n would push modes w i t h c y l i n d r i c a l n odal s u r f a c e s t o v e r y h i g h f r e q u e n c y . A pure c o a x i a l mode w i t h a m e t a l l i c c e n t r e c o n d u c t o r would f o l l o w the k A=0 l i n e and have a p r o p a g a t i o n v e l o c i t y o f c. The 1,1b,la p a r t o f the l o w e s t mode i s p r e d o m i n a n t l y c o a x i a l but does depend on the p r o p e r t i e s o f the d i e l e c t i c c o r e as d i s c u s s e d by B a r r y and i l l u s t r a t e d below. F i g . 31a shows the r e s u l t f o r £ z=600. The i n t e r c e p t o f the lo w e s t mode has r i s e n a l t h o u g h not by a f a c t o r o f 5 as would be expected f o r a p u r e l y d i e l e c t r i c mode. The s l o p e has i n c r e a s e d to 0.95. The e f f e c t o f d e c r e a s i n g R^/Ri i s shown i n F i g . 31b which i l l u s t r a t e s the case f o r R A=0.05 cm. The s e p a r a t i o n o f the d i e l e c t r i c and c o a x i a l modes i s now l e s s d i s t i n c t . The s e c t i o n o f the l o w e s t mode, l a to l b , i n F i g . 30b, i s analogous to the A mode observed by B a r r y . T h i s i m p l i e s t h a t f o r l o n g c r y s t a l s , the c o a x i a l A mode p l o t s h o u l d have a s l o p e o f about 90% of t h a t expected f o r a p u r e l y c o a x i a l mode. 1 0 r/R, Figure 32. Qualitative depiction of the v a r i a t i o n of E x across the d i e l e c t r i c rod diameter for selected points on mode plot of Figure 30b. 126 5.5b n=l A z i m u t h a l Dependence To check the e f f e c t o f the o u t e r c o n d u c t o r on a d i e l e c t r i c mode, such as the B mode, mode p l o t s were c a l c u l a t e d f o r the n=l wave g u i d e w i t h and w i t h o u t an o u t e r c o n d u c t o r . The problem w i t h o u t the o u t e r c o n d u c t o r has a l s o been d e a l t w i t h by J a w o r s k i (1978). Here t h e r e i s a n o d a l p l a n e a c r o s s which E z changes s i g n so t h a t , f o r i t s l o w e s t r a d i a l dependence, t h i s mode i s analogous to the B mode. The h i g h e r r a d i a l modes a r e not o f i n t e r e s t s i n c e they a r e a g a i n pushed to v e r y h i g h f r e q u e n c y f o r a f l a t c r y s t a l such as was used. The d e r i v a t i o n o f the c h a r a c t e r i s t i c e q u a t i o n f o r the n=l case appears i n Appendix 2. The c a l c u l a t e d mode p l o t s f o r the n=l d i e l e c t r i c wave g u i d e w i t h and w i t h o u t an o u t e r c o n d u c t o r w i t h the same parameters as i n F i g . 30a and F i g . 30b appear i n F i g s . 33a and 33b. We a g a i n see t h a t the r e g i o n s k*<0 are s i m i l a r . In the case w i t h an o u t e r c o n d u c t o r , we f i n d modes w i t h c o n s t a n t k£~ and modes w i t h c o n s t a n t k*. The modes w i t h c o n s t a n t k* can be i d e n t i f i e d as the l o w e s t wave g u i d e modes i n a c y l i n d r i c a l m e t a l tube o f r a d i u s R a. The modes w i t h c o n s t a n t kf have a s l o p e c l o s e to l/£± and are i d e n t i f i e d as d i e l e c t r i c modes. F i g . 34 shows the r a d i a l dependence o f E z a c r o s s a d i a m e t e r f o r the p o i n t 1 on the mode p l o t o f F i g . 33b. T h i s i s r o u g h l y analogous 127 Figure 33a. n=l d i e l e c t r i c wave guide mode plot for Ri =0.015 cm, 6^ =5, €*=3000 (no outer conductor) Figure 33b. n=l d i e l e c t r i c wave guide mode plot for Ri =0.015 cm, Rx =0.500 cm, £j.=5, £;c=3000. (outer conductor present) 128 -zo 1 -1 0 Figure 34. Q u a l i t a t i v e d e p i c t i o n of the v a r i a t i o n i n magnitude of Ez. across the diameter of the d i e l e c t r i c rod f o r point 1 on F i g . 33b. 129 Figure 35a. n=l d i e l e c t r i c wave guide mode plot for R^  =0.015 cm, Ri=0.05 cm, £^=5, £z=3000. (outer conductor present) Figure 35b. n=l d i e l e c t r i c wave guide mode plot for R1=0.015 cm, R^=0.500 cm, €x=5, £ z=600. (outer conductor present) 130 to the B mode. The d i e l e c t r i c and waveguide modes r e p e l each o t h e r but the a r e o t h e r w i s e independent. T h i s i s i l l u s t r a t e d i n F i g s . 35a and 35b. In F i g . 35a, Rj_ has been d e c r e a s e d by a f a c t o r o f 10. The waveguide modes have r i s e n out o f the range shown, but the d i e l e c t r i c modes a r e u n a f f e c t e d . In F i g . 35b, £ z has been d e c r e a s e d by a f a c t o r o f 5 and the i n t e r c e p t o f the l o w e s t d i e l e c t r i c mode has r i s e n by t h i s amount l e a v i n g the wave g u i d e mode u n a f f e c t e d . The i m p l i c a t i o n i s t h a t , o t h e r than f o r a c r o s s i n g o f the wave g u i d e mode, the d i e l e c t r i c mode, i n the absence o f end e f f e c t s , s h o u l d not be s t r o n g l y a f f e c t e d by the presence o f the o u t e r c o n d u c t o r . The f a c t t h a t the c o a x i a l and d i e l e c t r i c modeplots are p a r a l l e l i n B a r r y ' s work cannot be a t t r i b u t e d j u s t t o the e f f e c t o f the o u t e r c o n d u c t o r on the d i e l e c t r i c modes. I t i s a l s o apparent t h a t the s l o p e o f the A mode cannot be e n t i r e l y a t t r i b u t e d to the f a c t t h a t the c o r e o f the c o a x i a l r e s o n a t o r i s a d i e l e c t r i c . We expect t h a t these anomalies have to be a t t r i b u t e d to end e f f e c t s o f some s o r t and have t e s t e d t h i s by t r y i n g to measure d i e l e c t r i c resonances i n l o n g c r y s t a l s f o r which end e f f e c t s a r e o f l e s s i m p o r t a n c e . The r e s u l t s a r e d i s c u s s e d i n the next c h a p t e r . 131 CHAPTER VI R e s u l t s and D i s c u s s i o n o f D i e l e c t r i c Measurements on TTF-TCNQ 6.1 The Mode P l o t s As has been d i s c u s s e d , the r e a l p a r t s o f the d i e l e c t r i c c o n s t a n t f o r the b d i r e c t i o n can be o b t a i n e d from a p l o t o f the squares o f the resonance f r e q u e n c i e s v e r s u s the i n v e r s e squared l e n g t h s o f the c r y s t a l as i t i s c u t . For a h y p o t h e t i c a l c r y s t a l o beying the open c i r c u i t boundary c o n d i t i o n s , t h i s would a. 3-c o r r e s p o n d to p l o t t i n g f v e r s u s k z f o r the l o w e s t s e t o f modes. E x t r a p o l a t i n g to the o r d i n a t e a x i s s h o u l d then g i v e , u s i n g the c h a r a c t e r i s t i c e q u a t i o n , 5.9, f * = (k*+k*) (c74 *)/e\ The most u s e f u l model f o r e x t r a c t i n g € 'b i s t h e i n f i n i t e r e s o n a t o r w i t h the boundary c o n d i t i o n s s a t i s f i e d on the fo u r f a c e s but not i n the edge r e g i o n s . T h i s model uses Eqs. 5.28a, 5.28b, or 5.28c to y i e l d kyi and kyi . The s l o p e i s n o r m a l l y d e a l t w i t h by assuming kz = (mir/b) so t h a t the s l o p e becomes (cVm*£«) . B a r r y (1977) d e a l t w i t h f o u r modes i n d e t a i l . The B, C, and D modes were the t h r e e l o w e s t d i e l e c t r i c modes w i t h 0, 1, 132 and 2 e l e c t r i c f i e l d nodal p l a n e s p e r p e n d i c u l a r to the Y a x i s . In the Etrn n o t a t i o n , these a re r e f e r r e d to as E | f , E ) J L , and E l 3 r e s p e c t i v e l y . In the p r e s e n t work, c r y s t a l s a p p r o a c h i n g 1 cm i n l e n g t h have been used t o ob s e r v e modes w i t h 1 = 1 ; l«Cm^5, 1 = 2 ; Um<3, and 1=3; m=l. As w i l l be d e s c r i b e d below, the importance of end e f f e c t s appears t o d i m i n i s h f o r the h i g h e r modes. I t has a l s o been found t h a t the i d e n t i t y o f the c o a x i a l mode can be c o n f i r m e d by comparing the A mode i n l o n g c r y s t a l s w i t h the c o a x i a l mode observed u s i n g copper r e p l i c a s o f these c r y s t a l s . We w i l l f i r s t c o n s i d e r some o f the r e s u l t s c o n c e r n i n g the A mode. 133 6 . l a The A Mode Seven c r y s t a l s were a n a l y s e d i n d e t a i l . Some o f the mode p l o t s appear i n Appendix 3. I t can be seen t h a t the A mode f o r c r y s t a l s 4, 9, 10, and 12 does not y i e l d a s t r a i g h t l i n e . The A mode p l o t , f o r c r y s t a l 4 p a r t i c u l a r l y , d e v i a t e s toward the o r i g i n f o r l a r g e b. T h i s s u g g e s t s t h a t the c o a x i a l d i e l e c t r i c t r a n s m i s s i o n l i n e s o l u t i o n might be a p p r o p r i a t e i n the l o n g c r y s t a l l i m i t . In o r d e r to stu d y the e f f e c t of- the f i n i t e c r y s t a l l e n g t h , copper r e p l i c a s o f some o f the c r y s t a l s were made and mode p l o t s f o r p u r e l y c o a x i a l modes o b t a i n e d . The di m e n s i o n s o f the copper r e p l i c a s appear i n Table 3. The mode p l o t s appear i n F i g . 36. R e p l i c a C5 was mounted w i t h the q u a r t z f i b r e p e r p e n d i c u l a r to the r e p l i c a and a t t a c h e d a t the c e n t r e o f the r e p l i c a i n o r d e r to s t u d y the c o a x i a l resonance i n the absence o f g l u e a t the ends. U n f o r t u n a t e l y , the p o s i t i o n i n g o f r e p l i c a C5 was i m p r e c i s e u s i n g t h i s mount, and the s i g n i f i c a n c e o f the r e s u l t s o b t a i n e d was u n c e r t a i n . The e f f e c t o f the mounting epoxy on the resonances w i l l be d i s c u s s e d below. A l s o shown i n F i g . 36 i s the A mode f o r c r y s t a l 4. I t can be seen t h a t the A mode has a l e n g t h dependence s i m i l a r to the c o a x i a l modes and t h a t , f o r l a r g e b, both t y p e s approach the 2. 3. X l i n e g i v e n by f =c /(4b ) . On the b a s i s o f the d i e l e c t r i c t r a n s m i s s i o n l i n e c a l c u l a t i o n s , one would expect the l i m i t i n g TABLE 3 Dimensions o f copper r e p l i c a s used f o r c o a x i a l mode s t u d i e s R e p l i c a a (cm) c (cm) CI 0.049 0.0035 C2 0. 042 0. 007 C3 0. 040 0. 007 C4 diameter = 0.019 1800 1600 1400 ^ 1 2 0 0 o dooo & c <b & 8 0 0 600 400 200 O O Replica C2 A Replioa C3 • Replica U O R«PlKi C5 X Crystal U /f2=(c/2b)2 / o x e / O y • o ° I.  X I / • 8* / AO 2 4 6 8 10 12 1/t)2 (cm*2) Figure 36. Mode p l o t s f o r copper r e p l i c a s i n study of c o a x i a l modes. 136 s l o p e o f the A mode to be about 90% of t h a t f o r the p u r e l y c o a x i a l mode. The c r y s t a l s s t u d i e d were not l o n g enough to t e s t t h i s p o i n t . The d e v i a t i o n o f the p u r e l y c o a x i a l mode from f =c / ( 4 b ) can be r e p r e s e n t e d by a model i n which b i s r e p l a c e d by (b+A). For r e p l i c a s 2, 3, and 4, A was found to be (0.13±.01) cm. That A i s c o n s t a n t i m p l i e s t h a t , i n the range o f l e n g t h s s t u d i e d , the f i e l d s a t the end o f the c r y s t a l are a f f e c t e d by the presence o f the b o u n d a r i e s and n o t , d i r e c t l y , by the l e n g t h o f the c r y s t a l . The b e h a v i o u r o f the A mode i s thus found to be c o n s i s t e n t w i t h t h a t f o r a c o a x i a l mode. The d e v i a t i o n from a s l o p e o f about cV4 i s not i n t r i n s i c to TTF-TCNQ but i s l i k e l y j u s t a c h a r a c t e r i s t i c o f the c o a x i a l mode i n a r e s o n a t o r w i t h end e f f e c t s . 137 6.1b The D i e l e c t r i c Mode P l o t s The i n i t i a l r e s u l t s , such as those d i s p l a y e d i n the mode p l o t s f o r c r y s t a l s 4, 9, and 12, seemed to c o n f i r m t h a t the d i e l e c t r i c modes B, C, and D, d i d g i v e s t r a i g h t l i n e mode p l o t s w i t h a s l o p e s i m i l a r to t h a t o f the A mode p l o t . I t was, however, p o s s i b l e t o c a r r y out d e t a i l e d s t u d i e s u s i n g 2 c r y s t a l s , 18 and 19, w i t h l a r g e enough a and i n i t i a l b dim e n s i o n s to observe a number o f h i g h e r modes as w e l l as resonances f o r lower modes low v a l u e s o f k z = n/b. The mode p l o t s f o r a l l o f these c r y s t a l s appear i n Appendix 3. These r e s u l t s were o b t a i n e d a t 5K. For c r y s t a l 18, i t was p o s s i b l e to observe E^, resonances f o r 1=1 ; l£m^5 and 1=2 ; m=l. A t h i r d type o f resonance, l a b e l l e d B y on the mode p l o t , w i l l be d i s c u s s e d below. With the e x c e p t i o n o f B x, c r y s t a l 19 d i s p l a y e d a l l o f the above modes p l u s 1=2; m=2 and m=3 and 1=3 ; m=l. The m v a l u e s f o r the h i g h e r 1 modes a r e det e r m i n e d by n o t i n g t h a t the frequ e n c y becomes independent o f 1 as kz goes to z e r o . The most s t r i k i n g f e a t u r e o f the mode p l o t f o r c r y s t a l 18 i s the f a c t t h a t , f o r the h i g h e r modes i n p a r t i c u l a r , the s l o p e appears t o d e c r e a s e f o r s m a l l e r v a l u e s o f l / b . At h i g h e r v a l u e s o f l / b , the 1=1 mode p l o t s a l l seem to be p a r a l l e l to t h a t f o r the A mode. C r y s t a l 19, u n f o r t u n a t e l y , broke b e f o r e the s h o r t c r y s t a l b e h a v i o u r c o u l d be c o n f i r m e d but i t a l s o showed the tendency f o r 138 h i g h e r modes to approach the o r d i n a t e a x i s w i t h s m a l l e r s l o p e s . T a b l e 4 shows the v a l u e s o f £\ c a l c u l a t e d from each o f the d i e l e c t r i c modes assuming t h a t the l i m i t i n g s l o p e f o r l/ b - * 0 i s g i v e n by c al a/(4£l,). I t can be seen t h a t , f o r a g i v e n v a l u e o f 1, the d e r i v e d v a l u e o f £\ i n c r e a s e s w i t h i n c r e a s i n g m. T h i s t r e n d can be r a t i o n a l i z e d by a r g u i n g t h a t k* approaches 1 FT / ( 4 b ) o n l y i f the f i e l d s i n the c r y s t a l approach the l i m i t i n which t h e y approximate the open c i r c u i t boundary c o n d i t i o n s o l u t i o n s . For each d i m e n s i o n , t h i s l i m i t can be approached i n two ways. For a g i v e n mode, the p r o p a g a t i o n c o n s t a n t i n a g i v e n d i r e c t i o n w i l l go to zero as t h a t d i m e n s i o n g e t s l a r g e . T h i s i s the s i t u a t i o n as we approach l a r g e v a l u e s o f b. A l t e r n a t i v e l y , i f one t a k e s a g i v e n c r y s t a l , t h e t r u e p r o p a g a t i o n c o n s t a n t and the OCB s o l u t i o n i n a g i v e n d i r e c t i o n w i l l tend to approach each o t h e r f o r i n c r e a s i n g mode i n d i c e s i n t h a t d i r e c t i o n . I t i s not i m m e d i a t e l y o b v i o u s t h a t a b e t t e r a p p r o x i m a t i o n o f ky to the OCB s o l u t i o n s h o u l d improve the agreement between k z and i t s OCB v a l u e . I t was, however, f o u n d , as d i s c u s s e d i n c h a p t e r 4, t h a t s i t u a t i o n s i n which OCB c o n d i t i o n s were a p p l i e d to the fa c e p e r p e n d i c u l a r to Y y i e l d e d s o l u t i o n s f o r which k z a l s o approximated the OCB c o n d i t i o n s o l u t i o n . An o b v i o u s problem i s then p r e s e n t e d by the r e s u l t s f o r c r y s t a l 19 where, f o r c o r r e s p o n d i n g m v a l u e s , £* i s found to be s m a l l e r f o r 1=2 than f o r 1=1. T h i s i s not understood a t p r e s e n t . W i t h i n the 1=2 modes, however, the t r e n d f o r i n c r e a s i n g m i s s t i l l o b s e r v e d . TABLE 4 V a l u e s o f o b t a i n e d from d i e l e c t r i c resonance s t u d i e s f o r TTF-TCNQ c r y s t a l mode (E ^  ) 1 m 18 1 1 2.3 18 1 2 3.5 18 1 3 4.7 18 1 4 5.6 18 1 5 9.0 19 1 1 2.0 19 1 2 3.5 19 1 3 5.6 19 1 4 6.1 19 1 5 7.5 19 2 1 2.0 19 2 2 2.5 19 2 3 3.0 19 3 1 3.7 140 I t would appear, t h e n , t h a t one s h o u l d o b t a i n a lower l i m i t f o r £'a by u s i n g l o n g c r y s t a l s and h i g h e r modes. The v a l u e s o f £ 4 o b t a i n e d f o r the modes observed i n t h i s work do not seem to be a p p r o a c h i n g an upper l i m i t . On t h i s b a s i s , the a c t u a l v a l u e i s e xpected to be g r e a t e r than 9. With the o b s e r v a t i o n t h a t the l i m i t i n g mode p l o t s l o p e f o r l o n g c r y s t a l s i s not equ a l to the s l o p e a t s h o r t e r l e n g t h s , i t i s c l e a r t h a t the e x t r a p o l a t i o n back to the o r d i n a t e a x i s to o b t a i n £"{, must be approached c a u t i o u s l y . The b e s t v a l u e s o f w i l l l i k e l y be o b t a i n e d by e x t r a p o l a t i n g those mode p l o t s g i v i n g the h i g h e s t v a l u e s o f £'n . For g i v e n v a l u e s o f a and c , t h e v a l u e s o f kxi and kY, o b t a i n e d u s i n g Eqs. 5.28a, 5.28b, and 5.28c, a r e l a r g e l y i n s e n s i t i v e to k f . The v a l u e o f £'b o b t a i n e d i s thus i n v e r s e l y p r o p o r t i o n a l to the k<?" used. As i t i s c l e a r t h a t , f o r the lower modes, the v a l u e o f k j 1 o b t a i n e d i s a lower l i m i t , the 64 o b t a i n e d s h o u l d be an upper l i m i t . The l a r g e s t s o u r c e o f u n c e r t a i n t y i n e s t i m a t i n g £'b l i e s i n the measurement o f c* and the f a c t t h a t i t may not be c o n s t a n t over the whole c r y s t a l . c i s t y p i c a l l y about 5x10 cm and can be measured to about £10 3 cm. T h i s r e s u l t s i n an e r r o r i n £'b o f about 10%. F i g . 37 shows v a l u e s o f £'b f o r s e v e r a l c r y s t a l s p l o t t e d a g a i n s t f|( where f,( i s the square r o o t o f the mode p l o t • Crystal 4 O Crystal 9 7000 A Crystal 12 O Crystal 17 • Crystal 16 6000' • Crystal 19 5000 0 O o o 0 4000 o O O A O A • • mm 3000 • • • • • B 2000 \ p « — . , 1 15 20 25 30 35 40 frequency (GHz) Figure 3 7 . £b p l o t t e d against ffl f o r TTF-TCNQ c r y s t a l s studied s e v e r a l 142 i n t e r c e p t f o r each mode. C r y s t a l s 18 and 19 are r e p r e s e n t e d by s o l i d symbols. The o t h e r c r y s t a l s , because o f the s y s t e m a t i c e r r o r i n the s l o p e , a r e expected to g i v e upper l i m i t s t o €.[ . They do, i n f a c t , g i v e h i g h e r v a l u e s . On the b a s i s o f the r e s u l t s f o r c r y s t a l s 18 and 19, i t would appear t h a t t h e r e i s l i t t l e f r e q u e n c y dependence i n €'b from 15 GHz to 40GHz. The lower f r e q u e n c y v a l u e s a r e a g a i n , however, l e s s a c c u r a t e and sh o u l d be taken as upper l i m i t s so t h a t the l a c k o f f r e q u e n c y dependence may be a r e s u l t o f the s y s t e m a t i c e r r o r i n the s l o p e as d i s c u s s e d above. S e t t i n g t h i s a s i d e , we o b t a i n £t = ( 3 . 0 - 0 . 4 ) x l O 3 . I t s h o u l d be noted t h a t t h e r e a re some e r r a t i c r e s u l t s . The B mode f o r c r y s t a l 19, f o r example, g i v e s an anomalously h i g h v a l u e f o r £'b . I t i s p o s s i b l e t h a t such a n o m a l i e s a r i s e because o f the f a c t t h a t some modes may be more s e n s i t i v e than o t h e r s t o ma c r o s c o p i c i m p e r f e c t i o n s a t a p a r t i c u l a r p o i n t on the c r y s t a l . I t i s i n t e r e s t i n g to compare the f r e q u e n c y dependence o f the observed €'b w i t h t h a t p r e d i c t e d f o r a pinned charge d e n s i t y wave as g i v e n by Eq. 1.10. The r e a l p a r t o f € {u>) i s then g i v e n by (6.1) £ (us) = & + (kt.*-coy* + rxco T h i s model p r e d i c t s a change i n £'b o f about 260 i n g o i n g from 20GHz to 40GHz i f one uses the parameters g i v e n by E l d r i d g e and 143 B a t e s (1979) w i t h ( f ' ( 0 ) a d j u s t e d to g i v e £f'(20 GHz)=3000. While t h i s i s not i n c o n s i s t e n t w i t h the observed weak f r e q u e n c y dependence, the range o f f r e q u e n c i e s covered i s too s m a l l to a t t a c h much s i g n i f i c a n c e to the compa r i s o n . B e f o r e l e a v i n g the mode p l o t s , i t i s i n t e r e s t i n g to c o n s i d e r two o t h e r p o i n t s b r i e f l y . As has been mentioned, c r y s t a l 18 d i s p l a y e d a mode, l a b e l l e d Bx, w i t h an anomalously s m a l l s l o p e . T h i s has t e n t a t i v e l y been i d e n t i f i e d as a mode, analogous to the B mode, i n which the r e l a t i o n s h i p s o f the f i e l d s to the X and Y axes are r e v e r s e d . T h i s i d e n t i f i c a t i o n i s sup p o r t e d by the o b s e r v a t i o n t h a t the maximum c o u p l i n g a n g l e f o r By and D d i f f e r by n/2. Using t h i s i d e n t i f i c a t i o n , i t i s found t h a t the i n t e r c e p t c o r r e s p o n d s to €^=4177. T h i s i n d i c a t e s , a t l e a s t , t h a t the appearance o f such a mode i n t h i s f r e q u e n c y range may not be unr e a s o n a b l e f o r a c r y s t a l o f t h i s s i z e . There are two p o s s i b l e s o u r c e s f o r the v e r y s m a l l s l o p e . The f a c t t h a t t h e r e i s a component o f TT o r b i t a l o v e r l a p i n the c* d i r e c t i o n and not i n the a d i r e c t i o n s u g g e s t s t h a t £'c c o u l d be c o n s i d e r a b l y l a r g e r than e'^ . I t i s a l s o p o s s i b l e t h a t such a mode, w i t h the e l e c t r i c f i e l d f orming a l o o p p e r p e n d i c u l a r to the Y d i r e c t i o n , c o u l d have v e r y d i f f e r e n t end e f f e c t s from those encountered f o r the B mode. The l a c k o f u n d e r s t a n d i n g of end e f f e c t s f o r t h i s mode l i m i t s the i n f o r m a t i o n t h a t can be e x t r a c t e d from i t a t t h i s t i m e . The o t h e r i n t e r e s t i n g r e s u l t i n v o l v e s modes w i t h 1=2. I t 144 can be seen, i n the mode p l o t f o r c r y s t a l s 18 and 19, t h a t some o f t h e s e modes appear as d o u b l e t s . I t was found t h a t the s e p a r a t i o n o f the two resonances i s not c o n s t a n t as the c r y s t a l i s s h o r t e n e d . As the c r y s t a l i s f u r t h e r s h o r t e n e d , however, one seems to l o s e a c l e a r i n d i c a t i o n o f a doub l e resonance. The d o u b l i n g phenomenon was s t u d i e d i n some d e t a i l f o r the E * mode i n c r y s t a l 19. A sequence showing how the two resonances c o u p l e as a f u n c t i o n o f the d i s t a n c e o f the c r y s t a l c e n t r e from the s h o r t appears i n F i g . 38. I t can be seen t h a t t h e r e a r e p o s i t i o n s f o r which both resonances can be c o u p l e d . Maximum c o u p l i n g f o r the lower resonance i s a t about 9.4 mm from the s h o r t w h i l e the h i g h e r one c o u p l e s a t about 5.4 mm from the s h o r t . The r o t a t i o n a l dependence i s t h e same f o r both r e s o n a n c e s . The o r i g i n o f the doub l e resonances remains somewhat o f a my s t e r y . There i s no d i f f i c u l t y i n a c c e p t i n g t h a t a s t r o n g l y c o u p l e d resonance can have a f r e q u e n c y which depends on the p o s i t i o n i n the g u i d e . I t i s more d i f f i c u l t t o understand how a resonance c o u l d c o u p l e a t two d i f f e r e n t f r e q u e n c i e s f o r a s i n g l e p o s i t i o n o f the c r y s t a l . A p o s s i b l e c l u e i s i l l u s t r a t e d i n F i g . 39 where the c r y s t a l , a t the two maximum c o u p l i n g p o s i t i o n s , i s superimposed on a diagram r e p r e s e n t i n g the s t a n d i n g wave out to 1 gu i d e wave l e n g t h from the s h o r t . I f the i d e n t i f i c a t i o n o f t h i s mode as E ^ i s c o r r e c t , we can t h i n k o f the E 2 and E y f i e l d s as forming two l o o p s as shown. Because the Figure 38. Coupling of the double d i e l e c t r i c resonance i n c r y s t a l 19 at s e v e r a l distances of the c r y s t a l centre from the s h o r t . 146 F i g u r e 3 9 . Diagram showing r e l a t i o n s h i p o f the wave g u i d e f i e l d s to the proposed d i e l e c t r i c resonance f i e l d s f o r d o u b l e resonance 1 4 7 g u i d e wave l e n g t h does not c o r r e s p o n d to the wave l e n g t h i n the c r y s t a l , we expect t h a t when one o f the l o o p s i s i n a p o s i t i o n where i t c o u p l e s s t r o n g l y to the f i e l d s i n the g u i d e , the o t h e r l o o p i s r o u g h l y o r t h o g o n a l to the wave g u i d e f i e l d s . One can see t h a t the two maximum c o u p l i n g p o s i t i o n s c o r r e s p o n d to the p o s i t i o n s f o r one or the o t h e r l o o p to be s t r o n g l y c o u p l e d a t Both resonances a r e c o u p l e d when the c e n t r e o f the c r y s t a l i s a t ^/2. The r a t i o o f the c o u p l i n g s , a t t h i s p o s i t i o n , i s s i m i l a r to the r a t i o o f the maximum c o u p l i n g s . In the weak c o u p l i n g l i m i t , i t seems c l e a r t h a t t h e r e s h o u l d be o n l y one re s o n a n t f r e q u e n c y which s h o u l d not be dependent on the p o s i t i o n o f the c r y s t a l . Even i f we take the system to be s t r o n g l y c o u p l e d , one s i m p l y e x p e c t s the c o u p l i n g to add v e c t o r i a l l y and a f f e c t , a t most, the phase o f the resonance r e l a t i v e to the g u i d e f i e l d s . At p r e s e n t , t h e n , a s i d e from some apparent s i g n i f i c a n c e o f the r e l a t i o n s h i p between the f i e l d s i n the c r y s t a l and those i n the g u i d e , the o r i g i n o f the double resonances remains a myst e r y . 148 6.2 The Imaginary P a r t o f the D i e l e c t r i c C o n s tant I n f o r m a t i o n about the i m a g i n a r y p a r t o f the d i e l e c t r i c c o n s t a n t , £" =<±V<f/u), i s a c c e s s i b l e through the Q o f the res o n a n c e s . By s u b s t i t u t i n g the i m a g i n a r y d i e l e c t r i c c o n s t a n t , £i = £t + i £ ' b , i n t o Eq.5.9, one can o b t a i n the r e a l and i m a g i n a r y p a r t s o f the squared f r e q u e n c y i f we n e g l e c t any l o s s due to (6.2) Re(fZ)=SL ( k/ + k y 2 V +k* (6.3) I m ( f A ) = - c _ / _ J z L - \ ( k * +k* ) But the complex f r e q u e n c y f o r a l o s s y c a v i t y i s g i v e n by, (6.4) f = f e - i f 0 / ( 2 Q 0 ) so t h a t (6.5) f*=f* -f a V(2Q c ) - 2 i f e / ( 2 Q 0 ) ^ f e* - i f , / Q e I d e n t i f y i n g the r e a l and i m a g i n a r y p a r t s and assuming £b>>£'b r we have (6.6) f * =f A +f A 149 (6.7) f 0VQ o=f„ a el/e' where (6.8) f„a =(k? +k* ) c V ( 4 n \ ) (6.9) f* = c * kl / ( 4 T T ^ ) . 2- 5 f„ i s j u s t the mode p l o t i n t e r c e p t and i s the squared f r e q u e n c y o f a p a r t i c u l a r resonance. Eq. 6.7 can be r e a r r a n g e d to g i v e (6.10) £'l = ^ f t / ( f * Q0 ) . Q Q i s the i n t r i n s i c Q o f the r e s o n a t o r . I t i s r e l a t e d to the observed Q T and the c o u p l i n g Q, Q c, through (6.11) 1/Q T = ( 1 / Q 0 ) + ( 1 / Q t ) The r e f l e c t i o n from a c a v i t y i s g i v e n by (Hidy e t a l . ,1972) (6.12) R (<*;) = - = where P p i s the r e f l e c t e d power and i s the i n c i d e n t power. 150 On r e s o n a n c e , one has (1/Q r )JR (w„) = | ( 1 / Q 0 ) - (1/Q C) I , so t h a t (6.13) 1/Q0 = (1/2Q^) ( l i / R l ^ T ) where the upper s i g n i s f o r u n d e r c o u p l i n g and the lower f o r o v e r c o u p l i n g . <5^  has been e x t r a c t e d , i n t h i s way, f o r s i x c r y s t a l s and i s p l o t t e d a g a i n s t f r e q u e n c y i n F i g . 40. E r r o r b a r s a r e shown f o r a number o f r e s o n a n c e s . The l a r g e s t s o u r c e o f e r r o r l i e s i n o b t a i n i n g the w i d t h , and thus Q, from p o o r l y shaped r e s o n a n c e s . A p o s s i b l e s y s t e m a t i c e r r o r a r i s e s i n Eq. 6.10 where the £'b d e r i v e d from each f , i , f o r a g i v e n c r y s t a l , can be d i f f e r e n t . For the c o n d u c t i v i t i e s shown, the v a l u e o f £'b used i s the one a p p r o p r i a t e to f„ f o r the resonance i n q u e s t i o n . The most n o t a b l e f e a t u r e o f F i g . 40 i s t h a t , f o r lower f r e q u e n c i e s , the c o n d u c t i v i t i e s approach a s i n g l e v a l u e whereas, f o r f r e q u e n c i e s a p p r o a c h i n g 40 GHz, the c o n d u c t i v i t i e s a r e - i - / spread from about 0.2(«--cm) to 1.4(x?-cm) . There does appear to be a lower l i m i t which i s r o u g h l y f r e q u e n c y independent. C r y s t a l 4 and 12, l y i n g near t h i s l i m i t , were from the same b a t c h . An immediate q u e s t i o n r a i s e d by these r e s u l t s has to do w i t h the e f f e c t o f the epoxy used to mount the c r y s t a l s . T h i s does n o t , i n f a c t , appear to be a v a l i d s o u r c e o f c o n c e r n . There i s no c o r r e l a t i o n between the observed l o s s e s and the p Crystal a Crystal • Crystal O Crystal v Crystal x Crystal 4 9 12 17 18 19 o o a a • *| 7*« v v V X V x x x * XX o X o OQ> ° X CD o o 6b "2TT frequency C3Hz) 3 D "3b F i g u r e 40. P l o t o f c o n d u c t i v i t y v e r s u s f requency f o r s e v e r a l TTF-TCNQ c r y s t a l s s t u d i e d by d i e l e c t r i c r e s o n a n c e . 152 q u a l i t y o f a p p l i c a t i o n o f the epoxy. T h i s was c o n f i r m e d w i t h measurements i n which a q u a r t z f i b r e was a t t a c h e d , w i t h a l a r g e e xcess o f epoxy, to the bottom o f an a l r e a d y mounted c r y s t a l . The d e c r e a s e i n the Q o f the resonance was found to be from 0 to 30%. W h i l e € b^ i s p r o p o r t i o n a l to 1/Q, t h i s would not account f o r the range i n observed even i f the h i g h Vb c r y s t a l s were those w i t h excess epoxy. The lower l i m i t o f the observed c o n d u c t i v i t i e s l i e s s l i g h t l y below the v a l u e s observed by B a r r y (1977) i n the e a r l i e r measurements. The temperature dependence o f the c o n d u c t i v i t i e s has a l s o been s t u d i e d . l n c j i s p l o t t e d a g a i n s t 1/T f o r c r y s t a l 19 i n F i g . 41. The d a t a i s taken from the B mode f o r a number o f l e n g t h s o f the c r y s t a l . I t can be seen t h a t , f o r 14K < T < 25K, t h e r e i s an a c t i v a t i o n temperature o f about 73K. The o t h e r c r y s t a l s s t u d i e d y i e l d t e m p e r a t u r e s o f 71K and 55K. Below 14K, t h e r e i s a r e g i o n i n which <7j does not appear to f o l l o w an e r dependence. The average s l o p e i n t h i s r e g i o n i s about -10K. A f e a t u r e which i s p r e s e n t i n most o f the temperature dependence da t a i s a s m a l l l o c a l maximum around 10K. For lower f r e q u e n c y r e s o n a n c e s , t h i s can move up t o 14K. These r e s u l t s a r e n o t a b l y d i f f e r e n t from those o b t a i n e d u s i n g d.c. methods. T i e d j e (1975) found the c o n d u c t i v i t y below 10K to be l e s s l e s s than 0. 005 (Jl-cm)~' w i t h an a c t i v a t i o n t e m perature o f about 210K. B a r r y a t t r i b u t e d the excess c o n d u c t i v i t y t o a f r e q u e n c y dependent hopping c o n d u c t i v i t y 153 o b c -1 -2 0 o o ox *** c • Crystal 19 o 18.4 GHz o 19.3 GHz A 202 GHz O 21.0 GHz x 21 .8 GHz O 4 • o o o 0.1 0.2 1/T (Kr1 F i g u r e 41 . P l o t o f ln(«") v e r s u s 1/T f o r c r y s t a l 19 at s e v e r a l f r e q u e n c i e s . 154 a s s o c i a t e d w i t h i m p u r i t y s t a t e s i n the band gap. Mott and Davis (1971) c o n s i d e r such models i n d e t a i l . There are a number o f p o s s i b l e c o n t r i b u t i o n s to the c o n d u c t i v i t y . Two t h e r m a l l y a c t i v a t e d p r o c e s s e s i n v o l v e t h e r m a l l y a c t i v a t e d hopping w i t h i n the i m p u r i t y band and a c t i v a t e d e x c i t a t i o n from the i m p u r i t y to the c o n d u c t i o n band. The t h e r m a l l y a c t i v a t e d hopping w i l l n o r m a l l y be to the n e a r e s t neighbour and the c o n d u c t i v i t y w i l l have the u s u a l a c t i v a t e d temperature dependence. At v e r y low t e m p e r a t u r e s , v a r i a b l e range hopping can occur g i v i n g I n c p r o p o r t i o n a l to (T)~* . In a d d i t i o n to these mechanisms, Mott and D a v i s d i s c u s s two mechanisms which can o p e r a t e a t f i n i t e f r e q u e n c y to g i v e a temperature independent c o n d u c t i v i t y . The f i r s t i n v o l v e s a.c. hopping c o n d u c t i o n and g i v e s <y (w)<* W(ln(yFk/u))f where V ^ g i v e s the s t r e n g t h o f the e l e c t r o n - p h o n o n i n t e r a c t i o n . I f the i m p u r i t y s t a t e s a r e l o c a l i z e d near the Fermi energy, o p t i c a l t r a n s i t i o n s w i l l be more i m p o r t a n t and t h e r e w i l l be a c o n t r i b u t i o n (?{u))<X <*/(ln(I e /hco) f where I 0 i s r e l a t e d to the depth and volume o f the p o t e n t i a l w e l l i n which the i m p u r i t y i s l o c a l i z e d . 155 With t h i s background, i t seems p o s s i b l e to a t t r i b u t e the observed f r e q u e n c y dependence to o p t i c a l t r a n s i t i o n s from the i m p u r i t y t o the c o n d u c t i o n band. The temperature dependence observed below 10K i s p r o b a b l y a l s o r e l a t e d to the i m p u r i t y band a l t h o u g h i t i s not c l e a r whether i t i s t h e r m a l l y a c t i v a t e d n e a r e s t neighbour hopping or th e r m a l e x c i t a t i o n s to the c o n d u c t i o n band. The a c t i v a t e d regime above 14K i s l i k e l y due to t r a n s i t i o n s a c r o s s the semi-conductor gap o f TTF-TCNQ i t s e l f . The observed d.c. c o n d u c t i v i t y ( T i e d j e , 1975) i s comparable to the microwave c o n d u c t i v i t y around 20K. The d i f f e r e n c e i n d.c. and microwave a c t i v a t i o n temperature i s not understood s i n c e one would e x p e c t the t h e r m a l l y a c t i v a t e d i m p u r i t y c o n d u c t i o n to be p r e s e n t i n the d.c. measurement as w e l l as i n the microwave measurement. The presence o f the s m a l l l o c a l maximum between 10K and 14K has not been accounted f o r . I t would seem n e c e s s a r y , on the b a s i s o f these r e s u l t s , t o c o n s i d e r the c o n t r i b u t i o n to f'(0) from the observed f r e q u e n c y dependent c o n d u c t i v i t i e s . T h i s was done by u s i n g the Kramers-K r o n i g r e l a t i o n s and t e r m i n a t i n g the i n t e g r a l at 40GHz. The c o n t r i b u t i o n to £'(0) from the observed c o n d u c t i v i t i e s i s then g i v e n by o 156 The c o n d u c t i v i t y was t aken to be q u a d r a t i c w i t h f r e q u e n c y . For c r y s t a l s 17 and 18, the c o n t r i b u t i o n to £'(0) was found to be about 250. For c r y s t a l 12, the c o n t r i b u t i o n was about 50. T h i s , o f c o u r s e , n e g l e c t s the c o n t r i b u t i o n from the c o n d u c t i v i t y above 40GHz. I t can be seen t h a t the c o n t r i b u t i o n t o £'(0) from the o s c i l l a t o r s t r e n g t h a s s o c i a t e d w i t h the i m p u r i t y c o n d u c t i o n i s not n e g l i g i b l e . As a s o u r c e o f the spread i n €'b as observed i n t h i s e x p e r i m e n t , however i t i s p r o b a b l y not as i m p o r t a n t as the end e f f e c t s d i s c u s s e d p r e v i o u s l y . 157 CHAPTER V I I Microwave A b s o r p t i o n S t u d i e s o f TTF TCNQ The i n f r a - r e d a b s o r p t i o n e x p e r i m e n t s on TTF-TCNQ by E l d r i d g e and Bates (1979) have been i n t e r p r e t t e d as e v i d e n c e f o r a s t r o n g mode i n the neighbourhood o f 102 GHz. I t has been suggested t h a t t h i s proposed mode i s a s s o c i a t e d w i t h o s c i l l a t i o n s o f a pinned charge d e n s i t y wave. The r e l a t i o n o f such a pinned mode to the low f r e q u e n c y d i e l e c t r i c p r o p e r t i e s o f TTF-TCNQ has been d i s c u s s e d i n c h a p t e r I . The r e g i o n o f the spectrum of i n t e r e s t h e r e , r o u g h l y 3 cm"' to 4 cm ', h i n d e r s d i r e c t o b s e r v a t i o n o f the pinned mode a b s o r p t i o n i n t h a t t h i s r e g i o n l i e s j u s t below t h a t n o r m a l l y a c c e s s i b l e t o i n f r a - r e d t e c h n i q u e s . I t i s , however, j u s t w i t h i n the upper l i m i t o f microwave s o u r c e s a v a i l a b l e i n t h i s l a b o r a t o r y . The d i e l e c t r i c resonance t e c h n i q u e , d i s c u s s e d i n p r e c e d i n g c h a p t e r s , i s not u s e f u l a t these f r e q u e n c i e s . The major problem i n i n t e r p r e t i n g resonances a t these f r e q u e n c i e s i s the i d e n t i f i c a t i o n and s e p a r a t i o n o f t h e modes. T h i s might be a l l e v i a t e d , somewhat, by the use o f v e r y f i n e c r y s t a l s . However, the i n t e r p r e t a t i o n o f any mode p l o t o b t a i n e d would s t i l l be hampered by the l a c k o f u n d e r s t a n d i n g o f p o t e n t i a l l y s u b s t a n t i a l end e f f e c t s f o r such s m a l l c r y s t a l s . Another sou r c e o f d i f f i c u l t y i n i n t e r p r e t a t i o n o f a mode p l o t i s t h a t £'b would 158 be changing q u i c k l y i n the r e g i o n o f the pinned mode w i t h the r e s u l t t h a t the e x t r a p o l a t i o n t e c h n i q u e used to e x t r a c t €b' from the mode p l o t s would break down. I t has been proposed (Hardy, 1979) t h a t i t s h o u l d be p o s s i b l e to use a microwave analogue to the i n f r a r e d b o l o m e t r i c t e c h n i q u e s o f E l d r i d g e and B a t e s . In t h i s t e c h n i q u e , the a b s o r p t i o n o f i n c i d e n t r a d i a t i o n r e s u l t s i n a temperature r i s e i n the c r y s t a l . T h i s r i s e i n temperature i s d e c t e c t e d by m o n i t o r i n g the d.c. c o n d u c t i v i t y o f the c r y s t a l . In t h i s c h a p t e r , we w i l l d i s c u s s the i n i t i a l r e s u l t s from an experiment of t h i s type which i s now i n p r o g r e s s . The a c t u a l a b s o r p t i o n measurements have been made i n s e v e r a l microwave bands. W h i l e the c o n s t r u c t i o n o f the c r y s t a l mounts and e x e c u t i o n o f the experiment i s s i m i l a r i n a l l o f the bands used, the microwave p r o p e r t i e s o f the mount and the d e t a i l s o f the measurement are more c r i t i c a l f o r the h i g h f r e q u e n c y e x p e r i m e n t s . We w i l l c o n c e n t r a t e on the 75 to 110 GHz measurement and j u s t m e n t i o n , b r i e f l y , the q u a l i t a t i v e r e s u l t s from the 26 to 40 GHz and 60 to 90 GHz e x p e r i m e n t s . 159 7.1 The Experiment 7.1a The Spectrometer These e x p e r i m e n t s u t i l i z e d the computer c o n t r o l l e d microwave s p e c t r o m e t e r d e s c r i b e d i n Chapter IV. The 26 to 40 GHz s p e c t r o m e t e r was run i n the slow sweep mode i n which a p r e -c a l c u l a t e d v o l t a g e ramp i s o u t p u t , v i a a D/A c o n v e r t e r , t o the e x t e r n a l sweep i n p u t o f a W e i n s c h e l sweep o s c i l l a t o r c o n t r o l l i n g a backward wave o s c i l l a t o r . Data was c o l l e c t e d s i m u l t a n e o u l y v i a an A/D c o n v e r t o r and s t o r e d f o r p r o c e s s i n g . For the 60 to 90 GHz and 75 to 110 GHz r u n s , the v o l t a g e ramp was s u p p l i e d to a S i n g e r sweep o s c i l l a t o r c o n t r o l l i n g a Micro-Now BWO power s u p p l y . The microwave power was modulated at between 50 and 100 Hz u s i n g the AM i n p u t on the 26-40 GHz s o u r c e and u s i n g a Hughes model 44714H modulator f o r the h i g h e r bands. A v a r i a b l e a t t e n u a t o r was used a f t e r the s o u r c e to c o n t r o l the power. The c r y s t a l and microwave t e r m i n a t i o n were l o c a t e d i n a t u b u l a r can which c o u l d be pumped i n d e p e n d e n t l y o f the l i q u i d h e l i u m space i n the dewar. A carbon r e s i s t o r thermometer and a wound h e a t e r were a t t a c h e d to the wave g u i d e to a l l o w f o r temperature r e g u l a t i o n . 160 7.1b The C r y s t a l Mount There were s e v e r a l c o n s i d e r a t i o n s i n the d e s i g n o f the c r y s t a l mount. F i r s t l y , i n o r d e r t h a t the microwave f i e l d s be p o l a r i z e d a l o n g the a x i s o f i n t e r e s t , the c r y s t a l had to be mounted w i t h the b a x i s a c r o s s the c e n t r e o f the twave g u i d e and p a r a l l e l to the e l e c t r i c f i e l d s . The absorbed power was d e t e c t e d by m o n i t o r i n g the d.c. c o n d u c t i v i t y o f the c r y s t a l . One end o f the c r y s t a l was s h o r t e d to the g u i d e . The o t h e r end was a t t a c h e d to a l e a d which was i n s u l a t e d from, and taken out t h r o u g h , the j o i n t between two wave g u i d e f l a n g e s . I t was n e c e s s a r y t o c a p a c i t i v e l y s h o r t t h i s l e a d to the wave g u i d e i n o r d e r to m i n i m i z e the e x t r a c t i o n o f R.F. power a l o n g i t . T h i s was done by u s i n g , as the d.c. l e a d , mylar sheathed copper f o i l sandwiched between the g u i d e f l a n g e s . For the h i g h e r f r e q u e n c i e s , i t was i m p o r t a n t t h a t the g u i d e f l a n g e s mate p r o p e r l y . To f a c i l i t a t e t h i s , the f a c e o f the g u i d e f l a n g e was r e c e s s e d to a c c e p t the f o i l . I t was a l s o n e c e s s a r y t h a t the c r y s t a l mount be f l e x i b l e i n o r d e r to a l l o w f o r d i f f e r e n t i a l t h e r m a l c o n t r a c t i o n o f the c r y s t a l and g u i d e . T h i s was most s e r i o u s f o r the h i g h f r e q u e n c y mounts f o r which the c r y s t a l s used were n o r m a l l y v e r y t h i n . For t h i s mount, the c o n n e c t i o n between the c r y s t a l and the f o i l or f l a n g e was t h r o u g h g o l d w i r e s s i l v e r p a i n t e d onto the c r y s t a l . The h i g h f r e q u e n c y mount i s s k e t c h e d i n f i g u r e 42. 161 GOLD WIRE WAVE GUIDE FLANGE CRYSTAL F i g u r e 42 . High f r e q u e n c y f o r microwave i n TTF-TCNQ. mount a b s o r p t i o n stud i e s 162 7.1c D e t e c t i o n The c r y s t a l b i a s c i r c u i t was s i m p l y a 10M metal f i l m r e s i s t o r and a 9 V o l t Eveready 1222 b a t t e r y i n s e r i e s w i t h the c r y s t a l . The 10M& r e s i s t o r was a t low temperature and connected to the c r y s t a l t h r ough the waveguide. The v o l t a g e a c r o s s the 10MQ. r e s i s t o r was fed to an I t h a c o Dynatrac 391A l o c k - i n a m p l i f i e r . The analogue o u t p u t from the l o c k - i n a m p l i f i e r was fed to the computer f o r s t o r a g e . I t was a l s o n e c e s s a r y to c o r r e c t f o r v a r i a t i o n s i n the i n c i d e n t power. For the 26 to 40 GHz and 60 t o 90 GHz bands, t h i s was done by a t t a c h i n g a power meter or a c r y s t a l d e t e c t o r below the c r y s t a l and r e c o r d i n g the power a c r o s s the band i n a s e p a r a t e room temperature e x p e r i m e n t . Using these t r a c e s to r a t i o the raw d a t a gave s p e c t r a which were p r o p o r t i o n a l to the a b s o r p t i o n , except f o r a weakly f r e q u e n c y dependent f a c t o r c o r r e c t i n g f o r the change i n wave g u i d e v e l o c i t y a c r o s s the band. T h i s procedure has two d i s a d v a n t a g e s ; a) the i n c i d e n t power w i l l c e r t a i n l y change w i t h t emperature o f the c r y o s t a t and b) the response o f the c r y s t a l d e t e c t o r i s not v e r y f l a t w i t h frequency. Of c o u r s e , an a b s o l u t e measurement a l s o a l s o r e q u i r e s an a b s o l u t e measurement o f i n c i d e n t power as w e l l as a c a l i b r a t i o n o f the absorbed power s i g n a l . The n e c e s s a r y s t e p s to o b t a i n the a b s o l u t e a b s o r p t i o n have been c a r r i e d out f o r one c r y s t a l i n the 75 to 110 GHz band. The 163 incident power was detected in s i t u using a f i v e couple Au-Fe/Chromel-p thermopile with one end anchored to the wave guide termination below the c r y s t a l and the other end anchored to the c r y s t a l mount. The signal from the thermopile was detected with a Keithly 148 nanovoltmeter. The output of the BWO was also monitored using a d i r e c t i o n a l coupler and c r y s t a l detector. The "SYNSWEEP" option of "SWCONTROL" was altered to allow simultaneous input on three channels so that output from the l o c k - i n , nanovoltmeter, and c r y s t a l detector could be recorded together. The phase locking feature of the option was not used. The three channel modification of the program allowed continuous adjustment of the BWO gri d voltages with no ef f e c t on the ratio operation. The thermopile output was calibrated using a 200J1 heater wound on the terminator at the thermopile anchor. The thermopile signal corresponding to a given d.c. power input could be plotted. The assumption was then made that this c a l i b r a t i o n would be appropriate to the R.F. power absorbed. Because of the modulation, the peak R.F. power was, in fact, twice the observed power le v e l indicated by t h i s c a l i b r a t i o n . The signal from the c r y s t a l also had to be c a l i b r a t e d . The change in voltage across the 10MJ2 re s i s t o r was related to the change in the c r y s t a l resistance, ARC , by V=-V s A R C Rg / (R B +RC )* 164 where RB=10MSL, V B i s the b a t t e r y v o l t a g e , and Rc i s the c r y s t a l r e s i s t a n c e . T h i s r e s i s t a n c e depended s t r o n g l y on the d.c. b i a s c u r r e n t and was t y p i c a l l y 10M-fr t o 100MJ2 a t a t e mperature o f 10K. The c a l i b r a t i o n o f A R C t o i n c i d e n t microwave power was o b t a i n e d by p a s s i n g a d.c. c u r r e n t through the c r y s t a l and p l o t t i n g the observed r e s i s t a n c e a g a i n s t the power d e p o s i t e d i n the c r y s t a l t h i s p r o c e d u r e then assumes t h a t the h e a t i n g e f f e c t o f the d.c. power and microwave a b s o r p t i o n i s e q u i v a l e n t . Using the c u r v e o b t a i n e d , i t was p o s s i b l e to f i r s t f i n d the d.c. " o p e r a t i n g p o i n t " due to the b i a s i n g b a t t e r y . The s l o p e then gave AR<./ P. T h i s f i n a l l y a l l o w e d one to c a l i b r a t e the s i g n a l from the l o c k -i n a m p l i f i e r i n terms o f power absorbed. The r a t i o e d spectrum c o u l d then be c a l i b r a t e d i n terms o f a b s o r p t i o n . We note here t h a t i n the presence o f i n t r i n s i c non-ohmic e f f e c t s , the e q u i v a l e n c e o f d.c, and r . f h e a t i n g may not be v a l i d . T h i s p o i n t i s c u r r e n t l y under i n v e s t i g a t i o n . 165 7.2 R e s u l t s The use o f TTF-TCNQ c r y s t a l s as b o l o m e t e r s was f i r s t t e s t e d i n the 26 to 40 GHz band. By f i r s t c a l i b r a t i n g a v a r i a b l e a t t e n u a t o r and then u s i n g the a t t e n u a t o r to measure the s i g n a l as a f u n c t i o n o f i n c i d e n t power, i t was found t h a t the s i g n a l was l i n e a r w i t h power up to a t l e a s t 2mW i n c i d e n t power at 30 GHz. Two c r y s t a l s were s t u d i e d i n the 26 to 40 GHz band. Both s p e c t r a showed some s h a r p f e a t u r e s which were a t t r i b u t e d to d i e l e c t r i c r e s o n a n c e s . For one c r y s t a l s t u d i e d , the background a b s o r p t i o n was found to be r o u g h l y f l a t w h i l e , f o r the o t h e r , i t was found to i n c r e a s e by a f a c t o r o f 3 a c r o s s the band. T h i s b e h a v i o u r i s not i n c o n s i s t e n t w i t h the v a r i a t i o n i n the fre q u e n c y dependent c o n d u c t i v i t y a c r o s s t h i s band as observed by d i e l e c t r i c resonance. One c r y s t a l was s t u d i e d i n the 60 to 90 GHz band. The a b s o r p t i o n appeared to be d e c r e a s i n g s l i g h t l y from 60 t o 70 GHz a l t h o u g h the apparent presence o f s t r o n g s t a n d i n g waves i n both the r a t i o e d and the background s p e c t r a , i n t h i s r e g i o n , c a s t s some doubt on the s u c c e s s o f the r a t i o i n g i n t h i s i n s t a n c e . Above 70 GHz, the a b s o r p t i o n was f l a t w i t h the e x c e p t i o n o f one f e a t u r e near 90 GHz. T h i s i s b e l i e v e d to be a d i e l e c t r i c r esonance. The same c r y s t a l was a l s o s t u d i e d i n the 75 to 110 GHz 166 band. The n o n - l i n e a r i t y o f the r e s i s t a n c e v e r s u s d.c. power c a l i b r a t i o n has been a sou r c e o f some con c e r n i n t h a t the absorbed power c a l i b r a t i o n becomes s t r o n g l y dependent on the p o s i t i o n o f the b i a s i n g p o i n t . F i g u r e 43 shows an a b s o r p t i o n spectrum o b t a i n e d u s i n g the most r e l i a b l e R c v e r s u s power c u r v e o b t a i n e d . I t can be seen t h a t , i n a d d i t i o n to s e v e r a l f e a t u r e s between 85 and 100 GHz, t h e r e i s a prominent f e a t u r e a t about 107.5 GHz. I f we m e n t a l l y smooth the sh a r p s p i k e s which a r e l i k e l y due to h i g h e r modes a s s o c i a t e d w i t h the use o f 60 to 90 GHz wave g u i d e w e l l above 100 GHz, we f i n d a peak r a t i o (absorbed power/ i n c i d e n t power) of about 150X10 - 5, where we have i n c l u d e d the f a c t o r o f 2 a s s o c i a t e d w i t h the f a c t t h a t we have o n l y measured the average i n c i d e n t power. The r a t i o between 75 GHz and 85 GHz i s between 3X10 - 5 and 5Xlo'f . We can c o n v e r t these i n t o rough measures o f the c o n d u c t i v i t y i f we make some assumptions about the e l e c t r i c f i e l d s i n the c r y s t a l . The r e l a t i o n o f power down a wave g u i d e to the peak e l e c t r i c f i e l d , E„ , can be found i n e l e c t r o m a g n e t i s m t e x t s such as L o r r a i n and Corson (1970). They g i v e , i n MKS u n i t s , power=(E o x ab/4c/<.) ( l - ( y . / 2 b f )* where a i s the h e i g h t o f the wave g u i d e , b i s the w i d t h o f the wave g u i d e , c i s the speed o f l i g h t , and K i s the f r e e space 1501 100+ 50+ 85 m 95" 10^ 105" frequency (GHz) F i g u r e 4 3 . Spectrum showing the r a t i o o f absorbed to i n c i d e n t power f o r the TTF-TCNQ c r y s t a l s t u d i e d in the 75 to 110 GHz band . 168 wave l e n g t h . I f we assume t h a t the e f f e c t o f d e p o l a r i z i n g f i e l d s a r e removed by the a.c. s h o r t i n g of the c r y s t a l to the g u i d e , then the power absorbed by the c r y s t a l i s P9bs =a b w a t t s where a', b', and AZ a r e d i m e n s i o n s o f the c r y s t a l and C i s the c o n d u c t i v i t y i n (A.-M)' . T h i s i s c o r r e c t assuming a c r y s t a l l e n g t h e q u a l to the wave g u i d e h e i g h t . The d i m e n s i o n s o f the c r y s t a l used were 1.16 mm by 0.065 mm by 0.038 mm. The g u i d e d i m e n s i o n s were 1.5 mm by 3 mm. For these c o n d i t i o n s , we f i n d c T ( f ) (Jl-M)"' =2.1X10 3 ( l - ( 4 9 . 9 7 / f ) * ) 4 ( P ^ / P ^ . ) where f i s the f r e q u e n c y i n GHz.. Using t h i s f o r m u l a , we f i n d t h a t , between 75 and 85 GHz, trff) i s about 6.6X10^ (Jl-cm)"' and t h a t , near 107.5 GHz, CY i s about 2.8X10"* (J2-cm)"' . The r e s i s t a n c e a t the d.c. o p e r a t i n g p o i n t was 26 Mil which i m p l i e s a d.c. c o n d u c t i v i t y o f about 2X10 (jr-cm) . The 75 GHz c o n d u c t i v i t y appears to be o f t h i s o r d e r o f magnitude whereas one would expect i t to be a t l e a s t as l a r g e as the c o n d u c t i v i t y a t lower f r e q u e n c i e s . The t y p i c a l c o n d u c t i v i t y a t 40 GHz and 10K was between 0.3(J2-cm)-' and 1 (Jl-cmf1 . I f we accept t h a t the 75 GHz c o n d u c t i v i t y a t 10K s h o u l d be a t l e a s t 0.5 (-Jl-cm)"' , then the peak c o n d u c t i v i t y i s r a i s e d to about 20 (-0_-cm)"' . I t i s thus 169 i m p o r t a n t t o de t e r m i n e whether the apparent low c o n d u c t i v i t y a t 75 GHz i s r e a l or j u s t an a r t i f a c t o f the e x p e r i m e n t a l a n a l y s i s . I t i s o f i n t e r e s t t o c a l c u l a t e the expected peak c o n d u c t i v i t y r e q u i r e d f o r a mode o f w i d t h 5 GHz c e n t r e d a t 107 GHz to g i v e r i s e to the observed low f r e q u e n c y d i e l e c t r i c c o n s t a n t o f about 3000. T h i s c o n d u c t i v i t y i s found to be about 2400 (fl-cm)'1 . For a mode o f w i d t h 3 GHz, t h i s goes t o about 400 0 (fl- cm)"' . The observed peak c o n d u c t i v i t y i s c l e a r l y unable to account f o r the low f r e q u e n c y d i e l e c t r i c c o n s t a n t . The e f f e c t o f the d e p o l a r i z i n g f i e l d s , i f the c r y s t a l were not s h o r t e d to the w a l l s , would be to de c r e a s e the a b s o r p t i o n . For the p r e s e n t c r y s t a l , w i t h a d e p o l a r i z i n g f a c t o r o f about 5X 10 , the observed a b s o r p t i o n would be d e c r e a s e d by a f a c t o r o f about 1.6X10^ . I f t h i s were the c a s e , the observed f e a t u r e might be a b l e to account f o r the m i s s i n g o s c i l l a t o r s t r e n g t h . A g a i n , however, we p o i n t out t h a t d e p o l a r i z i n g e f f e c t s s h o u l d not be s i g n i f i c a n t i n the p r e s e n t experiment so t h a t the f i e l d s i n the c r y s t a l s h o u l d approximate those i n the g u i d e . Of c o u r s e f o r h i g h enough cr one g e t s a s t r o n g r e f l e c t i o n from the c r y s t a l , i n which case the e l e c t r i c f i e l d i n the g u i d e i s l e s s than t h a t o f the i n c i d e n t wave. I t i s not y e t c l e a r whether we are i n t h i s regime or n o t . At p r e s e n t , t h e n , the observed f e a t u r e a t 107.5 GHz i s not found t o have s u f f i c i e n t o s c i l l a t o r s t r e n g t h t o account f o r the high values of £^  observed at low frequency. 171 7.3 F u t u r e D i r e c t i o n s A b s o r p t i o n e x p e r i m e n t s o f t h i s type s t i l l seem to o f f e r some hope f o r d i r e c t o b s e r v a t i o n o f the pinned charge d e n s i t y wave i f i t s p i n n i n g f r e q u e n c y l i e s below 120 GHz. Experiments a r e p r e s e n t l y underway i n which the c a l i b r a t i o n o f the a b s o r p t i o n s i g n a l and the microwave p r o p e r t i e s o f the c r y s t a l mount w i l l be r e f i n e d . One b e n e f i t o f u s i n g a v a r i e t y o f c r y s t a l s and mounts w i l l be to a l l o w m o u n t - s p e c i f i c and d i e l e c t r i c resonances to be d i s t i n g u i s h e d . The p r e s e n t e x p e r i m e n t , t h e n , must be taken as i n c o n c l u s i v e . There appears to be some q u e s t i o n as t o the v a l i d i t y o f the a b s o r p t i o n to c o n d u c t i v i t y c o n v e r s i o n i n l i g h t o f the low c o n d u c t i v i t y o b t a i n e d a t 75 GHz. I t i s a l s o n e c e s s a r y t o i d e n t i f y t h e components o f the spectrum which a r e s p e c i f i c to c e r t a i n c r y s t a l s and mounts so t h a t the f e a t u r e s due to m i c r o s c o p i c p r o p e r t i e s o f TTF-TCNQ can be i s o l a t e d and st u d i e d . 172 CHAPTER V I I I Summary In the f i r s t p a r t o f t h i s work, the e l e c t r i c a l p r o p e r t i e s o f MEM(TCNQ)A were s t u d i e d i n the neighbourhood o f the dimer to monomer t r a n s i t i o n a t 60 ° C. The microwave and d.c. c o n d u c t i v i t i e s were found to be i n g e n e r a l agreement. J u s t below t h e t r a n s i t i o n , t h e microwave c o n d u c t i v i t y was found to be between 0.014 and 0.017 (Jl-cm)'' . Above the t r a n s i t i o n , ^ rose to between 14 and 32 (Jl-cm)'' . The a c t i v a t e d b e h a v i o u r o f the c o n d u c t i v i t y i n the d i m e r i z e d phase, as observed by d.c. methods, was c o n f i r m e d down to room t e m p e r a t u r e . Some d i s c u s s i o n was o f f e r e d r e g a r d i n g the d i f f i c u l t y i n h e r e n t i n i n t e r p r e t i n g t h e s e r e s u l t s i n terms o f the Hubbard models f o r the TCNQ c h a i n s . C l a r i f i c a t i o n o f the r o l e o f the o n - s i t e r e p u l s i o n i n d e t e r m i n i n g the p r o p e r t i e s o f t h i s m a t e r i a l w i l l have to come from o t h e r measurements such as t h e magnetic s u s c e p t i b i l i t y and thermopower. In the second p a r t o f t h i s work, the d i e l e c t r i c r e s o n a t o r s t u d i e s on TTF-TCNQ as done by B a r r y (1977) were extended. The main r e s u l t s were the c l a r i f i c a t i o n o f the b e h a v i o u r o f the c o a x i a l - l i k e mode and some u n d e r s t a n d i n g o f the s y s t e m a t i c e r r o r s i n t r o d u c e d by end e f f e c t s i n t o the e x t r a c t i o n o f d i e l e c t r i c c o n s t a n t s from the mode p l o t s . The low f r e q u e n c y v a l u e o f £'b was e s t i m a t e d to be (3. 0*0. 4)X10 3 . I t was suggested 173 t h a t the v a l u e o f i s a t l e a s t 9 and t h e r e were i n d i c a t i o n s t h a t €e* c o u l d be c o n s i d e r a b l y l a r g e r . The spread i n c o n d u c t i v i t i e s as o b t a i n e d i n the d i e l e c t r i c resonance measurements was f e l t to im p l y t h a t below 40 GHz, the fr e q u e n c y dependence o f the c o n d u c t i v i t y was s p e c i f i c t o p a r t i c u l a r c r y s t a l s and t h u s , l i k e l y , due to i m p u r i t y e f f e c t s . T h i s was to some e x t e n t c o n f i r m e d by the temperature dependence o f the c o n d u c t i v i t y . The f i n a l p a r t o f the work i n v o l v e d some p r e l i m i n a r y a t t e m p t s a t d i r e c t o b s e r v a t i o n o f the pinned charge d e n s i t y wave u s i n g microwave a b s o r p t i o n i n a TTF-TCNQ b o l o m e t e r . There i s s t i l l work needed, both i n r e f i n i n g the t e c h n i q u e and o b t a i n i n g s u f f i c i e n t d a t a to d i s t i n g u i s h between the e f f e c t s o f microwave resonances i n the ap p a r a t u s and i n t r i n s i c p r o p e r t i e s o f the c r y s t a l s . The r e s u l t s o f these e x p e r i m e n t s a r e await e d w i t h some a n t i c i p a t i o n . 174 APPENDIX A T i g h t B i n d i n g C a l c u l a t i o n s f o r a L i n e a r Chain w i t h a B a s i s We f i r s t c o n s i d e r a l i n e a r c h a i n o f atoms w i t h a l t e r n a t i n g s e p a r a t i o n s o f b and ( c - b ) . The u n i t c e l l d i m e n s i o n i s then c. The w a v e f u n c t i o n f o r the atom a t s i t e R i s a ( r - R ). We can c o n s t r u c t a wave f u n c t i o n , s a t i s f y i n g the B l o c h c o n d i t i o n s , o f the form, A . l \(r) = ( N f ^ e ' 1 1 1 " a f r - F ^ - z J where °^. = 1 or 2, z^=0, z A = b, and R n =nc. We thus have, (A.2a) ^ ( r ) = N* £ e a ( r - R n ) (A.2b) Vx(r) = N ~ k £ e < k ^ a ( r - R n -b) ** The o f f d i a g o n a l elements o f the H a m i l t o n i a n are g i v e n by (A.3a) (Y, (DTV ^ ( r ) ) = N'Xe'^ y ( r - R n )^a(r-R„'-b) dr (A. 3b) (^(r)7i%(r)) = N'XV^" ]a*(r-Rn -b)H a (r-R„' ) dr D e f i n i n g 175 (A.4) • -t1 = a ( r - R J 7 i a(r-Rn -b) dr and (A.5) - t a = j a * ( r - R n )?/ a(r-R„_, -b) dr . i kc we g e t ( A.6) ( tf, ( r ) ^ / Y * ( r ) ) = t t +tze and (A. 7) ( YA U ) /V V., ( r ) ) = t 1 +t J Le' i f t . D e f i n i n g E 0 , the d i a g o n a l element, we o b t a i n the e l e c t r o n i c energy, (A. 8) Ck = E0 1 y ' t j " +t£ +2t 1 t ^ c o s (kc) At k=0,^-£,= l ( t 1 + t i ) and a t k= rr/c , £K-£ = 1 11 1 - I . T h i s r e l a t i o n thus d e s c r i b e s two bands each o f w i d t h 2 t ^ s e p a r a t e d by a gap o f 2 | t 1 ~ t 3 k | . 176 APPENDIX B A n i s o t r o p i c D i e l e c t r i c Wave Guides w i t h and w i t h o u t Outer Conductors We w i l l b e g i n by r e v i e w i n g some o f the r e s u l t s g i v e n by B a r r y (1977), and then extend these t o the case where Ez c o n t a i n s an e ' * a z i m u t h a l dependence. We w i l l d e a l w i t h an i n f i n i t e d i e l e c t r i c rod o f r a d i u s R, w i t h i n a c i r c u l a r m e t a l l i c tube o f r a d i u s R^. The a x i s o f symmetry i s the Z a x i s . The d i e l e c t r i c rod has £y = £ x*£ z . Taking M. to be 1, the f i e l d s i n s i d e o f the g u i d e a re shown, by B a r r y , t o s a t i s f y B . l E,, = __i/ k zP|U + u) 1 ZEA k*( 2>r c r 2>4>/ k l \ * d<P c 2 r / B.3 H F = _ J L / £ H Z -<*ML± dEz\ kl [ ~d* c r TQ) B.4 =__i / kz + ^ « 9 E ^ Ki.* V r 2 4 c ^ r / where 177 B.5 k* =(ifVc * )e j L -kJ O u t s i d e o f the d i e l e c t r i c , the f i e l d s a r e the same but w i t h £ L r e p l a c e d by 1 and r e p l a c e d by B.6 k a = (w*/c * ) - k * The wave e q u a t i o n s f o r E^ and H z, i n s i d e the d i e l e c t r i c , can be shown to be B.7 c^F + 1 £F + d r * r 5 r and B. 8 djs + _1 9G + r * r 3 r where B.9 kj" = ( ^ / e j * * and B. 10 E2(r,cf>)°c F(r)e'' n 178 B - l l Hz(r,<t>)°< G ( r ) e " " ^ O u t s i d e o f the d i e l e c t r i c , and kj_ a r e r e p l a c e d by k A . The s o l u t i o n s t o Eqs. B.7 and B.8 are c y l i n d r i c a l B e s s e l f u n c t i o n s . In g e n e r a l , i n s i d e the d i e l e c t r i c , one uses B. 12 F ( r ) = A1 J n (k 1 r) B.13 G (r) = A ZJ„ (kj.r) and o u t s i d e o f the d i e l e c t r i c , one uses B - 1 4 F ( r ) = B . ^ (k^r) + c 1 H^jCk^r) B.15 G ( r ) = B A H ^ ( k A r ) + C ^ H ^ k ^ r ) One can then c o n s i d e r v a r i o u s s i t u a t i o n s . For n=0 and R a . = 0 t ?' o n e r e q u i r e s t h a t kj^ be l e s s than or equal to 0 i n o r d e r f o r t he wave to be g u i d e d . I t i s a l s o n e c e s s a r y t o have and equa l to 0 f o r the f i e l d s a t the c e n t r e to be f i n i t e . For i m a g i n a r y k A , i t i s c o n v e n i e n t to r e p l a c e H ^ ( k A r ) by - ( 2 i / r r ) K c (*r) *" a where t=-k^ . The s o l u t i o n s f o r n=0 can be s e p a r a t e d i n t o 1 7 9 T.E. and T.M. modes. Application of the boundary conditions allows one to construct equations for the c o e f f i c i e n t s . The condition for a n o n - t r i v i a l solution for these c o e f f i c i e n t s i s that the determinant of the c o e f f i c i e n t matrix be 0. This i s found to imply 1 B.17 ^K0(2TR1 Jtyk-jRj) + k 1 Jc(k1 R1 )K1(a'R1 )=0 When R^ i s made f i n i t e , one must include the terms involving and . k* can now be both posi t i v e and negative. One can s t i l l set Hz=0. For k* >0, E z outside of the d i e l e c t r i c i s given by B.18 E 2 ( r ) = ( B ^ ^ r ) + C 1 Y 0 ( k a r ) ) e ' A Z and the c h a r a c t e r i s t i c equation becomes {J c (k1 R7 )/k a}{Y 7 (k 2R 7 ) j a ( k A R j ) - J 7 (kxR1 )Ya (k^RA) } B. 19 - { ^ J 1 (klR1)/k1 }{Y0 (k^R^)J0 (k^R x)-J c,(k ; iR 1 )Y e (kxRA) } = 0 For k^<0, the f i e l d s outside of the d i e l e c t r i c are given in terms of I„(^r) and Kn (!Tr) . The c h a r a c t e r i s t i c equation i s then 180 {J0 R 1 ) /eT}{I 1 ( f R1 ) K C (<TR A)-K 1 ( / R ^ I , (/R^) } B. 20 -{ (£2/^1) J1 (k-jRj) } { I 0 ( ^ ) K 0 (2TR.,)-Io )K, (JTRA) } = 0 The s o l u t i o n s o f t h e s e c h a r a c t e r i s t i c e q u a t i o n s have been d i s c u s s e d i n Chapter V. One can then go on to d i s c u s s the case f o r n = l . H z cannot be s e t to 0 f o r t h i s case so the s o l u t i o n s become c o n s i d e r a b l y more c o m p l i c a t e d . For the d i e l e c t r i c wave g u i d e c a s e , i t i s a g a i n n e c e s s a r y t h a t kA be i m a g i n a r y . The e x t e r n a l f i e l d s a r e then c o n v e n i e n t l y d i s c u s s e d i n terms o f K 0 ( j r ) and ( / r ) . The r e s u l t i n g c h a r a c t e r i s t i c e q u a t i o n i s Eq. B.21 and appears on the f o l l o w i n g page. When R A i s made f i n i t e , k A can a g a i n be both r e a l and i m a g i n a r y . The boundary e q u a t i o n s a g a i n r e s u l t i n e q u a t i o n s f o r the c o e f f i c i e n t s which have n o n - t r i v i a l s o l u t i o n s o n l y i f the d e t e r m i n a n t o f the c o e f f i c i e n t m a t r i x i s 0. For the k^ >0 c a s e , the r e s u l t i n g c h a r a c t e r i s t i c e q u a t i o n , Eq. B.23, appears below. For the k^<0 c a s e , the c o n v e n i e n t r e p l a c e m e n t s f o r the B e s s e l f u n c t i o n s a r e B.22 K 1 (/r) f o r (k^r) K0 (yr) f o r J D ( k A r ) EQUATION B.21 182 EQUATION B.23 {k2Y0(k2R2)- j -X(k2R|}{k2JUCk2R2)--sM!^z2j w[{k2 J J C W - ^ S ) | Y L ( k ^ p - j k ^ k ^ j - ^ M ^ k A ) ] ! 183 I1 (arr) for Y 1 (k Ar) B.22 cont. I a U r) for Y e ( k A r ) . Terms of the form ( k A Z 0 ( k ^ r ) - ( Z ^ ( k a r ) / r ) ) are derivatives of Z 1 ( k A r ) and k a i s replaced by y while the the Bessel functions are replaced according to B.22. The solutions for this equation are also discussed in Chapter V. APPENDIX C D i e l e c t r i c Resonance Mode P l o t s f o r S e v e r a l TTF-TCNQ C r y s t a l s Figure 44. D i e l e c t r i c resonance mode plot for TTF-TCNQ c r y s t a l 4 F i g u r e 45. D i e l e c t r i c resonance mode p l o t f o r TTF-TCNQ c r y s t a l 9. crystal 12 0 2 4 6 8 10 12 14 1 / b2 (cm"2) Figure 46. D i e l e c t r i c resonance mode p l o t f o r TTF-TCNQ c r y s t a l 12 c r y s t a l 17 0 2 4 6 8 10 12 14 1/b 2 (cm 2) Figure 47. D i e l e c t r i c resonance mode plot for TTF-TCNQ c r y s t a l 17. 189 Figure 48. D i e l e c t r i c resonance mode plot for TTF-TCNQ c r y s t a l 18. crystal 19 0 1 2 3 1 / b2 (cm 2) igure 49. D i e l e c t r i c resonance mode plot for TTF-TCNQ c r y s t a l 19. 191 References A s h c r o f t , N.W., and Mermin, N.D., S o l i d State P h y s i c s . H o l t , Rhinehart, and Winston, New York(1976) B a l k a n s k i , M> i n O p t i c a l P r o p e r t i e s o f S o l i d s . F. 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