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Acoustic wave propagation in TTF-TCNQ 1977

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A C O U S T I C WAVE PROPAGAT ION IN T T F - T C N Q by J . THOMAS T I E D J E B . A . S c , U n i v e r s i t y o f T o r o n t o , 1973 M. S c . , 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 , 1975 A T H E S I S SUBM ITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PH I LOSOPHY THE FACULTY OF GRADUATE STUD I E S i n t h e D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMB IA J u l y , 1977 ( c ) J . Thomas T i e d j e In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 7/77 ABSTRACT D e t a i l e d measurements have been made o f the t e m p e r a t u r e dependence o f the v e l o c i t y o f t h r e e d i f f e r e n t modes o f sound p r o p a g a t i o n i n TTF- TCNQ c r y s t a l s , i n the range 7~300K. V a l u e s f o r the a and b a x i s Young's moduli and the s h e a r modulus C55 a r e i n f e r r e d from the sound v e l o c i t i e s . TTF-TCNQ. i s found t o be s t i f f e r p e r p e n d i c u l a r t o the c o n d u c t i n g d i r e c t i o n than p a r a l l e l t o i t . The e l a s t i c a n i s o t r o p y i s t y p i c a l o f c r y s t a l l i n e s o l i d s even though the a n i s o t r o p y o f the e l e c t r i c a l c o n d u c t i v i t y i s u n u s u a l l y large'. A s m a l l (1.5%) i n c r e a s e i n t h e v e l o c i t y o f e x t e n s i c n a l waves below the meta1 - i n s u l a t o r t r a n s i t i o n i s i n t e r p r e t e d as b e i n g due t o the d i s a p p e a r a n c e o f t h e c o n d u c t i o n e l e c t r o n s . A q u a n t i t a t i v e t h e o r y o f t h e low t e m p e r a t u r e v e l o c i t y anomaly l e a d s t o an a c c u r a t e e s t i m a t e o f t h e q -»• 0 e 1 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 . The sound v e l o c i t y measurements were made u s i n g an a c o u s t i c r e s o n a n c e t e c h n i q u e . Resonant modes o f v i b r a t i o n o f s i n g l e c r y s t a l s o f TTF-TCNQ were e x c i t e d e l e c t r o s t a t i c a l l y and d e t e c t e d c a p a c i t i v e l y u s i n g a UHF c a r r i e r s i g n a l . The d e t e c t i o n scheme i s shown t o be more s e n s i t i v e than c o n v e n t i o n a l d.c. b i a s e d c a p a c i t i v e p i c k u p s . A t h e o r e t i c a l s t u d y o f t h e e l e c t r o n i c c o n t r i b u t i o n t o t h e a t t e n u a t i o n o f sound i n one and two d i m e n s i o n a l m e t a l s and s e m i c o n d u c t o r s i s p r e - s e n t e d . The a t t e n u a t i o n i n one d i m e n s i o n a l m e t a l s i s shown to be a n o m a l o u s l y s m a l l . In b o t h one and two d i m e n s i o n a l m e t a l s , i n t h e quantum l i m i t the a t t e n u a t i o n depends s t r o n g l y on the d i r e c t i o n o f p r o p a g a t i o n o f t h e wave. A t r a n s p o r t e q u a t i o n s o l u t i o n t o t h e p r o b l e m of calculating the amplification of sound waves in a solid in the presence of a d.c. electric f i e l d is described. The treatment is much less complex than any that is currently available. i v TABLE OF CONTENTS Page ABSTRACT •• TABLE OF CONTENTS i v LIST OF FIGURES v i i LIST OF TABLES * ACKNOWLEDGEMENTS x i INTRODUCTION 1. The O r g a n i c C o n d u c t o r TTF-TCNQ 1 2. O r g a n i z a t i o n o f the T h e s i s 7 PART A: E l e c t r o n i c C o n t r i b u t i o n t o A t t e n u a t i o n and A m p l i f i c a t i o n o f A c o u s t i c Waves 9 I ATTENUATION IN METALS 10 1.1 I n t r o d u c t i o n 10 1.2 T r a n s p o r t E q u a t i o n Approach t o U l t r a s o n i c A t t e n u a t i o n 1 2 1.3 The Quantum L i m i t 2 2 \.k M e t a l s 2 ? ( i ) Three D i m e n s i o n a l M e t a l s 27 ( i i ) Two D i m e n s i o n a l M e t a l s 31 ( i i i ) One D i m e n s i o n a l M e t a l s 36 1.5 A n i s o t r o p y o f A t t e n u a t i o n ^ ( i ) One D i m e n s i o n a l M e t a l s ^5 ( i i ) Two D i m e n s i o n a l M e t a l s 50 1.6 Summary 55 II ATTENUATION IN SEMICONDUCTORS 57 2.1 Quantum L i m i t 57 2.2 T r a n s p o r t E q u a t i o n Approach 59 2.3 Met a l S e m i c o n d u c t o r T r a n s i t i o n 6A III AMPLIFICATION 71 3.1 I n t r o d u c t i o n 71 3.2 T r a n s p o r t E q u a t i o n 72 3.3 Energy T r a n s f e r 79 3.k C o n c l u s i o n ^5 V PART B: Measurements on TTF-TCNQ 8 7 I EXPERIMENTAL METHOD 88 1.1 C a p a c i t i v e Measurement T e c h n i q u e 88 ( i ) E l e c t r o n i c s 9° ( i i ) Sample S u p p o r t 100 1.2 S e n s i t i v i t y o f the Measurement T e c h n i q u e . 104 ( i ) Minimum D e t e c t a b l e Length Change .... ( i i ) A.C. Method 106 ( i i i ) D.C. Method 1 Q 9 II THE MODES OF VIBRATION OF TTF-TCNQ, CRYSTALS.. 1T» 2.1 Low Frequency Modes o f an E l o n g a t e d P l a t e 11** ( i ) F l e x u r a l Modes 1 1 5 ( i i ) T o r s i o n a l Modes 119 ( i i i ) Comments on a S h o r t P l a t e 121 ( i v ) E l o n g a t i o n a l Modes 123 (v) C r y s t a l l o g r a p h i c Symmetry 125 2.2 I n t e r p r e t a t i o n o f E x p e r i m e n t a l Mode Spectrum 130 ( i ) V i b r a t i n g Reed S u p p o r t 132 ( i i ) C e n t r a l P i n Su p p o r t 139 ( i i i ) Mode C o u p l i n g ]j*3 ( i v ) S u p p o r t Modes .. 2.3 V i b r a t i o n Damping 1^5 ( i ) Q Measurement 1~*-> ( i i ) T h e r m o e l a s t i c Damping 1^' ( i i i ) E l o n g a t i o n a l Modes 152 ( i v ) E f f e c t o f A i r on Resonance F r e - quency and Q 155 III INTERPRETATION OF TEMPERATURE DEPENDENCE OF SOUND VELOCITY 162 3.1 O v e r a l l Temperature Dependence 162 3-2 Low Temperature Anomaly 17° ( i ) Quantum L i m i t 1°° ( i i ) Thermodynamic L i m i t 191 ( i i i ) Comment on A c o u s t i c A b s o r p t i o n .... 199 SUMMARY 2 0 l t 1. The Main R e s u l t s o f t h i s Work 204 2. S u g g e s t i o n s f o r F u r t h e r Work 207 v i APPENDIX P a 9 e 1. Diode Temperature Sensor C a l i b r a t i o n 210 2 . C i r c u i t Diagrams 211 ( i ) Diode D e t e c t o r C i r c u i t 211 ( i i ) MOSFET P r e a m p l i f i e r 2 1 1 ( i i i ) Phase S h i f t e r 212 ( i v ) L o r e n t z i a n G e n e r a t o r 21M (v) S i g n a l A v e r a g e r Input A m p l i f i e r 2 1H 3- Thermal E x p a n s i o n C o r r e c t i o n 215 BIBLIOGRAPHY 2 1 6 v i i LIST OF FIGURES F i g u r e Page 1. TTF and TCNQ M o l e c u l e s 2 2. . TTF-TCNQ C r y s t a l S t r u c t u r e 3 3. TTF-TCNQ b A x i s C o n d u c t i v i t y 5 k. Temperature Dependence o f E l e c t r o n i c Energy Gap 6 5 - C r o s s e c t i o n o f t h e Fermi Sphere Showing the I n t e r - a c t i o n S u r f a c e 25 6. A c o u s t i c A t t e n u a t i o n as a F u n c t i o n o f q£ f o r 2 and 3D M e t a l s 35 7. Fermi S u r f a c e and I n t e r a c t i o n S u r f a c e i n a 1D M e t a l . . 3 7 8. A n g u l a r Dependence o f A t t e n u a t i o n i n a 1D Meta l i n the Quantum L i m i t ^6 9. A n g u l a r Dependence o f A t t e n u a t i o n i n 1D " e t a ' f o r A = 8 *»9 10. A n g u l a r Dependence o f A t t e n u a t i o n i n a 2D Meta l i n t h e Quantum L i m i t 52 11. A n g u l a r Dependence o f A t t e n u a t i o n i n a 2D M e t a l f o r A = 8 53 12. I n t e r a c t i o n S u r f a c e s - M e t a l l i c Band 66 13. I n t e r a c t i o n S u r f a c e s - S e m i c o n d u c t o r Band 67 1A. A t t e n u a t i o n Near a Meta1-Semi c o n d u c t o r T r a n s i t i o n i n a 1D Meta l i n the Quantum L i m i t 68 15. D r i v e and D e t e c t o r C i r c u i t 89 16. D r i v e and B r i d g e D e t e c t o r C i r c u i t 89 17. B l o c k Diagram o f E l e c t r o n i c Equipment 93 18. Sample M o u n t i n g C o n f i g u r a t i o n s 95 19. P h o t o g r a p h s o f Mounted Samples 96 20. Cut-Away View o f Sample Box 97 v i i i F i g u r e Page 21. P h o t o g r a p h o f the O u t s i d e o f the Sample H o l d e r 98 22. P h o t o g r a p h o f the I n s i d e o f t h e Sample H o l d e r 99 23. S h i e l d e d E l e c t r o d e 101 2k. S p e c i a l Notched E l e c t r o d e 101 25- C r y o s t a t f o r Low Temperature Measurements 102 26. C a p a c i t i v e V i b r a t i o n D e t e c t o r - A.C. Method 106 21. C a p a c i t i v e V i b r a t i o n D e t e c t o r - D.C. Method 109 28. N o i s e E q u i v a l e n t C i r c u i t 110 29. Contour Map o f S e n s i t i v i t y o f D.C. Method 112 30. F l e x u r a l Mode Shapes 117 31. Modes o f a Square C a n t i l e v e r P l a t e ....122 32. Modes o f a R e c t a n g u l a r Free P l a t e 122 33- Arrangement o f TTF and TCNQ. M o l e c u l e s i n the a c P l a n e 124 3**. E x p e r i m e n t a l F l e x u r a l Mode Spectrum 131 35. Low Frequency F l e x u r a l and T o r s i o n a l Mode D i s p e r s i o n Diagram 133 36. E f f e c t o f D.C. B i a s V o l t a g e on F l e x u r a l Resonance Frequency 135 37. L o n g i t u d i n a l Mode D i s p e r s i o n Diagram 138 38. F l e x u r a l and T o r s i o n a l Mode C r o s s i n g 1*2 39. I n t e r f e r e n c e w i t h F a c F l e x u r a l Mode I*4** i«0. W i g g l e s 1*7 k\. F l e x u r a l Mode Damping 1*9 k2. L o n g i t u d i n a l Mode Dampi ng 151 ix Figure Page 43- Sound Velocity and Attenuation Near the Metal- Semiconductor Transition 153 44. Air Entrained by a Flexural Mode 156 45. Air Entrained by a Torsional Mode 157 46. Temperature Dependence of b Axis Elongational and Torsional Mode Velocities 163 47. Temperature Dependence of b Axis Flexural Mode Velocity 164 48. Temperature Dependence of a Axis Flexural and Torsional Mode Velocities 165 49. Low Temperature Anomaly in the Young's Modulus Velocity 177 50. Excitation Spectrum for a Non-Interacting 1D Tight-Binding Band 179 51. Dyson's Equation 1d2 52. Electron Gas Polarization Diagram 183 53. Hartree Polarizabi1ity for a 1D Tight-Binding Band...184 54. Acoustic Phonon Dispersion for Experimental Electron-Phonon Coupling Constant 187 55- Acoustic Phonon Dispersion for an Electron-Phonon Coupling Constant which gives T c = 54K 188 56. Band Structure of TTF-TCNQ. 192 57- Strain Dependence of the Density of Occupied States..193 58. Low Temperature Longitudinal Mode Dampi ng 201 LIST OF TABLES T a b l e Page I TTF-TCNQ. M a t e r i a l P a r a m e t e r s *7 II A t t e n u a t i o n f o r 1, 2, 3D M e t a l s i n a D.C. E l e c t r i c F i e l d 81 III F l e x u r a l Mode Par a m e t e r s 116 IV F l e x u r a l Mode E l a s t i c C o n s t a n t s ..128 V Room Temperature Sound V e l o c i t i e s f o r TTF-TCNQ, \h\ VI E f f e c t o f A i r on F l e x u r a l and T o r s i o n a l Mode F r e q u e n c i e s 158 VII Young's M o d u l i f o r V a r i o u s M a t e r i a l s 162 VII I G r i i n e i s e n C o n s t a n t - f o r V a r i o u s M a t e r i a l s 17* IX P r e s s u r e D e r i v a t i v e o f B u l k Modulus f o r V a r i o u s M a t e r i a l s 176 X Summary o f the E x p e r i m e n t a l R e s u l t s 206 ACKNOWLEDGEMENTS I am g r a t e f u l t o my s u p e r v i s o r Rudi H a e r i n g f o r the c o n s i d e r a b l e amount o f time and e f f o r t he has sp e n t h e l p i n g me w i t h t h i s p r o j e c t . H i s p h y s i c a l i n t u i t i o n and common sense were i n v a l u a b l e , to say the l e a s t . The c a p a c i t i v e measurement t e c h n i q u e was s u g g e s t e d by W a l t e r Hardy, and h i s h e l p was e s s e n t i a l i n making i t work. The quantum approach t o the i n t e r p r e t a t i o n o f the low t e m p e r a t u r e v e l o c i t y anomaly was s u g g e s t e d by E. T o s a t t i . A l l a s p e c t s o f the i n t e r p r e t a t i o n o f the t e m p e r a t u r e dependence o f the sound v e l o c i t i e s were worked o u t i n c o l l a b o r a t i o n w i t h Manfred J e r i c h o . 1 b e n e f i t e d from many u s e f u l d i s c u s s i o n s w i t h B. B e r g e r s e n and W. I. F r i e s e n . I w o u l d l i k e t o thank L. W e i l e r f o r s u p p l y i n g me w i t h TTF-TCNQ samples and f o r e x p l a n a t i o n s o f v a r i o u s c h e m i s t r y p r o b l e m s . The TTF-TCNQ c r y s t a l s used i n the e x p e r i m e n t were s y n t h e s i z e d and grown by Y. Hoyano. I t i s a p l e a s u r e t o thank Susan H a e r i n g f o r t a k i n g , d e v e l o p i n g and p r i n t i n g the pho t o g r a p h s i n t h i s t h e s i s . L o r e Hoffmann s i m p l i f i e d the f i n a l p r o d u c t i o n p r o c e s s w i t h her e f f i c i e n t t y p i n g and t h o u g h t f u l a p p r o a c h t o the o r g a n i z a t i o n o f the thes i s. I am g r a t e f u l t o the N a t i o n a l R e s e a r c h C o u n c i l f o r f i n a n c i a l 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 INTRODUCTION 1. The O r g a n i c Conductor TTF-TCNQ T e t r a t h i o f u l v a l i n i u m (TTF) t e t r a c y a n o q u i n o d i m e t h a n i d e (TCNQ) i s an e l e c t r i c a l l y c o n d u c t i n g o r g a n i c s o l i d composed o f t h e TTF and TCNQ m o l e c u l e s shown i n F i g . 1. The m a t e r i a l has s e v e r a l unusual p r o p e r - t i e s . F i r s t i t i s a b e t t e r c o n d u c t o r than a l m o s t any o t h e r o r g a n i c m a t e r i a l known. In f a c t , a t 60K i t s c o n d u c t i v i t y ( l O 4 ( f i - c m ) " 1 ) i s comparable w i t h mercury a t room t e m p e r a t u r e . The a v a i l a b i l i t y o f good o r g a n i c c o n d u c t o r s opens up the p o s s i b i l i t y o f making m a t e r i a l s w i t h d e s i r a b l e e l e c t r o n i c p r o p e r t i e s by c h e m i c a l m o d i f i c a t i o n o f t h e c o n s t i t u e n t m o l e c u l e s . P o t e n t i a l a p p l i c a t i o n s i n c l u d e new o r improved e l e c t r o n i c d e v i c e s and h i g h e r t e m p e r a t u r e s u p e r c o n d u c t o r s . A second unusual f e a t u r e o f TTF-TCNQ i s t h a t i t s c o n d u c t i v i t y i s v e r y a n i s o t r o p i c . The a n i s o t r o p y a r i s e s from the n a t u r e o f the c r y s t a l s t r u c t u r e ; as shown i n F i g . 2, the l a r g e f l a t TTF and TCNQ m o l e c u l e s a r e a r r a n g e d i n s e g r e g a t e d s t a c k s . The m o l e c u l a r o r b i t a l s f o r n e i g h - b o u r i n g m o l e c u l e s on the same s t a c k o v e r l a p much more s t r o n g l y than the m o l e c u l a r o r b i t a l s f o r m o l e c u l e s on d i f f e r e n t s t a c k s . The r e s u l t i s t h a t e l e c t r o n s a r e a b l e t o move more f r e e l y a l o n g the c h a i n s than p e r p e n d i c u l a r t o the c h a i n s . The c o r r e s p o n d i n g a n i s o t r o p y i n the con- d u c t i v i t y i s l a r g e enough t h a t the m a t e r i a l may be r e g a r d e d as a n e a r l y one d i m e n s i o n a l m e t a l . Theory p r e d i c t s a number o f unique p r o p e r t i e s f o r one d i m e n s i o n a l c o n d u c t o r s . One o f t h e s e c h a r a c t e r i s t i c s , namely the P e i e r l s i n s t a - F i g . 1 TTF and TCNQ M o l e c u l e s 3 Fig. 2 - Photograph of a model of the TTF-TCNQ. crystal structure b i l i t y , c a u s e s a s t r u c t u r a l phase t r a n s i t i o n i n w h i c h the one dimen- s i o n a l metal changes i n t o a s e m i c o n d u c t o r . E l e c t r i c a l c o n d u c t i v i t y measurements r e v e a l t h a t TTF-TCNQ does undergo a phase t r a n s i t i o n o f t h i s t y p e a t low t e m p e r a t u r e s (see F i g . 3). X-ray and n e u t r o n s c a t t e r i n g e x p e r i m e n t s seem t o c o n f i r m t h a t the phase t r a n s i t i o n i s a P e i e r l s t r a n s i t i o n . However, below the m e t a l - s e m i c o n d u c t o r t r a n s i t i o n t h e r e i s a t l e a s t one and p o s s i b l y as many as t h r e e ( D j u r e k e t a l 1977) a d d i t i o n a l phase t r a n s i t i o n s . The two most pr o m i n e n t phase t r a n s i t i o n s , n ear 38K and 5^K, a r e b e s t i l l u s t r a t e d by the t e m p e r a t u r e dependence o f the e l e c t r o n i c e nergy gap i n the s e m i c o n d u c t i n g phase as shown i n F i g . 4 ( T i e d j e 1975). The phase t r a n s i t i o n s a r e r e f l e c t e d i n s h a r p i n c r e a s e s i n the e l e c t r o n i c e n e r g y gap 2 A , as a f u n c t i o n o f t e m p e r a t u r e . The n a t u r e o f t h e s e phase t r a n s i t i o n s i s not w e l l u n d e r s t o o d . Most o f t h e i n i t i a l work on TTF-TCNQ was s t i m u l a t e d by the ob- s e r v a t i o n i n a few samples o f TTF-TCNQ o f an a n o m a l o u s l y h i g h con- d u c t i v i t y peak (Coleman e t a l 1973) j u s t above the meta1 - i n s u l a t o r t r a n s i t i o n . The anomalous c o n d u c t i v i t y was i n t e r p r e t e d a t the t i m e as due t o s u p e r c o n d u c t i n g f l u c t u a t i o n s enhanced by the o n s e t o f the P e i e r l s d i s t o r t i o n . No one has succeeded i n d u p l i c a t i n g t h e s e measure- ments, a l t h o u g h t h e r e have been many a t t e m p t s . For r e v i e w s o f r e c e n t work on TTF-TCNQ and r e l a t e d m a t e r i a l s see B u l a e v s k i i (1975), Andre e t a l (1976) and K e l l e r (1977). 5 Fig. 3 - Temperature dependence o f t h e TTF-TCNQ b a x i s c o n d u c t i v i t y . Sample § 2 k came from the same b a t c h as many o f the samples on w h i c h a c o u s t i c measurements were made. 6 O o r o o H 7? 4^ O ACT) ( K ) B O O cn O r o o o 1 1 1 X © CO > > r~ r ~ X V m m s\ X X @ x — — © X e x © X @ X © X © © X @ X • © u. © X c S X © X © X © X © X © X ® X — ©X GX © < T 1 1 1 F i g . k - Temperature dependence o f t h e e l e c t r o n i c e n ergy gap (2A) i n the i n s u l a t i n g phase o f TTF-TCNQ. N o t i c e the change i n s l o p e n e a r 52K and 38K. 7 2. O r g a n i z a t i o n o f the T h e s i s The remainder o f t h i s t h e s i s i s d i v i d e d i n t o two p a r t s . In P a r t A we i n v e s t i g a t e the e f f e c t o f t h e d i m e n s i o n a l i t y o f the e l e c t r o n gas on the e l e c t r o n i c c o n t r i b u t i o n t o the a t t e n u a t i o n o f sound i n m e t a l s and s e m i c o n d u c t o r s . B oth e l e c t r o m a g n e t i c and de- f o r m a t i o n p o t e n t i a l c o u p l i n g between the e l e c t r o n s and the sound wave a r e c o n s i d e r e d . The dependence o f the a t t e n u a t i o n o f h i g h f r e q u e n c y a c o u s t i c waves on the d i r e c t i o n o f p r o p a g a t i o n o f the wave i s c a l c u l a t e d f o r one and two d i m e n s i o n a l m e t a l s . An im- p r o v e d t r a n s p o r t t h e o r y o f the a m p l i f i c a t i o n o f sound i n the p r e s e n c e o f a d.c. e l e c t r i c f i e l d i s a l s o p r e s e n t e d . In P a r t B we d i s c u s s some e x p e r i m e n t a l measurements on the p r o - p a g a t i o n o f sound i n c r y s t a l s o f TTF-TCNQ.. The l a r g e a n i s o t r o p y i n the e l e c t r i c a l p r o p e r t i e s o f TTF-TCNQ a r e i l l u s t r a t e d by e l e c t r i c a l c o n d u c t i v i t y measurements (Hardy e t a l 1976) d i e l e c t r i c c o n s t a n t measurements (Cohen e t a l 1976) and m o l e c u l a r o r b i t a l c a l c u l a t i o n s ( B e r l i n s k y e t a l 197*). S i m i l a r l y t h e t e m p e r a t u r e dependence o f the l a t t i c e c o n s t a n t s ( B l e s s i n g and Coppens 197*) and the n a t u r e o f the b o n d i n g i n the c r y s t a l s u g g e s t t h a t the l a t t i c e may be e l a s t i c a l l y h i g h l y a n i s o t r o p i c as w e l l . In o r d e r t o measure the a n i s o t r o p y i n the e l a s t i c p r o p e r t i e s and t o h e l p c l a r i f y the n a t u r e o f the low tem- p e r a t u r e phase t r a n s i t i o n s d e t a i l e d measurements have been made o f th e t e m p e r a t u r e dependence o f the v e l o c i t y and a t t e n u a t i o n o f sound i n TTF-TCNQ. 8 P a r t B i s d i v i d e d i n t o t h r e e c h a p t e r s . In the f i r s t c h a p t e r the c a p a c i t i v e t e c h n i q u e t h a t was used t o e x c i t e and d e t e c t a c o u s t i c r e - sonances i n s i n g l e c r y s t a l s o f TTF-TCNQ i s d e s c r i b e d i n d e t a i l . An a n a l y s i s o f t h e s e n s i t i v i t y o f c a p a c i t i v e d i s p l a c e m e n t d e t e c t o r s i s i n c l u d e d . The second c h a p t e r e x p l a i n s how the v i b r a t i o n s p e c t r u m can be used t o d e t e r m i n e a number o f d i f f e r e n t e l a s t i c c o n s t a n t s f o r TTF-TCNQ. The p r i n c i p a l damping mechanisms f o r samples v i b r a t i n g i n a i r and i n a vacuum a r e d i s c u s s e d . The f i n a l c h a p t e r c o n t a i n s an i n t e r p r e t a t i o n o f t h e t e m p e r a t u r e dependence o f the v e l o c i t y o f sound. A s m a l l low t e m p e r a t u r e anomaly i n t he v e l o c i t y i s i n t e r p r e t e d as b e i n g due t o an e l e c t r o n i c con- t r i b u t i o n t o the e l a s t i c m o d u l i . A p o s s i b l e e x p l a n a t i o n r o r an i n - c r e a s e i n damping o f some o f t h e a c o u s t i c modes a t low t e m p e r a t u r e s i s p r o p o s e d . P A R T A ELECTRONIC CONTRIBUTION TO ATTENUATION AND AMPLIFICATION OF ACOUSTIC WAVES 10 CHAPTER I A t t e n u a t i o n 1.1 I n t r o d u c t i o n The e l e c t r o n i c c o n t r i b u t i o n t o u l t r a s o n i c a t t e n u a t i o n has been s t u d i e d e x t e n s i v e l y i n t h r e e d i m e n s i o n a l m e t a l s ( P i p p a r d 1955. Cohen et a l I 9 6 0 , R i c e and Sham 1 9 7 0 ) . The r e c e n t d i s c o v e r y o f h i g h l y a n i s o t r o p i c q u a s i one and two d i m e n s i o n a l m e t a l s has s t i m u l a t e d I n t e r e s t In t h e p r o p e r t i e s o f e l e c t r o n i c s y s t e m s o f r e d u c e d d i m e n - s i o n a l i t y ( G l a s e r 197*, W i l s o n e t a l 1975). In t h i s c h a p t e r we e x t e n d t h e t h e o r y o f u l t r a s o n i c a t t e n u a t i o n so t h a t i t a p p l i e s t o s u c h s y s t e m s We p r e s e n t g e n e r a l e x p r e s s i o n s f o r t h e a t t e n u a t i o n c o n s t a n t o f l o n g i - t u d i n a l and t r a n s v e r s e w a v e s , and we o b t a i n l i m i t i n g e x p r e s s i o n s v a l i d In t h e h y d r o d y n a m i c a l and i n t h e quan tum m e c h a n i c a l l i m i t s . We show t h a t t h e u l t r a s o n i c a t t e n u a t i o n i n o ne a n d two d i m e n s i o n a l s y s t e m s d i f f e r s s i g n i f i c a n t l y f r om t h e a t t e n u a t i o n i n t h r e e d i m e n s i o n a l s t r u c t u r e s . The d i f f e r e n c e i s p a r t i c u l a r l y s i g n i f i c a n t i n t h e c a s e of one d i m e n s i o n a l s y s t e m s w h e r e e n e r g y and momentum s e l e c t i o n r u l e s a r e d i f f i c u l t t o s a t i s f y . I t I s w e l l known t h a t t h e a b s o r p t i o n o f s ound i n m e t a l s depends on t h e r e l a t i o n b e t w e e n t h e mean f r e e p a t h o f an e l e c t r o n a t t h e 2jL q In Fe rm i s u r f a c e , £ , and t h e wave l e n g t h o f t h e sound w a v e , X = t h i s c h a p t e r , t h e B o l t z m a n n t r a n s p o r t e q u a t i o n i s u sed t o c a l c u l a t e t h e a t t e n u a t i o n a s a f u n c t i o n o f f r e q u e n c y f o r a r b i t r a r y v a l u e s o f ql. Quantum m e c h a n i c a l p e r t u r b a t i o n t h e o r y i s a l s o u sed t o d e r i v e t h e a t t e n u a t i o n i n t h e quantum l i m i t . A l t h o u g h o t h e r c o n t r i b u t i o n s t o t h e a t t e n u a t i o n w i l l be p r e s e n t In r e a l m e t a l s , o n l y t h e e l e c t r o n i c c o n t r i b u t i o n w i l l be c o n s i d e r e d here. W i t h t h i s l i m i t a t i o n t h e u l t r a s o n i c a t t e n u a t i o n p r o b l e m i s f o r m u l a t e d f o r a t h r e e d i m e n s i o n a l e l e c t r o n gas u s i n g t h e B o l t z m a n n e q u a t i o n i n t h e r e l a x a t i o n t i m e a p p r o x i m a t i o n . The e l e c t r o n s a r e assumed t o i n t e r a c t w i t h t h e l a t t i c e t h r o u g h " c o l l i s i o n s " , s e l f - c o n s i s t e n t e l e c t r o m a g n e t i c f i e l d s , and a s c a l a r d e f o r m a t i o n p o t e n t i a l . Then w i t h t h e d e f o r m a t i o n p o t e n t i a l i n t e r a c t i o n a l o n e , t h e h i g h f r e q u e n c y (q£ » 1) l i m i t i s r e d e r i v e d u s i n g quantum m e c h a n i c a l p e r t u r b a t i o n t h e o r y . The two a p p r o a c h e s m e n t i o n e d above a r e t h e n s p e c i a l i z e d t o t h e t h r e e , t w o , and one d i m e n s i o n a l e l e c t r o n i c s y s t e m s . In e a c h c a s e t h e a t t e n u a t i o n i s c a l c u l a t e d e x p l i c i t l y f o r f r e e e l e c t r o n s . i n a l l c a s e s t h e e l e c t r o n s a r e assumed t o be c o n t a i n e d i n a t h r e e d i m e n s i o n a l c r y s t a l l a t t i c e . 1.2 T r a n s p o r t E q u a t i o n Approach to U l t r a s o n i c A t t e n u a t i o n The Boltzmann equation, i n the r e l a x a t i o n time approximation, j s given by (see e.g. Ziman 1972) sound wave, and f i s the e'ectron d i s t r i b u t i o n f u n c t i o n . De- formation p o t e n t i a l c o u p l i n g i s included at a l a t e r stage. The e l e c t r o n s tend to r e l a x towards the l o c a l e q u i l i b r i u m d i s t r i b u t i o n assume that the e l e c t r o n s r e l a x to an e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n centered at the l o c a l l a t t i c e v e l o c i t y . A l s o , s i n c e the s c a t t e r i n g processes are l o c a l , the e l e c t r o n d e n s i t y i s not a f f e c t e d . For t h i s reason the l o c a l e q u i l i b r i u m chemical p o t e n t i a l must be de- termined s e l f - c o n s i s t e n t l y from the as yet unknown l o c a l e l e c t r o n d e n s i t y . The v e l o c i t y o f an e l e c t r o n In an a r b i t r a r y band s t r u c t u r e ' s » — = ^ Z-k*-^)' Hence the energy of an e l e c t r o n in a frame of reference moving w i t h the l o c a l l a t t i c e v e l o c i t y u_ i s given by [1] where E Is the s e l f - c o n s i s t e n t e l e c t r i c f i e l d generated by the f u n c t i o n f , through s c a t t e r i n g . F o l l o w i n g H o l s t e i n (1959) we e'(k) = e(k) - f i k - u Adding the term 7ik;u_ to the e l e c t r o n energy corresponds to t i l t i n g the band s t r u c t u r e In space. To f i r s t order t h t s corresponds to a Fermi s u r f a c e s h i f t e d i n k_space by . Thus, the l o c a l e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n Is T (k, x, t ) - f . (e'(k); v(x, t))« f . - (*k- u + n ^ where u(x, t ) i s the s e l f - c o n s i s t e n t chemical p o t e n t i a l mentioned above, f 0 i s the e q u i l i b r i u m e l e c t r o n d i s t r i b u t i o n i n the absence of e sound wave, and rsi i s the . o s c i l l a t i n g component .of the e l e c t r o n number d e n s i t y . The Boltzmann equation may now be s o l v e d to f i r s t order by l i n e a r i z i n g . Let f = f + f i where f\ « ^ X i s the o o s c i l l a t i n g component of the d i s t r i b u t i o n f u n c t i o n t h a t i s induced by the sound wave. Equation [1] can now be w r i t t e n ~ »" fl + | i Z k E OO-afj - e|-V ke(k) §{° T h i s s o l u t i o n to f i r s t order I n E , n^, u, and f j , a l l o f which a r e p r o p o r t i o n a l to the amplitude o f the sound wave, I s [2] f x « 3u T e v-E - t i k-u - •—• n x 3n 1 , 1 3f, 1 - lux + I a j vT 3e The e l e c t r o n i c c u r r e n t d e n s i t y i s d e f i n e d by J —c 2e (2^) I 3 I d 3 k V f i [3] mu a- E — E • — - n i e S R where S => ̂  i s the phase v e l o c i t y o f the sound wave, and 2e 2x [4a] a - 7 ^ 3 J d 3k 1 - iojx ^ f.3£.\ iojx + i g / v r \ 9 e / 2 e 2 r t i [4b] I = - ^ 7 3 ~ d 3 k 1 - i u T + i g j y j I 3e J ^ a - f i r TT£P d 3k 1 - i OJX + i q j v r ^ 3c y The t o t a l c u r r e n t a s s o c i a t e d w i t h the sound wave c o n s i s t s o f the sum of the ion c u r r e n t and the e l e c t r o n i c c u r r e n t which Is Induced 15 by the s e l f - c o n s i s t e n t e l e c t r i c f i e l d . The t o t a l c u r r e n t d e n s i t y Is J - J n e u — — e — I f we r e s t r i c t our a t t e n t i o n to monovalent m e t a l s , the e l e c t r o n number d e n s i t y n w i l l equal the ion number d e n s i t y . The c o n t i n u i t y equation f o r e l e c t r o n s Is to n j e + q_*J.e = 0 o r [S] J e „ - " nie'S where J e | , Is the component of J e p a r a l l e l to g_. The s e l f - c o n s i s t e n t e l e c t r i c f i e l d i s determined from the t o t a l c u r r e n t d e n s i t y by the Maxwell equations ( K i t t e l 1963). The r e s u l t i s [6a] and [6b] E „ - hlL (j „ + n e u„ J L The n e x t s t e p Is t o o b t a i n e x p r e s s i o n s f o r t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d and c u r r e n t In t e r m s o f t h e l a t t i c e v e l o c i t y . The p r o b l e m can be s i m p l i f i e d by a s s u m i n g t h a t t h e sound w a v e v e c t o r q_ I s In t h e d i r e c t i o n o f a p r i n c i p a l a x i s o f t h e e l e c t r o n c o n s t a n t e n e r g y s u r f a c e s . In t h i s c a s e t h e c o n d u c t i v i t y t e n s o r s a r e a l l d i a g o n a l , and t h e component s c a n be t r e a t e d s e p a r a t e l y . i n t h e c a s e o f l o n g i t u d i n a l waves (qjru), [5] c a n be u sed t o e l i m i n a t e n^ f r o m [3] t o g i v e . mu G E XX XX w h e r e a ' = =— , E ' - -z - — , and t h e w a v e v e c t o r q Is In XX I "" K XX 1 — K ~LM X X t h e x d i r e c t i o n . F o r l o n g i t u d i n a l w a v e s , o n l y t h e x components o f _J , E_ and u_ a r e n o n - z e r o . Now i f [6a] i s s u b s t i t u t e d f o r E , t h e n E 1 knio x x x x [8] J = - n e u 1 J e x x Oo 0) ^TTIO' 1 + XX. CO n e^T whe re a 0 ° . N o t i c e t h a t J a - n e u .when co « a ' ~ E 1 . m e x x x x x In t h i s ( p e r f e c t s c r e e n i n g ) l i m i t t h e Ion c u r r e n t Is e x a c t l y ma tched by t h e e l e c t r o n c u r r e n t . An e x p r e s s i o n f o r E t s o b t a i n e d by s u b - X s t l t u t l n g [8] I n t o [7]: [9] n e u x x E« xx XX 1 + 0) XX bl E ' xx Oo In t h e t r a n s v e r s e wave c a s e (q_1 u ) , t h e r e i s no e l e c t r o n d e n s i t y o s c i l l a t i o n h e n c e . mu M O l J - o E - E — ^ - L A U J ey y y y y y e x w h e r e . a i s i n - t h e x d i r e c t i o n , and t h e l a t t i . c e - w a v e i s p o l a r i z e d In t h e y d i r e c t i o n . S u b s t i t u t e [6b] i n t o [10] t h e n J = - n e u ey y y _ U) \c M. (o \c/ yy and E » - y n e u y y yy 0O «»> W yy yy i - y y The s c r e e n i n g o f t h e t r a n s v e r s e waves I s l e s s e f f i c i e n t t han t h e s c r e e n i n g o f t h e l o n g i t u d i n a l wave b e c a u s e t h e c h a r g e c a r r i e r s I n t e r - 18 a c t through magnetic f o r c e s r a t h e r than through the s t r o n g e r e l e c t r i c f o r c e s . The f a c t o r s o f ( —| r e f l e c t t h i s d i f f e r e n c e . The expressions may be w r i t t e n more compactly by obser v i n g that the c o n t r i b u t i o n of the conduction e l e c t r o n s to the d i e l e c t r i c constant o f a metal w i t h c o n d u c t i v i t y cr, Is 4TTI Thus, f o r l o n g i t u d i n a l waves [11] x n e u xx CO XX - 1 and [12] J =• - n e u ex x 1 + r*1 -"xx XX where e' = 1 + a* xx u xx A l s o f o r t r a n s v e r s e waves n e u . . /_\2 y HIT i / S Y t u ( c l Eyy w [13] y y - l and [14] 19 where «„. =» 1 - Q a y y The work done on t h e e l e c t r o n s p e r u n i t t i m e by t h e sound wave Is ( B l o u n t 1959) where t h e e l e c t r o n i c e n e r g y band has been assumed t o be p a r a b o l i c . The f i r s t t erm In [15] I s t h e ohmlc l o s s due t o t h e p r e s e n c e o f t h e s e l f - c o n s i s t e n t f i e l d . The se c o n d t e r m i s a v i s c o u s d r a g e f f e c t , w h i c h r e s u l t s f r o m t h e e l e c t r o n - l a t t i c e c o l l i s i o n s w here- by t h e e l e c t r o n s r e a c h l o c a l e q u i l i b r i u m w i t h t h e l a t t i c e . T h i s e f f e c t does n o t depend on t h e c h a r g e o f t h e e l e c t r o n s , and I s p r e s e n t even In t h e a b s e n c e o f t h e s e l f - c o n s i s t e n t f i e l d . However, s i n c e i n t e r n a l e l e c t r i c f i e l d s f o r c e t h e e l e c t r o n s t o f o l l o w t h e l a t t i c e wave c l o s e l y , t h e i n c l u s i o n o f a s e l f - c o n s i s t e n t f i e l d g r e a t l y r e d u c e s t h e c o l l i s i o n d r a g t e r m . As a r e s u l t t h e f i r s t t e r m In [15] d o m i n a t e s , e x c e p t a t v e r y h i g h f r e q u e n c i e s . The a t t e n u a t i o n o f t h e sound wave Is f o u n d by d i v i d i n g t h e power 1 i 2 a b s o r b e d p e r u n i t v o l u m e , P, by t h e e n e r g y f l u x Tr P |u_| S v/here p I s t h e d e n s i t y o f t h e l a t t i c e . Thus P The a t t e n u a t i o n o f l o n g i t u d i n a l and t r a n s v e r s e waves I s f o u n d by s u b s t i t u t i n g [ 1 1 ] , [12] and [ 1 3 ] , [14] r e s p e c t i v e l y , I n t o [ 1 5 ] , F o r l o n g i t u d i n a l waves we f i n d [16a] a p S x nm Re whereas f o r t r a n s v e r s e waves [16b] where .€ = 1 + for? 0) \cj " The l a s t two r e s u l t s a r e new e x p r e s s i o n s f o r u l t r a s o n i c a t t e n u a t i o n . So f a r , t h e c o u p l i n g between t h e sound wave and t h e e l e c t r o n s has been assumed t o be due t o c o l l i s i o n s and t o a s e l f - c o n s i s t e n t e l e c t r o m a g n e t i c f i e l d . A d e f o r m a t i o n p o t e n t i a l t e n s o r c a n a l s o c o u p l e t h e e l e c t r o n s t o t h e wave. I f t h e d e f o r m a t i o n p o t e n t i a l i s a s c a l a r , i t a f f e c t s o n l y t h e i n t e r a c t i o n o f e l e c t r o n s w i t h l o n g i t u d i n a l waves, and does n o t change t h e i n t e r a c t i o n w i t h t r a n s v e r s e waves. I f a s c a l a r d e f o r m a t i o n p o t e n t i a l , C, Is I n t r o d u c e d i n t o t h e Bo l t z m a n n e q u a t i o n ( s e e H a r r i s o n I960, o r T u c k e r and Rampton 1972), and t h e e x p r e s s i o n f o r t h e power d i s s i p a t i o n i s s u i t a b l y m o d i f i e d , I t c a n be shown t h a t t h e a t t e n u a t i o n o f a c o u s t i c waves I s g i v e n by c where D = ,, ? determines the r e l a t i v e Importance o f e l e c t r o m a g n e t i c and deformation p o t e n t i a l c o u p l i n g . For example, I f C » — j j * r - , the deformation p o t e n t i a l w i l l dominate. The three terms i n [17] may be i n t e r - p reted as f o l l o w s . The f i r s t term i s due to the s e l f - c o n s i s t e n t f i e l d p l u s the screened c o l l i s i o n c o u p l i n g . The l a s t term i s the screened deformation p o t e n t i a l a l o n e , and the second term i s a cr o s s term which . I n.c uudas „con t r 5 bu t i ons-. from . - a l l - »three-<mechan«sras. We now develop an a l t e r n a t i v e method which i s v a l i d i n the quantum l i m i t , and in c l u d e s o n l y the deformation p o t e n t i a l i n t e r - a c t i o n . 1.3 The Quantum L i m i t I f t h e e n e r g y t i i i ) o f t h e e x t e r n a l l y i m p r e s s e d phonon i s l a r g e r t h a n t h e u n c e r t a i n t y i n t h e e l e c t r o n i c e n e r g y ^ due t o t h e c o l l i s i o n s , t h e n one i s s a i d t o be w o r k i n g i n t h e quantum l i m i t . In t h i s l i m i t quantum m e c h a n i c a l p e r t u r b a t i o n t h e o r y can be used t o e v a l u a t e the a t t e n u a t i o n o f a sound wave. The f o l l o w i n g a d d i t i o n a l a s s u m p t i o n s a r e made: 1. The e l e c t r o n s a r e f r e e , e x c e p t f o r d i m e n s i o n a l i t y c o n s t r a i n t s . 2. The phonon d i s p e r s i o n i s l i n e a r . 3. The phonon e n e r g y fica i s much s m a l l e r than t h e Fermi e n e r g y . 4.. The . e l e c t ron-phonon, i n t e r a c t i o n . i s ..adequately .des c r i bed by t h e s t a n d a r d i n t e r a c t i o n H a m i l t o n i a n where C i s a s c a l a r d e f o r m a t i o n p o t e n t i a l , a ^ i s a phonon c r e a t i o n o p e r a t o r , and c * i s an o p e r a t o r w h i c h c r e a t e s an e l e c t r o n w i t h w a v e v e c t o r k_ and s p i n a. Most o f t h e r e m a i n d e r o f t h i s s e c t i o n i s m a t e r i a l w h i c h has been d e s c r i b e d i n d e t a i l e l s e w h e r e ( K i t t e l 1963, T u c k e r & Rampton 1972). I t i s i n c l u d e d h e r e t o p r o v i d e a framework f o r d e a l i n g w i t h t h e one and two d i m e n s i o n a l s y s t e m s . The p r o b a b i l i t y p e r u n i t t i m e t h a t a phonon o f w a v e v e c t o r c|_ I s a b s o r b e d by. an e l e c t r o n Is and t h e p r o b a b i l i t y t h a t a phonon q I s e m i t t e d I s - 1>u - e ( k - g j j w here t h e r a t e s a r e n o r m a l i z e d t o u n i t volume, and f o ( k ) , n (q) a r e t h e e q u i I i b r i u m e l e c t r o n and phonon d i s t r i b u t i o n f u n c t i o n s r e s p e c t i v e l y . A c o u s t i c a t t e n u a t i o n can be d e f i n e d as t h e n e t r a t e o f a b s o r p t i o n of phonons In t h e mode q_ d i v i d e d by t h e phonon f l u x n(q_)S. Thus where we have assumed t h a t kT » t i w . T h i s e x p r e s s i o n may be e v a l u a t e d by c o n v e r t i n g t h e summation t o an I n t e g r a l . Then riot 1 c 2 ( T I d 3k ^ f 0 ( k ) . ~ f 0(k+o u)^fi ̂ e ( k ) + fiu> - e(k + o j j It Is of I n t e r e s t to I n v e s t i g a t e which e l e c t r o n i c s t a t e s c o n t r i b u t e to the a t t e n u a t i o n . Wavevector ("momentum") must be conserved In any phonon emission o r ab s o r p t i o n process, otherwise the ma t r i x element of the electron-phonon I n t e r a c t i o n i s zer o . In an a b s o r p t i o n p r o c e s s , f o r example, k_ l =k . + CL. Furthermore, i n the golden r u l e approximation used here, the t r a n s i t i o n r a t e Is zero unless energy Is a l s o conserved. Thus e(k,') = e(k) + TiSq For f r e e e l e c t r o n s the energy c o n s e r v a t i o n requirement reduces t o TiSq m 2m * The energy and momentum c o n s e r v a t i o n requirements d e f i n e a set o f e l e c t r o n i c s t a t e s which can I n t e r a c t w i t h the sound wave. These s t a t e s l i e In the v i c i n i t y of a s u r f a c e i n k space, across which an e l e c t r o n s c a t t e r s i n any phonon a b s o r p t i o n or emission process (PIppard 1963). A c r o s s e c t l o n of t h i s I n t e r a c t i o n s u r f a c e Is shown In F i g . 5» f o r a three dimensional Fermi sphere. C l e a r l y the major . 2 5 - ' INTERACTION SURFACE F i g . 5 - C r o s s e c t i o n o f the Fermi s p h e r e showing t he i n t e r a c t i o n s u r f a c e c o n t r i b u t i o n t o t h e s c a t t e r i n g r a t e w i l l come fr o m t h e n e i g h b o u r h o o d o f t h e Fermi s u r f a c e where t h e r e a r e empty e l e c t r o n i c s t a t e s f o r t h e e l e c t r o n s t o s c a t t e r I n t o . As w i l l be shown In a l a t e r s e c t i o n , t h i s s i t u a t i o n Is d r a s t i c a l l y a l t e r e d In one d i m e n s i o n a l m e t a l s . 27 1.4 Metals (i) Three Dimensional Metals A c o u s t i c a t t e n u a t i o n i n t h r e e d i m e n s i o n a l m e t a l s has been s t u d i e d e x t e n s i v e l y . The r e s u l t s w i l l be p r e s e n t e d h e r e f o r c o m p a r i s o n w i t h t h e one and two d i m e n s i o n a l c a s e s . The t r a n s - p o r t e q u a t i o n method w i l l be used f i r s t . F or f r e e e l e c t r o n s , and kT « Cp , t h e c o n d u c t i v i t y t e n s o r £ ='JL, i s g i v e n by xx 1 - i COT a 3 - j j t a n " 1 (A - cox) + t a n " 1 (A + COT)J . I , h + (A - ( ,vr) 2\ I + T l 0 9 \ l + (A + M T J a j J _ = _J> 3_ yy z z 1 - i COT 2a 3 (̂ )[ t a n " 1 (A-COT) + t a n " 1 (A+COT) l o g p + (A - COT)2] |_1 + (A + COT)2J where a = — , A=q-£, and ̂ v ^ r i s t h e mean f r e e p a t h o f an e l e c t r o n 1" I COT T » p a t t he Fermi s u r f a c e . The o t h e r components o f t h e c o n d u c t i v i t y t e n s o r a r e z e r o . The sound wave i s assumed t o p r o p a g a t e i n t h e x d i r e c t i o n . The o n l y n o n - z e r o component o f R_ i s [19] A 2 I COT ( 1 - i COT ) 28 Furthermore, If S «Vp, then the conductlvfty expression reduces to [20a] a - r—v 2— ^ T  1 1 xx 1-IOJT a 3 j a - tan" 1 A - I A(i)T 1+A2 [20b] a °o = v - ? 2 - r ^ r ( a 2 + 1) f t a n " 1  1 J yy zz 1 — icur 2a3 J ' V . T+A^y " a The attenuation of longitudinal and transverse waves is found by substituting [19], [20a] and [20b] into [16a] and [16b]. In the low frequency l i m i t (A « 1) where the screening is perfect {to « a„, a' for longitudinal waves, cu « ( l ) 2 ° - ' ( c) 2°yy for transverse wavesj the attenuation In the absence of deformation potential coupling Is h nm , ? al= 15 PST A „ 1 nm A2 t 5 pSx for longitudinal and transverse waves respectively. For higher frequencies in which A > 1, but the screening is s t i l l perfect, the following more general expressions may be used: [21] nm *l P S T A 2 tan" 1 A 3 (A - tan"l AJ" - 1 nm 't pSx 2A3 3 [(1+A2)tan"l A - A] -] In t h e h i g h f r e q u e n c y l i m i t (A » 0 where t h e s c r e e n i n g I s p e r f e c t , t h e a t t e n u a t i o n I s al" o" p T q V F h nm t 3ir pS M F For c o m p a r i s o n w i t h t h e r e s u l t o f t h e quantum c a l c u l a t i o n , t h e e x - p r e s s i o n f o r t h e a t t e n u a t i o n , w h i c h i n c l u d e s t h e d e f o r m a t i o n p o t e n t i a l may a l s o be e v a l u a t e d i n t h e quantum l i m i t . In t h i s l i m i t t h e r e - l a x a t i o n t i m e i s a l l o w e d t o become l a r g e so t h a t A » 1, and u 2-»- » u, where to i s t h e plasma f r e q u e n c y . "When"these c o n d i t i o n s a r e P * P s a t i s f i e d , t h e e x p r e s s i o n [17] f o r t h e a t t e n u a t i o n i n t h e p r e s e n c e o f d . d e f o r m a t i o n p o t e n t i a l , r e d u c e s t o nm r, S u b s t i t u t i n g t h e a p p r o p r i a t e l i m i t o f t h e c o n d u c t i v i t y [ 2 0 a ] , i n t o t h e l a s t e q u a t i o n , we o b t a i n 2 where R3 \6iu\e2- / / Z Is t h e t h r e e d i m e n s i o n a l Fermi-Thomas s c r e e n i n g l e n g t h . I t Is I n t e r e s t i n g t o n o t e t h a t (\ + (q R a ) " 2 ] Is t h e mvp Fermi-Thomas d i e l e c t r i c c o n s t a n t v a l i d f o r q « — — . t i In t h e quantum l i m i t , t h e a t t e n u a t i o n may a l s o be f o u n d by I n t e - g r a t i n g [ 1 8 ] . The r e s u l t I s r__, ir nm # v. . W al= 6" pT I IT) q v F A c o m p a r i s o n o f t h e l a s t r e s u l t w i t h [22] r e v e a l s two t h i n g s . F i r s t t h e quantum c a l c u l a t i o n does n o t t a k e i n t o a c c o u n t t h e s c r e e n i n g o f t h e sound wave by t h e c o n d u c t i o n e l e c t r o n s . S e c o n d l y , t h e e l e c t r o - m a g n e t i c c c u p l . i n g .mechanism...Is.equlvalen.t...to.a d e f o r m a t i o n p o t e n t i a l o f s t r e n g t h — . T h i s i s j u s t t h e p o t e n t i a l a r i s i n g f r o m a c h a r g e d e n s i t y o s c i l l a t i o n o f t h e f o r m ne . I t f o l l o w s f r o m [22] t h a t when qR3 « 1 t h e s e l f c o n s i s t e n t f i e l d c o u p l i n g i s e q u i v a l e n t 2 e t o a d e f o r m a t i o n p o t e n t i a l o f s t r e n g t h 3* F . The e l e c t r o m a g n e t i c e l e c t r o n - l a t t i c e c o u p l i n g n o r m a l l y d o m i n a t e s i n m e t a l s a t a l l r e a s o n a b l e f r e q u e n c i e s . To su m m a r i z e , t h e quantum a p p r o a c h g i v e s t h e t r a n s p o r t r e s u l t I f an e l e c t r o m a g n e t i c c o u p l i n g e n e r g y o f - ^ 2 — Is added t o t h e d e f o r m a t i o n p o t e n t i a l and t h e e n t i r e I n t e r - a c t i o n p o t e n t i a l i s s c r e e n e d by d i v i d i n g by t h e Fermi-Thomas d i e l e c t r i c c o n s t a n t . ( i i ) Two D i m e n s i o n a l M e t a l s By two dimensional metals we mean metals In which the e l e c t r o n i c energy depends on k and k but not on k . For example, the Fermi x y z surf a c e of a two dimensional f r e e e l e c t r o n gas i s a c y l i n d e r , centered on the k a x i s . A s s o c i a t e d w i t h the two dimensional nature z of the e l e c t r o n gas, there are three modes of sound propagation In a d d i t i o n to the l o n g i t u d i n a l and t r a n s v e r s e waves propagating in the conducting plane and p o l a r i z e d i n the conducting plane. F i r s t there is a transverse wave t r a v e l l i n g in the conducting plane and p o l a r i z e d p e r p e n d i c u l a r to the conducting plane. Secondly there are two modes, one l o n g i t u d i n a l and one t r a n s v e r s e , propagating p e r p e n d i c u l a r to the conducting plane. Waves p o l a r i z e d in the non-conducting d i r e c t i o n cannot d e l i v e r energy to the e l e c t r o n s in the l i n e a r approximation considered here. Hence these waves are not attenuated. The remaining s p e c i a l wave i s the transverse wave propagating in the non-conducting d i r e c t i o n . R e f e r r i n g back to the expression f o r the c o n d u c t i v i t y tensor [ 4 a ] , we see that f o r t h i s mode q*v = o, hence a = a - -r^r— and a = - 1 — xx yy 1-IIOT zz This value f o r the c o n d u c t i v i t y s u b s t i t u t e d Into [16b] y i e l d s the a t t e n u a t i o n [24] a t = nm pSx (cox) : 1 + ( C O T ) 2 f - 1 which Is much sm a l l e r than the corresponding three dimensional r e s u l t In al1 1imlts. tn c a l c u l a t i n g t h e a t t e n u a t i o n o f .the two modes t r a v e l l i n g tn t h e c o n d u c t i n g p l a n e and p o l a r i z e d i n t h e c o n d u c t i n g p l a n e , we use t h e same p r o c e d u r e as t n t h e t h r e e d i m e n s i o n a l c a s e . The n o n - z e r o components o f t h e c o n d u c t i v i t y t e n s o r f o r a two d i m e n s i o n a l f r e e e l e c t r o n s y s t e m w i t h kT « C p a r e g i v e n by [25a] a - - r - ^ - ~ { 1 - \ [25b] o-„ = — / / i 2 + f -1 1 yy 1-IUT a2 \ ) where t h e symbols have t h e same meaning as i n t h e t h r e e d i m e n s i o n a l p r o b l e m . The o n l y n o n - z e r o component o f I s R c A 2 xx x itox ( 1 - i c u t ) 2oa w h i c h d i f f e r s o n l y i n t h e f a c t o r o f 2 from t h e c o r r e s p o n d i n g t h r e e d i m e n s i o n a l r e s u l t . The a t t e n u a t i o n o f t h e l o n g i t u d i n a l and t r a n s v e r s e modes t s f o u n d by s u b s t i t u t i n g t he l a s t t h r e e r e s u l t s i n t o [16a] and [16b] r e s p e c - t i v e l y . In t h e low f r e q u e n c y (A « 1) p e r f e c t s c r e e n i n g l i m i t , t h e a t t e n u a t i o n i s 1 nm . 2 a l a a t "h~ psT ' F o r a r b i t r a r y A, and p e r f e c t s c r e e n i n g A 2 nm l 2 6 ] °c = °t ~ p!7 2(/T+A^-l) ] In t h e l a r g e A l i m i t t h e l a s t r e s u l t s i m p l i f i e s t o 1 nm a l ~ \ = 2 pT q V F F o r p u r p o s e s o f c o m p a r i s o n w i t h t h e r e s u l t o f t h e quantum c a l c u l a t i o n , t h e a t t e n u a t i o n o f l o n g i t u d i n a l waves In t h e l o n g r e l a x a t i o n t i m e , o r quantum 1 I m l t , I s / C + / | T R N E 2 \ V r o - 7 i _ * n m I q 1 q V F W a l ~ 2 pT I ^ j ( l + ( q R 2 ) " 2 ^ where R 2 = ( l ( 7 T n e 2 J ' s t n e t w o d i m e n s i o n a l Fermi-Thomas s c r e e n i n g ^ l e n g t h . The a t t e n u a t i o n i n t h i s l i m i t may a l s o be f o u n d by e v a l u a t i n g [18] f o r a two d i m e n s i o n a l e l e c t r o n g a s , w h i c h g i v e s W al = 2 PT fe) q V F ' As In t h e t h r e e d i m e n s i o n a l c a s e t h e quantum r e s u l t [28] Is I d e n t i c a l 3* with the Boltzmann equation result [27], except that the latter in- cludes electromagnetic coupling and screening. A graph of the attenuation of longitudinal acoustic waves in two and three dimensional metals, calculated from eqns. [21] and [26] as a function of A (= at) is shown in Fig. 6. The similarity of the two curves in Fig. 6 suggests the following interpolation formula for the inverse tangent function: This formula is asymptotically exact for large and small x and de- viates from the exact value of tan _ 1x by a maximum of abcut 1.1% for x = 2 . 35 F i g . 6 - A c o u s t i c a t t e n u a t i o n as a f u n c t i o n o f at f o r 3D and 2D m e t a l s i n u n i t s o f n m V r as c a l c u l a t e d from [21] and [26] PS ( i i i ) One D i m e n s i o n a l M e t a l We now i n v e s t i g a t e a c o u s t i c a t t e n u a t i o n i n one d i m e n s i o n a l m e t a l s . By one d i m e n s i o n a l , we mean t h a t the e l e c t r o n i c e n e r g y depends o n l y on k and not on k o r k . As a r e s u l t t h e Fermi s u r f a c e c o n s i s t s x y z o f two p a r a l l e l p l a n e s , p e r p e n d i c u l a r t o t h e a x i s . As i n t h e two d i m e n s i o n a l m e t a l , t h e r e a r e f i v e d i s t i n c t c a s e s d e p e n d i n g on t h e r e l a t i v e o r i e n t a t i o n o f t h e c o n d u c t i n g a x i s , t h e sound p r o p a g a t i o n d i r e c t i o n and t h e p o l a r i z a t i o n v e c t o r . Only two o f t h e waves i n t e r a c t w i t h t h e e l e c t r o n s . One i s a t r a n s v e r s e wave p o l a r i z e d a l o n g t h e c o n d u c t i n g a x i s , and t h e o t h e r i s a l o n g i t u d i n a l wave t r a v e l l i n g a i o n g t h e c o n d u c t i n g a x i s . The a t t e n u a t i o n o f t h e f i r s t o f t h e s e i s i d e n t i c a l t o t h e a t t e n u a t i o n o f t h e s i m i l a r mode i n t h e two d i m e n s i o n a l m e t a l ; hence i t i s a l s o g i v e n by [24]. In t h e c a s e o f t h e l o n g i t u d i n a l wave p r o p a g a t i n g a l o n g t h e c o n - d u c t i n g a x i s , i t i s i n s t r u c t i v e t o c o n s i d e r t h e quantum l i m i t f i r s t . The c a l c u l a t i o n i s s i m i l a r t o t h e two d i m e n s i o n a l p r o b l e m e x c e p t t h a t t h e e l e c t r o n i c e nergy e(k) i s a f u n c t i o n o f k̂  o n l y . The r e s u l t , however, i s q u i t e d i f f e r e n t . The r e a s o n f o r the d i f f e r e n c e i s most e a s i l y u n d e r s t o o d by e x a m i n i n g the i n t e r a c t i o n s u r f a c e . As has been d e s c r i b e d above, t h e i n t e r a c t i o n s u r f a c e d e f i n e s t h e e l e c t r o n j< s t a t e s , w h i c h a r e a l l o w e d by momentum and energy c o n s e r v a t i o n t o 37 F i g . 7 - Fermi s u r f a c e and i n t e r a c t i o n s u r f a c e i n a one d i m e n s i o n a l metal interact with a sound wave of wavevector q «kp. Unlike the inter- action surface for the two and three dimensional cases, the inter- action surface for one dimension does not intersect the Fermi sur- face (see Fig.7). Although this feature does not restrict electrons from scattering with phonons of wavevector ~ 2kp in the present case where q « kp , we expect a much reduced attenuation, which approaches zero at low temperatures. In fact, this conclusion is borne out by evaluating [18] in the S « v p limit: This expression differs from the corresponding two and three dimen- / £ F \ - E F sional results by the presence of the factor! j e r/j" . Clearly ct£ approaches zero as T goes to zero. The quantum limit may also be extracted from the transport inte- grals. However, the A,tor ->• » limit must be taken with care. Since the acoustic attenuation expression contains a real part operator, a small real part may be significant even when the imaginary part is much larger. The non-zero part of the conductivity tensor for longi- tudinal sound waves in a one dimensional metal may be written [29] This integral may be evaluated in the tor •> °° limit by using the functional relation Urn \ ^ = p -L- + i IT 6 (v - S) WT » V - S M + — ) where P means principal part. The real part of the conductivity (imaginary part of the integral) may now be found exactly, and the imaginary part can be approximated by treating ̂ - | ^ as a delta function at ± Vp. The result is - °° 0 - -7-xx A kT - i — Once again there is a simple relation between R_ and g^. The only non-zero component of R_ is A* °- R = - x x x iiox (1 - icox) Oo When the last two results are substituted into the attenuation equation, [17], we obtain x eF (c + ^f\ qv F nm / F\ _ - r=- / q 2 , _L [30] «£= Z i r ^ l j ^ J e kT . 2 e F where R, =(-^—i—ol is the one dimensional Fermi-Thomas screening l 2 i T n e 2 | length. The transport method once again gives the same result [30], as the quantum method [29]» barring the expected failure of the quantum calculation to include electromagnetic coupling and screening. The extreme quantum limit that is considered above (cox >> 1) is unlikely to be attainable in practice. In this limit the delta function in [18] implies s t r i c t energy conservation in electron phonon col l i s i o n processes, and the attenuation arises from the thermal broadening of the Fermi surface. If the condition cox >> 1 is re- laxed to the weaker condition A >> 1, the f i n i t e c o l l i s i o n time may be taken into account by replacing the delta function in [18] by the Lorentzian TL irx (e(k) + fuo - e(k + q ^ + ^ -1 The ultrasonic attenuation, then arises from the intersection of the Fermi surface with the t a i l of the Lorentzian and is given by , , n m /c \ 2 ^ a l = pTT fa) ' Aside from a factor of two, this expression differs from the corresponding two dimensional result [28] by a factor of A " 1 . This factor reflects the non-intersection of the interaction and Fermi surfaces. The attenuation [31] may also be obtained using the transport equation method, by neglecting the self-consistent f i e l d and c o l l i s i o n drag, in favour of the deformation potential 41 As pointed out earlier, the self-consistent f i e l d , election sound wave coupling, is expected to dominate in metals at least in the low frequency limit. We f i r s t try to obtain an expression analogous to the two dimensional result [26], using the transport equation method. However, i f the same approximations are made as in the two (or three) dimensional cases, the attenuation in a one dimensional electron gas is identically zero. To obtain a non-zero result we relax the zero temperature approximation and calculate the transport tensors to order [ — J . The transport integrals ( f depend on temperature through the temperature dependence of the Fermi function. As the temperature rises from zero, the unit step in the Fermi function broadens to a width of order kT and the chemical potential increases from the Fermi energy to y = e. Her) Provided that both of these effects are included, the non-zero component of the conductivity tensor is found to be TT2 /kT\ 2 to f i r s t order in 6 where 0 = — / — j « 1. Similarly A 2 ° [33] R = . • r — (1 - 9) L J x IU)T(I-IIOT) cr Substituting [32] and [33] into [17] we obtain [34] a 3 PST n m 1 + A2 for a free electron gas. This expression is valid for arbitrary A , provided that the perfect screening limit holds « a0, and that A is not so large that the extreme quantum limit applies. The transport equation method, in the relaxation time approximation, may also be used to calculate the ultrasonic attenuation for an arbitrary band structure, although the general expression which results, is much more complicated than [17]. However, in the perfect screening limit the expression for the attenuation is once again quite simple. We then find that the attenuation of longitudinal waves travelling in the x direction is given by [35] where terms of order have been ignored, and the transport tensors £ , ^ and are given by [4a], [4b] and [4c] respectively. Similarly a = n e ^ T where m f (k)d 3k is the total number of electrons in the conduction band. Notice that the m dependence of [35] is f i c t i t i o u s since both a G and £ are defined to be proportional to m"1. The one dimensional transport integrals are sufficiently simple that the temperature dependent part of the attenuation may be evaluated for an arbitrary band structure, in the perfect screening limit. If the band structure in the vicinity of the Fermi surface is given by e(k) = e p + f,(k - k p) v p + 2 m * F where 1 i>£ V F f. 8k k = k r and (m*)" 1 = f p " k = k r then the temperature dependence of the Fermi level is V = e F 12 e p where c p = j m* v p 2 . The ultrasonic attenuation is given by ir 2 nm"' [36] ^ - - g - ^ r / * k F \ / k T \ 2 A 5 where A = qvpx as before and the total number of carriers in the band is n = DpfikpVp where D p is the density of states at the Fermi surface. The last result closely resembles the free electron result [34]. The non-intersection of the Fermi surface with the interaction surface is reflected in the fact that the attenuation [ 3 4 ] becomes frequency independent for A » 1, instead of approaching a linear frequency dependence. The same behaviour has already been noted in [31] for deformation potential coupling. On the other hand, the absence of the leading term in the ultrasonic attenuation in the A « 1 limit cannot be explained on the basis of the energy and wavenumber selection rules. In fact i t is easy to show, using the col l i s i o n broadened energy conservation requirement that for A << 1 electrons at the Fermi surface are just as likely to interact with the sound wave as the electrons on the interaction surface. The reason for the small attenuation in the one dimensional metal for A << 1 lies in the absence of the higher order relaxation times Tj j = 2, 3» • • •» discussed by Bhatia and Moore (i960)'. Further- more the expression [34] (or [36]) should be taken only as an order of magnitude estimate of the attenuation since another term of /kT\2 order J — \ has been omitted by assuming the sound wave to be \ e F / isothermal [Akhiezer, Kaganov and Liubarskii (1957)]- 1.5 Anisotropy of Attenuation In one and two dimensional metals it is possible to adjust the position of the interaction surface simply by changing the angle of propagation of the acoustic wave relative to the conducting direction (or plane). In the quantum limit, when the interaction s u r f a c e i n t e r s e c t s an extremum o f t h e Fermi s u r f a c e t h e r e i s a l a r g e enhancement i n the sound a b s o r p t i o n . For t h i s r eason the a t t e n u a t i o n o f a c o u s t i c waves w i l l be v e r y a n i s o t r o p i c i n one and two d i m e n s i o n a l m e t a l s , i n the quantum l i m i t . In t h i s s e c t i o n we assume f o r con- v e n i e n c e t h a t the a c o u s t i c wave i s c o u p l e d t o the e l e c t r o n s o n l y t h r o u g h a d e f o r m a t i o n p o t e n t i a l C. ( i ) One D i m e n s i o n a l M e t a l The i n t e r a c t i o n s u r f a c e has been d e f i n e d above as the s u r f a c e i n k space f o r w h i c h e (k+q_) - e (k_) = f i q S w i t h q « kp. In a one dimen- s i o n a l metal the e l e c t r o n i c e n e r g i e s depend o n l y on k x so t h a t the d e f i n i t i o n o f the i n t e r a c t i o n s u r f a c e reduces t o e ( k x + q cos 6) - e ( k x ) = fiqS f o r a one d i m e n s i o n a l metal i n w h i c h the a c o u s t i c wave p r o p a g a t e s a t an a n g l e 0 r e l a t i v e t o t h e c o n d u c t i n g d i r e c t i o n . Note t h a t c h a n g i n g t h e d i r e c t i o n o f q_ r e l a t i v e t o the c o n d u c t i n g a x i s i s m a t h e m a t i c a l l y e q u i v a l e n t t o r e p l a c i n g the speed o f sound S by S/cos 0. T h i s d i - r e c t i o n dependence o f the a p p a r e n t v e l o c i t y o f sound has a d r a s t i c e f f e c t on the u l t r a s o n i c a t t e n u a t i o n i n the quantum l i m i t . F i r s t we c o n s i d e r the c o l 1 i s i o n l e s s quantum regime i n w h i c h COT » 1. As was p o i n t e d o u t i n the p r e v i o u s s e c t i o n , i n t h i s l i m i t the a b s o r p - t i o n p r o c e s s e s a r i s e from the i n t e r s e c t i o n o f the e x p o n e n t i a l t a i l o f t h e F e r m i - D i r a c d i s t r i b u t i o n f u n c t i o n w i t h the energy c o n s e r v i n g 6 46 e o Fig. 8 - Angular dependence of ultrasonic attenuation in 1D metal C ^ calculated from [37] for T = 300K normalized to aQ - j ~ j (^~J VF The inset is an enlargement of the peak attenuation near 9 = 90°. function. Evaluating [18] we find r v m tD\ * nm / C \ 2 eF q v F . 2 [37] a„(6) = — — h—) -,-= sech z L 1 I 2 pS UEp J kT cos 6 This result reduces to [29] as 8 + 0 and goes to zero as 6 ->- TT/2. At intermediate values of 6 the attenuation goes through a large maximum when the interaction surface touches the Fermi surface. The 0 dependence of the absorption is shown in Fig. 8. In this figure, TTF-TCNQ material parameters taken from Table I have been used. TABLE I - TTF-TCNQ Material Parameters . (T^t - « F ) / » 2.8x10 2 1 cm"3 1.62 g/cm3 4x10 5 cm/S m V F e F TTF Band 8 mQ 0.5x107 0.5x103 b axis scattering time (60K) TCNQ Band 107 cm/S 103 K 5x10' •lU Berl insky et al (1972*) 48 The a b s o r p t i o n peak o c c u r s a t 9 m a x = c o s " (S/vp) and has an a n g u l a r w i d t h o f o r d e r S/vp r a d i a n s . The peak a b s o r p t i o n i s a JL / C \2 11 11 f o r kT > fito and a JL JDJE / C \ 2 E F f o r kT < tito. The f i r s t r e s u l t i s a f a c t o r o f o r d e r ( e p / k T ) ( v p / S ) l a r g e r than t h e c o r r e s p o n d i n g t h r e e d i m e n s i o n a l r e s u l t [23] r e f l e c t i n g the f a c t t h a t i n a one d i m e n s i o n a l metal the i n t e r a c t i o n s u r f a c e can i n t e r s e c t a l l o f one s h e e t o f t h e Fermi s u r f a c e a t once, r a t h e r than j u s t a narrow r i n g as i n a t h r e e d i m e n s i o n a l m e t a l . When t h e c o n d i t i o n cox » 1 i s r e l a x e d t o the weaker c o n d i t i o n A » 1, the a t t e n u a t i o n i n a one d i m e n s i o n a l metal comes f r o m t h e i n t e r s e c t i o n o f t h e Fermi s u r f a c e w i t h the t a i l s o f the broadened e n e r g y c o n s e r v i n g " 6 " f u n c t i o n . I f we r e p l a c e the 6 f u n c t i o n i n [18] by a L o r e n t z i a n , as d i s c u s s e d i n the p r e c e e d i n g s e c t i o n , then i n the low t e m p e r a t u r e l i m i t where the Fermi s u r f a c e i s s h a r p l y d e f i n e d , a t r i v i a l i n t e g r a t i o n l e a d s t o [38] 49 F i g . 9 " A n g u l a r dependence o f u l t r a s o n i c a t t e n u a t i o n i r f a ID metal c a l c u l a t e d from [58] w i t h A = 8. 50 f o r t he dependence o f the a t t e n u a t i o n on the p r o p a g a t i o n d i r e c t i o n . T h i s e x p r e s s i o n reduces t o the e a r l i e r r e s u l t s a t 8 = 0 ( [ 3 1 ] ) and TT/2, and has a peak a t 6 m a x = COS _ 1 ( 1 /A). The peak v a l u e o f the a t t e n u a t i o n i s much s m a l l e r than i n the c o l 1 i s ion l e s s (tor » 1) regime. The peak a t t e n u a t i o n i s a nm -cmax p S x (2e w h i c h i s a f a c t o r o f o r d e r A l a r g e r than the c o r r e s p o n d i n g t h r e e d i m e n s i o n a l r e s u l t . A p l o t o f a ^ ( ^ ) f ° r A = 8 i s shown i n F i g . 9- ( i i ) Two D i m e n s i o n a l M e t a l s The u l t r a s o n i c a t t e n u a t i o n i n two d i m e n s i o n a l m e t a l s i s a l s o a n i - s o t r o p i c i n the quantum l i m i t . In two d i m e n s i o n a l m e t a l s t h e r e i s a l a r g e peak i n the a t t e n u a t i o n when t h e i n t e r a c t i o n s u r f a c e moves o u t t o t o u c h t h e s u r f a c e o f the Fermi c y l i n d e r t a n g e n t i a 11y. T h i s peak o c c u r s when t h e a c o u s t i c wave p r o p a g a t i o n d i r e c t i o n i s n e a r l y p e r p e n d i c u l a r t o the c o n d u c t i n g p l a n e . We now use [17] t o c a l c u l a t e the a t t e n u a t i o n o f an a c o u s t i c wave p r o p a g a t i n g a t an a n g l e 6 t o the c o n d u c t i n g p l a n e i n a two d i m e n s i o n a l m e t a l . In the extreme quantum l i m i t i n w h i c h OJT » 1 and energy i s s t r i c t l y c o n s e r v e d i n e l e c t r o n - p h o n o n c o l l i s i o n p r o c e s s e s / n \ _ 1 nm a £ ( 9 ) = 2 p T s u , F 1 - V p 2 C O S 2 0 £p 1 " V p 2 c o s 2 8 E i q v F COS0 f o r kT > tico. In t h i s e x p r e s s i o n when the argument o f the s q u a r e r o o t i s n e g a t i v e the r e s u l t i s ta k e n to be z e r o . W i t h t h i s p r o v i s o t he a t t e n u a t i o n i s z e r o a t 9 = TT/2 and matches t h e p r e v i o u s r e s u l t [28] a t 6 = 0. For cose > S / v F 2 \ ~i [39] at(Q) - ( c o s 2 9 - j o £ ( 0 ) where ct^(O) i s g i v e n by [ 2 8 ] . For 9 ne a r 9 m a x = c o s _ 1 ( S / v p ) the a t t e n u a t i o n r e a c h e s a peak o f V F a£max = - S - \ ^ / a £ ( 0 ) " A n o n - z e r o t e m p e r a t u r e vkT > tico) broadens the sh a r p Fermi s u r f a c e and reduces the peak a t t e n u a t i o n t o % i a x ~ ~ I k T al{0)' When kT > tico t he a t t e n u a t i o n as a f u n c t i o n o f 9 i s g i v e n by ' e r \ i £4°1 a £ ( 9 ) = c ^ ( k T ) a£ ( 0 ) J x ^ sech 2(x-e)dx wi t h 6 I Vp2COs20) kT The i n t e g r a l i n [40] must be done n u m e r i c a l l y . However, f o r cos9 > S/ 52 50 3000 T = I00K 40 2000 30 a, (0) 1000 p n i 0 89.82 89.84 89.86 89.88 6 (°) 10 0 J L 0 Fig. 10 - Angular dependence of ultrasonic attenuation in a 2D metal calculated from [40]. The inset shows the peak attenuation with an expanded horizontal scale. 53 F i g . 11 - A n g u l a r dependence o f u l t r a s o n i c a t t e n u a t i o n i n a 2D metal c a l c u l a t e d from [ 4 1 ] w i t h A = 8. and a t t e m p e r a t u r e s low compared to t h e Fermi t e m p e r a t u r e the l a s t r e s u l t ( [ 4 0 ] ) i s e q u a l t o [39] t o a good a p p r o x i m a t i o n . A p l o t o f the a t t e n u a t i o n as a f u n c t i o n o f a n g l e i s shown i n F i g . 10, u s i n g e F = lO^K and S/v p = 1 0 " 3 . I t i s u n l i k e l y t h a t t h e extreme quantum l i m i t (COT » 1) can be a c h i e v e d i n p r a c t i c e . I f the c o n d i t i o n COT » 1 i s r e l a x e d t o the more r e a l i s t i c c o n d i t i o n q£ » 1 then t h e r e i s no l o n g e r s t r i c t e n e rgy c o n s e r v a t i o n i n e l e c t r o n - p h o n o n s c a t t e r i n g p r o c e s s e s . In t h i s c a s e the 6 f u n c t i o n i n [18] i s r e p l a c e d by a L o r e n t z i a n as d e s c r i b e d above. In the low t e m p e r a t u r e l i m i t the two d i m e n s i o n a l i n t e g r a t i o n i n [18] may be done e x a c t l y f o r a r b i t r a r y d i r e c t i o n o f sound p r o p a g a t i o n . .The r e s u l t i s r,.., /„ x 2 t a n - 1 ( A c o s O ) 1 j . , n S [41] a £ ( 6 ) = - [ — J ^ T + A 2 c o s 2 e j A «£«>) • T h i s e x p r e s s i o n reduces t o t h e c o l 1 i s i o n l e s s r e s u l t [40] a t 8 = 0,TT/2; however, j u s t as i n the one d i m e n s i o n a l case c o l l i s i o n s d r a s t i c a l l y reduce the a c o u s t i c a b s o r p t i o n peak n e a r 6 = TT/2. A p l o t o f a ^ ( Q ) as a f u n c t i o n o f 0 i s shown i n F i g . 11 f o r A = 8. The peak a t t e n u a t i o n i s a £ m a x = 0 - 2 2 6 A a £ ( 0 ) w h i c h o c c u r s a t an a n g l e 8 m a x = c o s - 1 ( 1 . 8 2 5 / A ) w i t h r e s p e c t t o the c o n d u c t i n g p l a n e . J u s t as i n the one d i m e n s i o n a l c a s e the peak attenuation at 6 = 0 m = v, in the two dimensional metal is a factor of order A larger than the corresponding isotropic attenuation in a three dimensional metal. 1.6 Summary We have extended the theory of ultrasonic attenuation in metals so that this theory may be applied to metals whose electronic band structures are one or two dimensional. Our results are valid for arbitrary values of A and include electron-phonon coupling via a scalar deformation potential as well as coupling via collisions and via the self-consistent electric f i e l d which arises from the response of the electrons to the sound wave. We have shown that the ultra- sonic attenuation in one and two dimensional systems diff e r s i g n i f i - cantly from we 11-known results for three dimensional systems. In particular the attenuation is shown to be anomalously small and strongly temperature dependent for metallic one dimensional systems. In addition the attenuation is shown to be highly anisotropic in one and two dimensional metals in the quantum limit. There are other applications of the theory in addition to the application to quasi one and two dimensional metals. For example, it may be applied to the attenuation resulting from an accumulation layer i n an MOS junction, or to the attenuation associated with layered metal-insulator heterostructures. The theory also applies to a three dimensional electron gas in a strong magnetic f i e l d . In this case, one dimensional behavior results from the quantization of the electron moti.on i n the p l a n e p e r p e n d i c u l a r t o the a p p l i e d f i e l d . The Fermi s u r f a c e f o r a p a r t i c u l a r Landau l e v e l may be " t u n e d " t o match the i n t e r a c t i o n s u r f a c e by v a r y i n g t h e m a g n e t i c f i e l d . In the l i m i t A » 1 t h i s r e s u l t i n l a r g e peaks i n t h e a c o u s t i c a t t e n u a t i o n as a f u n c t i o n o f m a g n e t i c f i e l d . These " g i a n t quantum o s c i l l a t i o n s " have been o b s e r v e d i n g a l l i u m by S h a p i r a and Lax (1965). T h i s m a g n e t i c f i e l d t u n i n g o f the Fermi s u r f a c e i s a n a l o g o u s t o c h a n g i n g t h e p o s i t i o n o f t h e i n t e r a c t i o n s u r f a c e i n a one d i m e n s i o n a l metal by v a r y i n g the d i r e c t i o n o f p r o p a g a t i o n o f t h e a c o u s t i c wave. The p r i n c i p a l e x p e r i m e n t a l o b s t a c l e t o o b s e r v i n g the a n i s o t r o p y o f t h e a t t e n u a t i o n i n one and two d i m e n s i o n a l m e t a l s i s i n o b t a i n i n g a l o n g enough e l e c t r o n mean f r e e p a t h and h i g h enough f r e q u e n c y t o a c h i e v e t h e A > 1 l i m i t . CHAPTER I I A t t e n u a t i o n i n S e m i c o n d u c t o r s 2.1 Quantum L i m i t In t h i s c h a p t e r the t e c h n i q u e s t h a t have a l r e a d y been a p p l i e d t o m e t a l s w i l l be used t o c a l c u l a t e the a c o u s t i c a t t e n u a t i o n due t o e l e c t r o n s i n n-type s e m i c o n d u c t o r s . Both d e f o r m a t i o n p o t e n t i a l and e l e c t r o m a g n e t i c c o u p l i n g w i l l be c o n s i d e r e d , and an e n e r g y i n d e p e n d e n t s c a t t e r i n g time i s assumed. T h i s a s s u m p t i o n was u n n e c e s s a r y i n the p r e v i o u s c h a p t e r because i n m e t a l s o n l y the s c a t t e r i n g time f o r Fermi energy e l e c t r o n s i s i m p o r t a n t . The f i r s t s t e p i s to use p e r t u r b a t i o n t h e o r y to c a l c u l a t e the n e t phonon a b s o r p t i o n r a t e i n the quantum regime where A » 1. The o n l y d i f f e r e n c e from the m e t a l l i c a p p r o a c h i s i n the d e f i n i t i o n o f the e l e c t r o n d i s t r i b u t i o n f u n c t i o n f ( k ) . For m e t a l s f (k_) = ^exp j^e (k_) - C p ^ / k T ^ + 1^ _ 1 whereas f o r s e m i c o n d u c t o r s f (k) = exp J^u-e (k_) ^/kT^ where u i s the c h e m i c a l p o t e n t i a l . In the metal c a s e £ p » kT w h i l e f o r s e m i c o n d u c t o r s u<0. I f we assume the energy bands t o be p a r a b o l i c , the e l e c t r o n energy may be decomposed i n t o a sum o f t h r e e p a r t s where , n\2 and 1TI3 a r e e f f e c t i v e masses. T h i s d e c o m p o s i t i o n makes i t p o s s i b l e t o w r i t e the a t t e n u a t i o n a = as a p r o d u c t o f sums [ 1 ] a T r C 2 q 7^ X where q_ || k x . T h i s a b i l i t y to f a c t o r i z e the e l e c t r o n d i s t r i b u t i o n f u n c t i o n means t h a t the a t t e n u a t i o n a l o n g the p r i n c i p a l d i r e c t i o n s i s i ndependent o f the d i m e n s i o n a l i t y o f the e l e c t r o n gas . The d imen - s i o n a l i t y o n l y a f f e c t s the form o f the e x p r e s s i o n f o r the cha r ge c a r r i e r d e n s i t y . The e x p r e s s i o n in [1] may be e v a l u a t e d by c o n v e r t i n g the summations i n t o i n t e g r a l s in the s t a n d a r d way. In two l i m i t i n g c a se s the i n t e - g r a t i o n s may be pe r f o rmed e x p l i c i t l y . For S « v , and f o r S » v th where v , = (2kT/m)^ i s the thermal v e l o c i t y o f the e l e c t r o n s . T h i s t h t h e r m a l v e l o c i t y t a k e s the p l a c e i n s e m i c o n d u c t o r s o f the Fermi ve- l o c i t y i n m e t a l s . S i m i l a r l y t he i m p o r t a n t mean f r e e p a t h i n s e m i - c o n d u c t o r s i s £ = V j - p 1 - At room t e m p e r a t u r e and u s i n g the f r e e e l e c t r o n mass v , - 10 7cm/S, so t h a t S/v . - 30. In o r d e r f o r t h e sound v e l o c i t y t h t h t o be o f the same o r d e r as the therm a l e l e c t r o n v e l o c i t y t he t e m p e r a t u r e must be lowered t o about 0.3K- In c a s e s i n w h i c h the t h e r m a l v e l o c i t y o f the e l e c t r o n s i s comparable t o the speed o f sound a n u m e r i c a l i n t e - g r a t i o n i s r e q u i r e d t o e v a l u a t e the a t t e n u a t i o n . 2.2 T r a n s p o r t E q u a t i o n Approach In the same way as f o r m e t a l s the 3 o i t z m a n n t r a n s p o r t e q u a t i o n can be used t o c a l c u l a t e the a c o u s t i c a t t e n u a t i o n f o r a r b i t r a r y v a l u e s o f qZ. In what f o l l o w s the sound wave i s assumed t o p r o p a g a t e a l o n g x w h i c h i s assumed t o be a c o n d u c t i n g d i r e c t i o n f o r the one and two d i m e n s i o n a l s e m i c o n d u c t o r s . The o u t s t a n d i n g d i f f e r e n c e between the s e m i c o n d u c t o r and metal r e s u l t s i s t h e l a c k o f dependence on dimen- s i o n a l i t y i n the s e m i c o n d u c t o r c a s e because o f the absence o f a Fermi s u r f a c e . The f i r s t s t e p i s t o e v a l u a t e the t r a n s p o r t t e n s o r s [4a] and [4c] o f C h a p t e r I. The f r e q u e n c y dependent c o n d u c t i v i t y t e n s o r has two d i s t i n c t n o n - z e r o components f o r two and t h r e e d i m e n s i o n a l semicon- d u c t o r s . Only one o f t h e s e components i s z e r o i n one d i m e n s i o n . They a r e where a = A(1 - i t o x ) - 1 and $ ( z ) i s the e r r o r f u n c t i o n d e f i n e d by 2 ' z a r e v a l i d f o r COT < 1 o n l y . A l s o exp ( - t 2 ) d t . The e x p r e s s i o n s f o r the c o n d u c t i v i t y o R = A 2 p x x x icoT (1 - JCOT) 2a i f the sound wave c o u p l e s t o t h e e l e c t r o n s p r i m a r i l y t h r o u g h a s e l f - c o n s i s t e n t e l e c t r o m a g n e t i c f i e l d t he a t t e n u a t i o n o f l o n g i t u d i n a l and t r a n s v e r s e waves i s found by s u b s t i t u t i n g the above t r a n s p o r t t e n s o r s i n t o [16a] and [16b] o f C h a p t e r I. In the low f r e q u e n c y (A « 1) p e r f e c t s c r e e n i n g l i m i t t h e a t t e n u a t i o n i s [ 6 ] a£ = pT7 f o r l o n g i t u d i n a l waves and 1 nm , p f o r t r a n s v e r s e waves. The t r a n s p o r t e q u a t i o n a p p r o a c h may a l s o be a p p l i e d i n the quantum l i m i t i n w h i c h A » 1. J u s t as f o r t h e one d i m e n s i o n a l metal the A -> 0 0 l i m i t must be ta k e n w i t h c a r e i n o r d e r t o a v o i d l o s i n g a s m a l l r e a l o r i m a g i n a r y p a r t . The i n t e g r a t i o n i s most e a s i l y done i n r e c t a n g u l a r c c o r d i n a t e s as f o l l o w s : S i n c e (- 3f/9e) can be f a c t o r e d the k y and k z i n t e g r a t i o n s a r e s t r a i g h t f o r w a r d . The k x i n t e g r a t i o n i s done u s i n g l i m 1 n 1 , • ! • / c\ —> r t 7-7— = P —7 + nr 6(v„ - S) T-*- 0 0 q ( v - S) - I/T qCvy " s ' The p r i n c i p a l p a r t i n t e g r a l can be done a n a l y t i c a l l y i n two l i m i t s . For S » v. t h # exp (-41)* I ^ ] xx 1 - icoT a , v t h v v - h and f o r S « v , t h a L v2 v . J X X 1 - icox a L The Oyy component o f the c o n d u c t i v i t y t e n s o r i s needed t o c a l c u l a t e the a t t e n u a t i o n o f t r a n s v e r s e waves. I t i s c a l c u l a t e d i n a s i m i l a r way. For S « v . th a = a 'o 1 yy z z 1 - iojx a > , . 2S " and f o r S » v t h a = a yy z z q ° i e x / _s i \ 1 - i t oT a [ u e x p \ v 2 h y + i 'th These e x p r e s s i o n s may be s u b s t i t u t e d d i r e c t l y i n t o [16a] and [16b] i n C h a p t e r I, t o f i n d the a c o u s t i c a t t e n u a t i o n . In the p e r f e c t s c r e e n i n g l i m i t i n the absence o f a d e f o r m a t i o n p o t e n t i a l , the a t t e n u a t i o n f o r the d i f f e r e n t c a s e s i s g i v e n below. Fo r S « v 4 , /r7 nm t h t TT L and f o r S » v t h [8] at 2 ^ ^ S" > S V t h exp ' t h / a. - — 1  V ? h 2 ~ S 2 ~ A £ A c o m p a r i s o n o f t h e s e r e s u l t s ([7] and [8]) w i t h [2] and [3] r e v e a l s t h a t i n the quantum l i m i t , e l e c t r o m a g n e t i c c o u p l i n g i s e q u i v a l e n t t o a d e f o r m a t i o n p o t e n t i a l e q u al t o kT. T h i s r e l a t i o n s h i p i s a n a l o g o u s t o the e q u i v a l e n c e i n m e t a l s o f e l e c t r o m a g n e t i c c o u p l i n g t o a de- f o r m a t i o n p o t e n t i a l e q u a l t o the Fermi e n e r g y , as d i s c u s s e d i n Chap t e r I. In s e m i c o n d u c t o r s the c a r r i e r d e n s i t y may be s m a l l enough t h a t the s c r e e n i n g i s i n c o m p l e t e and the e l e c t r o m a g n e t i c c o u p l i n g ( 4 i r n e 2 / q 2 ) i s s m a l l compared t o d e f o r m a t i o n p o t e n t i a l o r p i e z o e l e c t r i c c o u p l i n g . When the d e f o r m a t i o n p o t e n t i a l i s the dominant c o u p l i n g mechanism, th e a t t e n u a t i o n o f l o n g i t u d i n a l waves i s d e t e r m i n e d by the l a s t term i n eqn. [17] o f C h a p t e r I. We c o n s i d e r the quantum l i m i t f i r s t . S u b s t i t u t i n g the a p p r o p r i a t e t r a n s p o r t t e n s o r s c a l c u l a t e d above we f i n d f o r S « v , t h and f o r S » v , t h where R = (kT/4-nne 2)^ i s t h e Debye-Huckel s c r e e n i n g l e n g t h . The r e - s u l t s [9] and ]10] above a r e i d e n t i c a l t o the e x p r e s s i o n s [2] and [3] o b t a i n e d u s i n g quantum p e r t u r b a t i o n t h e o r y e x c e p t t h a t the d e f o r m a t i o n p o t e n t i a l C i s r e p l a c e d by a s c r e e n e d d e f o r m a t i o n p o t e n t i a l . The f a c t o r s (1 + ( q k ) " 2 ) and ( l - i o p 2 / u , 2 ) a r e d i e l e c t r i c c o n s t a n t s f o r a c l a s s i c a l ( n on-degenerate) e l e c t r o n gas i n the low f r e q u e n c y and h i g h 64 f r e q u e n c y (S » v ^ ) l i m i t s r e s p e c t i v e l y ( K i t t e l 1963) - The e x p r e s s i o n [9] has been o b t a i n e d by S p e c t o r (1966). The t r a n s p o r t e q u a t i o n method can a l s o be used to c a l c u l a t e the a t t e n u a t i o n i n the low f r e q u e n c y l i m i t (A « 1) when the d e f o r m a t i o n p o t e n t i a l i s the dominant c o u p l i n g mechanism. A f t e r e x p a n d i n g the c o n d u c t i v i t y e x p r e s s i o n [4] t o l o w e s t o r d e r i n A and s u b s t i t u t i n g the e x p a n s i o n i n t o the l a s t term i n eqn. [17] o f C h a p t e r I we f i n d t h a t The r e s u l t s [9] and [11] above w i l l be r e d e r i v e d i n C h a p t e r I I I below i n c o n n e c t i o n w i t h a c o u s t i c a m p l i f i c a t i o n i n the p r e s e n c e o f a d.c. e l e c t r i c f i e l d . The r e l a t i o n s h i p between the a t t e n u a t i o n o f l o n g i t u - d i n a l waves and t r a n s v e r s e waves i s d i s c u s s e d i n t h a t c h a p t e r , a l o n g w i t h a d e s c r i p t i o n o f how the d e f o r m a t i o n p o t e n t i a l r e s u l t s a r e m o d i f i e d i f t h e r e i s a p i e z o e l e c t r i c i n t e r a c t i o n . 2.3 M eta1-Semiconductor T r a n s i t i o n We have j u s t shown t h a t the e l e c t r o n i c c o n t r i b u t i o n t o the a t t e n u a t i o n o f a sound wave p r o p a g a t i n g a l o n g a c o n d u c t i n g d i r e c t i o n i n a s e m i c o n - d u c t o r i s i ndependent o f the d i m e n s i o n a l i t y o f the s e m i c o n d u c t o r . On the o t h e r hand i n C h a p t e r I above, the a t t e n u a t i o n o f sound i n a one d i m e n s i o n a l metal was found t o be a n o m a l o u s l y s m a l l . These r e s u l t s s u g g e s t t h a t when a one d i m e n s i o n a l c o n d u c t o r undergoes, a t r a n s i t i o n [11] f r o m a m e t a l l i c t o a s e m i c o n d u c t i n g s t a t e (TTF-TCNQ f o r example) the e l e c t r o n i c c o n t r i b u t i o n t o the u l t r a s o n i c a t t e n u a t i o n may i n - c r e a s e r a t h e r than d e c r e a s e as might be i n t u i t i v e l y e x p e c t e d . In o r d e r t o t r e a t the p r o b l e m o f a m e t a 1 - s e m i c o n d u c t o r t r a n s i t i o n i t i s n e c e s s a r y t o d r op the a s s u m p t i o n o f f r e e e l e c t r o n e n e r g y bands. In i t s p l a c e we assume a one d i m e n s i o n a l band s t r u c t u r e c o n s i s t i n g o f a s i n g l e h a l f - f i l l e d t i g h t - b i n d i n g band g i v e n by [12] e ( k ) = - e p cos k b where b i s the l a t t i c e c o n s t a n t a l o n g the c o n d u c t i n g d i r e c t i o n and 2 e p i s the b a n d w i d t h . The e f f e c t o f a m e t a 1 - s e m i c o n d u c t o r t r a n s i t i o n i s t o open a gap i n the m i d d l e o f t h e band so t h a t i n t h e semiconduc- t i n g phase the e n e r g y band i s g i v e n by [13] e ( k ) = ± / cosz kb + A Z ( T ) where 2 A(T) i s a t e m p e r a t u r e dependent e l e c t r o n i c e nergy gap. The + s i g n a p p l i e s f o r (k| > kp and the - s i g n f o r |k( < kp. I t i s v e r y d i f f i c u l t t o c a l c u l a t e the a t t e n u a t i o n i n t h i s s i t u a t i o n f o r a r b i t r a r y v a l u e s o f A. The quantum l i m i t on the o t h e r hand i s more t r a c t a b l e . We w i l l c o n s i d e r t h i s l i m i t o n l y . The d e f o r m a t i o n p o t e n t i a l c o u p l i n g C i s assumed t o be a c o n s t a n t , i ndependent o f k and the s i z e o f the e nergy gap, eventhough i t i s n o t c l e a r how good t h i s a s s u m p t i o n i s , p a r t i c u l a r l y f o r e l e c t r o n i c s t a t e s c l o s e t o the gap. As d i s c u s s e d i n t he p r e v i o u s c h a p t e r , i n the extreme quantum l i m i t sound wave a t t e n u a t i o n r e s u l t s from e l e c t r o n - p h o n o n s c a t t e r i n g p r o c e s s e s i n w h i c h both energy and momentum a r e s t r i c t l y c o n s e r v e d . The e l e c t r o n i c s t a t e s a l l o w e d by energy and momentum c o n s e r v a t i o n t o p a r t i c i p a t e i n e l e c t r o n - p h o n o n s c a t t e r i n g p r o c e s s e s d e f i n e a s u r f a c e i n k space known as the i n t e r a c t i o n s u r f a c e . In a one d i m e n s i o n a l m e t a l , when the sound wave p r o p a g a t i o n i s i n the c o n d u c t i n g d i r e c t i o n , t he i n t e r - a c t i o n s u r f a c e i s w e l l s e p a r a t e d from the Fermi s u r f a c e . As a r e s u l t t h e a t t e n u a t i o n i s a n o m a l o u s l y s m a l l . In a m e t a l l i c t i g h t - b i n d i n g band t h e r e a r e two d i s t i n c t i n t e r a c t i o n s u r f a c e s r e f l e c t i n g the f a c t t h a t t he band c o n t a i n s b o t h p o s i t i v e and n e g a t i v e c u r v a t u r e p o r t i o n s . One o f the i n t e r a c t i o n s u r f a c e s i s near the o r i g i n i n k_ space and the o t h e r i s near the zone boundary, as i l l u s t r a t e d i n F i g . 1 2 . e ( k ) *- k F i g . 12 - P o s i t i o n s o f I n t e r a c t i o n S u r f a c e s - M e t a l l i c Band I f the c o n t r i b u t i o n t o the a t t e n u a t i o n from b o t h p a r t s o f the band a r e i n c l u d e d then [14] a _ ? nm- / C V 'I PS \eF) (" iS)Q V F a c c o r d i n g t o eqn. [18] o f C h a p t e r I. In t h i s e x p r e s s i o n m" = "h 2/Epb> 2 i s the e f f e c t i v e mass f o r an e l e c t r o n near the bottom o f the band and v c = erb/"h i s the v e l o c i t y o f a Fermi e l e c t r o n . The above e x p r e s s i o n r r i s s i m i l a r t o the c o r r e s p o n d i n g r e s u l t [29] g i v e n i n the p r e v i o u s c h a p t e r f o r a f r e e e l e c t r o n gas. When an energy gap opens up a t t h e Fermi s u r f a c e two a d d i t i o n a l p l a c e s i n the band become a v a i l a b l e where t h e energy and momentum s e l e c t i o n r u l e s can be s a t i s f i e d . As i l l u s t r a t e d i n F i g . 13 t h e s e new i n t e r a c t i o n s u r f a c e s a r e c l o s e t o t h e Fermi l e v e l ( o r c h e m i c a l e(k) *» k _7T b TT b F i g . 13 - P o s i t i o n s o f I n t e r a c t i o n S u r f a c e s - S e m i c o n d u c t i n g Band 68 F i g . 14 - U l t r a s o n i c a t t e n u a t i o n as a f u n c t i o n o f t e m p e r a t u r e near a m e t a l - s e m i c o n d u c t o r t r a n s i t i o n i n a ID c o n d u c t o r . The a t t e n u a t i o n was c a l c u l a t e d from [15] and i s n o r m a l i z e d t o a Q 2ntrr pS Vep, (i)2 q v f • p o t e n t i a l ) i f the energy gap i s not too l a r g e , and hence can p r o v i d e a s i g n i f i c a n t c o n t r i b u t i o n t o t h e a t t e n u a t i o n o f sound waves. The a t t e n u a t i o n i n the s e m i c o n d u c t i n g phase can be c a l c u l a t e d by e v a l u a t i n g [18] o f C h a p t e r I u s i n g [13] f o r the e l e c t r o n i c energy band. The r e s u l t i s 2 assuming q « kp. T h i s t i m e the e f f e c t i v e mass m" = A graph o f the t e m p e r a t u r e dependence o f t h e a t t e n u a t i o n p r e d i c t e d by [15] i s shown i n F i g . 14 f o r a one d i m e n s i o n a l metal w h i c h undergoes a m e t a l - s e m i c o n d u c t o r t r a n s i t i o n a t 50K. The energy gap i s assumed t o have a B C S - l i k e t e m p e r a t u r e dependence below t h e t r a n s i t i o n t e m p e r a t u r e . The a t t e n u a t i o n i n the m e t a l l i c phase i s e x t r e m e l y s m a l l i n t h e c o l 1 i s i o n l e s s UT » 1 l i m i t . I f the c o n d i t i o n WT » 1 i s r e l a x e d t o A » 1, then the a t t e n u a t i o n i n the m e t a l l i c phase w i l l be comp a r a b l e t o [ 3 1 ] . Even though the m e t a l l i c a t t e n u a t i o n w i l l now be much l a r g e r than i n the c o l 1 i s i o n l e s s regime i t w i l l s t i l l be s m a l l compared t o the peak a t t e n u a t i o n i n the s e m i c o n d u c t i n g phase. On the o t h e r hand, th e s e m i c o n d u c t o r phase a t t e n u a t i o n i s n o t s i g n i f i c a n t l y a f f e c t e d by r e l a x i n g the WT » 1 c o n d i t i o n t o A » 1, s i n c e the i n t e r a c t i o n s u r f a c e i s a l r e a d y c l o s e t o the Fermi l e v e l , w i t h o u t any c o l l i s i o n b r o a d e n i n g . In p r a c t i c e t h e low f r e q u e n c y A « 1 l i m i t i s much more l i k e l y t o be p h y s i c a l l y r e a l i z e a b l e than the quantum l i m i t , p a r t i c u l a r l y i n a m a t e r i a l s u c h as TTF-TCNQ. where the e l e c t r o n mean f r e e p a t h i s o n l y a few l a t t i c e c o n s t a n t s . From e a r l i e r work i n C h a p t e r I we e x p e c t the e l e c t r o - m a g n e t i c c o u p l i n g mechanism t o be f a r l a r g e r than the d e f o r m a t i o n p o t e n t i a l c o u p l i n g a t low f r e q u e n c i e s . In t h i s c a s e the m e t a l l i c a t t e n u a t i o n i s v e r y low so t h a t one might a l s o e x p e c t t o see an i n c r e a s e i n the low f r e q u e n c y a t t e n u a t i o n , t o something a p p r o a c h i n g t h e t h r e e d i m e n s i o n a l v a l u e , i n g o i n g from the metal t o the sem i - c o n d u c t o r . CHAPTER I I I A m p l i f i c a t i o n 3.1 I n t r o d u c t i o n I t i s w e l l known t h a t under c e r t a i n c i r c u m s t a n c e s u l t r a s o n i c waves may be a m p l i f i e d i n m e t a l s o r s e m i c o n d u c t o r s i f t h e c o n d u c t i o n e l e c t r o n s have a d.c. d r i f t v e l o c i t y ( H u t s o n , McFee and White 1961> Vrba and H a e r i n g 1973). The a m p l i f i c a t i o n o f sound waves may be r e g a r d e d as a n e g a t i v e a t t e n u a t i o n t h a t can o c c u r i n the p r e s e n c e o f a d.c. e l e c t r i c f i e l d . A c c o r d i n g l y the methods d e s c r i b e d i n C h a p t e r I f o r c a l c u l a t i n g t h e a t t e n u a t i o n may be a p p l i e d t o the a m p l i f i c a t i o n p r oblem. In g e n e r a l t h e a t t e n u a t i o n and hence the a m p l i f i c a t i o n o f sound waves by e l e c t r o n s depends on the r e l a t i o n between the e l e c t r o n mean f r e e p a t h t an<H t h e sound w a v e l e n g t h 2-rr/q. The t h e o r y o f a c o u s t i c a m p l i f i c a t i o n i n t h e p r e s e n c e o f a d.c. e l e c t r i c f i e l d has been worked o u t by W e i n r e i c h (1956) and White (1962) f o r the low f r e q u e n c y {ql « 1) l i m i t and by P i p p a r d (1963) f o r the h i g h f r e q u e n c y l i m i t [qt « 1). S p e c t o r (1962) has used the Boltzmann e q u a t i o n t o produce a t h e o r y t h a t i s v a l i d f o r a r b i t r a r y qt. The S p e c t o r (1962) t h e o r y i s v e r y complex f o r the f o l l o w i n g r e a s o n . When a d.c. e l e c t r i c f i e l d i s i n t r o d u c e d i n t o the t r a n s p o r t t r e a t m e n t o f u l t r a s o n i c a t t e n u a t i o n a l a r g e number o f a d d i t i o n a l terms o f e q u a l o r d e r a r e g e n e r a t e d , none o f w h i c h can be n e g l e c t e d . In t h i s c h a p t e r we o u t l i n e a t h e o r y where the d.c. e l e t r i c f i e l d i s t a k e n i n t o a c c o u n t by s h i f t i n g t he d i s t r i b u t i o n f u n c t i o n i n k space. T h i s a p p r o a c h e l i m i n a t e s the need t o d e a l w i t h a l a r g e number o f new terms a r i s i n g 72 from t h e d.c. f i e l d , and v a s t l y s i m p l i f i e s t he problem. U s i n g the new a p p r o a c h , we a r e a b l e t o c o n f i r m S p e c t o r ' s r e s u l t s w h i c h have n e v e r been v e r i f i e d p r e v i o u s l y . In a d d i t i o n o u r a p p r o a c h i s v a l i d f o r s t r o n g d.c. e l e c t r i c f i e l d s i n the same way t h a t the method o f S p e c t o r (19&8) i s v a l i d f o r s t r o n g f i e l d s . In t h i s c h a p t e r we c a l c u l a t e the a t t e n u a t i o n ( a m p l i f i c a t i o n ) o f sound waves i n n - t y p e s e m i c o n d u c t o r s and t h r e e d i m e n s i o n a l n e a r l y - f r e e - e l e c t r o n m e t a l s , i n the p r e s e n c e o f a d.c. e l e c t r i c f i e l d . The c a l c u l a t i o n f o l l o w s the t r a n s p o r t e q u a t i o n approach d e v e l o p e d i n C h a p t e r I. In b o t h t h e metal and s e m i c o n d u c t o r the c o n d u c t i o n e l e c t r o n s a r e m o d e l l e d by a f r e e e l e c t r o n gas and n o n - e l e c t r o n i c c o n t r i b u t i o n s t o t h e a t t e n u a t i o n a r e i g n o r e d . In t h e metal a s e l f - c o n s i s t e n t e l e c t r i c f i e l d i s used t o c o u p l e the sound wave t o the c o n d u c t i o n e l e c t r o n s . In the s e m i c o n d u c t o r a d e f o r m a t i o n p o t e n t i a l t e n s o r i s assumed t o be t h e dominant c o u p l i n g mechanism. F i r s t , the p r o b l e m w i l l be s e t up f o r a r b i t r a r y e l e c t r o n s t a t i s t i c s and b o t h s e l f - c o n - s i s t e n t f i e l d and d e f o r m a t i o n p o t e n t i a l c o u p l i n g . 3.2 T r a n s p o r t E q u a t i o n In the. p r e s e n c e o f a sound wave d e s c r i b e d by a l o c a l l a t t i c e v e l o c i t y û  « exp [ i (qx-iot) ] , t h e Boltzmann t r a n s p o r t e q u a t i o n f o r e l e c t r o n s i s [1] 73 where f i s the e l e c t r o n d i s t r i b u t i o n f u n c t i o n , i s a s e l f - c o n - s i s t e n t e l e c t r i c f i e l d and £ i s a d e f o r m a t i o n p o t e n t i a l t e n s o r . In the r e l a x a t i o n time a p p r o x i m a t i o n used here s c a t t e r i n g p r o c e s s e s c a use the e l e c t r o n d i s t r i b u t i o n f u n c t i o n t o r e l a x t o the l o c a l e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f . As i n C h a p t e r I, S e c t i o n 1.2, f can be a p p r o x i m a t e d by [2] 7(v,r_,t) = f 0 ( v - u ; y ( r , t ) ) 9 f o / 2 e F \ where f i s the e q u i l i b r i u m e l e c t r o n d i s t r i b u t i o n f u n c t i o n i n the absence o f a sound wave, p i s a s e l f - c o n s i s t e n t c h e m i c a l p o t e n t i a l , Cp i s the e l e c t r o n Fermi e n e r g y , n i s t h e e q u i l i b r i u m e l e c t r o n number d e n s i t y and n 1 i s a s m a l l o s c i l l a t o r y component o f t h e e l e c t r o n dens i t y . A d.c. e l e c t r i c f i e l d w i l l be i n t r o d u c e d i n t o e q n . [ l ] by p o s t u l a t i n g t h a t the s o l e e f f e c t o f the f i e l d i s t o s h i f t t he e q u i l i b r i u m e l e c t r o n d i s t r i b u t i o n by the a v e r a g e d r i f t v e l o c i t y = - —• JE Q where EQ i s the d.c. f i e l d . T h i s a s s u m p t i o n i s c o r r e c t t o f i r s t o r d e r i n the e l e c t r i c f i e l d , . In t h i s a p p r o x i m a t i o n , the d.c. f i e l d may be i n t r o d u c e d by r e - d e f i n i n g the l o c a l e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f by 9f d [3] f ( v , r , t ) * ^ - ^ ( . ( v - ^ - u + l ^ n , ) f o r a t h r e e d i m e n s i o n a l metal where is a doppler s h i f t e d e q u i l i b r i u m d i s t r i b u t i o n function. The Boltzmann equation [ 1 ] with f defined by [3] may now be solved to f i r s t order by s u b s t i t u t i n g where f° is the d.c. part of the perturbation from equilibrium, and fj 1 tt exp [ i (qx-cot) ] is the a.c. part of the d i s t r i b u t i o n function. We f i n d that f? - f d " f 1 o o and [4] f} 9e qq-C.-u mu\ / \ 2 e F ex ET + " T = ' V " L " 7 T n l .— 1 eico ex/ \ dy 3 n \ ^1-iiox+iqvx^ - 1 . The e l e c t r o n i c contribution to the a.c. current is given by [5] J = 2 e e " ~ f2lrT3 J v f} d 3k. Now define a doppler s h i f t e d a.c. component to the d i s t r i b u t i o n function by ex lE i + qq/£*u_ mu e i to et (̂ 1 - i (to-io^Jx + iqvx^ 2 e F 8e where wd = a.- Then the c u r r e n t e x p r e s s i o n [5] may be r e w r i t t e n as and J = J - n i e v — e — e 1 —< where [6] J d — e eiio mu n i e S d R d d 3 k f i =T2fy3 j d 3 k f ^ d 2 e 2 r £ = J 2 ^ J d 3 k v v 1 - i (to-Wj)x + i q v x V de - ~ 6Tr 3n S d d d k 3f 1 - Ru-cô x + iqvx \ 3 (- ° and S = (co-to^j)/q i s the d o p p l e r s h i f t e d sound phase v e l o c i t y . The t o t a l a.c. c u r r e n t d e n s i t y due t o the sound wave i s the sum o f t h e e l e c t r o n i c c u r r e n t and the background i o n c u r r e n t . Thus [7] J = J + n e u — — e — where i s t h e t o t a l c u r r e n t , and n i s t h e i o n number d e n s i t y . i n a monovalent metal ( n - t y p e s e m i c o n d u c t o r ) the i o n ( i o n i z e d i m p u r i t y ) number d e n s i t y w i l l e q u a l t he e q u i l i b r i u m e l e c t r o n number d e n s i t y . In t he p r e s e n c e o f a sound wave the i o n number d e n s i t y w i l l c o n t a i n a s m a l l o s c i l l a t o r y component g i v e n by nu/S where u/S i s t h e s t r a i n i n d u c e d by t h e sound wave. The o s c i l l a t o r y term i n the e l e c t r o n number d e n s i t y may be found by u s i n g t h e c o n t i n u i t y e q u a t i o n f o r e l e c t r o n s tonje + g . * ^ = 0. T h i s e q u a t i o n can a l s o be w r i t t e n i n terms o f the d o p p l e r s h i f t e d q u a n t i t i e s (to-io,)nie + q ' J ^ = 0 d — e or [9] n i e S d = - / u where J^.. i s the component o f p a r a l l e l t o q_. A l s o t h e e l e c t r i c f i e l d w h i c h i s c o n s i s t e n t w i t h the t o t a l c u r r e n t may be found from M a x w e l l ' s e q u a t i o n s t o be [ 1 0 a ] E N - ^ L / j e | | + n e u H ) and [10b] The nex t s t e p i n o b t a i n i n g the net power f l o w from the sound wave t o t h e e l e c t r o n gas i s t o o b t a i n e x p r e s s i o n s f o r the s e l f - c o n s i s t e n t e l e c t r i c f i e l d and c u r r e n t i n terms o f the l o c a l l a t t i c e v e l o c i t y . For a f r e e e l e c t r o n g a s , and a sound w a v e v e c t o r q_ i n the x d i r e c t i o n , t h e c o n d u c t i v i t y t e n s o r i s d i a g o n a l . In t h i s c a s e t h e v e c t o r n o t a t i o n may be dropped and the components can be treated independently. For longitudinal waves (q_ II u) eqn. [9] can be used to eliminate n^ from [6]. Thus q2C u [11] ex = old E l x + XX X eito mu ex where ald = I (1 - Rd ).. Only the x components of Ej and u_ are X X X non-zero for longitudinal waves. Substitute [10a] for E then X [12] where t ex n e u T c^d x _x 4 d L °o fori o l d (l + D S X \ XX)_ id X 1 + •^LL a l a is a d i e l e c t r i c constant, y ~ 1 -toy x . • q_'v to n?.r and D = 1* " X o . Also i f [12] is substituted into [11] 'o m xx ^ T r n e 2 we find the following expression for the total driving f i e l d f e l t by the electrons: [13] E l x + q 2C u XX X e i to n e u d d ,ld 1 + D xx ya oJ The self-consistent e l e c t r i c f i e l d and current associated with a transverse sound wave may be found in exactly the same way. When a transverse wave is polarized in the y direction and propagates in the x direction there are e l e c t r i c currents in both the x and y directions, Using the electron continuity equation and [6], we find that the doppler shifted current in the x direction i s , q 2 c w t t u ex xy y e i to I f t he c u r r e n t and e l e c t r i c f i e l d a r e r e q u i r e d t o s a t i s f y the Max- w e l l e q u a t i o n [10a] w h i c h i n t h i s c a s e reduces t o [15] E AiTi l x co ex t h e n the t o t a l c u r r e n t i n the x d i r e c t i o n i s [16] ex ,ld <x d (Q2Z U xy y e i c o y S i m i l a r l y , the s e l f - c o n s i s t e n t c u r r e n t i n the y d i r e c t i o n i s , [17] J = a E l v ey y \ l v ex mu \ v , S ex where i s t h e y component o f t h e e l e c t r o n d r i f t v e l o c i t y v^, and J i s g i v e n by [ 1 6 ] . By u s i n g the s e l f - c o n s i s t e n c y r e q u i r e m e n t [10b] 6X we f i n d t h a t [18] J ey n e u / i , • / C \ 2 J ° D \ V ^ J y_ / *t7T i /S\^ d _ y dy ex e t I co \c) % o0J s t t 4in /S\2 d where e = 1 - — 1 a . The e l e c t r i c f i e l d i s found by s u b s t i t u t i n g y co \c J y [18] back i n t o [ 1 7 ] . Thus i»iri / S \ 2 [19] E l y co et \c a \ v, n e u (1 - J S ex We a r e now ready t o e v a l u a t e the energy t r a n s f e r between the sound wave and the e l e c t r o n gas. 3.3 Energy T r a n s f e r The power t r a n s f e r r e d f r o m the sound wave t o the e l e c t r o n s i s The f i r s t term i s t h e work done per u n i t time by t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d and d e f o r m a t i o n p o t e n t i a l g r a d i e n t on t h e e l e c t r o n s . The second term i n [20] i s due t o the i n t e r a c t i o n o f t h e l a t t i c e wave w i t h t h e e l e c t r o n s t h r o u g h c o l l i s i o n s (see C h a p t e r I) and i s i m p o r t a n t o n l y f o r h i g h f r e q u e n c i e s (co ~ o ^ ) . N o t i c e t h a t a l l d.c. f i e l d s and c u r r e n t s have been o m i t t e d from [20]. In t h e l i n e a r a p p r o x i m a t i o n c o n s i d e r e d h e r e , d.c. q u a n t i t i e s c o n t r i b u t e terms i n the power e x p r e s s i o n w h i c h e i t h e r have a z e r o time a v e r a g e o r r e - p r e s e n t the ohmic l o s s e s a s s o c i a t e d w i t h the d r i f t i n g e l e c t r o n d i s t r i b u t i o n . The a t t e n u a t i o n ( a m p l i f i c a t i o n ) f a c t o r f o r u l t r a s o n i c waves i s o b t a i n e d f r o m [20] by d i v i d i n g by t h e a c o u s t i c e n e r g y f l u x . Thus where a i s the a t t e n u a t i o n . R a t h e r than e v a l u a t i n g t h e a t t e n u a t i o n w i t h both s e l f - c o n s i s t e n t f i e l d and d e f o r m a t i o n p o t e n t i a l c o u p l i n g , we c o n s i d e r two l i m i t i n g c a s e s . In t h e f i r s t l i m i t , the d e f o r m a t i o n p o t e n t i a l i s assumed t o be n e g l i g i b l e compared w i t h the comp a r a b l e e l e c t r o m a g n e t i c c o u p l i n g e n e r g y — . T h i s a s s u m p t i o n i s v a l i d up t o h i g h f r e q u e n c i e s [20] a = P / ( J r p | u | 2 ) (- 10 Ghz) i n m e t a l s . In the second l i m i t i n g c a s e the d e f o r m a t i o n 4 i m e 2 p o t e n t i a l i s assumed t o be much g r e a t e r than and hence i s th e dominant c o u p l i n g mechanism. T h i s i s a good a s s u m p t i o n a t h i g h f r e q u e n c y i n s e m i c o n d u c t o r s . F i r s t we c o n s i d e r l o n g i t u d i n a l waves i n m e t a l s and n e g l e c t the d e f o r m a t i o n p o t e n t i a l . The a t t e n u a t i o n o f l o n g i t u d i n a l waves i s found by s u b s t i t u t i n g [10a] and [13] i n t o [ 2 0 ] . I f t h e d e f o r m a t i o n An i p o t e n t i a l i s s e t e q u a l t o z e r o and e = 1 + r o to aQ, then [21] nm al " P S T Re r i d ' ITd" T h i s e x p r e s s i o n s h o u l d be compared w i t h [16a] i n S e c t i o n 1.2, t h e a n a l o g o u s e x p r e s s i o n i n the absence o f a d.c. f i e l d . The a t t e n u a t i o n may be e v a l u a t e d by s u b s t i t u t i n g the a p p r o p r i a t e t r a n s p o r t t e n s o r s g i v e n i n S e c t i o n 1.4, i n t o [ 2 1 ] . For a t h r e e d i m e n s i o n a l metal t o t h e l o w e s t o r d e r i n S/vp, f o r to « o x and A = at, the r e s u l t i s r „ „ n nm T A 2 t a n " 1 A ,.. s t W al = JST [ 3 (A - t a n-l - A T ° " y ) " \ f o r a r b i t r a r y A, where u = c p v , /to. For A « 1 eqn. [22] reduces t o nm -u + A 2 (1 and f o r A » 1 nm IT . /. \ S i m i l a r l y , by s u b s t i t u t i n g the t r a n s p o r t t e n s o r s f o r the one and two d i m e n s i o n a l m e t a l s from S e c t i o n 1.4 i n t o eqn. [21 ] , one o b t a i n s t h e a t t e n u a t i o n f o r t h e lower d i m e n s i o n a l i t y m e t a l s i n a d.c. e l e c t r i c f i e l d . The r e s u l t s a r e summarized i n T a b l e II f o r a r b i t r a r y A i n the co « oJ, l i m i t . The t h r e e d i m e n s i o n a l r e s u l t [22] i s i d e n t i c a l w i t h the e x p r e s s i o n s o b t a i n e d by S p e c t o r (1962), i n the a p p r o p r i a t e 1imi t s . TABLE I I A t t e n u a t i o n i n 1, 2 and 3 D i m e n s i o n a l M e t a l s i n a D.C. E l e c t r i c F i e l d / nm . . Y 1D 2D 3D A 2 ( 1 - u ) A 2 t a n - 1 A (1 - u) ~ V 2 ( / I + A z - 1 ) 3(A - t a n " 1 A In the absence o f a d e f o r m a t i o n p o t e n t i a l , the d.c. e l e c t r i c f i e l d has no s i g n i f i c a n t e f f e c t on the a t t e n u a t i o n o f t r a n s v e r s e waves. To show t h i s , s u b s t i t u t e [17] and [18] i n t o [20], then a. = nm pSx Re i k . 4TT i / S \ 2 where tz =1 [ — J a . To l o w e s t o r d e r i n S/v,. the a t t e n u a t i o n i s o co \ c / o F the same as i n the z e r o f i e l d c a s e d i s c u s s e d i n S e c t i o n 1.4. In c o n v e n t i o n a l m e t a l s u must be v e r y s m a l l compared w i t h one i n o r d e r t o a v o i d u n r e a l i s t i c a l l y l a r g e ohmic power d i s s i p a t i o n . T h e r e - 82 f o r e , i t f o l l o w s from [22] t h a t u n l e s s u > 1 can be a c h i e v e d , a c o u s t i c a m p l i f i c a t i o n t h r o u g h i n t e r a c t i o n w i t h a d.c. f i e l d i s i m p r a c t i c a l i n m e t a l s , e x c e p t p o s s i b l y a t low f r e q u e n c i e s ( A « 1 ) . However, i n t h i s c a s e t h e r e i s an a d d i t i o n a l c o m p l i c a t i o n . In the low f r e q u e n c y l i m i t i n the p r e s e n c e o f a l a r g e d.c. f i e l d the a t t e n u a t i o n becomes s e n s i t i v e t o a second o r d e r term i n J such as — e t h e a c o u s t o e l e c t r i c c u r r e n t ( M i k o s h i b a , 1959). For example i f u - 1 then the s e l f - c o n s i s t e n t f i e l d ET_ - (u/S) A 2 .E . S i n c e the s t r a i n u/S < 1 0 ~ 5 and A « 1, the s e l f - c o n s i s t e n t e l e c t r i c f i e l d w i l l be much s m a l l e r than t h e d.c. e l e c t r i c f i e l d E q . The l a r g e d.c. f i e l d c o u p l e d w i t h a s m a l l s econd o r d e r d.c. term i n c o u l d g i v e r i s e t o a s i g n i f i c a n t c o n t r i b u t i o n t o t h e a t t e n u a t i o n , not i n c l u d e d i n [ 2 2 ] . F o r maximum a c o u s t i c g a i n one w o u l d l i k e a m a t e r i a l w i t h a l a r g e e l e c t r o n - s o u n d wave c o u p l i n g , h i g h m o b i l i t y c a r r i e r s , and a r e l a t i v e l y low c a r r i e r d e n s i t y t o l i m i t the ohmic l o s s e s . These c h a r a c t e r i s t i c s a r e a v a i l a b l e i n some s e m i c o n d u c t o r s . In n o n - p i e z o e l e c t r i c s e m i c o n d u c t o r s we e x p e c t the d e f o r m a t i o n p o t e n t i a l t o be t h e most i m p o r t a n t c o u p l i n g mechanism between the e l e c t r o n s and the sound wave. For d e f o r m a t i o n p o t e n t i a l c o u p l i n g the a t t e n u a t i o n o f l o n g i t u d i n a l waves i s found by s u b s t i t u t i n g [13] i n t o [20] and a ssuming D » 1. Thus xx ^Trne 2 [23] a p S t nm D 2 xx Re In o r d e r t o e v a l u a t e t h i s e x p r e s s i o n one needs t o know the t r a n s p o r t t e n s o r components a d and R d, w h i c h a r e d e f i n e d by the i n t e g r a l s f o l l o w i n g [ 6 ] . A f t e r d o i n g the i n t e g r a t i o n s w i t h Boltzmann s t a t i s t i c s f o r t he e l e c t r o n s as i n S e c t i o n 2.2, we f i n d t h a t i n the l i m i t coxy < 1 [24] 1 • ~3 i a I - iw y t a z - / i r " exp where a = q£/(l - icoyt) and the mean f r e e p a t h £ i s d e f i n e d i n terms o f a therm a l v e l o c i t y as £ = (2kT/m)^x, and *(x) = 7 r exp ( - t 2 ) d t In the quantum l i m i t i n w h i c h q£, cox » 1, t h e c o n d u c t i v i t y i s [25] d  2 ° o °x = "qT / r f t h 'th where v , = (kT/M) 2and v , » S has been assumed. In most s e m i c o n d u c t o r s t h t h the th e r m a l v e l o c i t y o f the e l e c t r o n s v ^ i s much l a r g e r than the speed o f sound S f o r t e m p e r a t u r e s above l i q u i d h e l i u m t e m p e r a t u r e . The x com- ponent o f t h e d i f f u s i o n v e c t o r R d i s r e l a t e d t o the c o n d u c t i v i t y by [26] R = L J x q £ 2 iwyx (1 - toyx) 2aQ and the o t h e r components o f R_ a r e z e r o . The a t t e n u a t i o n o f l o n g i t u d i n a l waves can now be e v a l u a t e d i n the low f r e q u e n c y l i m i t (q£ « 1) and i n the quantum l i m i t (q£ » 1) by s u b s t i t u t i n g [24] and [25] r e s p e c t i v e l y i n t o [23] and [ 2 6 ] . In the q£ « 1 l i m i t 84 /C \ 2 (qR) 1* (1 - y ) W al " pSx \~kT (l + M ) 2 ) 2 + ( ^ ) 2 ( 1 - M ) : where R = [kT/(4-rme 2)] 2 i s the Debye-H'uckel s c r e e n i n g l e n g t h and (4irne 2/m)^ i s the plasma f r e q u e n c y . S i m i l a r l y i n the q£» 1 0) P 1 imi t t 2 8 ^ a£ ~ T P~S7 VTTJ (, + ( q R ) - 2 ) 2 f o r S « v l L. t h The above p r o c e d u r e w i l l now be a p p l i e d t o t r a n s v e r s e waves i r . s e m i c o n d u c t o r s . The a t t e n u a t i o n o f t r a n s v e r s e waves i s found by s u b s t i t u t i n g the c u r r e n t e x p r e s s i o n s [16] and [18] and the e l e c t r i c f i e l d e x p r e s s i o n s [15] and [19] i n t o the power e q u a t i o n [ 2 0 ] . In the c a s e D » 1, the c u r r e n t and f i e l d p a r a l l e l t o the p r o p a g a t i o n d i r e c t i o n g i v e the i m p o r t a n t c o n t r i b u t i o n t o the a t t e n u a t i o n . The a t t e n u a t i o n i s g i v e n by [29] cr = D 2 _ L J t pSx xy Re A co m p a r i s o n o f t h i s r e s u l t w i t h the c o r r e s p o n d i n g r e s u l t f o r l o n g i - t u d i n a l waves, [ 2 3 ] , r e v e a l s t h a t the two a r e i d e n t i c a l e x c e p t t h a t t h e d i a g o n a l component C o f the d e f o r m a t i o n p o t e n t i a l t e n s o r i s xx r e p l a c e d by the o f f d i a g o n a l component C . A c c o r d i n g l y the ql « 1 xy and qt » 1 l i m i t s g i v e n i n [27] and [28] r e s p e c t i v e l y , a l s o a p p l y t o t r a n s v e r s e waves p r o v i d e d t h a t C i s r e p l a c e d by C xx xy A l t h o u g h t h e p r e c e e d i n g a t t e n u a t i o n e x p r e s s i o n s have been d e r i v e d assuming d e f o r m a t i o n p o t e n t i a l e l e c t r o n - s o u n d wave c o u p l i n g , t h e y may be r e a d i l y m o d i f i e d t o i n c l u d e a p i e z o e l e c t r i c c o u p l i n g c o n s t a n t . In p i e z o e l e c t r i c m a t e r i a l s such as CdS where the p i e z o e l e c t r i c i n t e r - a c t i o n i s much l a r g e r than the d e f o r m a t i o n p o t e n t i a l i n t e r a c t i o n a t a l l a t t a i n a b l e f r e q u e n c i e s , t he d e f o r m a t i o n p o t e n t i a l f a c t o r qC can be r e p l a c e d by ed , where d i s the p i e z o e l e c t r i c c o n s t a n t . When xy xy t h i s r e p l a c e m e n t i s made the e x p r e s s i o n f o r the a t t e n u a t i o n i n the low f r e q u e n c y l i m i t [27] reduces t o t h a t o b t a i n e d by Whi t e (1962) u s i n g a d i f f e r e n t method. Note t h a t i n the regime i n w h i c h t h e d e f o r m a t i o n p o t e n t i a l ( o r p i e z o e l e c t r i c ) c o u p l i n g i s dominant, u l t r a s o n i c waves a r e a m p l i f i e d when the e l e c t r o n d r i f t v e l o c i t y i n the d i r e c t i o n o f the sound p r o p a g a t i o n i s g r e a t e r than the sound phase v e l o c i t y . T h i s a m p l i f i c a t i o n has been o b s e r v e d i n CdS (Vrba and H a e r i n g , 1973) and the measured a c o u s t i c g a i n i s c o n s i s t e n t w i t h eqn.[27] (Hughes, 1975). 3-4 C o n c l u s i o n To summarize, we have o u t l i n e d a t r a n s p o r t e q u a t i o n a p proach t o the pro b l e m o f a c o u s t i c a m p l i f i c a t i o n i n a d.c. e l e c t r i c f i e l d , t h a t i s much l e s s complex than any t h a t i s c u r r e n t l y a v a i l a b l e ( S p e c t o r 1962, 1968). The new t r e a t m e n t d u p l i c a t e s t he r e s u l t s o f the e a r l i e r work and i s s i m p l e enough t o be r e a d i l y g e n e r a l i z e d t o a p p l y t o one and two d i m e n s i o n a l m e t a l s . F u r t h e r m o r e the method i s not r e s t r i c t e d t o 86 m e t a l s and can be e a s i l y a p p l i e d t o s e m i c o n d u c t o r s , u n l i k e the e a r l i e r t r e a t m e n t . In our c a l c u l a t i o n t h e r e i s no need to make any s p e c i a l a s s u m p t i o n s about the d i r e c t i o n o f the d.c. d r i f t f i e l d . P A R T B MEASUREMENTS ON TTF-TCNQ CHAPTER I E x p e r i m e n t a l Method 1.1 C a p a c i t i v e Measurement T e c h n i q u e C a p a c i t i v e t r a n s d u c e r s have been w i d e l y used f o r e x c i t i n g and d e t e c t i n g s m a l l a m p l i t u d e v i b r a t i o n s (Barmatz and Chen 1974, C a n t r e l l and B r e a z e a l e 1977, McGuigan e t a l 1977). The two main reasons f o r the p o p u l a r i t y o f c a p a c i t i v e t r a n s d u c e r s a r e t h e i r p r a c t i c a l s i m p l i c i t y and h i g h s e n s i t i v i t y t o s m a l l d i s p l a c e m e n t s . In the c o n v e n t i o n a l c a - p a c i t i v e d i s p l a c e m e n t d e t e c t o r , a l a r g e d.c. b i a s v o l t a g e i s a p p l i e d between the t e s t o b j e c t and a nearby e l e c t r o d e . O s c i l l a t i o n s i n the p o s i t i o n o f the t e s t o b j e c t w i l l m odulate t h e c a p a c i t a n c e between t h e t e s t o b j e c t and the e l e c t r o d e and cause a c u r r e n t t o f l o w i n a l a r g e s e r i e s r e s i s t o r . The v o l t a g e s i g n a l on t h e r e s i s t o r becomes p r o g r e s s i v e l y s m a l l e r and h a r d e r t o measure as the f r e q u e n c y o f o s c i l l a t i o n o f t h e t e s t o b j e c t i s low e r e d and the p i c k u p c a p a c i t a n c e d e c r e a s e s . In t h i s p aper we d e s c r i b e an a l t e r n a t i v e c a p a c i t i v e d e t e c t i o n scheme w h i c h does no t s i g n i f i c a n t l y l o s e s e n s i t i v i t y a t low f r e q u e n c i e s and low c a p a c i t a n c e v a l u e s . The new ap p r o a c h t o the c a p a c i t i v e v i b r a t i o n p i c k u p was d e s i g n e d t o make a c c u r a t e sound v e l o c i t y and a b s o r p t i o n measurements i n s i n g l e c r y s t a l s o f TTF-TCNQ. A c a p a c i t i v e t e c h n i q u e was chosen t o a v o i d h a v i n g t o make good low l o s s a c o u s t i c bonds t o the s m a l l and somewhat i r r e g u l a r TTF-TCNQ c r y s t a l s . In o r d e r t o be a b l e t o see l o n g i t u d i n a l modes.in the e l o n g a t e d TTF-TCNQ p l a t e l e t s , the p i c k u p t r a n s d u c e r must 89 F ig . 16 - D r i v e and b r i d g e d e t e c t o r c i r c u i t o p e r a t e e f f e c t i v e l y w i t h a t o t a l c a p a c i t a n c e o f l e s s than 0.1 p f . In the rem a i n d e r o f t h i s s e c t i o n we d e s c r i b e the new c a p a c i t i v e d i s p l a c e m e n t m e a s u r i n g t e c h n i q u e , and the e l e c t r o s t a t i c v i b r a t i o n e x c i t a t i o n scheme. ( i ) E l e c t r o n i c s A c o u s t i c r e s o n a n c e s were e x c i t e d i n the samples e l e c t r o s t a t i c a l l y by r e l y i n g on the f o r c e between the p l a t e s o f a ch a r g e d c a p a c i t o r . As shown s c h e m a t i c a l l y i n F i g . 15 the l e f t hand end o f the sample forms one s i d e o f t h e c a p a c i t o r and a nearby e l e c t r o d e forms t h e o t h e r s i d e . When t h i s c a p a c i t o r i s d r i v e n by the o s c i l l a t o r shown as e^ i n F i g . 15, t h e r e i s a p e r i o d i c f o r c e on t h e sample. T h i s o s c i l l a t o r - a f r e q u e n c y s y n t h e s i z e r - i s o p e r a t e d i n the f r e q u e n c y range 0-10 MHz. To i n c r e a s e t h e f o r c e on the sample t h e s y n t h e s i z e r o u t p u t v o l t a g e i s s t e p p e d up by a f a c t o r o f f o u r t o a maximum o f 110 V peak-to-peak by a Va r i - L , L F - 4 5 2 wideband t r a n s f o r m e r . To f u r t h e r i n c r e a s e t h e d r i v i n g f o r c e and t o reduce t h e r e l a t i v e i m p o r t a n c e o f t h e second harmonic component o f t h e d r i v i n g f o r c e , a d.c. b i a s o f up t o hOO V can be superimposed on the a.c. s i g n a l a c r o s s the d r i v e c a p a c i t o r . The v i b r a t i o n o f the sample i s d e t e c t e d by u s i n g an r f c a r r i e r s i g n a l t o measure changes i n a p i c k u p c a p a c i t a n c e . In the s i m p l e s t c a s e o f a p a r a l l e l p l a t e c a p a c i t o r , t he c a p a c i t a n c e w i l l be i n v e r s e l y p r o p o r t i o n a l t o the d i s t a n c e between the sample and the p i c k u p e l e c t r o d e . When the sample moves the d e t e c t o r c a p a c i t a n c e w i l l change l e a d i n g t o a c o r r e s p o n d i n g change i n i t s impedance. T h i s impedance v a r i a t i o n may be d e t e c t e d u s i n g the c i r c u i t shown s c h e m a t i c a l l y i n F i g . 15- The c a r r i e r s i g n a l g e n e r a t o r marked e c i n F i g . 15 o p e r a t e s i n the f r e q u e n c y band 300-1000 MHz, and i t s o u t p u t i s c o n n e c t e d t o the sample. When the sample v i b r a t e s , t he v a r y i n g impedance o f t h e p i c k u p c a p a c i t o r , a m p l i t u d e modulates t h e r f s i g n a l f l o w i n g t h r o u g h i t . The a m p l i t u d e m o d u l a t i o n on the c a r r i e r i s r e c o v e r e d by a S c h o t t k y b a r r i e r d i o d e d e t e c t o r and low pass f i l t e r ( see A p p e n d i x § 1.(i) f o r c i r c u i t d i a g r a m ) . Hence any d i s p l a c e m e n t o f the sample i s r e f l e c t e d i n the d i o d e o u t p u t v o l t a g e . A l t h o u g h t h i s s i m p l e d e t e c t i o n scheme i s adequate f o r many p u r p o s e s , t h e s e n s i t i v i t y can be improved by u s i n g a b r i d g e c i r c u i t as shown s c h e m a t i c a l l y i n F i g . 16. In t h i s c i r c u i t t he r f s i g n a l t r a n s m i t t e d t h r o u g h t h e sample p i c k u p c a p a c i t o r i s compared .with t h e s i g n a l t h r o u g h a s t a t i o n a r y dummy c a p a c i t o r . I f t h e sample and dummy c a p a c i t a n c e s a r e e q u a l t h e o u t p u t from the b a l a n c e d / u n b a l a n c e d wideband t r a n s f o r m e r ( V a r i - L , HF-122) w i l l be z e r o . Of c o u r s e the i n d u c t i v e and r e s i s t i v e components o f the impedance i n the two arms must a l s o be e q u a l f o r a n u l l o u t p u t . In p r i n c i p l e a tuned t r a n s f o r m e r w o u l d g i v e b e t t e r s e n s i t i v i t y , however, a wideband t r a n s f o r m e r was used i n the TTF-TCNQ measurements because i t i s more c o n v e n i e n t when d e a l i n g w i t h a v a r i e t y o f samples o f d i f f e r e n t s i z e and shape. The o u t p u t o f the t r a n s f o r m e r i s f o l l o w e d by a low n o i s e UHF ampl i f ier.. ( A v a n t e k , AMM-1010) and then by a d i o d e d e t e c t o r . The b r i d g e improves t h e s e n s i t i v i t y o f the d e t e c t o r f o r two main r e a s o n s . The f i r s t r eason i s t h a t the r f s i g n a l g e n e r a t o r n o i s e i s c a n c e l l e d o u t when the b r i d g e i s b a l a n c e d , s i n c e t h i s n o i s e i s the same i n bo t h arms. S e c o n d l y the s i g n a l i s a s m a l l d e v i a t i o n from a n u l l whereas w i t h o u t the b r i d g e the s i g n a l i s a s m a l l r i p p l e on top o f a l a r g e r f c a r r i e r . S i n c e the b r i d g e o u t p u t s i g n a l i s s m a l l i t may be a m p l i f i e d i n a low n o i s e r f a m p l i f i e r b e f o r e d e t e c t i o n . T h i s p r e l i m i n a r y a m p l i f i c a t i o n reduces the i m p o r t a n c e o f d i o d e n o i s e . The c l o s e p r o x i m i t y o f the d e t e c t o r e l e c t r o d e t o the d r i v e e l e c t r o d e , p a r t i c u l a r l y f o r s m a l l s a m p l e s , can l e a d t o problems w i t h e l e c t r i c a l p i c k u p . Even though b o t h e l e c t r o d e s a r e s h i e l d e d t h e r e i s a s u b s t a n t i a l d i r e c t c a p a c i t i v e p i c k u p from the d r i v e e l e c t r o d e t o t h e d e t e c t o r e l e c t r o d e . T h i s s p u r i o u s p i c k u p can cause p r o b l e m s ; however, i t may be f i l t e r e d o u t w i t h a s h o r t e d s e c t i o n o f c o a x i a l c a b l e as shown s c h e m a t i c a l l y i n F i g s . 15 and 16. The l e n g t h o f the c o a x i a l c a b l e i s chosen so t h a t i t i s a q u a r t e r w a v e l e n g t h l o n g a t the f r e q u e n c y o f the r f c a r r i e r . The q u a r t e r wave s h o r t a c t s as an open c i r c u i t a t t he c a r r i e r f r e q u e n c y and as a s h o r t c i r c u i t a t the lower d r i v e f r e q u e n c y . I f a q u a r t e r wave s h o r t i s c o n n e c t e d t o the b r i d g e o u t p u t , t h e d i r e c t p i c k u p i s e l i m i n a t e d b e f o r e i t can cause p r o b l e m s . A s i m i l a r c o a x i a l s h o r t a t the o u t p u t o f the r f g e n e r a t o r p r e v e n t s any b a c k f l o w o f the d r i v e s i g n a l i n t o the r f o s c i l l a t o r . As d e s c r i b e d above, when the d r i v e o s c i l l a t o r i s t u r n e d on the sample w i l l v i b r a t e and g e n e r a t e a s i g n a l t h a t i s p r o p o r t i o n a l t o the v i b r a t i o n a m p l i t u d e a t the o u t p u t o f the d i o d e d e t e c t o r . T h i s s i g n a l i s d e t e c t e d u s i n g a h e t e r o d y n e d e t e c t i o n scheme, o u t l i n e d i n the b l o c k d i a g r a m i n D.C. BIAS VOLTAGE RF SIGNAL GENERATOR HP 8640B STEP UP TRANSFORMER FREQUENCY SYNTHESIZER HP 3330 A ,4. PHASE SHIFTER SAMPLE AND BRIDGE ASSEMBLY LOW NOISE RF AMP. V DIODE DETECTOR AMPLIFIER MIXER LP F AMPL ILTER IFIER i SIGNAL - AVERAGER - 0 F i g . 17 Block Diagram of Electronic Equipment F i g . 17, w h i c h we now d e s c r i b e i n d e t a i l . The f i r s t s t e p i s t o a m p l i f y the d i o d e o u t p u t by about 40 db u s i n g a PAR 114 ( p l u g - i n 119) low n o i s e preamp. The PAR preamp i s used when t h e d r i v e f r e q u e n c y i s i n the range 100 Hz t o 1 MHz. For d r i v e f r e q u e n c i e s between 1 MHz and 10 MHz a MOSFET a m p l i f i e r (see A p p e n d i x § 1 . ( i i ) f o r c i r c u i t diagram) was used, f o l l o w e d by Avantek wideband a m p l i f i e r s UA 105 and UA 106. The t o t a l g a i n f o r t h i s c o m b i n a t i o n was a l s o about 40 db. The a m p l i f i e d s i g n a l then goes i n t o a d o u b l e b a l a n c e d m i x e r ( M i n i c i r c u i t s ZAD-6) t o g e t h e r w i t h a r e f e r e n c e s i g n a l f r o m t h e d r i v e o s c i l l a t o r . S i n c e t h e r e f e r e n c e s i g n a l and the s i g n a l coming from the v i b r a t i n g sample a r e n o r m a l l y a t t h e same f r e q u e n c y , the m i x e r w i l l have a d.c. o u t p u t whose a m p l i t u d e w i l l depend on the r e l a t i v e phase o f the two i n p u t s i g n a l s . The m i x e r o u t p u t w i l l be a maximum f o r z e r o phase d i f f e r e n c e and z e r o f o r 90° phase d i f f e r e n c e . C l e a r l y t h e d e t e c t o r d i o d e and a m p l i f i e r s w i l l i n t r o d u c e v a r i o u s u n s p e c i f i e d phase s h i f t s i n t o the s i g n a l coming from the sample. To compensate, the phase o f the r e f e r e n c e s i g n a l can be a d j u s t e d by a phase s h i f t e r (see A p p e n d i x § 1.. ( i i i ) f o r c i r c u i t d i a g r a m ) . The m i x e r o u t p u t i s a m p l i f i e d (see A p p e n d i x § 1. (v) f o r t h e a m p l i f i e r c i r c u i t diagram) and f e d i n t o a N i c o l e t 535 s i g n a l a v e r a g e r . In o p e r a t i o n the f r e q u e n c y s y n t h e s i z e r s t e p s a u t o m a t i c a l l y t h r o u g h a p r e s e l e c t e d f r e q u e n c y i n t e r v a l i n 100 o r 1000 s t e p s a t a r a t e o f 1, 3 o r 10 ms per s t e p . The s i g n a l a v e r a g e r s t o r e s t h e m i x e r o u t p u t a t each s t e p so t h a t s u c c e s s i v e f r e q u e n c y sweeps may be a c c u m u l a t e d t o improve t h e s i g n a l t o n o i s e r a t i o . 95 SCALE i—-} 1 mm TOP VIEW 9mm F i g . 18 - Sample mounting c o n f i g u r a t i o n s 96 Fig. 19 - Photographs of mounted samples Scale: 8 x actual size 97 WORM GEAR DRIVESHAFT DIODE TEMPERATURE SENSOR COPPER BOX HEATER POST SPUR GEAR TRANSFORMER SAMPLE SHIELDED DETECTOR ELECTRODE SCALE: } 1 i cm F i g . 20 - Cut-away vi e w o f sample box 98 Fig. 21 - Photograph of the outside of the sample holder with the door plate removed. F i g . 22 - P h o t o g r a p h o f the i n s i d e o f the sample h o l d e r . Note the a a x i s sample between the e l e c t r o d e s . In practice one observes a series of peaks in the output as a function of the drive frequency. These peaks correspond to acoustic resonances in the sample or its support. The peak width, amplitude and centre frequency of these resonances are the quantities of inter- est. (i i) Sample Support In this section we describe the sample support mechanism and the ancillary low temperature apparatus. In a l l cases the TTF-TCNQ. samples were glued to a conducting support with Dupont 4929 silver paint which was diluted with 2-butoxyethy1 acetate to lengthen the drying time. Two basic mounting configurations were used as shown in Fig.1B and the photographs ; n F!g.1S- in the "vibrating reed" configuration (Fig.18a) one end of the sample is glued to a brass support. In the other configuration (Fig.l8b) the sample is attached to a pointed tungsten wire at the centre of its broad crystallographic a b face. In this case the glue contact spot is typically about .15 mm in diameter. A variety of different size tungsten wires were used from .003" to .015" in diameter. A point was etched on the wire electro- l y t i c a l l y , in a IM NaOH solution. The mounted sample is held in a copper box between moveable shielded electrodes. A cut-away view of a sample mounted in the box is shown in Fig.20, and photographs of the outside and interior of the box are shown in Figs.21 and 22. Its main upper part was machined out of a single piece of copper to reduce rf leakage and improve the mechanical integrity 101 0-80 BRASS SCREW NYLON INSULATION GUIDE TAB BRASS SHIELD F i g . 23 - S h i e l d e d e l e c t r o d e S c a l e : 8 x a c t u a l s i z e F i g . 2k - S p e c i a l n o t c h e d e l e c t r o d e f o r smal1 samples 102 JLXSL ©t at S U P P O R T F L A N G E S T A I N L E S S S T E E L V A C U U M C A N LIQUID N I T R O G E N C O A X I A L C A B L E S A N D W O R M G E A R D R I V E R O D S H E L I U M E X C H A N G E G A S S A M P L E H O L D E R LIQUID H E L I U F i g . 25 - C r y o s t a t f o r low t e m p e r a t u r e measurements o f t h e e l e c t r o d e d r i v e mechanism. The e l e c t r o d e s can be removed from the box, so t h a t d i f f e r e n t shape e l e c t r o d e s can be i n s t a l l e d t o match the s i z e o f the sample and the t y p e o f mode t o be e x c i t e d . Two o f the e l e c t r o d e s used a r e shown i n F i g s . 2 3 and 2k. The n o t c h e d e l e c t r o d e i n F i g . 2 k was used w i t h v e r y s m a l l samples. The s p a c i n g between the e l e c t r o d e and the sample i s a d j u s t e d t o .01 - .10 mm by t u r n i n g n y l o n s p u r g e a r s t h r e a d e d o n t o t h e 0-80 t h r e a d e d b r a s s s h a f t o f the e l e c t r o d e . The s p u r g e a r s i n t u r n a r e d r i v e n by worms. One r o t a t i o n o f t h e worm gear moves th e e l e c t r o d e by .013 rnm, so t h a t f i n e a d j u s t m e n t s can be made t o t h e s p a c i n g between the sample and the e l e c t r o d e s . In a d d i t i o n t o the e l e c t r o d e d r i v e mechanism t h e c opper box a l s o c o n t a i n s the t r a n s f o r m e r and dummy sample needed f o r the b r i d g e de- t e c t i o n scheme d e s c r i b e d above (see F i g . 2 0 ) . The e n t i r e a s sembly i s suspended by t h r e e 3 mm s e m i - r i g i d c o a x i a l c a b l e s i n s i d e an e v a c u a t e d s t a i n l e s s s t e e l c a n , as shown i n F i g . 2 5 . In a d d i t i o n , t h e r e a r e t h r e e p i e c e s o f t h i n w a l l e d s t a i n l e s s s t e e l t u b i n g c o n n e c t e d t o the worm g e a r s . The t h r e e p i e c e s o f t u b i n g p r o t r u d e o u t o f the top o f the vacuum can t h r o u g h 0 r i n g s e a l s , a l l o w i n g a d j u s t m e n t o f the p o s i t i o n s o f t h e e l e c t r o d e s when the sample chamber i s e v a c u a t e d . For low tem- p e r a t u r e measurements h e l i u m exchange gas i s i n t r o d u c e d i n t o the can and a l l o w e d t o c i r c u l a t e i n s i d e the c o p p e r box w h i c h c o n t a i n s the sample. As l o n g as the p r e s s u r e i s £ 0 . 5 T o r r , the exchange gas does no t a p p r e c i a b l y damp th e sample v i b r a t i o n . D u r i n g a low t e m p e r a t u r e run the vacuum can i s n o r m a l l y suspended above the l i q u i d l e v e l , i n a glass dewar containing l i q u i d helium (Fig.25). In this mode of operation four l i t r e s of l i q u i d helium w i l l last about 2k hours. The diode temperature sensor and heater on the top of the copper sample box (Fig.20) are used in conjunction with a Lakeshore Cryo- tronics diode temperature controller to maintain the copper box temperature constant to < 10 mK. The diode (#D2755) was calibrated commercially and this c a l i b r a t i o n is given in the Appendix § 1. It was checked at l i q u i d helium, l i q u i d nitrogen and roughly at room temperature. In addition the diode was used to make four probe d.c. conductivity measurements as a function of temperature on a TTF-TCNQ c r y s t a l . The measured phase tr a n s i t i o n temperatures were consistent with e a r l i e r measurements (Tiedje 1975). 1.2 S e n s i t i v i t y of the Measurement Technique (i) Minimum Detectable Length Change The s e n s i t i v i t y of the apparatus to small displacements of the sample was determined by driving an a axis sample in i t s fundamental flexural mode with a known driving voltage. When a force F is applied to the free end of a cantilever beam of length £, according to elementary beam theory the resonant displacement of the free end is AH - °- F £ 2 Ad - - — where E is the Young's modulus, I is the area moment of i n e r t i a of the crossection of the beam, and Q. is the quality factor of the m e c h a n i c a l r e s o n a n c e . The minimum d e t e c t a b l e d i s p l a c e m e n t i s d e t e r - mined from the s i g n a l t o n o i s e r a t i o (SNR) f o r a known d r i v i n g f o r c e F by A d m i n = Ad/SNR. The f o r c e on the end o f the beam can be e s t i m a t e d from the f o r c e on the p l a t e s o f a c h a r g e d p a r a l l e l p l a t e c a p a c i t o r w h i c h i s _ _ eoA V 2 where A i s t h e a r e a o f the p l a t e s , d i s the p l a t e s e p a r a t i o n and V i s t h e v o l t a g e on the c a p a c i t o r . . We c o n s i d e r Sample #22 w i t h d i m e n s i o n s 0.335x0.204x0.018 mm as an example. T h i s sample had i t s fundamental f l e x u r a l mode a t 118 k h z , w i t h an a i r damped 0. o f 100 a t room t e m p e r a t u r e . The d r i v e and de- t e c t o r e l e c t r o d e s c o v e r e d the bottom t h i r d o f the sample and were s e p a r a t e d from the sample by a p p r o x i m a t e l y 0.08 and 0.02 mm r e - s p e c t i v e l y . U s i n g the room t e m p e r a t u r e a a x i s Young's modulus o f 3 - l x l O 1 1 dynes/cm 2 (see C h a p t e r II b e l o w ) , the measured d r i v i n g v o l t a g e (10V) and s i g n a l t o n o i s e r a t i o ( 1 7 6 ) , we c a l c u l a t e t h a t the minimum d e t e c t a b l e d i s p l a c e m e n t i s A d m ; n - 8 x 1 0 - 1 1 cm. For t h e SNR o f 176 q u o t e d above, t h e s i g n a l a v e r a g e r o u t p u t n o i s e bandwidth was 1.2 Hz. A c c o r d i n g t o the p a r a l l e l p l a t e f o r m u l a , the p i c k u p c a p a c i t o r had a c a p a c i t a n c e o f 9 x 1 0 ~ 3 p f . ( i i ) A.C. Method In t h i s s e c t i o n we examine some o f the p r a c t i c a l l i m i t a t i o n s t o the s e n s i t i v i t y o f the c a p a c i t i v e d i s p l a c e m e n t m e a s u r i n g t e c h n i q u e . We c o n s i d e r the s i m p l i f i e d model o f t h e d e t e c t o r c i r c u i t shown i n the s c h e m a t i c i n Fig.26 below. F i g . 26 - C a p a c i t i v e V i b r a t i o n D e t e c t o r - A.C. Method The n o i s e s o u r c e e n i s assumed t o be due t o Johnson n o i s e i n the r e - s i s t i v e e lement i n the LC r e s o n a n t c i r c u i t , i n w h i c h C i s the p i c k u p c a p a c i t o r . We have i g n o r e d the g e n e r a t o r n o i s e , because i n p r i n c i p l e i t can be c a n c e l l e d o u t by a s u i t a b l e b r i d g e d e t e c t i o n scheme. We a l s o assume t h a t the e f f e c t i v e impedance o f both the g e n e r a t o r and the a m p l i f i e r i n p u t can be a d j u s t e d t o any d e s i r e d v a l u e by impedance m a t c h i n g u s i n g t r a n s f o r m e r s , f o r example. The m a t c h i n g c i r c u i t r y has been o m i t t e d i n F i g . 26 f o r c l a r i t y . For optimum n o i s e p e r f o r m a n c e most r f a m p l i f i e r s r e q u i r e a s o u r c e impedance about e q u a l t o t h e i r i n p u t impedance. T h i s c o n d i t i o n w i l l be s a t i s f i e d a t resonance p r o v i d e d R a = R n + R q. F u r t h e r m o r e , we assume t h a t the a m p l i t u d e o f the r f c a r r i e r s i g n a l i s l i m i t e d by the power d i s s i p a t i o n i n the sample, o r e q u i v a l e n t l y by the power d i s - s i p a t i o n i n the r e s i s t o r R n. In t h i s s i t u a t i o n the optimum t r a n s f o r m e d g e n e r a t o r impedance i s z e r o . N o r m a l l y one d e t e c t s changes i n t h e p i c k u p c a p a c i t a n c e by m o n i t o r i n g the a m p l i t u d e o f the t r a n s - m i t t e d r f s i g n a l . In t h i s c a s e the c i r c u i t i s most s e n s i t i v e when f r e q u e n c y o f the LC c i r c u i t and Qg i s the q u a l i t y f a c t o r f o r t h e e l e c t r i c a l r e s o n a n c e . When a l l o f the above c o n d i t i o n s a r e met t h e minimum d e t e c t a b l e change i n c a p a c i t a n c e i s where T i s t h e t e m p e r a t u r e o f the r e s i s t o r R n, T a i s the n o i s e t e m p e r a t u r e o f the a m p l i f i e r (Motchenbacher and F i t c h e n 1973) and Av i s the o u t p u t n o i s e bandwidth o f the d e t e c t o r . In e q n . [ l ] ^ i s a f a c t o r w h i c h s e t s the d e s i r e d d e t e c t i o n t h r e s h o l d f o r a c a p a c i t a n c e change. For example when ? = 1, a s i g n a l i s c o n s i d e r e d t o be d e t e c t e d i f i t has an a m p l i t u d e equal t o the rms n o i s e l e v e l . In t h i s s e c t i o n we use r, = 2 . P i s the power d i s s i p a t e d i h R n. An e x p r e s s i o n s i m i l a r t o [ 1 ] has been o b t a i n e d by B r a g i n s k i i and Manukin (1 9 7 7 ) - t h e g e n e r a t o r f r e q u e n c y where coe i s the r e s o n a n t [ 1 ] AC _2_ [k(T + T a ) A v ] C C Q e p l In practice the power dissipation is not usually a limitation. The limiting factor is more likely to be the capacitor breakdown voltage. In this case, when the breakdown voltage is V m a x, the minimum detectable capacitance change is C (ioe C Qe)i Vn d 'max where Ad/d is the corresponding fractional change in the spacing of the capacitor plates. An order of magnitude estimate for the maximum Q_e of an LC resonant ci r c u i t at room temperature is 100, taking into account coupling losses. Much higher Q.e's could be obtained using superconducting circuits. For example, Q_e > 1 0 6 can be achieved for superconducting LC circuits and even higher Qe's are possible in superconducting cavity resonators (Hartwig 1973). However, we will consider room temperature circuits only. It is interesting to compare the experimental sensitivity to the theoretical limit [ 2 ] . If the following values are used: T = 300K, T a = 170K (Avantek AMM-1010 amplifier), toe = 4 x 10 9 s" 1, C = 9 x 1 0 - 3 pf, Qe = 100 and V m a x/d = 1 0 s V/cm (Cantrell and Breazeale 1974) then the limiting displacement is Ad - 2 x 1 0 " 1 3 /Av cm. Experimentally the rf f i e l d on the detector capacitor is likely to be at least an order of magnitude less than 1 0 5 V/cm. Accordingly the experimental sensitivity of 8 x 1 0 - 1 1 ZA\7 cm is probably only about an order of magnitude less than the limiting value predicted from [ 2 ] , using the actual rf f i e l d . By judicious selection of the resonance frequency we, (coe C Q e ) - 1 ~ 10 can be achieved at room temperature for a wide range of pickup capa- citance. For (we C Q e ) - 1 = 10, Ad = 10" l l f /A~V cm. Braginskii et al (1971) have b u i l t a detection system capable of measuring Ad = 3x10" l l + /Kv cm with C - 2000 pf and oj e = 3 x l 0 7 s" 1. ( i i i ) D.C. Method For comparison purposes we now consider the d.c. analogue of the capacitive displacement measuring technique described above. In the d.c. approach, a constant bias voltage rather than an a.c. s i g n a l , is used as a probe to measure changes in the pickup capacitance. A schematic of the detector < i r c u i t is shown in Fig. 27 below. Changes in the pickup capacitance C a l t e r the stored charge and induce currents in the bias r e s i s t o r RD. Accordingly any vibration of the sample is reflected in a voltage on the bias r e s i s t o r . C Fig. 27 - Capacitive Vibration Detector - D.C. Method 110 There are two main noise sources in the circuit shown in Fig. 27- They are the Johnson noise in the bias resistor and the amplifier noise. In order to estimate the sensitivity of the detector some information about the noise performance of the amplifier input stage is required. For vibration frequencies up to 10 MHz at least, the best available device for the input stage is a junction f i e l d effect transistor (JFET). To be specific, we will consider a Siliconix 2N4867 transistor which is a good low noise commercially available devi ce. A noise equivalent circuit for the capacitive detector using an amplifier with a JFET input stage, is shown in Fig. 28 below. jioVVC C Rb i b Cj R; i n Fig. 28 - Noise Equivalent Circuit The Norton equivalent circuit has been used for the bias resistor noise generator. The mean-square resistor noise current T^2 is 4kTAv/RD- The c u r r e n t s i g n a l g e n e r a t e d by the o s c i l l a t i n g c a p a c i t a n c e i s r e p r e - s e n t e d by an e q u i v a l e n t c u r r e n t s o u r c e jcoACV, where co i s the m e c h a n i c a l v i b r a t i o n a n g u l a r f r e q u e n c y . At midband the rms v o l t a g e n o i s e e n f o r the 2N4867 JFET i s 2 x 1 0 - 9 V and the c u r r e n t n o i s e i n i s 3 x 1 0 - 1 5 A /Kv. Below 30 hz e n i s dominated by 1/f n o i s e and above 10 khz i n i n c r e a s e s 1 i n e a r l y wi t h co. From the e q u i v a l e n t c i r c u i t i n F i g . 28 we f i n d t h a t the minimum d e t e c t a b l e c a p a c i t a n c e change i s f o r a 1 hz b a n d w i d t h . The h i g h e s t s e n s i t i v i t y i s a c h i e v e d by u s i n g the l a r g e s t p o s s i b l e b i a s r e s i s t o r up t o a r e s i s t a n c e o f [co(C + C ; ) ] " 1 o r where t h e c u r r e n t n o i s e i n d o m i n a t e s , a t w h i c h time the s e n s i t i v i t y becomes independent o f R^. A r e a s o n a b l e upper l i m i t f o r R^ i s 1 0 8 fi. We assume the i n p u t r e s i s t a n c e o f the JFET i s l a r g e r than t h i s and t h a t Cj = 5 pf- I f we s u b s t i t u t e t h e c i r c u i t p a r a m e t e r s g i v e n above i n t o [3] and use C = 9 x 1 0 - 3 pf and co = 7-4x10 5 s - 1 c o r r e s p o n d i n g t o the f i r s t f l e x u r a l mode o f Sample #22 then the minimum d e t e c t a b l e d i s p l a c e m e n t i s Ad - 1 0 " 1 0 cm when the bias f i e l d on the c a p a c i t o r i s 1 0 5 V/cm. A c o n t o u r map i n the (co,C) p l a n e o f the d i s p l a c e m e n t m e a s u r i n g s e n - s i t i v i t y o f t h e d.c. c a p a c i t i v e t e c h n i q u e i s shown i n F i g . 29. From th e c o n t o u r map we c o n c l u d e t h a t the d.c. method works b e s t f o r h i g h v i b r a t i o n f r e q u e n c i e s and l a r g e p i c k u p c a p a c i t a n c e s . In t h i s regime F i g . 29 - Minimum d e t e c t a b l e d i s p l a c e m e n t Ad f r e q u e n c y f and p i c k u p c a p a c i t a n c e method. The c o n t o u r l a b e l s a r e i n as a f u n c t i o n o f C u s i n g the d.c. c e n t i m e t e r s . i t s s e n s i t i v i t y i s comparable w i t h the a.c. method where Ad - 10~ 1 ̂ *̂ cm. However, the d.c. method i s l e s s s a t i s f a c t o r y f o r low v i b r a t i o n f r e q u e n c i e s and s m a l l samples f o r w h i c h the p i c k u p c a p a c i t a n c e i s n e c e s s a r i l y smal1. CHAPTER I I The Modes o f V i b r a t i o n o f TTF-TCNQ C r y s t a l s 2.1 Low Frequency Modes o f an e l o n g a t e d p l a t e The a c o u s t i c mode s p e c t r u m o f an e l a s t i c body i s e x c e e d i n g l y complex even f o r o b j e c t s w i t h s i m p l e shapes. I t i s f e a s i b l e t o c a l c u l a t e the a c o u s t i c resonance f r e q u e n c i e s f o r o n l y a few s p e c i a l c a s e s . P r o v i d e d t h a t t h e m a t e r i a l i s e l a s t i c a l l y i s o t r o p i c , t he i n f i n i t e medium, the i n f i n i t e t h i n p l a t e , the i n f i n i t e c y l i n d e r , and t h e s p h e r e a r e s o l u b l e . For a r b i t r a r y c r y s t a l l o g r a p h i c symmetry o n l y the i n f i n i t e medium i s s o l u b l e , a l t h o u g h the o t h e r c a s e s a r e s o l u b l e f o r c e r t a i n t y p e s o f e l a s t i c a n i - s o t r o p y . I t i s i n t e r e s t i n g t o n o t e t h a t the a c o u s t i c r e s o n a t o r p r o b l e m i s much more complex than the comparable e l e c t r o m a g n e t i c r e s o n a t o r p r o blem. An i n t u i t i v e e x p l a n a t i o n i s t h a t t h e r e a r e o n l y two e l e c t r o - m a g n e t i c waves p o s s i b l e i n an i n f i n i t e medium, namely two l i n e a r l y p o l a r i z e d t r a n s v e r s e waves, whereas i n the a c o u s t i c c a s e t h e r e a r e t h r e e waves p o s s i b l e - two t r a n s v e r s e waves and one l o n g i t u d i n a l wave. The a c o u s t i c r e s o n a t o r we a r e i n t e r e s t e d i n h e r e i s an e l o n g a t e d , a p p r o x i m a t e l y r e c t a n g u l a r p a r a l l e l o p i p e d w i t h m o n o c l i n i c symmetry. A l t h o u g h i t i s n o t p o s s i b l e t o c a l c u l a t e a l l o f the resona n c e f r e q u e n c i e s o f a TTF-TCNQ c r y s t a l , i t i s p o s s i b l e t o o b t a i n some v e r y good a p p r o x i - m a t i o n s f o r the low f r e q u e n c y modes. To b e g i n w i t h l e t us assume t h a t TTF-TCNQ i s e l a s t i c a l l y i s o t r o p i c , i n o r d e r t o s i m p l i f y the d i s c u s s i o n . The c o m p l i c a t i o n s a r i s i n g f rom e l a s t i c a n i s o t r o p y w i l l be d i s c u s s e d l a t e r , a l t h o u g h i t w i l l t u r n o u t t h a t TTF-TCNQ. i s not f a r from b e i n g i s o t r o p i c , e l a s t i c a l l y . ( i ) F l e x u r a l Modes The l o w e s t f r e q u e n c y mode o f t h e e l o n g a t e d p l a t e l e t i s a b e n d i n g ( " f l e x u r a l " ) mode w i t h d i s p l a c e m e n t s p e r p e n d i c u l a r t o the broad f a c e o f the p l a t e l e t . S i n c e the TTF-TCNQ p l a t e l e t s a r e g e o m e t r i c a l l y s i m i l a r t o an o r d i n a r y p l a s t i c r u l e r , t he low f r e q u e n c y modes can be r e a d i l y v i s u a l i z e d w i t h the h e l p o f a r u l e r . The r u l e r has t h r e e d i f f e r e n t t y p e s o f modes w h i c h c o u l d be c l a s s i f i e d as f l e x u r a l . The l o w e s t f r e q u e n c y t y p e ( l a b e l l e d F^ ) c o r r e s p o n d s t o be n d i n g a l o n g the l e n g t h o f the r u l e r w i t h d i s p l a c e m e n t s p e r p e n d i c u l a r t o t h e broad f a c e . The second t y p e ( F ^ ) i s the same as t h e f i r s t e x c e p t t h a t t h e d i s p l a c e m e n t i s p e r p e n d i c u l a r t o the edge o f t h e r u l e r . The t h i r d t y p e (F ) i s a bend a c r o s s the w i d t h o f the r u l e r w i t h d i s p l a c e m e n t s 3 C p e r p e n d i c u l a r t o the broad f a c e . The r a t i o n a l e b e h i n d the mode l a b e l l i n g scheme w i l l become c l e a r l a t e r . The p r o p a g a t i o n o f a f l e x u r a l wave a l o n g a beam whose l o n g a x i s i s p a r a l l e l t o y, i s d e s c r i b e d by the wave e q u a t i o n i n the l i m i t t h a t t h e w a v e l e n g t h i s l o n g compared t o the beam t h i c k - n e s s . In e q u a t i o n [1] u i s t h e d i s p l a c e m e n t o f the beam c e n t r e l i n e f r om the e q u i l i b r i u m p o s i t i o n , A i s t h e c r o s s e c t i o n a l a r e a o f t h e [1] EI 9 % = 0 pA ay1* beam, and E is its Young's modulus. The area moment of inertia I is the moment of inertia of the beam crossection about a line through the beam centre line which is perpendicular to both the direction of the displacement u and the long axis of the beam. Consider the foot long ruler again as an example. If t is its thickness and w its width then I = t3w/12 for the soft F b c modes and I = w3t/12 for the s t i f f F^g modes. Equation [1] also applies to flexural modes in thin plates. For an isotropic plate with Poisson's ratio v and a width much larger than a flexural wavelength, the Young's modulus E in [1] is replaced by the plate modulus E/(l - v 2 ) . The flexural resonance frequencies are determined by looking for the solutions of [1] which satisfy the boundary conditions or the ends of the beam. The resonance frequencies are given by (Timoshenko 1974) [2] f = _L / H fe-\2 L Z J Tn 2TT / pA \l) where £ is the length of the beam and mn is the n t n root of a transceden- tal equation which is specified by the boundary conditions. The boundary conditions of interest here are "c1amped-free" for the vibrating reed (cantilever beam) configuration and "free-free" for the central pin support. For these boundary conditions the f i r s t four mn values are given in Table III TABLE I I I Flexural Mode Parameters iriQ m i rri2 m 3 Clamped-Free 1.875 4.694 7.855 10.996 Free-Free 0 4.730 7-853 10.996 (n + JT)TT 1.571 4.712 7.854 10.996 and the c o r r e s p o n d i n g mode shapes a r e shown i n F i g . 30. n Clamped-Free n F r e e - F r e e F i g . 30 - F l e x u r a l Mode Shapes For l a r g e n, m n a s y m p t o t i c a l l y a p p r o a c h e s (n + i ) ir . As p o i n t e d o u t e a r l i e r t he one d i m e n s i o n a l wave e q u a t i o n [1] i s o n l y a good a p p r o x i m a t i o n i f t h e f l e x u r a l w a v e l e n g t h i s l o n g compared t o the beam t h i c k n e s s . F u r t h e r m o r e the a p p r o x i m a t i o n w i l l b r e a k down i f t he shear modulus i s v e r y s m a l l . The impo r t a n c e o f the s h e a r modulus i s b e s t i l l u s t r a t e d by c o n s i d e r i n g the extreme c a s e o f a pad o f paper where the s h e a r f o r c e between s h e e t s i s n e a r l y z e r o . I n t h i s c a s e the f l e x u r a l r i g i d i t y o f the pad o f paper i s dominated by the shear modulus between sheets and is therefore close to zero. In less extreme cases the contribution of the shear modulus is negligible, and the beam rigidity is determined entirely by the compression of the concave side of the beam and the extension of the convex side. Nevertheless, because of the chainlike nature of the crystal structure of TTF-TCNQ it has been suggested that the interchain shear moduli may be unusually small (Barmatz et al 197**). The shear modulus and f i n i t e beam thickness can be taken into account by adding some more terms to [1] (Timoshenko 1974). If the modified differential equation is solved for the resonance fre- quencies, one finds that the additional terms in the equation reduce the effective Young's modulus for the n t n flexural mode in [2], by the factor (Goens 1931) [3] 1 + K (I)' m* | where K ~ 1 and G is a shear modulus. The beam thickness t is measured in the direction of the beam displacement during bending. In order to obtain an exact expression for the resonance frequency of even the lowest frequency flexural mode one would have to solve a three dimen- sional differential equation and look for solutions which match the boundary conditions over the entire surface of the beam rather than just at its ends. In the samples which are supported by the central pin, the post at the centre provides an additional constraint which is not easily dealt with 119 u s i n g s i m p l e beam t h e o r y . However, f o r the l o w e s t f l e x u r a l mode a s e m i q u a n t i t a t i v e e s t i m a t e o f t h e e f f e c t o f the c e n t r a l s u p p o r t can be o b t a i n e d from the n u m e r i c a l c a l c u l a t i o n s o f S o u t h w e l l (1922) f o r c e n t r a l l y p i n n e d d i s c s . The e f f e c t o f even a v e r y s m a l l p i n c o n t a c t a r e a i s t o r a i s e t he z e r o f r e q u e n c y n = 0 mode o f a f r e e - f r e e beam up t o about 80% o f t h e n = 1 mode f r e q u e n c y . The f l e x u r a l modes w i t h a mode a t the c e n t r e and the h i g h e r f r e q u e n c y f l e x u r a l modes a r e not a f f e c t e d v e r y much by the p i n . ( i i ) T o r s i o n a l Modes We now c o n s i d e r t h e low f r e q u e n c y t o r s i o n a l (T) modes o f t h e e l o n g a t e d p l a t e l e t ( o r r u l e r ) . The d i s p l a c e m e n t o f the p l a t e l e t f o r the t o r s i o n a l modes we a r e i n t e r e s t e d i n h e r e , i s a volume con- s e r v i n g s t r a i n i n w h i c h n e i g h b o u r i n g c r o s s e c t i o n s p e r p e n d i c u l a r t o the l o n g a x i s o f the sample a r e t w i s t e d r e l a t i v e t o one a n o t h e r . The p r o p a g a t i o n o f t o r s i o n a l waves a l o n g a beam w h i c h i s l o n g com- p a r e d t o i t s l a t e r a l d i m e n s i o n s i s d e s c r i b e d by the wave e q u a t i o n (Landau and L i f s h i t z 1970) m 924> = C 92<j> 1 J 9 t 2 p i 3 y 2 where the l o n g a x i s o f the beam i s a l o n g y as b e f o r e , and <j> i s the a n g l e o f r o t a t i o n o f a c r o s s e c t i o n . Ip i s t h e a r e a moment o f i n e r t i a o f a c r o s s e c t i o n about i t s c e n t r e , and C and p a r e the t o r s i o n a l r i g i d i t y and d e n s i t y o f the beam r e s p e c t i v e l y . The t o r s i o n a l r esonance f r e q u e n c i e s f o r a beam o f l e n g t h t a r e g i v e n by l5' fn - M; M F o r the c l a m p e d - f r e e boundary c o n d i t i o n s n i s an odd i n t e g e r and f o r the f r e e - f r e e boundary c o n d i t i o n s n i s an even i n t e g e r . U n l e s s the c r o s s e c t i o n o f the beam i s c i r c u l a r , p l a n e c r o s s e c t i o n s become warped under t o r s i o n . Because o f t h i s f e a t u r e , the t o r s i o n a l r i g i d i t y i s a c o m p l i c a t e d f u n c t i o n o f the shape o f the c r o s s e c t i o n (Timoshenko 1951 ) . The t o r s i o n a l r i g i d i t i e s f o r some s i m p l e shapes a r e g i v e n by Landau and L i f s h i t z ( 1970 ) . For a beam w i t h a r e c t a n g u l a r c r o s s e c t i o n o f w i d t h w and t h i c k n e s s t < 0 .2 w. C i s g i v e n by (Timoshenko 1951) [«] " ' t ( ' ^ ; ) ' In the t « w l i m i t , t he t o r s i o n a l r e s o n a n c e f r e q u e n c i e s i n [5] reduce t o f = — M~ — n w / p 21 where G i s the s h e a r modulus, as b e f o r e . In t h e o r y , i f the l a t e r a l d i m e n s i o n s o f the beam a r e comparable t o a t o r s i o n a l w a v e l e n g t h then a c o r r e c t i o n f a c t o r a n a l o g o u s t o [3] i s needed i n the e x p r e s s i o n f o r the t o r s i o n a l resonance f r e q u e n c i e s . In p r a c t i c e the c o r r e c t i o n f a c t o r does not change the f r e q u e n c y much even when the l a t e r a l d i m e n s i o n i s e q u a l t o a h a l f w a v e l e n g t h . F u r t h e r m o r e an a p p r o x i m a t e t h e o r e t i c a l c a l c u l a t i o n o f the c o r r e c t i o n f a c t o r i s a p p a r e n t l y i n d i s a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s (Behrens 1968). T h i s c o r r e c t i o n w i l l be i g n o r e d h e r e . ( i i i ) Comments on a S h o r t P l a t e We now d e s c r i b e q u a l i t a t i v e l y t he low f r e q u e n c y modes o f a p l a t e i n w h i c h the w i d t h o f t h e p l a t e i s comp a r a b l e t o i t s l e n g t h . T h i s s i t u a t i o n i s o f i n t e r e s t because i t i l l u s t r a t e s what happens t o the modes o f an e l o n g a t e d p l a t e when the w a v e l e n g t h i s comparable t o i t s w i d t h . In a d d i t i o n some measurements were made on n e a r l y s q u a r e p l a t e l e t s c u t o f f the end o f s t a n d a r d e l o n g a t e d TTF-TCNQ c r y s t a l s . The low f r e q u e n c y modes o f a p l a t e a r e u s u a l l y i l l u s t r a t e d by C h l a d n i f i g u r e s , w h i c h a r e the p a t t e r n s o f nodal l i n e s o f the modes ( W a l l e r 1961, L e i s s a 1969)• The e x p e r i m e n t a l l y d e t e r m i n e d p a t t e r n s o f nodal l i n e s f o r the f i r s t few modes o f a s q u a r e c a n t i l e v e r p l a t e and a r e c t a n g u l a r f r e e p l a t e a r e shown i n Fig.31 and 32 r e s p e c t i v e l y . Observe t h a t the f i r s t mode i n Fig.31 appears t o be p u r e l y f l e x u r a l , the second one t o r s i o n a l and the t h i r d one f l e x u r a l a g a i n . The f o u r t h mode l o o k s l i k e a t r a n s - v e r s e f l e x u r a l mode, but i t i s not p u r e l y f l e x u r a l because o f the edge clamp. A l t h o u g h the f r e q u e n c y o f t h i s mode cannot be c a l c u l a t e d a c c u r a t e l y u s i n g the s i m p l e beam f o r m u l a s , the f r e q u e n c y o f the f i r s t two modes can be c a l c u l a t e d w i t h r e a s o n a b l e a c c u r a c y from [2] and [ 5 ] , s i n c e the v i b r a t i o n w a v e l e n g t h f o r b o t h modes i s a t l e a s t f o u r t i m e s the w i d t h o f the "beam". In a d d i t i o n the f i r s t two modes a r e w e l l s e p a r a t e d from 2.63 6.27 8.14 9.22 R e l a t i v e Frequency F i g . 31 " Modes o f a Square C a n t i l e v e r P l a t e 0 1 2 F i g . 32 -Modes o f a R e c t a n g u l a r (2:1) Free P l a t e t h e i r n e i g h b o u r s i n f r e q u e n c y , making them easy t o i d e n t i f y e x p e r i - mental l y . An e x a m i n a t i o n o f the nodal p a t t e r n s i n Fig.32 shows t h a t j u s t as f o r the c a n t i l e v e r p l a t e , the h i g h e r f r e q u e n c y modes a r e l i k e l y t o be a complex m i x t u r e o f t o r s i o n and f l e x . I t i s d i f f i c u l t t o o b t a i n i n f o r m a t i o n about the e l a s t i c moduli from t h e s e modes because o f t h e i r c o m p l i c a t e d n a t u r e . We now t e m p o r a r i l y l e a v e the s u b j e c t o f t o r s i o n a l and f l e x u r a l v i b r a t i o n s and i n v e s t i g a t e a n o t h e r s i m p l e t y p e o f v i b r a t i o n o f a l o n g p l a t e . ( i v ) E l o n g a t i o n a l Modes The f i n a l t y p e o f mode w h i c h we w i l l c o n s i d e r i s t h e l o n g i t u d i n a l s t r e t c h o r e l o n g a t i o n a l (L) mode. The p r o p a g a t i o n o f an e x t e n s i o n a l wave a l o n g a rod o r i e n t e d p a r a l l e l t o the y a x i s , i s d e s c r i b e d by t h e wave e q u a t i o n m liiL = £ iiy. L / J 8t2 p 9y2 where u i s t h e d i s p l a c e m e n t o f the rod a l o n g i t s a x i s . The Young's modulus E appears i n [7] r a t h e r than a b u l k wave l o n g i t u d i n a l modulus because the rod i s f r e e t o expand o r c o n t r a c t l a t e r a l l y d epending on whether i t i s b e i n g compressed o r e x t e n d e d l o n g i t u d i n a l l y . As b e f o r e , the resonance f r e q u e n c i e s a r e o b t a i n e d by l o o k i n g f o r s o l u t i o n s o f the wave e q u a t i o n w h i c h s a t i s f y the boundary c o n d i t i o n s F i g . 33 - Arrangement o f TTF and TCNQ m o l e c u l e s i n the a c p l a n e . The s o l i d d o t s a r e t i p p e d up above the p l a n e . on the ends of the beam. For free-free boundary conditions the re- sonance frequencies are For short beams and high frequencies the longitudinal acoustic wave- length may be comparable with the transverse dimensions of the beam. In this s i t u a t i o n , the one dimensional wave equation [7] is no longer a good approximation just as the one dimensional equations for the flexural and torsional waves are no longer a good approximation in the same l i m i t . Provided the wavelength is not too short, the elon- gational resonance frequencies may be corrected by dividing the Young's modulus E by the factor (Love 19̂ 4) for a beam with Poisson's ra t i o v and a rectangular crossection. (v) Crystallographic Symmetry Up to now we have assumed the vibrating plate or beam to be made of an isotropic material. However, the TTF-TCNQ samples on which the measurements were made are not isotr o p i c ; rather, they are monoclinic crystals (space group P2j/C ). The positions of the TTF and TCNQ molecules in the crystallographic a c plane are shown in Fig. 33 This notation is explained by Henry and Lonsdale (1951). ( B l e s s i n g and Coppens 1 9 7 4 ) . F i g . 2 shows the s e g r e g a t e d s t a c k i n g arrangement o f the m o l e c u l e s i n the b d i r e c t i o n w h i c h i s l a r g e l y r e - s p o n s i b l e f o r the unusual e l e c t r i c a l p r o p e r t i e s o f TTF-TCNQ. A l l p f t h e UBC grown TTF-TCNQ c r y s t a l s have the unique ( b ) a x i s p a r a l l e l t o the l o n g a x i s o f the c r y s t a l and the a a x i s p a r a l l e l t o the broad t r a n s v e r s e d i m e n s i o n s . The c a x i s i s about 14.5° away from b e i n g p e r p e n d i c u l a r t o the o t h e r two a x e s ; hence i t i s not q u i t e p e r - p e n d i c u l a r t o the broad a b f a c e o f t h e c r y s t a l . T h i s f e a t u r e a l l o w s a two f o l d a m b i g u i t y i n the d i r e c t i o n o f the c a x i s i n a r e a l c r y s t a l and opens up the p o s s i b i l i t y t h a t what appears t o be a s i n g l e c r y s t a l may a c t u a l l y be t w i n n e d . For c o n v e n i e n c e we w i l l use t h e r e c i p r o c a l l a t t i c e v e c t o r c" w h i c h i s d e f i n e d t o be p e r p e n d i c u l a r t o a and b, i n s t e a d o f c when d i s c u s s i n g the v i b r a t i o n modes i n r e l a t i o n t o the c r y s t a l l o g r a p h i c symmetry a x e s . M o n o c l i n i c a c o u s t i c r e s o n a t o r s a r e not easy t o d e a l w i t h t h e o r e t i c a l l y , because i n g e n e r a l 13 independent e l a s t i c c o n s t a n t s need t o be c o n s i d e r e d ( A u l d 1 9 7 3 ) . In a beam made o f a m o n o c l i n i c m a t e r i a l , the f l e x u r a l modes a r e c o u p l e d t o the t o r s i o n a l modes i n a complex way, by c e r t a i n e l a s t i c c o n s t a n t s . However, i f t h e TTF-TCNQ c a x i s were p e r p e n d i c u l a r t o a ( o r the c r y s t a l were s u i t a b l y t w i n n e d ) , the f l e x u r a l modes and t o r s i o n a l modes would be u n c o u p l e d . To make the v i b r a t i o n p r o b l e m manageable we w i l l assume t h a t c i s p e r p e n d i c u l a r t o a. T h i s amounts t o assuming t h a t the samples have o r t h o r h o m b i c symmetry and t h a t the c r y s t a l l o g r a p h i c symmetry axes a r e a l i g n e d a l o n g t h e symmetry axes o f t h e sample c r y s t a l . The a p p r o x i m a t i o n i s p r o b a b l y not u n r e a s o n a b l e s i n c e the c a x i s i s o n l y 14.5° away from b e i n g p e r p e n d i c u l a r t o the a a x i s . In g e n e r a l an o r t h o r h o m b i c m a t e r i a l has n i n e e l a s t i c c o n s t a n t s , w h i c h may be broken down i n t o t h r e e Young's m o d u l i , t h r e e P o i s s o n ' s r a t i o s and t h r e e s h e a r m o d u l i . The e l a s t i c moduli a r e most con- v e n i e n t l y d e f i n e d i n terms o f t h e 6X6 c o m p l i a n c e m a t r i x s j j ( A u l d 1973). The Young's moduli E a , and E c f o r the t h r e e c r y s t a l l o - g r a p h i c symmetry d i r e c t i o n s a r e e q u a l t o s ^ " 1 , s ^ 1 and S g " 1 r e s p e c t i v e l y when the a b and c" axes a r e a l i g n e d p a r a l l e l t o t h e x y and z a x e s . S i m i l a r l y t h e t h r e e s h e a r m o d u l i C 4 4 , C55 and 0 5 5 a r e e q u a l t o s ^ " 1 , s ^ 1 and S g " 1 r e s p e c t i v e l y ( L e k h n i t s k i i 1963). The e x p r e s s i o n s f o r the resonance f r e q u e n c i e s o f i s o t r o p i c beams can be r e a d i l y g e n e r a l i z e d t o a p p l y t o o r t h o r h o m b i c beams. L e t us c o n s i d e r the f l e x u r a l modes f i r s t . The Young's modulus i n t h e f r e q u e n c y e x p r e s s i o n [2] s h o u l d be the modulus a l o n g t h e d i r e c t i o n w h i c h the beam i s compressed and e x t e n d e d d u r i n g v i b r a t i o n . For example, the r e l e v a n t Young's modulus f o r the F b c and F b modes i s E b . The s h e a r modulus w h i c h goes i n t o t he c o r r e c t i o n f a c t o r [3] i s b e s t d e s c r i b e d w i t h the h e l p o f the pad o f paper a n a l o g y . I f t h e beam were a s t a c k o f w e a k l y i n t e r a c t i n g s h e e t s , bent i n i t s s o f t e s t d i r e c t i o n , the a p p r o p r i a t e s h e a r modulus t o put i n t o t he c o r r e c t i o n f a c t o r would be the modulus a g a i n s t s l i d i n g o f the s h e e t s on top o f one a n o t h e r i n a d i r e c t i o n p a r a l l e l t o the l o n g a x i s o f the beam. For the F^c modes, c ^ i s t h e a p p r o p r i a t e s h e a r modulus. The e l a s t c o n s t a n t s w h i c h a p p l y t o the t h r e e e x p e r i m e n t a l l y o b s e r v e d t y p e s o f f l e x u r a l modes a r e summarized i n T a b l e IV. TABLE IV F l e x u r a l Mode E l a s t i c C o n s t a n t s Mode f Young's Modulus E Shear Modulus ^ G F b c E b c66 E a c55 . r e f e r t o eqn. [2] and [3] The t o r s i o n a l r i g i d i t y has been c a l c u l a t e d f o r an o r t h o r h o m b i c beam w i t h a r e c t a n g u l a r c r o s s e c t i o n i n the books by Hearmon (1961) and L e k h n i t s k i i (1963). The r e s u l t s may be r e a d i l y adapted t o t h e two d i f f e r e n t t y p e s o f t o r s i o n a l modes w h i c h were s t u d i e d e x p e r i m e n t a l l y . In t he f i r s t t y p e o f mode ( T a mode) t h e t o r s i o n a x i s i s p a r a l l e l t o the a a x i s . In t h i s c a s e the t o r s i o n a l r i g i d i t y [6] s h o u l d be r e p l a c e d by r _ r t 3 w / 192 t / c66\*\ [10] C " c66 — M ^ ^ ( c ^ J J f o r t < 0.2 w and c 6£ ~ c^^. The t h i c k n e s s t i s t o be measured i n the c" ( t h i n ) d i r e c t i o n , and the w i d t h i s measured i n the b d i r e c t i o n . S i m i l a r l y , w h e n the t o r s i o n a x i s i s p a r a l l e l t o b (T^ mode) r -, r t 3 w /1 1 9 2 t / c 6 6 \ * \ W C = C 6 6 — ^ " T T w(ci?) ) where t i s measured i n the c" as b e f o r e , and the w i d t h w i s measured i n t h e a d i r e c t i o n . I t i s t r i v i a l l y e a sy t o g e n e r a l i z e the e x p r e s s i o n f o r the e l o n g a t i o n a l mode f r e q u e n c i e s t o an o r t h o r h o m b i c beam. One need o n l y r e p l a c e the i s o t r o p i c Young's modulus E i n [ 8 ] by t h e b a x i s Young's modulus E^. A l t h o u g h o n l y the b a x i s modes were measured e x p e r i m e n t a l l y , the f r e - q u e n c i e s o f e l o n g a t i o n a l modes f o r beams w i t h l o n g axes i n the a o r c d i r e c t i o n c o u l d be c a l c u l a t e d i n an e x a c t l y a n a l o g o u s way. S i m i l a r l y , t h e c o r r e c t i o n f a c t o r [ 9 ] f o r t h e b a x i s mode s h o u l d be r e p l a c e d by (Behrens 1968) r , ^ i 1 j. 1 / n i i A 2 / 2 2 . 2 .2\ [12] 1 + J (2IJ ( V12 w + v23 t j f o r an o r t h o r h o m b i c beam where v 1 2 and v 2 3 a r e P o i s s o n ' s r a t i o s d e f i n e d i n terms o f t h e e l e m e n t s o f the c o m p l i a n c e m a t r i x by v ^ / l i a = - S i 2 and V 2 3 / E c = " s23- B e f o r e g o i n g on t o d e s c r i b e some o f t h e e x p e r i m e n t a l r e s u l t s we f i r s t d i s c u s s a more d r a s t i c a p p r o x i m a t i o n t o the e l a s t i c symmetry o f TTF-TCNQ. T h i s a p p r o x i m a t i o n w i l l be u s e f u l l a t e r on i n o b t a i n i n g an e s t i m a t e f o r the b u l k modulus from the e x p e r i m e n t a l d a t a . The s a l i e n t f e a t u r e o f the TTF-TCNQ s t r u c t u r e i s the l i n e a r s t a c k i n g o f the m o l e c u l e s a l o n g the b d i r e c t i o n . A c c o r d i n g l y one might e x p e c t the e l a s t i c p r o p e r t i e s t o be d i f f e r e n t d e p e n d i n g on whether the s t r a i n i s p a r a l l e l o r p e r - p e n d i c u l a r t o the m o l e c u l a r s t a c k s . The h i g h e s t symmetry c r y s t a l s y s t e m f o r w h i c h t h i s d i s t i n c t i o n i s p o s s i b l e i s the hexagonal s y s t e m , i n w h i c h t h e r e a r e f i v e i n d e p e n d e n t e l a s t i c c o n s t a n t s . A c r y s t a l b e l o n g i n g t o t h e o r t h o r h o m b i c s y s t e m w i l l have hexagonal symmetry i f two o f the o r t h o r h o m b i c symmetry d i r e c t i o n s a r e e q u i v a l e n t T h i s r e l a t i o n s h i p i m p l i e s t h a t when b i s the p r e f e r r e d d i r e c t i o n i n a hexagonal m a t e r i a l , t he a c p l a n e i s e l a s t i c a l l y i s o t r o p i c . A n e c e s s a r y c o n d i t i o n f o r t h e hexagonal symmetry t o be a good a p p r o x i - m a t i o n i s t h a t t h e d i f f e r e n c e between t h e a and c a x i s Young's moduli be s m a l l , a t l e a s t compared t o the d i f f e r e n c e between the a and b a x i s Young's m o d u l i . The i d e a t h a t the a and c d i r e c t i o n s a r e a p p r o x i rnat e l y e q u a l by c o m p a r i s o n , i s s u p p o r t e d by r e c e n t room t e m p e r a t u r e c o m p r e s s i b i l i t y measurements (Debray e t a l 1977) and thermal e x p a n s i o n d a t a ( B l e s s i n g and Coppens 1974). In c o n c l u s i o n a hexagonal model i s the s i m p l e s t a p p r o x i m a t i o n t o t h e s t r u c t u r e o f TTF-TCNQ t h a t s t i l l i n c l u d e s t h e e s s e n t i a l a n i s o t r o p y o f the m a t e r i a l . 2.2 I n t e r p r e t a t i o n o f E x p e r i m e n t a l Mode Spectrum The e x p r e s s i o n s d e r i v e d i n the p r e v i o u s s e c t i o n f o r the v i b r a t i o n f r e q u e n c i e s o f e l o n g a t e d p l a t e s w i l l be used t o i n t e r p r e t t he e x p e r i - mental mode s p e c t r u m o f TTF-TCNQ c r y s t a l s . As we have a l r e a d y p o i n t e d o u t t h e r e a r e s e v e r e m a t h e m a t i c a l d i f f i c u l t i e s i n c a l c u l a t i n g t he v i b r a t i o n f r e q u e n c i e s o f a r e c t a n g u l a r r e s o n a t o r made o f a m o n o c l i n i c 10 khz 30 khz 50 khz E x p e r i m e n t a l f l e x u r a l mode s p e c t r u m . The numbered marks be low t he e x p e r i m e n t a l t r a c e i n d i c a t e the f r e q u e n c i e s o f t he f l e x u r a l modes computed f r om [2] u s i n g the measured sample d imen s i on s and a sound v e l o c i t y t o f i t t he second f l e x u r a l mode. The n o i s e l e v e l i s c omparab le t o t he t h i c k n e s s o f t he l i n e . m a t e r i a l . However, a more s e r i o u s p r a c t i c a l l i m i t a t i o n t o the a c c u r a t e c a l c u l a t i o n o f the r e s o n a n t f r e q u e n c i e s i s the somewhat i r r e g u l a r geometry o f the a v a i l a b l e TTF-TCNQ c r y s t a l s . For example, the sample t h i c k n e s s ( c * d i m e n s i o n ) w h i c h i s n o r m a l l y the l e a s t u n i f o r m d i m e n s i o n , t y p i c a l l y t a p e r s o f f s u b s t a n t i a l l y n ear the ends. I f the ends a r e c u t o f f w i t h a r a z o r b l a d e a good c r y s t a l w i l l n o t v a r y i n t h i c k n e s s by more than about 10% o v e r i t s l e n g t h . In a d d i t i o n , the s i l v e r p a i n t clamp on the end o f t h e sample i n the v i b r a t i n g reed c o n f i g u r a t i o n , i s n e i t h e r p e r f e c t l y r i g i d n or p e r f e c t l y u n i f o r m . S i m i l a r l y the c e n t r a l s u p p o r t p o i n t f o r the l o n g i t u d i n a l l y mounted samples p e r t u r b s t h e f r e e - f r e e boundary c o n d i t i o n s . For a l l o f t h e s e r e a s o n s we e x p e c t t o see d e v i a t i o n s from the i d e a l i z e d r esonance f r e - q u e n c i e s g i v e n i n S e c t i o n 2.1 above. ( i ) V i b r a t i n g Reed S u p p o r t The v i b r a t i n g reed s u p p o r t c o n f i g u r a t i o n was used t o s t u d y f l e x u r a l and t o r s i o n a l modes. An e x p e r i m e n t a l f l e x u r a l mode s p e c t r u m i s g i v e n i n F i g . 3 4 , w h i c h shows the f i r s t f o u r f l e x u r a l modes a l o n g w i t h a t h e o r e t i c a l s p e c t r u m o b t a i n e d from [2] by a d j u s t i n g t h e Young's modulus t o f i t the second e x p e r i m e n t a l f l e x u r a l mode. The f i r s t r e sonance i n Fig.34 appears t o be weaker than the second one because i t has a lower Q (see S e c t i o n 2.3 below) and the a m p l i f i e r g a i n i s s m a l l e r a t low f r e q u e n c i e s . The resonance l i n e s a r e a n t i s y m m e t r i c because the phase o f the r e f e r e n c e i n p u t t o the m i x e r has been s e t t o d e t e c t the component o f the sample response w h i c h i s o u t o f phase w i t h the d r i v i n g f o r c e . 133 400 300- f ( k h z ) 2 0 0 - I00h 3 4 5 ( ir/2 units) F i g . 35 - Low frequency f l e x u r a l and t o r s i o n a l modes of a v i b r a t i n g reed. By i n c r e a s i n g the s y s t e m g a i n , i t was p o s s i b l e t o see the next f i v e h a r monics i n the s e r i e s shown i n F i g . 3 ^ . In a d d i t i o n , s i x low f r e - quency 1^ t o r s i o n a l and f l e x u r a l modes were i d e n t i f i e d . By ana- l o g y w i t h the c o n t i n u o u s d i s p e r s i o n c u r v e s f o r a c o u s t i c waves p r o - p a g a t i n g a l o n g an i n f i n i t e l y l o n g sample, d i s p e r s i o n c u r v e s can a l s o be p l o t t e d f o r the modes o f a f i n i t e l e n g t h sample as a s e r i e s o f d i s c r e t e p o i n t s . The low f r e q u e n c y mode d i s p e r s i o n d i a g r a m f o r a sample i n the v i b r a t i n g reed c o n f i g u r a t i o n i s shown i n Fig.35. In a d d i t i o n t o the modes p l o t t e d i n Fig.35, a l a r g e number o f u n i d e n t i - f i e d h i g h e r f r e q u e n c y modes a r e o b s e r v e d e x p e r i m e n t a l l y up t o about 1 MHz. Three p i e c e s o f i n f o r m a t i o n a r e h e l p f u l i n i d e n t i f y i n g t he v a r i o u s f l e x u r a l and t o r s i o n a l modes. F i r s t , t he t o r s i o n a l modes can be s e - p a r a t e d from t h e f l e x u r a l modes by t h e i r t e m p e r a t u r e dependences. The t o r s i o n a l mode f r e q u e n c i e s depend on a s h e a r v e l o c i t y w h i c h has a weaker t e m p e r a t u r e dependence than t h e Young's modulus v e l o c i t y w h i c h d e t e r m i n e s the f l e x u r a l f r e q u e n c i e s (see Ch a p t e r I I I b e l o w ) . S e c o n d l y , the two d i f f e r e n t t y p e s o f f l e x u r a l modes ( F ^ a and F b c ) may be d i s t i n g u i s h e d by the way they c o u p l e t o the d r i v e and d e t e c t o r e l e c t r o d e s . For example the F^ a f l e x u r a l modes a r e p r e f e r e n t i a l l y e x c i t e d i f the a x i s o f the d r i v e and d e t e c t o r e l e c t r o d e s i s a l i g n e d p a r a l l e l t o the a a x i s . N e e d l e s s t o s a y , no m a t t e r w h i c h t y p e o f mode i s p r e f e r e n t i a l l y e x c i t e d i t i s a l m o s t i m p o s s i b l e t o a v o i d a s l i g h t e x c i t a t i o n o f a l l the o t h e r t y p e s o f modes t h r o u g h n o n - i d e a l sample and e l e c t r o d e geometry. F i n a l l y , once the modes have been 135 B I A S V O L T A G E F i g . 36 ~ E f f e c t o f d . c . b i a s v o l t a g e o n f u n d a m e n t a l F b c f l e x u r a l mode r e s o n a n c e f r e q u e n c y . i d e n t i f i e d as F b a , F b c o r T b, the f r e q u e n c y e q u a t i o n s [2] and [5] can be used t o a s s i g n harmonic numbers and as a check on the F b a , F b c and T"b i d e n t i f i c a t i o n . The resonance f r e q u e n c i e s o f a t h i n v i b r a t i n g reed can be a r t i f i - c i a l l y reduced i f a l a r g e d.c. b i a s v o l t a g e i s a p p l i e d t o the d r i v e o r d e t e c t o r e l e c t r o d e s (Barmatz and Chen 197*0. Because o f the d ~ 2 dependence o f t h e f o r c e on the p l a t e s o f a ch a r g e d p a r a l l e l p l a t e c a p a c i t o r , t h e s p r i n g c o n s t a n t f o r a bent reed w i l l have an e l e c t r i c a l component as w e l l as an e l a s t i c component. In the t e c h n i q u e d e s c r i b e d h e r e no d.c. b i a s f i e l d i s r e q u i r e d on the d e t e c t o r c a p a c i t o r . However, a d.c. b i a s i s used a t the d r i v e e l e c t r o d e t o i n c r e a s e the d r i v i n g f o r c e on the sample. The e f f e c t o f t h i s b i a s v o l t a g e on the f r e q u e n c y o f the. fundamental F b c f l e x u r a l mode i s shown i n Fig.36. The d.c. f i e l d has a much s m a l l e r e f f e c t on the t o r s i o n a l and h i g h e r h armonic f l e x u r a l modes. As l o n g as t h e l o n g a x i s o f the reed i s a l i g n e d p a r a l l e l w i t h the c r y s t a 1 l o g r a p h i c b a x i s , t he low f r e q u e n c y modes g i v e no i n f o r m a t i o n about the Young's moduli p e r p e n d i c u l a r t o the b a x i s . In o r d e r t o measure the a a x i s Young's modulus, t h i n s l i c e s were c u t o f f the end o f normal TTF-TCNQ. c r y s t a l s p e r p e n d i c u l a r t o the b a x i s . The shape o f the f o u r s l i c e s s t u d i e d e x p e r i m e n t a l l y ranged from n e a r l y s q u a r e t o r e c t a n g u l a r w i t h the a d i m e n s i o n t w i c e as l o n g as the b d i m e n s i o n . One end (be* p l a n e ) o f t h e sample was g l u e d t o a s u p p o r t t o produce a s m a l l v i b r a t i n g reed w i t h i t s l o n g a x i s i n the a d i r e c t i o n . Only t h e f i r s t t h r e e o r f o u r modes c o u l d be c o n f i d e n t l y i n t e r p r e t e d i n terms o f the mode p a t t e r n s shown i n F i g . 3 1 o f S e c t i o n 2.1 ( i i i ) above. The f i r s t and t h i r d r e s onances a r e F a c f l e x u r a l modes, whose f r e q u e n c i e s a r e d e t e r m i n e d by the a a x i s Young's modulus E a as o u t l i n e d i n T a b l e IV. S e c t i o n 2.1 (v) above. S i m i l a r l y , the second mode i s a T a t o r s i o n a l mode whose f r e q u e n c y i s d e t e r m i n e d by the s h e a r modulus c^g as i n d i c a t e d i n eqn. [11]. The measured sound v e l o c i t i e s a r e summarized i n T a b l e IV. The s i g n a l t o n o i s e r a t i o f o r t h e fundamental f l e x u r a l mode was g r e a t e r than 100 w i t h a 1 Hz n o i s e b a n d w i d t h even f o r the s m a l l e s t sample s t u d i e d , w h i c h was a 2 u g , 0 . 3 mm l o n g s l i c e . S i n c e the a a x i s samples were s i g n i f i c a n t l y l e s s u n i f o r m than the b i g g e r b a x i s samples t h e e x p e r i m e n t a l v a l u e f o r E a i s n o t as a c c u r a t e as the e x p e r i m e n t a l v a l u e f o r E^. An o t h e r e x p e r i m e n t a l d i f f i c u l t y a r o s e on c o o l i n g t h e a a x i s s a m p l e s . U n l e s s the l e n g t h (a d i m e n s i o n ) o f t h e sample was s i g n i f i c a n t l y b i g g e r than i t s w i d t h (b d i m e n s i o n ) , d i f f e r e n t i a l thermal c o n t r a c t i o n o f t h e bond a t t h e end would s p l i t the sample i n h a l f a l o n g i t s l e n g t h . Thermal c r a c k i n g o f the sample shows up as a l a r g e i r r e p r o d u c i b 1 e d i s - c o n t i n u i t y i n the t e m p e r a t u r e dependence o f the resonance f r e q u e n c y , and i n a s p l i t t i n g o f a s i n g l e resonance l i n e i n t o a d o u b l e t o r m u l t i p l e t . The t h e r m a l l y i n d u c e d c r a c k s a r e v i s i b l e i f the c r y s t a l i s examined under a m i c r o s c o p e . Of t h e f o u r a a x i s v i b r a t i n g reeds measured o n l y one would c y c l e down t o h e l i u m t e m p e r a t u r e and back t o room t e m p e r a t u r e w i t h o u t b r e a k i n g . I t i s p o s s i b l e t h a t some s m a l l f r a c t u r e s were a l s o produced near the bond i n the l o n g e r b a x i s v i - b r a t i n g reed samples d u r i n g t h e r m a l c y c l i n g . 2.0 1.5 1.0 0 / S A M P L E # 1 6 . / L MODE A y © / / / / / / / / / / / / / / [/_ 0 1 2 3 4 5 6 7 WAVENUMBER <jrU units) Fig. 37 - Longitudinal mode dispersion ( i i ) C e n t r a l P i n Su p p o r t The problems w i t h 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 may be l a r g e l y a v o i d e d i f the sample i s s u p p o r t e d by a v e r y s m a l l s i l v e r p a i n t c o n t a c t on the end o f a p o i n t e d w i r e , as d e s c r i b e d i n C h a p t e r I , S e c t i o n 1.1 ( i i ) . T h i s s u p p o r t c o n f i g u r a t i o n i s a f a v o u r a b l e one f o r e x c i t i n g e l o n g a t i o n a l a c o u s t i c modes i n t h e sample. For a l o n g sample w i t h a u n i f o r m c r o s s - s e c t i o n , the e l o n g a t i o n a l resonance f r e q u e n c i e s a r e g i v e n by [8] p r o - v i d e d one i g n o r e s the s m a l l p i n c o n t a c t a t the c e n t r e o f the broad a b f a c e o f the sample. T h i s a p p r o x i m a t i o n i s e x p e c t e d t o be b e s t f o r the odd numbered modes s i n c e t h e s e modes have a node a t the s u p p o r t p i n . I f the e x p e r i m e n t a l e l o n g a t i o n a l mode f r e q u e n c i e s a r e p l o t t e d as a f u n c t i o n o f the wavenumber 2-nn/Z, where L i s the l e n g t h o f the sample, t h e n one o b t a i n s t h e d i s c r e t e d i s p e r s i o n c u r v e shown i n F i g . 3 7 . Eqn. [8] p r e d i c t s a l i n e a r dependence o f the resonance f r e q u e n c y on wavenumber. A c l o s e e x a m i n a t i o n o f Fig.37 revea1s t h a t the even h a r m o n i c s a r e s l i g h t l y below the s t r a i g h t l i n e and the odd harmonics a r e s l i g h t l y above the s t r a i g h t l i n e . In a d d i t i o n , the even harmonics tend t o be more h e a v i l y damped than the odd h a r m o n i c s . We a t t r i b u t e t h e s e d i f f e - r e n c e s between the even and odd har m o n i c s t o the e f f e c t o f the c e n t r a l p i n s u p p o r t . The e f f e c t o f t h e f i n i t e l e n g t h c o r r e c t i o n f a c t o r [9] ( o r [12]) i s t o reduce the f r e q u e n c y o f the s e v e n t h harmonic by 1-2%. Beyond the f o u r t h h a r m o n i c , more than one f r e q u e n c y i s p l o t t e d i n F i g . 3 7 f o r each wavenumber. The r e a s o n f o r t h e m u l t i p l i c i t y i s t h a t t h e r e a r e s e v e r a l modes o f n e a r l y e q u a l s t r e n g t h near the f r e q u e n c y where a l o n g i t u d i n a l resonance s h o u l d be. Pre s u m e a b l y , as the wave- length of the longitudinal mode becomes shorter, the non-uniformities in the sample dimensions become progressively more important and the longitudinal modes are coupled more strongly to other acoustic cavity modes. The other acoustic modes which could be coupled to the longi- tudinal modes are high frequency harmonics of the b axis flexural and torsional modes and transverse F a c type flexural modes. Moreover, the sample whose longitudinal resonance frequencies are shown in Fig. 37 begins to support transverse shear wave resonances at frequencies corresponding to n > 7. Above 2 MHz the acoustic mode spectrum de- generates into a closely spaced set of heavily damped resonances. Of course a l l of these modes are affected in some complicated way by the central support. This complexity of the high frequency acoustic mode spectrum provides a practical upper limit to the frequency range for which the acoustic resonance method is useful. The measurements could probably be extended to higher frequencies by studying acoustic pulse propagation along the sample rather than looking at resonances. A comparison of eqns. [2] and [8] in light of the discussion on the crystalline anisotropy in Section 2.1 (v) above, reveals that the elongational and b axis flexural resonance frequencies are determined by the same elastic constant to f i r s t approximation - namely Eb. The Young's modulus velocity (ED/p)5 has been measured for eight TTF-TCNQ. samples by substituting the experimentally determined flexural and longitudinal resonance frequencies into [2] and [8] respectively. Within our experimental error there is no systematic softening of the F b c type flexural modes as would be expected from [3], if the shear modulus C L ^ were anomalously small as has been suggested by Barmatz e t a l (197*0 and I s h i g u r o e t a l (1977)- The shear s o f t e n i n g o f t he h i g h e r f r e q u e n c y type modes i s c o n s i s t e n t w i t h the s h e a r modulus C 5 5 t h a t i s d e t e r m i n e d from the t o r s i o n a l modes. Even though the f l e x u r a l modes can be used t o d e t e r m i n e the same e x t e n s i o n a l v e l o c i t y ( E ^ / p ) ^ as the l o n g i t u d i n a l mode, the l o n g i t u d i n a l one g e n e r a l l y g i v e s a more a c c u r a t e e s t i m a t e o f t h i s v e l o c i t y . The reaso n i s t h a t t he l o n g i t u d i n a l resonance i s l e s s s e n s i t i v e t o v a r i a t i o n s i n the sample t h i c k n e s s ( c * d i m e n s i o n ) w h i c h i s the l e a s t u n i f o r m dimen- s i o n . The room t e m p e r a t u r e sound v e l o c i t i e s as d e t e r m i n e d from t h e a c o u s t i c resonance f r e q u e n c i e s o f f i f t e e n d i f f e r e n t s a m p l e s , a r e summarized i n T a b l e V. The b a x i s e x t e n s i o n a l v e l o c i t y ( E ^ / p ) ^ and th e s h e a r v e l o c i t y ( c g g / p ) ^ a r e c o n s i s t e n t w i t h i n e l a s t i c n e u t r o n s c a t t e r i n g measurements o f S h a p i r o e t a l (1977). The a a x i s e x t e n - 1 s i o n a l v e l o c i t y ( E a / p ) ? i s c o n s i s t e n t w i t h r e c e n t c o m p r e s s i b i l i t y measurements o f Debray e t a l (1977). The d e t a i l e d t e m p e r a t u r e depen- dence o f t h e s e v e l o c i t i e s i s d i s c c u s e d i n the n e x t c h a p t e r . TABLE V Room Temperature Sound V e l o c i t i e s i n TTF-TCNQ Mode V e l o c i t y ( 1 0 s cm/s) b a x i s e x t e n s i o n a l 2.8 ± .1 a a x i s e x t e n s i o n a l k.k ± .5 egg s h e a r 1.7 ± .2 100 140 180 220 T (K) Fig. 38 - Flexural and torsional mode crossing ( i i i ) Mode C o u p l i ng B e f o r e g o i n g on t o d i s c u s s the t e m p e r a t u r e dependence o f the sound v e l o c i t i e s i n d e t a i l , we f i r s t o u t l i n e some o f the e f f e c t s o f e x t r a n e o u s mode c o u p l i n g on the t e m p e r a t u r e dependence o f the resonan c e f r e q u e n c i e s . As p o i n t e d o u t i n the n e x t c h a p t e r , the b a x i s Young's modulus v e l o c i t y has a s t r o n g e r t e m p e r a t u r e dependence than the s h e a r v e l o c i t y . For t h i s r e a s o n , i f a mode w h i c h depends on the Young's modulus i s a t n e a r l y t h e same f r e q u e n c y as a t o r s i o n a l mode a t one t e m p e r a t u r e the two modes may c r o s s as the t e m p e r a t u r e i s changed. A mode c r o s s i n g o f t h i s t y p e i s shown i n F i g . 38. The t o r s i o n a l mode s t a r t s o f f j u s t above a nearby f l e x u r a l mode a t h i g h t e m p e r a t u r e s . As t h e t e m p e r a t u r e i s low e r e d the f r e q u e n c y o f the t o r s i o n a l mode moves below the f l e x u r a l mode because o f t he d i f f e r e n c e i n t h e i r t e m p e r a t u r e dependences. C o u p l i n g between the two modes p r e v e n t s them from a c t u a l l y i n t e r s e c t i n g . The c o u p l i n g c o u l d be caused by the o f f d i a g o n a l component S26 ' n the m o n o c l i n i c c o m p l i a n c e m a t r i x (Hearmon 1961) o r by asymmetry i n the sample o r s u p p o r t . The t o r s i o n a l mode i n Fig.38 i s t h e fundamental t o r s i o n a l mode and the f l e x u r a l mode i s p r o b a b l y a symm e t r i c F b c t y p e f l e x u r a l mode w i t h t h r e e nodes. These r e s o n a n c e s were o b s e r v e d w i t h t h e sample s u p p o r t e d i n the l o n g i t u d i n a l mode c o n f i g u r a t i o n . Mode c o u p l i n g can be a pr o b l e m i n making a c c u r a t e measurements o f the t e m p e r a t u r e dependence o f the sound v e l o c i t y . The p r o b l e m seems t o be p a r t i c u l a r l y s e v e r e f o r e l o n g a t i o n a l modes i n the t e m p e r a t u r e range between 20K and 52K, where the Young's modulus has an a n o m a l o u s l y s t r o n g t e m p e r a t u r e dependence compared w i t h t h e s h e a r modulus (see n e x t c h a p t e r ) . 93.0 T F i g . 39 - F a c f l e x u r a l mode w i t h i n t e r f e r e n c e from an u n i d e n t i f i e d mode o f the s u p p o r t The s t r o n g t e m p e r a t u r e dependence makes i t more l i k e l y t h a t the r e - sonance o f i n t e r e s t w i l l c r o s s some o t h e r mode w i t h a weaker tem- p e r a t u r e dependence. F i g . 39 shows a t y p i c a l mode c r o s s i n g o f t h i s t y p e . Here an F a c t y p e mode i s c r o s s e d near 39K by an u n i d e n t i f i e d mode, t h a t i s p r o b a b l y r e l a t e d t o the s u p p o r t . ( i v ) S u pport Modes I t i s v i r t u a l l y i m p o s s i b l e t o a v o i d some i n t e r f e r e n c e w i t h t h e sample resonances from the modes o f the s u p p o r t . However, i t i s p o s s i b l e t o a v o i d s u p p o r t modes o v e r a l i m i t e d f r e q u e n c y range. For example no s u p p o r t modes were o b s e r v e d among the low f r e q u e n c y f l e x u r a l and t o r s i o n a l modes o f b a x i s c r y s t a l s clamped i n the v i b r a t i n g reed c o n f i g u r a t i o n . On t h e o t h e r hand, as i l l u s t r a t e d i n Fi g . 3 9 t h e r e were some problems w i t h i n t e r f e r e n c e w i t h t h e f l e x u r a l modes o f t h e a a x i s s l i c e s p r o b a b l y because t h e s e modes were a t h i g h e r f r e q u e n c i e s . I n t e r - f e r e n c e from t h e modes o f the t u n g s t e n s u p p o r t w i r e f o r the l o n g i t u d i n a l mounted sam p l e s , was a l l e v i a t e d by u s i n g a t h i c k s u p p o r t p o s t (0.015" d i a m e t e r ) so t h a t the s u p p o r t modes were w i d e l y spaced and a t r e l a t i v e l y h i g h f r e q u e n c i e s . In any c a s e the sample modes and s u p p o r t modes can be e a s i l y d i s t i n g u i s h e d a t h i g h t e m p e r a t u r e s , by the d r a m a t i c d i f f e r e n c e i n t h e i r t e m p e r a t u r e dependences. 2.3 V i b r a t i o n Damping ( i ) Q_ Measurement The a c o u s t i c resonances can be used t o d e t e r m i n e the a b s o r p t i o n o f sound as w e l l as the v e l o c i t y o f sound. The a b s o r p t i o n i s p r o p o r t i o n a l t o the w i d t h o f the resonance l i n e . A c o n v e n i e n t measure o f the ab- s o r p t i o n i s the q u a l i t y f a c t o r Q_ d e f i n e d by Q"1 = A f / f where f i s t h e resonance f r e q u e n c y and Af i s t h e " f u l l w i d t h a t h a l f maximum" o f the s y m m e t r i c ( i n phase) a m p l i t u d e r e s p o n s e . E q u i v a l e n t l y Af i s the s e p a r a t i o n between extrema i n the a n t i s y m m e t r i c (out o f phase) r e s o n a n t r e s p o n s e . The Q_ i s r e l a t e d t o the i n t e n s i t y a t t e n u a t i o n f a c t o r a d i s c u s s e d i n P a r t A above by a = Q._1q where q i s t h e sound wavenumber. The Q_ was measured e x p e r i m e n t a l l y by comparing the s y m m e t r i c r e - sponse o f the sample w i t h a s y n t h e t i c l o r e n t z i a n on a d u a l beam o s c i l l o s c o p e . The s y n t h e t i c l o r e n t z i a n was g e n e r a t e d by sweeping a v o l t a g e c o n t r o l l e d o s c i l l a t o r t h r o u g h t h e r e s o n a n c e f r e q u e n c y o f a tuned c i r c u i t (Q_ ~ 70) as shown i n the c i r c u i t d i a g r a m i n A p p e n d i x § 1 . ( v ) . By a d j u s t i n g the a m p l i t u d e o f the f r e q u e n c y sweep o f the v o l t a g e c o n t r o l l e d o s c i l l a t o r , t h e a p p a r e n t w i d t h o f t h e s y n t h e t i c l i n e c o u l d be a d j u s t e d t o match t h e w i d t h o f the sample r e s o n a n c e . A f t e r c a l i b r a t i o n the s y n t h e t i c l o r e n t z i a n p r o v i d e d a c o n v e n i e n t means f o r m e a s u r i n g l i n e w i d t h t o a r e l a t i v e a c c u r a c y o f 1%. In o r d e r t o make a c c u r a t e l i n e w i d t h measurements one must be c a r e f u l t o a v o i d d i s t o r t i n g the sample resonance l i n e s . For example i f t h e d r i v e o s c i l l a t o r i s swept too q u i c k l y t h r o u g h t h e a c o u s t i c r e s o n a n c e , a r i n g i n g phenomena known i n NMR as " w i g g l e s " (Abragam 1961) w i l l o c c u r . An extreme example o f w i g g l e s i s shown i n F i g . kO. H o r i z o n t a l s c a l e : 1 Hz/cm Sweep r a t e : 3-3 Hz/s F i g . 40 - W i g g l e s ( i i ) T h e r m o e l a s t i c Damping Heat c o n d u c t i o n i s an i m p o r t a n t l o s s mechanism f o r the low f r e - quency f l e x u r a l modes o f TTF-TCNQ. c r y s t a l s . As d i s c u s s e d by Zener (1948) and B h a t i a ( 1 9 6 7 ) , whenever the i s o t h e r m a l and a d i a b a t i c e l a s t i c moduli (see Ch a p t e r I I I ) a r e n o t e q u a l , t h e r m a l c o n d u c t i o n between c o m p r e s s i o n s and r a r e f a c t i o n s w i l l cause a c o u s t i c damping. In c o n v e n t i o n a l m e t a l s t h e r m a l c o n d u c t i o n does not cause s i g n i f i c a n t damping o f l o n g i t u d i n a l waves a t f r e q u e n c i e s below about 10 Ghz. However, the damping can be s u b s t a n t i a l f o r f l e x u r a l modes o f t h i n p l a t e s where the compressed p a r t o f the p l a t e i s c l o s e t o the ex- . panded p a r t . The t h e r m o e l a s t i c damping a r e c t a n g u l a r c r o s s e c t i o n i s o f a f l e x u r a l resonance o f a reed w i t h g i v e n by ( B h a t i a 1967) [13] Q. ^ — f 2 + f 2 • f o - 2 " r T where E s and Ey a r e the a d i a b a t i c and i s o t h e r m a l Young's moduli r e - s p e c t i v e l y , f i s t h e resonance f r e q u e n c y and t i s the reed t h i c k n e s s , measured a l o n g c" f o r the F D C modes. The thermal d i f f u s i v i t y D i s the r a t i o o f the th e r m a l c o n d u c t i v i t y K t o the s p e c i f i c heat a t c o n s t a n t volume, C y. To o b t a i n a n u m e r i c a l e s t i m a t e f o r the ab- s o r p t i o n we a p p r o x i m a t e ( E s - E j ) / E s by ( B s - B j ) / B s where B g (By) i s the a d i a b a t i c ( i s o t h e r m a l ) b u l k modulus, and use By/B s = C v/Cp d e r i v e d by B h a t i a (1567) (see a l s o eqn. [2] i n Ch a p t e r I I I b e l o w ) . W i t h the e x p r e s s i o n f o r the d i f f e r e n c e between Cp and C v g i v e n by Landau and L i f s h i t z (1969) and u s i n g t h e Gr'uneisen a p p r o x i m a t i o n d i s c u s s e d i n S e c t i o n 3-1 o f Chapt e r I I I below, we o b t a i n ~ Ey C - C v [14] % s = P C p = YaT where y ~ 2.6 i s a Gr'uneisen c o n s t a n t and a i s the volume e x p a n s i o n c o e f f i c i e n t . From eqns [13] and [14] we can e s t i m a t e Q"1 f o r the f l e x u r a l modes u s i n g p u b l i s h e d thermal e x p a n s i o n ( S c h a f e r e t a l 1975, B l e s s i n g and Coppens 197*0, b a x i s thermal c o n d u c t i v i t y (Salamon e t a l 1975) and s p e c i f i c heat d a t a (Craven e t a l 197*0. For example, c o n s i d e r Sample #13 w h i c h was 0.038 mm t h i c k and had i t s fundamental f l e x u r a l r e s o n a n c e 2 5 2 0 15 I (IO"4) 10 0 T T SAMPLE 13 MODE NUMBERS 1 2 3 4 oo oo o 4 A x * o o 0°° * e © o O O © © ©o e © ° © o o o _L 1 2 0 4 0 T ( K ) 6 0 F i g . 41 - Damping o f f i r s t f o u r F b c f l e x u r a l modes a t 2.1 khz. For t h i s mode we e s t i m a t e Q - 1 = 8x10"^ u s i n g a therm a l r e l a x a t i o n f r e q u e n c y f Q = 21 khz. The e x p e r i m e n t a l l y d e t e r m i n e d a b s o r p t i o n i s shown i n F i g . 1*1 as a f u n c t i o n o f t e m p e r a t u r e f o r the f i r s t f o u r f l e x u r a l modes. A l t h o u g h the e s t i m a t e d v a l u e o f Q"1 a t 55K o f 8x10 - l t i s c l o s e t o t h e measured v a l u e o f 13x10 - l + f o r the fundamental f l e x u r a l mode a t 55K, the dependence o f the damping on the f l e x u r a l mode f r e q u e n c y i s wrong. W i t h f Q = 21 khz i n [13] the a b s o r p t i o n w i l l be l a r g e r f o r the second f l e x u r a l mode whereas e x p e r i m e n t a l l y i t i s o b s e r v e d t o be s m a l l e r . In a d d i t i o n , above 55K the damping o f the second, t h i r d and f o u r t h harmonics i s o n l y w e a k l y t e m p e r a t u r e dependent (see F i g . 4 1 ) . To h e l p e x p l a i n t h e s e f e a t u r e s o f t h e e x p e r i m e n t a l da"a we make th e f o l l o w i n g o b s e r v a t i o n s . F i r s t t he th e r m a l r e l a x a t i o n r a t e f D w i l l i n c r e a s e w i t h d e c r e a s i n g t e m p e r a t u r e a t l e a s t as f a s t as T - 1 . T h i s t e m p e r a t u r e dependence s h o u l d be v a l i d down t o 30K, a t l e a s t . S e c o n d l y Y a T f Q w i l l be a weak f u n c t i o n o f t e m p e r a t u r e . The ex- p e r i m e n t a l d a t a i n F i g . 4 1 may now be a c c o u n t e d f o r q u a l i t a t i v e l y by p o s t u l a t i n g t h a t f Q s a t i s f i e s f i < f Q < f 2 where f i ( f 2 ) a r e the f i r s t (second) f l e x u r a l r esonance f r e q u e n c i e s . F u r t h e r m o r e , i f we o b s e r v e t h a t the damping i s n e a r l y e q u a l f o r the f i r s t two modes a t 40K, we can e s t i m a t e f Q a t t h i s t e m p e r a t u r e . T h i s e s t i m a t e f o r f G (5 khz) i m p l i e s t h a t t he therm a l c o n d u c t i v i t y i n the c" d i r e c t i o n i s 0.02 W/cm-K. T h i s v a l u e compares w i t h 0.12 wa/cm-K f o r the b a x i s thermal c o n d u c t i v i t y measured by Salamon e t a l (1375) a t the same t e m p e r a t u r e . S i m i l a r l y f c = 5 khz i m p l i e s t h a t Q"1 = 12x10 - t t 6 4 I 4) 2 0 1 1 1 1 1 SAMPLE ' #23 • 4 r i P A W © y»\ .® © 1 I / ii / o f » ^ o © "V* © *• 1 1 1 1 1 0 50 100 150 200 250 300 T (K) hi - Damping o f fundamental l o n g i t u d i n a l mode. The d i s c o n t i n u i t y a t 205K a r o s e when a measurement was r e p e a t e d a f t e r a l l o w i n g t h e sample t o remain a t low t e m p e r a t u r e s o v e r n i g h t . a t kOK, w h i c h i s about a f a c t o r o f two b i g g e r than the o b s e r v e d a b s o r p t i o n . C l e a r l y , a l l o f t h e s e c a l c u l a t i o n s a r e v e r y rough. N e v e r t h e l e s s , one can c o n c l u d e t h a t t h e r m a l c o n d u c t i o n w i l l have a s i g n i f i c a n t e f f e c t on the damping o f the f l e x u r a l modes. I t s h o u l d be p o s s i b l e t o d e t e r m i n e the t r a n s v e r s e t h e r m a l d i f f u s i v i t y , by c a r e f u l measurements o f the damping o f f l e x u r a l modes. F i n a l l y , we n o t e t h a t t h e r m o e l a s t i c damping may a c c o u n t f o r the anomalous f r e q u e n c y dependence o b s e r v e d by Barmatz e t a l (1975) i n the damping o f f l e x u r a l modes i n 2H-TaSe 2. ( i i i ) E l o n g a t i o n a l Modes The room t e m p e r a t u r e damping o f the fundamental e l o n g a t i o n a l mode i n TTF-TCNQ. i s t y p i c a l l y an o r d e r o f magnitude, s m a l l e r than the room t e m p e r a t u r e damping o f the fundamental f l e x u r a l mode. In a d d i t i o n the a b s o r p t i o n f o r the e l o n g a t i o n a l modes i n c r e a s e s w i t h harmonic number u n l i k e the f i r s t few f l e x u r a l modes. The t e m p e r a t u r e dependence o f t h e a c o u s t i c a b s o r p t i o n f o r the fundamental l o n g i t u d i n a l mode o f Sample #23 i s shown i n F i g . kl. A l t h o u g h t h e magnitude and d e t a i l e d t e m p e r a t u r e dependence o f the e x p e r i m e n t a l l y o b s e r v e d l o n g i t u d i n a l mode damping a r e n o t c o m p l e t e l y r e p r o d u c i b l e between d i f f e r e n t samples and d i f f e r e n t runs w i t h the same sample, c e r t a i n g r o s s f e a t u r e s a r e a l w a y s p r e s e n t . There i s a broad minimum i n t h e v i c i n i t y o f 60K where Q ~ I O 4 . As the sample i s c o o l e d t h r o u g h the metal i n s u l a t o r t r a n s i t i o n the a b s o r p t i o n b e g i n s t o i n c r e a s e , r e a c h i n g a peak i n the range 3O-40K where i t i s a f a c t o r o f 3"6 t i m e s l a r g e r than the a b s o r p t i o n near 60K. As the t e m p e r a t u r e i s lowered s t i l l f u r t h e r the a b s o r p t i o n d e c r e a s e s a g a i n . 153 — © U i f ) O o o E^23 2 8 o ° 8 8 o o o o o o o o o o o o o o © Vc •0 46 4 8 5 0 52 54 56 Fig. A3 - Sound velocity and attenuation near the metal-insulator trans i t i on In Sample #23 t h e r e i s a l s o a 10K wide a b s o r p t i o n peak near 150K and a very narrow (< 0.5K wide) peak i n the damping j u s t below t he m e t a l - i n s u l a t o r t r a n s i t i o n . The d e t a i l s o f the t e m p e r a t u r e dependence o f t h e damping and the v e l o c i t y o f sound n e a r the m e t a l - i n s u l a t o r t r a n s i t i o n a r e shown i n F i g . 43. The t e m p e r a t u r e dependence o f the v e l o c i t y o f sound i s d i s c u s s e d i n d e t a i l i n the n e x t c h a p t e r . A l - though n e i t h e r the narrow a b s o r p t i o n peak n e a r the t r a n s i t i o n n o r the w i d e r maximum n e a r 150K were o b s e r v e d i n any o t h e r sample, i t i s q u i t e p o s s i b l e t h a t they were m i s s e d by not t a k i n g measurements a t f i n e enough t e m p e r a t u r e i n t e r v a l s . The a b s o r p t i o n peak near the m e t a l - i n s u l a t o r t r a n s i t i o n i s r e m i n i s c e n t o f a s i m i l a r f e a t u r e o b s e r v e d near t h e incommensurate c h a r g e d e n s i t y wave t r a n s i t i o n i n 2H-TaSe2 by Barmatz e t a l (1975). A l t h o u g h the peak near 150K may be due t o some e x t r a n e o u s e f f e c t , i t i s t e m p t i n g t o t r y t o r e l a t e i t t o the d i s - a p p e a r a n c e o f the 2kp s c a t t e r i n g o f d i f f u s e X - r a y s , o b s e r v e d by Khanna e t a l (1977) near 150K. An upper l i m i t t o the c o n v e n t i o n a l e l e c t r o n i c c o n t r i b u t i o n t o the a b s o r p t i o n can be o b t a i n e d by u s i n g the e x p r e s s i o n s d e r i v e d i n P a r t A, C h a p t e r I, f o r the a t t e n u a t i o n o f sound i n t h r e e d i m e n s i o n a l m e t a l s , o r the e x p r e s s i o n f o r the peak a t t e n u a t i o n j u s t below t he m e t a l - s e m i c o n d u c t o r t r a n s i t i o n , d i s c u s s e d i n P a r t A, S e c t i o n 2.3- From the b a n d s t r u c t u r e ( B e r l i n s k y e t a l 1974), d.c. c o n d u c t i v i t y , and c r y s t a l s t r u c t u r e we e s t i m a t e t h e e l e c t r o n i c e f f e c t i v e mass nr = 6 in-,, the Fermi v e l o c i t y V p - 107 cm/s, the e l e c t r o n i c s c a t t e r i n g time x ~ 5x10~ll*s a t 60K and the c a r r i e r d e n s i t y n = 2.8x1021 cm" 3. W i t h t h e s e v a l u e s f o r the m a t e r i a l p a r a m e t e r s , the e l e c t r o n i c con- t r i b u t i o n t o the damping o f a 300 khz l o n g i t u d i n a l mode i s c a l c u l a t e d t o be Q - 1 - 10~ 9. S i n c e the measured a b s o r p t i o n i s o f o r d e r 10 - 1*, we c o n c l u d e t h a t t h e o b s e r v e d damping i s n o t due t o the c o n d u c t i o n e l e c t r o n l o s s mechanism d i s c u s s e d i n P a r t A. I f TTF-TCNQ. remained m e t a l l i c a t low t e m p e r a t u r e s i t i s p o s s i b l e t h a t t h i s e l e c t r o n i c l o s s mechanism wou l d e v e n t u a l l y become i m p o r t a n t a t low t e m p e r a t u r e s , as i t does i n o r d i n a r y m e t a l s . A l t h o u g h t h e r m a l c o n d u c t i o n i s p r o b a b l y t h e dominant l o s s mechanism f o r the low f r e q u e n c y f l e x u r a l modes, i t s c o n t r i b u t i o n t o the damping o f t h e l o n g i t u d i n a l modes i s c o m p l e t e l y n e g l i g i b l e . D i s l o c a t i o n damping ( B h a t i a 1 9 6 7 ) and c o u p l i n g t o low Q s u p p o r t modes a r e p r o b a b l y i m p o r t a n t s o u r c e s o f l o s s f o r the l o n g i t u d i n a l modes. An a d d i t i o n a l damping mechanism i s s u g g e s t e d by the model used i n t h e n e x t c h a p t e r t o e x p l a i n the t e m p e r a t u r e dependence o f the sound v e l o c i t y . We comment on t h i s l o s s mechanism a t t h e end o f t h e n e x t c h a p t e r . ( i v ) E f f e c t o f A i r on Resonance Frequency and Q B e f o r e g o i n g on t o d i s c u s s t h e t e m p e r a t u r e dependence o f the sound v e l o c i t y i n TTF-TCNQ we b r i e f l y o u t l i n e t he e f f e c t o f a i r a t one a t - mosphere on the r e s o n a n t f r e q u e n c i e s and Q o f the v i b r a t i n g sample. The s h i f t i n the r e s o n a n t f r e q u e n c y i s due t o the mass o f e n t r a i n e d a i r t h a t accompanies the v i b r a t i n g sample. When the sample v i b r a t i o n f r e q u e n c y i s low enough t h a t the c o r r e s p o n d i n g w a v e l e n g t h o f sound i n a i r i s l o n g compared t o the t r a n s v e r s e ( a ) d i m e n s i o n o f the sample, the s u r r o u n d i n g a i r may be t r e a t e d as an i n c o m p r e s s i b l e non- v i s c o u s f l u i d . F or t y p i c a l TTF-TCNQ c r y s t a l s t h i s c o n d i t i o n i s w e l l s a t i s f i e d up t o about 50 khz. In t h i s l i m i t the a i r e n t r a i n e d by an F| 3 C f l e x u r a l mode may be a p p r o x i m a t e d by a c y l i n d e r w i t h i t s a x i s a l o n g b and i t s d i a m e t e r e q u a l t o the w i d t h o f the c r y s t a l , as shown i n F i g . 44. c" 3. M^taftT. 1" III n e n t r a i ned a i r a c" sample c r o s s e c t i o n Fig.44 - A i r E n t r a i n e d by a F l e x u r a l Mode I t i s easy t o show t h a t the e f f e c t i v e i n c r e a s e i n the mass o f the sample per u n i t l e n g t h l e a d s t o a r e d u c t i o n i n the F l e x u r a l ( F D C ) r e s o n a n t f r e q u e n c y by the f a c t o r [15] 1 + TT W p a i r 4 t p where p •„ and p a r e t h e d e n s i t i e s o f a i r and the sample r e s p e c t i v e l y . The same r e s u l t has been o b t a i n e d by L i n d h o l m e t a l (1965) u s i n g a more s o p h i s t i c a t e d a p p r o a c h . L i n d h o l m e t a l (1965) a l s o c a l c u l a t e the c o r r e s p o n d i n g mass l o a d i n g f o r t o r s i o n a l modes and o b t a i n the c o r r e c t i o n f a c t o r T h i s f a c t o r may be a p p r o x i m a t e d by c a l c u l a t i n g the a d d i t i o n a l a x i a l moment o f i n e r t i a c o n t r i b u t e d by two c y l i n d e r s o f a i r p a r a l l e l t o t h e l o n g a x i s o f the sample ( b ) and w i t h d i a m e t e r s e q u a l t o h a l f t h e w i d t h o f the sample as shown i n F i g . 45. The r e s u l t i n g c o r r e c t i o n f a c t o r i s o e n t r a i ned a i r a c" sample c r o s s e c t i o n Fig.45 - A i r E n t r a i n e d by a T o r s i o n a l Mode th e same as [16] e x c e p t t h a t the n u m e r i c a l f a c t o r 3̂ /32 i s r e p l a c e d by 9TT/64. The f r e q u e n c y s h i f t s caused by a i r l o a d i n g can be measured by l o o k i n g f o r a change i n resonance f r e q u e n c y when the sample chamber a -a- Is e v a c u a t e d . For example c o n s i d e r Sample #14. A c o m p a r i s o n be- tween the o b s e r v e d e f f e c t o f a i r on the resonance f r e q u e n c y and the p r e d i c t i o n s o f [15] and [16] i s shown i n T a b l e VI f o r the second and f i f t h F b c f l e x u r a l modes and the second and t h i r d T b t o r s i o n a l modes. In T a b l e VI the a i r i s assumed t o be a t one atmosphere w i t h d e n s i t y 0.00129 g/cm 3. The o b s e r v e d f r e q u e n c y s h i f t s a r e i n r e a s o n a b l e agreement w i t h the p r e d i c t e d v a l u e s . TABLE VI E f f e c t o f A i r a t One Atmosphere on F l e x u r a l and T o r s i o n a l Mode F r e q u e n c i Mode Frequency (khz) Frequency Theory S h i f t (hz) Exp t . F b c 2 5-3 19 25 F b c 5 44.6 162 120 \ 2 47-2 65 100 T b 3 73-5 101 100 The e l o n g a t i o n a l modes a r e a t too h i g h a f r e q u e n c y f o r the a i r t o be t r e a t e d as an i n c o m p r e s s i b l e n o n - v i s c o u s f l u i d . The p r i m a r y s o u r c e o f a i r e n t r a i n m e n t i n a l o n g i t u d i n a l mode i s the v i s c o u s boundary l a y e r w h i c h a t t a c h e s i t s e l f t o the br o a d ab s u r f a c e o f the c r y s t a l because o f the n o n - z e r o v i s c o s i t y o f t h e a i r . The e f f e c t i v e t h i c k n e s s o f t h e boundary l a y e r i s 6/2 (Landau and L i f s h i t z 1959) where 6 = (^n/ojp^ ; i s a s k i n d e p t h f o r s h e a r waves i n a f l u i d w i t h v i s c o s i t y n. I f we t a k e n = 1.8x10 - l + p o i s e (g/cm 2-s) a t room tem- p e r a t u r e then 6/2 - 2x10-t*cm f o r a 300 khz mode. T h i s l a y e r o f a i r w i l l reduce the r e s o n a n t f r e q u e n c y o f a 300 khz e l o n g a t i o n a l mode by 12 Hz. E x p e r i m e n t a l l y the change i n resonance f r e q u e n c y i s u s u a l l y o b s e r v e d t o be l a r g e r than 12 Hz. F u r t h e r m o r e the s h i f t does not a l w a y s have the same s i g n f o r d i f f e r e n t s a m p l e s . T h i s b e h a v i o u s i s i n t e r p r e t e d as b e i n g due t o a change i n c o u p l i n g be- tween the e l o n g a t i o n a l mode and o t h e r nearby modes caused by the i n c r e a s e i n Q when the a i r i s removed. The Q o f a l o s s l e s s sample v i b r a t i n g i n a i r i s d e t e r m i n e d by power l o s s due t o v i s c o u s h e a t i n g o f the a i r and r a d i a t e d a c o u s t i c e n e r g y . The a i r damping may be c a l c u l a t e d a p p r o x i m a t e l y f o r s t a n d a r d s i z e samples above 50 khz. In t h i s c a s e the v i s c o u s l o s s i s r e l a t i v e l y s m a l l compared t o t h e r a d i a t i o n l o s s and the i n t e r i o r d i m e n s i o n s o f the sample c o n t a i n e r a r e l a r g e compared t o the sound w a v e l e n g t h , so t h a t w a l l s can be i g n o r e d . F i r s t we c o n s i d e r the h i g h f r e q u e n c y l i m i t i n w h i c h the w a v e l e n g t h o f sound i n a i r i s s m a l l compared t o bo t h the a and b d i m e n s i o n s o f t h e sample. T h i s l i m i t i s a p p l i c a b l e a t t y p i c a l l o n g i t u d i n a l mode f r e q u e n c i e s . The r a d i a t e d sound power i s g i v e n by A I = P a i r v s A A where v s i s the v e l o c i t y o f sound i n a i r and u n i s the normal component o f the v e l o c i t y o f the s u r f a c e e l ement AA. I f we assume t h a t the en- t i r e s u r f a c e a r e a g e n e r a t e s sound, p a r t l y because o f s u r f a c e roughness and p a r t l y because t h e r e may be c o u p l e d l a t e r a l m o tions o f the sample, then u n = u and [17] P tot 2 p a i r where Q_/2TT i s d e f i n e d as t h e energy s t o r e d d i v i d e d by the energy d i s s i p a t e d p er c y c l e . The a c t u a l a i r damped Q. o f Sample #10 was 1 7 6 whereas [17] p r e d i c t s a Q o f 1*»3. In a vacuum the measured Q. was 3100. In the i n t e r m e d i a t e f r e q u e n c y range i n w h i c h the sound w a v e l e n g t h i n a i r i s l o n g e r than the a d i m e n s i o n o f the sample but s t i l l s h o r t e r than the b d i m e n s i o n , the sample may be m o d e l l e d by an i n f i n i t e c y l i n d e r w i t h d i a m e t e r w. The sound power r a d i a t e d by an i n t e r - m e d i a t e f r e q u e n c y f l e x u r a l mode may be a p p r o x i m a t e d by the power r a d i a t e d by a t r a n s v e r s e l y o s c i l l a t i n g c y l i n d e r , g i v e n by (Landau and L i f s h i t z 1959) p e r u n i t l e n g t h where u i s the v e l o c i t y o f the c y l i n d e r . The c o r r e s - p o n d i n g e x p r e s s i o n f o r the Q i s [18] P a 1 r P 1 I f we s u b s t i t u t e v a l u e s a p p r o p r i a t e t o the f i f t h F b c f l e x u r a l mode (44.6 khz) o f Sample #14, then [18] y i e l d s Q = 993- In the absence o f a i r the o b s e r v e d Q i s 1500. A d d i n g this measured i n t r i n s i c l o s s t o the c a l c u l a t e d a i r l o s s l e a d s t o a n e t a i r damped Q o f 597- T h i s compares w i t h an e x p e r i m e n t a l v a l u e o f 500. A l t h o u g h o u r c a l c u l a t i o n s o f t h e e f f e c t o f a i r on the damping and resonance f r e q u e n c i e s o f v i b r a t i n g TTF-TCNQ samples a r e o n l y a p p r o x i m a t e , the p h y s i c a l o r i g i n o f the o b s e r v e d e f f e c t s appear t o be w e l l u n d e r s t o o d . CHAPTER I I I I n t e r p r e t a t i o n o f Temperature Dependence o f Sound V e l o c i t y 3.1 O v e r a l l Temperature Dependence B e f o r e d i s c u s s i n g the t e m p e r a t u r e dependence o f the e l a s t i c m o d u l i , we f i r s t compare the measured e l a s t i c c o n s t a n t s (see T a b l e V, i n the p r e v i o u s c h a p t e r ) and a n i s o t r o p y o f TTF-TCNQ w i t h some common m a t e r i a l T a b l e V l l c o n t a i n s a l i s t o f Young's moduli f o r c o v a l e n t ( S i ) , m e t a l l i (Au, Pb, Na), i o n i c (NaCl) and van d e r Waals (Ar) s o l i d s . TABLE V l l Young's M o d u l i F o r V a r i o u s M a t e r i a l s M a t e r i a l Young's Modulus T(K) ( 1 0 1 1 dynes/cm 2) S i 13-1 300 Au 4.65 0 NaCl 4.37 300 TTF-TCNQ a 4.3 0 b 2.0 0 Pb 2.05 0 Na 0.241 90 Ar 0.117 82 H u n t i n g t o n (1958), Ki t t e l ( 1 9 7 1 ) , Gewurtz e t a l (1972) 2.6 T T F - T C N Q b AXIS VELOCITIES 2.5 2.4 ELONGATIONAL MODE 1 _S_ample 23 2.or G © O O 2.2̂ 2.1 2.0 TORSIONAL MODE • Sample 21 c c 0 50 100 150 T ( K ) 200 250 300 g. hG - Temperature dependence o f b a x i s e l o n g a t i o n a l and t o r s i o n a l mode v e l o c i t i e s 164 2.85H 2.80 r- E U lO O > 2 . 7 5 h 2.70 0 2 0 4 0 6 0 8 0 100 120 F i g . 47 - Temperature dependence o f b a x i s Young's modulus v e l o c i t y o b t a i n e d from an F b c mode 165 2.6 TTF-TCNQ a AXIS VELOCITIES 5.3 5.2 E o tn O 2.5 2.4 >- O O £ 2.3f < o oo 2.2f-or o 2.1 ELONGATIONAL VELOCITY TORSIONAL MODE e e 9 Sample 22 5.1 e o If) O 15.0 t i — o o 4.9 LU > < O 14.8 < o UJ 14.7 2.0F 4.6 0 50 100 150 T (K) 200 250 300 F i g . 48 - Temperature dependence o f a a x i s Young's modulus v e l o c i t y and s h e a r v e l o c i t y o b t a i n e d from F a c and T a modes r e s p e c t i v e l y The b o n d i n g i n the a d i r e c t i o n i n TTF-TCNQ i s e x p e c t e d t o be a t l e a s t p a r t l y i o n i c and the c o r r e s p o n d i n g Young's modulus i s comparable w i t h the i o n i c s o l i d N a C l . S i m i l a r l y i n the b d i r e c t i o n where we ex- p e c t some m e t a l l i c component t o the b o n d i n g , t h e TTF-TCNQ Young's modulus i s comparable t o l e a d , a s o f t m e t a l . C e r t a i n l y w i t h i n t he c a t e g o r y o f m e t a l l i c s o l i d s and t o a l e s s e r e x t e n t the i o n i c s o l i d s t h e r e i s a w i d e range o f e l a s t i c c o n s t a n t s . N e v e r t h e l e s s , i t i s c l e a r t h a t the e l a s t i c c o n s t a n t s o f TTF-TCNQ a r e comparable t o t h o s e f o r o t h e r common m a t e r i a l s . An a n o m a l o u s l y s o f t a a x i s modulus might be e x p e c t e d i f t h e c o n d u c t i n g m o l e c u l a r s t a c k s were w e a k l y c o u p l e d . However, both the a and b a x i s moduli a r e s i g n i f i c a n t l y b i g g e r than the modulus f o r a w e a k l y bound s o l i d such as Argon a t 8 2 K (see T a b l e V l l ) . Some i d e a o f t h e e l a s t i c a n i s o t r o p y i s o b t a i n e d by t a k i n g t h e r a t i o o f t h e Young's modulus a l o n g a t o t h e Young's modulus a l o n g b. T h i s r a t i o i s about 2 . 2 f o r TTF-TCNQ, whereas f o r z i n c ( h e x a g o n a l ) t h e r a t i o o f t h e Young's moduli p a r a l l e l and p e r p e n d i c u l a r t o the hexagonal symmetry a x i s , i s 3-37. By t h i s measure z i n c i s more a n i s o t r o p i c e l a s t i c a l l y than TTF-TCNQ. A com- p a r i s o n w i t h t h e e l a s t i c c o n s t a n t s o f o t h e r c r y s t a l l i n e m a t e r i a l s c o n f i r m s t h a t t h e e l a s t i c a n i s o t r o p y o f TTF-TCNQ i s more o r l e s s t y p i c a l o f n o n - c u b i c c r y s t a l l i n e m a t e r i a l s . Now l e t us c o n s i d e r the t e m p e r a t u r e dependence o f t h e v e l o c i t y o f sound i n TTF-TCNQ. The t e m p e r a t u r e dependence o f t h e b a x i s Young's modulus v e l o c i t y ( E ^ / p ) ^ i s shown i n Figs.46 and 47- The d a t a i n F i g s . 4 6 and47were o b t a i n e d from an e l o n g a t i o n a l mode and an F [ 3 C f l e x u r a l mode r e s p e c t i v e l y . Fig.48 shows the t e m p e r a t u r e dependence o f the a a x i s Young's modulus v e l o c i t y ( E g / p ) ^ as d e t e r m i n e d f r o m an F a c f l e x u r a l mode. As d i s c u s s e d i n C h a p t e r I I , S e c t i o n 2.1 two d i f f e r e n t t o r s i o n a l modes can be used t o measure t h e s h e a r v e l o c i t y ( c 6 € / p ) ^ , t o a good a p p r o x i m a t i o n . The t e m p e r a t u r e dependence o f t h i s s h e a r v e l o c i t y i s shown i n Figs.46 and 48. The s h e a r v e l o c i t y d a t a i n Fig.46 was o b t a i n e d from a Tu, t o r s i o n a l mode and the d a t a i n Fig.48 f r o m a T a mode. A l l o f t h e above sound v e l o c i t y measurements have been c o r r e c t e d f o r therm a l e x p a n s i o n u s i n g t h e c o r r e c t i o n o f J e r i c h o e t a l (1977) g i v e n i n the A p p e n d i x § 3. This c o r r e c t i o n was i n f e r r e d from t h e X-ray d a t a o f B l e s s i n g and Coppens (1974) and t h e b a x i s thermal e x p a n s i o n measure- ments o f S c h a f e r e t a l (1975). The t e m p e r a t u r e dependence o f the e l a s t i c c o n s t a n t s f o l lows d i r e c t i y from the t e m p e r a t u r e dependence o f the sound v e l o c i t y and the d e n s i t y . The main f e a t u r e s i n the v e l o c i t y r e s u l t s a r e t h e k i n k i n the Young's modulus v e l o c i t i e s n e a r 52K, and the s t r o n g t e m p e r a t u r e dependence i n the h i g h e r t e m p e r a t u r e r e g i o n . B e f o r e d i s c u s s i n g t h e anomaly n e a r 52K, we f i r s t c o n s i d e r t h e l a r g e o v e r a l l t e m p e r a t u r e dependence o f t h e v e l o c i t i e s . The v e l o c i t y r e s u l t s i m p l y t h a t the e l a s t i c moduli de- c r e a s e by ~ 40% between OK and room t e m p e r a t u r e . A l t h o u g h t h i s may seem l i k e a l a r g e t e m p e r a t u r e dependence when compared w i t h c o n v e n t i o n a l m e t a l s where the e l a s t i c c o n s t a n t s t y p i c a l l y change by o n l y a few p e r - c e n t o v e r t h i s r a nge, a l a r g e t e m p e r a t u r e dependence would be e x p e c t e d from the r e l a t i v e l y low s o l i d i f i c a t i o n t e m p e r a t u r e f o r TTF-TCNQ o f 498K ( W e i l e r 1977). In f a c t , i f one compares the t e m p e r a t u r e dependence o f t h e e l a s t i c moduli f o r a v a r i e t y o f s o l i d s from OK up t o j u s t below t h e i r m e l t i n g p o i n t a l a r g e r e d u c t i o n i n modulus (~ 50%) i s n o r m a l l y o b s e r v e d . We now c o n s i d e r t h e b u l k modulus s i n c e the t e m p e r a t u r e dependence o f t h i s modulus i s the e a s i e s t t o c a l c u l a t e . I f we make the hexagonal a p p r o x i m a t i o n t o t h e c r y s t a l l o g r a p h i c symmetry o f TTF-TCNQ, d e s c r i b e d i n C h a p t e r I I , S e c t i o n 2 . 1 , then the e x p e r i m e n t a l Young's moduli may be used t o o b t a i n a rough e s t i m a t e o f the b u l k modulus. In t h e hexago- n a l a p p r o x i m a t i o n t h e b u l k modulus i s g i v e n by P o i s s o n ' s r a t i o s . I f the P o i s s o n ' s r a t i o s a r e r e s t r i c t e d t o l i e between 0 and i (Landau and L i f s h i t z 1970 ) t h e b u l k modulus must be between ^fT" + a r , d i n f i n i t y . O b v i o u s l y t h i s i s not a v e r y p r e c i s e e s t i m a t e . A l t h o u g h v and v' have not been measured e x p e r i m e n t a l l y , n e v e r t h e l e s s , i f we a r b i t r a r i l y s e t v = v 1 = 0 . 2 then the room t e m p e r a t u r e v a l u e s f o r the e l a s t i c c o n s t a n t s g i v e n i n T a b l e V o f Ch a p t e r II combined w i t h t h e t e m p e r a t u r e dependence p r e s e n t e d e a r l i e r i n t h i s c h a p t e r s u g g e s t t h a t B - 2 x 1 0 1 1 dynes/cm 2 a t OK. T h i s e s t i m a t e f o r the z e r o t e m p e r a t u r e b u l k modulus compares w i t h a room t e m p e r a t u r e b u l k modulus o f 0 . 9 4 x 1 0 1 1 dynes/cm 2 measured by Debray e t a l ( 1 9 7 7 ) . Even though i t i s n o t p o s s i b l e t o a r r i v e a t a v e r y a c c u r a t e e s t i m a t e f o r B from the e x p e r i m e n t a l d a t a on E a and E b , s i n c e a l l o f t h e measured e l a s t i c moduli have s i m i l a r o v e r a l l t e m p e r a t u r e dependences i t i s r e a s o n a b l e t o assume the b u l k modulus f o l l o w s where E a and E b a r e t h e a and b a x i s Young's moduli and v and v' a r e t h e o t h e r moduli as a f u n c t i o n o f t e m p e r a t u r e . In g e n e r a l i t i s e x t r e m e l y d i f f i c u l t t o c a l c u l a t e the b u l k modulus o f a s o l i d , as i t means d e r i v i n g an e x p r e s s i o n f o r the p r e s s u r e o f a s o l i d as a f u n c t i o n o f i t s volume, i n o t h e r words an e q u a t i o n o f s t a t e . In the c o n v e n t i o n a l a d i a b a t i c a p p r o x i m a t i o n the e q u a t i o n o f motion o f t h e l a t t i c e , and hence the sound v e l o c i t y , i s d e t e r m i n e d by t h e depen- dence o f t h e e l e c t r o n i c e nergy e i g e n v a l u e s on the p o s i t i o n o f t h e n u c l e i and by the d i r e c t e l e c t r o s t a t i c i n t e r a c t i o n between the n u c l e i . The d i r e c t i n t e r a c t i o n between d i f f e r e n t n u c l e i i s p r o b a b l y s m a l l compared t o t h e i n t e r a c t i o n between n u c l e i and c o r e e l e c t r o n s and between c o r e e l e c t r o n s on d i f f e r e n t n u c l e i . A c c o r d i n g l y i f one knew how t h e e l e c t r o n e nergy e i g e n v a l u e s depended on the p o s i t i o n s o f t h e i o n s one c o u l d c a l - c u l a t e a l l o f the e l a s t i c c o n s t a n t s . In one s p e c i a l c a s e , namely the a l k a l i m e t a l s , t h i s can be done w i t h a r e a s o n a b l e degree o f a c c u r a c y . In t h i s c a s e t h e o u t e r e l e c t r o n s can be c l o s e l y a p p r o x i m a t e d by f r e e e l e c t r o n s and t h e n u c l e i and c o r e e l e c t r o n s can be i g n o r e d because they a r e t h o r o u g h l y s c r e e n e d between d i f f e r e n t s i t e s . The f r e e e l e c t r o n n a t u r e o f the e l e c t r o n energy s t a t e s does not change when the l a t t i c e i s s t r a i n e d . The o n l y e f f e c t o f the s t r a i n i s t o cause a change i n the Fermi l e v e l n e c e s s a r y t o m a i n t a i n l o c a l c h a r g e n e u t r a l i t y . Thus, t h e dependence o f t h e e l e c t r o n i c e nergy s t a t e s on the p o s i t i o n o f the i o n s i s known and the e l a s t i c moduli can be c a l c u l a t e d ( K i t t e l 1971). However, f o r a l m o s t any m a t e r i a l o t h e r than the a l k a l i m e t a l s more than one e l e c t r o n energy band needs t o be c o n s i d e r e d and the bands change i n c o m p l i c a t e d ways w i t h s t r a i n . A l t h o u g h the c o n t r i b u t i o n o f particular bands to the elastic moduli can s t i l l be calculated, i t is no longer reasonable to try to calculate the moduli from f i r s t principles. Instead the normal procedure is to start with a para- meterized intermolecular potential. We will follow this procedure here. In general the isothermal bulk modulus for any system is given by (Landau and Lifshitz 1969) where F is the free energy of the system, and V is its volume. Simi- larly the adiabatic bulk modulus is given by where U is the internal energy, and the volume derivatives are taken at constant entropy S. In order to calculate the temperature dependence of B.j. and B s we need F and U as functions of temperature and volume. The standard expressions are (Girifalco 1973) [la] F = F 0 + [lb] U = U Q + where the intermolecular potential has been expanded to third order in t h e volume s t r a i n . The t h i r d o r d e r term i s n e c e s s a r y because o f th e r m a l e x p a n s i o n , as w i l l be c l e a r l a t e r . The therm a l phonon f r e - q u e n c i e s u)q w i l l depend on volume i n some c o m p l i c a t e d way. We a p p r o x i m a t e t h e i r volume dependence by to. (V) = % (V D) (V G/V)" where y ' s a volume independent G r u n e i s e n c o n s t a n t . In t h i s a p p r o x i - mation the volume e x p a n s i o n c o e f f i c i e n t a(T) i s g i v e n by ( C a l l e n 196O) m m Y C P ( T ) YC v(T) [2] A ( T ) = V ( TT B 7T TT ~ V(T)B t(T) where Cp and C y a r e the. s p e c i f i c h e a t s a t c o n s t a n t p r e s s u r e and volume r e s p e c t i v e l y . By i n t e g r a t i n g [2] we f i n d [3] V(T) « V o T ^ S V o T ^ where U i s the i n t e r n a l energy o f the l a t t i c e . fitoq /fW q\ U o I - J L c o t h (^j q In eqn.[3] we have assumed t h a t Cp - C v and t h a t t h e t e m p e r a t u r e depen- dence o f the volume and b u l k modulus i s s m a l l compared t o the t e m p e r a t u r e dependence o f t h e s p e c i f i c h e a t . These a p p r o x i m a t i o n s improve a t low t e m p e r a t u r e s . U s i n g t h e d e f i n i t i o n o f the i s o t h e r m a l b u l k modulus and the ex- p r e s s i o n f o r the f r e e e nergy [ l a ] , one can now w r i t e down the i s o - t h e r m a l b u l k modulus as a f u n c t i o n o f t e m p e r a t u r e i n terms o f t h e p a r a m e t e r s y and A. Both o f t h e s e p a r a m e t e r s a r e m a n i f e s t a t i o n s o f the anharmonic p a r t o f the i n t e r m o l e c u l a r f o r c e c o n s t a n t s . I t i s u s e f u l t o e x p r e s s A i n terms o f t h e p r e s s u r e d e r i v a t i v e o f t h e b u l k modulus s i n c e t h i s q u a n t i t y can be measured d i r e c t l y . The p r e s s u r e d e r i v a t i v e o f By i s o b t a i n e d from by d i v i d i n g by By. The second term i n [4] i s e x p e c t e d t o be a t most comparable w i t h the t o t a l tempera Lure dependent p a r t o f By whereas A w i l l be shown t o be > 10B f o r TTF-TCNQ. To a good a p p r o x i m a t i o n the second term i n [4] can be n e g l e c t e d and 8By/3P = A/By. In terms o f 9By/3P the i s o t h e r m a l b u l k modulus i s [5a] B (T) = B Q + 1 v o The f i r s t term i n the s q u a r e b r a c k e t s i n [5a] tends t o make the l a t t i c e s t i f f e r a t h i g h t e m p e r a t u r e s . T h i s term a r i s e s from t h e phonon p r e s s u r e . The n e g a t i v e 8By/8P term comes from the thermal e x p a n s i o n combined w i t h the s o f t e n i n g o f the i n t e r m o l e c u l a r p o t e n t i a l w i t h i n c r e a s i n g volume. S i m i l a r l y , t h e a d i a b a t i c b u l k modulus i s found by t a k i n g p a r t i a l de- r i v a t i v e s o f the i n t e r n a l energy w i t h r e s p e c t t o volume, m a i n t a i n i n g (y+1) 8 By TP~ U - Y T t h e e n t r o p y c o n s t a n t . In the Gr'uneisen a p p r o x i m a t i o n , the e n t r o p y may be kep t f i x e d by a l l o w i n g the t e m p e r a t u r e t o be a f u n c t i o n o f volume i n such a way t h a t Wq/T i s independent o f volume. S i n c e t h e e n t r o p y i s a f u n c t i o n o f Wq/T o n l y , i t w i l l be c o n s t a n t i f w^/T i s c o n s t a n t . A p r o c e d u r e s i m i l a r t o t h a t used i n o b t a i n i n g the i s o - t h e r m a l modulus then l e a d s t o [5b] B S(T) = B Q + N o r m a l l y u l t r a s o n i c t e c h n i q u e s measure a d i a b a t i c m o d u l i . Whether i t i s the a d i a b a t i c o r i s o t h e r m a l modulus w h i c h i s a p p r o p r i a t e depends on the r e l a t i o n between the p e r i o d o f the sound wave and the ti m e f o r the r m a l r e l a x a t i o n between a r a r e f a c t i o n and c o m p r e s s i o n i n the wave. In the TTF-TCNQ. e x p e r i m e n t the a d i a b a t i c moduli d e t e r m i n e the r e s o - nance f r e q u e n c y o f a l l o f the modes s t u d i e d w i t h the p o s s i b l e e x c e p t i o n o f t h e fundamental f l e x u r a l mode and i t s f i r s t few h a r m o n i c s , d e p e n d i n g on the geometry o f the sample. In a f l e x u r a l mode the compressed and expanded p a r t s o f the sample a r e s e p a r a t e d by a d i s t a n c e equal t o t h e sample t h i c k n e s s o n l y , even though the mode f r e q u e n c y may be r e l a t i v e l y low. In t h i s . c a s e the i s o t h e r m a l modulus a p p l i e s when the r e s o n a n t f r e q u e n c y i s s m a l l compared t o the th e r m a l r e l a x a t i o n f r e q u e n c y f D d i s c u s s e d i n S e c t i o n 2.3 ( i i ) and the a d i a b a t i c modulus a p p l i e s a t h i g h e r f r e q u e n c i e s . Y+1 8P Eqn. [ 2 ] combined w i t h o u r e s t i m a t e o f Bs(0) and the p u b l i s h e d t h e r - mal e x p a n s i o n and s p e c i f i c heat d a t a , e n a b l e us t o e s t i m a t e y. From the X-ray s t r u c t u r e d a t a ( B l e s s i n g and Coppens 1974) we e s t i m a t e ct(T) = 1.6 a^Cl) where % ( ! " ) i s the b a x i s l i n e a r e x p a n s i o n c o e f f i c i e n t measured by S c h a f e r e t a l (1975). U s i n g t h i s e s t i m a t e f o r a(T) we o b t a i n y = 2.56, w h i c h i s not an u n r e a s o n a b l e v a l u e . The G r u n e i s e n c o n s t a n t s f o r a v a r i e t y o f d i f f e r e n t m a t e r i a l s a r e g i v e n i n T a b l e V I M . TABLE V I I I G r U n e i s e n C o n s t a n t s f o r V a r i o u s M a t e r i a l s M a t e r i a l G r U n e i s e n y T(K) S i 0.4': 300 Au 3.0 0 NaCl 1.55 300 TTF-TCNQ 2.6 0 Na 1.14 90 Ar 2.7 82 t D a n i e l s (1963), Gewurtz e t a l (1972) Now t h a t we have an e s t i m a t e f o r y [ 2 ] can be used t o p r e d i c t t he heat c a p a c i t y beyond the 12K range n e a r 55K measured by Craven e t a l (1974). The heat c a p a c i t y w h i c h r e s u l t s i s t y p i c a l o f m o l e c u l a r s o l i d s ( L o r d 1941). Near room t e m p e r a t u r e the p r e d i c t e d heat c a p a c i t y i s 35R and i n c r e a s i n g a p p r o x i m a t e l y l i n e a r l y a t 0.05 R/K. R i s the gas con- s t a n t p er mole o f TTF-TCNQ f o r m u l a u n i t s . The l a r g e v a l u e o f the heat 175 c a p a c i t y c l e a r l y i n d i c a t e s the i m p o r t a n c e o f i n t r a m o l e c u l a r and l i b r a t i o n a l d e g rees o f freedom. These low f r e q u e n c y E i n s t e i n modes o f t h e TTF and TCNQ. m o l e c u l e s dominate Cp above 20K. I t i s a l s o p o s s i b l e t o make a d i r e c t c o m p a r i s o n between the tem- p e r a t u r e dependence o f the b u l k modulus and the therm a l e x p a n s i o n co- e f f i c i e n t . D i f f e r e n t i a t i n g [5b] w i t h r e s p e c t t o volume and n e g l e c t i n g t h e t e m p e r a t u r e dependence o f 9B S/9P we g e t [6] J _ l i s „ B,. dT 9 B Q Y+1 a(T) I t may not be a very good a p p r o x i m a t i o n t o n e g l e c t t h e t e m p e r a t u r e de- pendence o f 9B S/9P ( D a n i e l s 1963), however, t h e r e i s n e i t h e r e x p e r i m e n t a l d a t a nor a r e a s o n a b l e model a v a i l a b l e t o d e s c r i b e i t s t e m p e r a t u r e depen- dence. I f we make t h e p r e v i o u s l y s t a t e d a s s u m p t i o n t h a t the t e m p e r a t u r e dependence o f the b u l k modulus f o l l o w s t h e t e m p e r a t u r e dependence o f the measured e l a s t i c m o d u l i , then i n c o n j u n c t i o n w i t h the e x p e r i m e n t a l v e l o c i t y d a t a , [6] may be used t o e s t i m a t e 9B S/9P. U s i n g t h e b a x i s v e l o c i t y d a t a , t o g e t h e r w i t h a = 1.6 a b and y = 2.56 we f i n d 9B S/9P ~ 15~17 - In l i g h t o f the a p p r o x i m a t i o n s we have made, t h i s number compares w e l l w i t h 9B-J-/9P ~ 12 i n f e r r e d from t h e p r e s s u r e measurements o f Debray e t a l (1977). The p r e s s u r e d e r i v a t i v e o f the b u l k modulus i s g i v e n i n T a b l e IX f o r a number o f m a t e r i a l s . Between 52 K and about 200K f o r the b a x i s e x t e n s i o n a l modes and o v e r a w i d e r t e m p e r a t u r e range f o r the o t h e r modes, the shape o f the v e l o c i t y TABLE IX t P r e s s u r e Dependence o f B u l k Modulus t M a t e r i a l 9 B T I F S i 5.3 Au 6.1 NaCl 5.7 TTF-TCNQ 15-17 Na 3-3 A r 8.5 D a n i e l s (1963), Paul and Warschauer (1963) c u r v e s i s c o n s i s t e n t w i t h [ 6 ] . (Pvecall t h a t i n o u r a p p r o x i m a t i o n v - 1 9v/9T = ( 2 B ) " 1 9B/9T, where v i s a sound v e l o c i t y . ) We t h e r e - f o r e s u g g e s t t h a t i f TTF-TCNQ remained m e t a l l i c down t o OK, the t e m p e r a t u r e dependence o f the v e l o c i t i e s would be o f t h e g e n e r a l form shown by the dashed l i n e i n F i g . 48. For t h i s r eason we i n t e r - p r e t the anomaly near 52K i n the e x t e n s i o n a l mode v e l o c i t i e s as a s t i f f e n i n g i n the modulus above a background w h i c h r e p r e s e n t s the c o n t r i b u t i o n t o the e l a s t i c c o n s t a n t s from anharmonic e f f e c t s . The o t h e r p o s s i b l e i n t e r p r e t a t i o n i s t o r e g a r d the anomaly as a broad s o f t e n i n g between 40K and 120K. T h i s i n t e r p r e t a t i o n i s r e j e c t e d be- cause the r e s u l t i n g t e m p e r a t u r e dependence does not match the therm a l e x p a n s i o n d a t a q u i t e as w e l l , and s e c o n d l y t h e r e i s no o t h e r e x p e r i - mental e v i d e n c e f o r such a broad t r a n s i t i o n t e m p e r a t u r e r e g i o n (>80K). 177 T ( K ) F i g . kS - Enlargement o f low t e m p e r a t u r e anomaly i n the Young's modulus v e l o c i t y . The dashed l i n e i s o b t a i n e d from eqn . [ 6 ] i n the t e x t . The b r e a k i n the c u r v e near k2K r e s u l t s from a s p l i t t i n g o f the f i r s t l o n g i t u d i n a l mode caused by i n t e r f e r e n c e from a n o t h e r mode, p r o b a b l y a harmonic o f t h e fundamental f l e x u r a l o r t o r s i o n a l modes. S u m m a r i z i n g , we c o n c l u d e from the l a r g e v a l u e f o r the heat c a p a c i t y and from the i n t e r d e p e n d e n c e o f the heat c a p a c i t y , t h e r m a l e x p a n s i o n and b u l k modulus t h a t the t e m p e r a t u r e dependence o f a l l t h r e e q u a n t i t i e s i s dominated by l i b r a t i o n s and i n t r a m o l e c u l a r modes above ~ 20K. 3-2 Low Temperature Anomaly We now d i s c u s s the low t e m p e r a t u r e anomaly i n t h e sound v e l o c i t y , t a k i n g the p o i n t o f v i e w t h a t i t i s a s t i f f e n i n g i n the e l a s t i c modulus above the background t e m p e r a t u r e dependence. The v e l o c i t y anomaly i s shown i n F i g . k S . The low t e m p e r a t u r e v e l o c i t y d a t a can be summarized as f o l l o w s . Below t h e m e t a l - i n s u l a t o r t r a n s i t i o n t h e r e i s an anomalous i n c r e a s e i n v e l o c i t y f o r modes w h i c h i n v o l v e a volume change. T h i s i n c r e a s e r e a c h e s a maximum o f about 1.5% above the e x t r a p o l a t e d back- ground a t OK. The f r a c t i o n a l i n c r e a s e i s a b o u t - t h e same f o r the Young's modulus modes i n the a and b d i r e c t i o n s and much s m a l l e r o r even a b s e n t i n the s h e a r modes. In the remainder o f t h i s c h a p t e r we show how t h e s e e x p e r i m e n t a l r e s u l t s may be i n t e r p r e t e d i n terms o f the c o n t r i b u t i o n o f t h e c o n d u c t i o n e l e c t r o n s t o the sound v e l o c i t y . T h i s s e c t i o n i s d i v i d e d i n t o two p a r t s . In the f i r s t p a r t we show t h a t an 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 n the h i g h f r e q u e n c y quantum l i m i t l e a d s t o a s o f t e n i n g o f t h e sound v e l o c i t y i n the m e t a l l i c phase. In t h e second p a r t we c o n s i d e r the t i g h t - b i n d i n g band s t r u c t u r e o f TTF-TCNQ i n d e t a i l and show t h a t the e l e c t r o n - p h o n o n c o u p l i n g can a l s o l e a d t o a s o f t e n i n g i n t h e m e t a l l i c phase a t z e r o f r e q u e n c y . F i g . 50 - E x c i t a t i o n s p e c t r u m f o r a n o n - i n t e r a c t i n g t i g h t - b i n d i n g e l e c t r o n band (TCNQ. band) and an u n c o u p l e d a c o u s t i c phonon b r a n c h . ( i ) Quantum L i m i t In the quantum l i m i t (see P a r t A, C h a p t e r I , S e c t i o n 1.3) the e f f e c t o f t he c o n d u c t i o n e l e c t r o n s on the sound v e l o c i t y may be c a l c u l a t e d by t r e a t i n g t h e 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 as a p e r t u r b a t i o n on the un- c o u p l e d e l e c t r o n and phonon system. The u n p e r t u r b e d energy l e v e l s f o r th e e l e c t r o n s a r e d e s c r i b e d by a s i n g l e p a r t i c l e t i g h t - b i n d i n g band. C o n s i s t e n t w i t h t he t i g h t - b i n d i n g a p p r o x i m a t i o n ( B a r i s i c 1972), the u n p e r t u r b e d phonons e x i s t i n a l a t t i c e o f n e u t r a l ' i o n s ' . Thus t h e r e i s a phonon mode ( a c o u s t i c phonon) w h i c h p r o p a g a t e s down t o z e r o f r e - quency w i t h a l i n e a r d i s p e r s i o n i n t h e u n p e r t u r b e d system. T h i s be- h a v i o u r c o n t r a s t s w i t h n e a r l y f r e e e l e c t r o n models where the unper- t u r b e d l a t t i c e c o n s i s t s o f c h a r g e d i o n s , and t h e c o r r e s p o n d i n g un- p e r t u r b e d phonon f r e q u e n c y i s an i o n i c plasma f r e q u e n c y . The e x c i t a t i o n s p e c t r u m f o r the n o n - i n t e r a c t i n g one d i m e n s i o n a l e l e c t r o n - p h o n o n s y s t e m a t z e r o t e m p e r a t u r e i s shown i n Fig. 5 0 . Only one low f r e q u e n c y phonon b r a n c h i s shown f o r c l a r i t y , a l t h o u g h i n g e n e r a l f o r any d i r e c t i o n o f p r o p a g a t i o n t h e r e a r e two o t h e r b r a n c h e s . The sound v e l o c i t y d e t e r m i n e s the s l o p e o f the phonon b r a n c h and the Fermi v e l o c i t y d e t e r m i n e s t h e s l o p e o f the e l e c t r o n b r a n c h near q = 0. These s l o p e s a r e drawn a p p r o x i m a t e l y t o s c a l e i n Fig. 5 0 . The e l e c t r o n Fermi v e l o c i t y shown i s f o r the TCNQ band c a l c u l a t e d by B e r l i n s k y e t a l (1974). The shaded a r e a i s the l o c u s o f e(k+q) - c ( k ) w i t h k as a parameter and the a d d i t i o n a l r e q u i r e m e n t t h a t t he i n i t i a l s t a t e k be f u l l and the f i n a l s t a t e k+q be empty. e ( k ) i s the energy o f an e l e c t r o n in t h e s t a t e k. 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 H ; n t (see P a r t A, C h a p t e r I , S e c t i o n 1.3) w i l l c o u p l e a phonon w i t h w a v e v e c t o r q t o a l l the e l e c t r o n i c e x c i t a t i o n s w i t h t h e same w a v e v e c t o r . In terms o f F i g . 50, the e l e c t r o n i c e x c i t a t i o n s l y i n g a l o n g a v e r t i c a l l i n e d i r e c t l y above the phonon o f i n t e r e s t a r e c o u p l e d t o t h e phonon. To second o r d e r i n p e r t u r b a t i o n t h e o r y t h i s c o u p l i n g reduces t h e phonon f r e q u e n c y t o m * * o 1 V l < k l " i n t l k + q > l 2 [7] T.o.q - T»a>q - ^ £ F ( k + q ) - e ( k ) - ' The u n p e r t u r b e d phonon f r e q u e n c y i s co° and t h e occupancy n(q) o f the phonon mode q i s assumed to s a t i s f y n(q) » 1. We s u b s t i t u t e t h e s q u a r e d m a t r i x e l e m e n t s o f H i n t " 9 I 1 - f i ( a q " a - q ) C k - q a °k a a g i v e n i n P a r t A, S e c t i o n 1.3 where 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 g = C//B^~ . B S i s the a d i a b a t i c b u l k modulus and C i s a d e f o r m a t i o n p o t e n t i a l . W i t h t h e s e s u b s t i t u t i o n s [7] reduces t o r c l . * o V f ( k ) - f(k+g) [8] tlco q = flag " 9 2 - f - I e(k+q) - e(k) assuming t h a t t h e sound v e l o c i t y i s much s m a l l e r than the e l e c t r o n Fermi v e l o c i t y . A l t h o u g h second o r d e r p e r t u r b a t i o n t h e o r y i s p r o b a b l y t h e s i m p l e s t way o f a p p r o x i m a t i n g the e f f e c t 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 on t h e phonon f r e q u e n c i e s , a more g e n e r a l t e c h n i q u e i s a l s o a v a i l a b l e . In the Green's f u n c t i o n f o r m a l i s m , the phonon f r e q u e n c y O J ^ i s a p o l e o f the p e r t u r b e d phonon Green's f u n c t i o n D ( q ) . An e x p r e s s i o n f o r D(q) can be o b t a i n e d w i t h the h e l p o f Dyson's e q u a t i o n shown s c h e m a t i c a l l y i n F i g . 5 1 - F i g . 51 _ Dyson's E q u a t i o n T h i s e q u a t i o n may be r e a r r a n g e d t o g i v e [9] D ( q ) - 1 = D ^ q ) " 1 - £ n (q) . The l a s t term i n c l u d e s the phonon s e l f - e n e r g y II (q). We now s u b s t i t u t e ( A b r i k o s o v e t a l 1963) f o r the f i n i t e t e m p e r a t u r e phonon Green's f u n c t i o n and an i d e n t i c a l e x p r e s s i o n f o r D(q) e x c e p t the s u p e r s c r i p t z e r o i s removed. I f we s e t ico = Wq i n [9] then [10] o, q 2 = o)q°2 [ l + g 2 H ( q ) ] . T h i s e x p r e s s i o n f o r the p e r t u r b e d phonon f r e q u e n c i e s i s v a l i d t o a r b i t r a r y o r d e r i n p e r t u r b a t i o n t h e o r y . The f i r s t term i n the d i a - grammatic e x p a n s i o n f o r II (q) i s k k + q n°(q) F i g . 52 - E l e c t r o n Gas P o l a r i z a t i o n Diagram T h i s l o w e s t o r d e r p o l a r i z a t i o n i n s e r t i o n o r " b u b b l e d i a g r a m " i s v e r y w e l l known. In t h e low f r e q u e n c y l i m i t i t can be shown t h a t ( F e t t e r and Walecka 1971, Doniach and Sondheimer 197*0 11 ( q ) " I e ( k + q ) - E ( k ) * The summation on the r i g h t i s the H a r t r e e p o l a r i z a b i 1 i t y x°(q) o f t h e e l e c t r o n gas. I f t h i s e x p r e s s i o n i s s u b s t i t u t e d i n t o [10] we F i g . 53 - H a r t r e e p o l a r i z a b i 1 i t y f o r the one d i m e n s i o n a l TCNQ. band d i s c u s s e d in the t e x t r e g a i n the second o r d e r p e r t u r b a t i o n t h e o r y r e s u l t [8] f o r the p e r - t u r b e d phonon f r e q u e n c i e s . I t i s not d i f f i c u l t t o e v a l u a t e the p o l a r i z a b i 1 i t y x°(q) f ° r a one d i m e n s i o n a l t i g h t b i n d i n g band o f e l e c t r o n s . The s t a t i c p o l a r i - z a b i l i t y f o r a m e t a l l i c band a t t e m p e r a t u r e s low compared w i t h t h e Fermi t e m p e r a t u r e i s where N(e) i s t h e e l e c t r o n i c d e n s i t y o f s t a t e s , Cf i s the Fermi energy and kj. the Fermi w a v e v e c t o r . A graph o f x°(q) as a f u n c t i o n o f q i s shown i n Fig. 5 3 u s i n g the same t i g h t b i n d i n g band p a r a m e t e r s used i n t h e e x c i t a t i o n s p e c t r u m shown i n F i g . 5 0 . The l o g a r i t h m i c s i n g u l a r i t y i n x°(cl) a t q = 2k^ i s r e s p o n s i b l e f o r the l a r g e Kohn anomaly i n t h e phonon s p e c t r u m and t h e r e s u l t i n g c h a r g e d e n s i t y wave o r P e i e r l s t r a n s i t i o n i n one d i m e n s i o n a l c o n d u c t o r s . The d i v e r g e n c e i n x°(cl) i s p r e s e n t a t z e r o t e m p e r a t u r e i n any m a t e r i a l w i t h p a r a l l e l s e c t i o n s o f Fermi s u r f a c e . Note t h a t x 0^) f o r a t i g h t - b i n d i n g band does not approach z e r o f o r l a r g e q u n l i k e the n e a r l y f r e e e l e c t r o n c a s e (Andre e t a l 1976). A l t h o u g h the s i n g u l a r i t y i n x°(cl) ' s i m p o r t a n t i n d e t e r m i n i n g t h e phonon s p e c t r u m near 2k^, we a r e more i n t e r e s t e d i n t h e l o n g wave- l e n g t h phonons w i t h q -»• 0. In the l o n g w a v e l e n g t h l i m i t , the change i n the sound v e l o c i t y i m p l i e d by [8] i s [11] 2 s i n f f u In The i n t e g r a l i n [12] a p p r o a c h e s N(e^.) i n the m e t a l l i c phase a t low t e m p e r a t u r e s and goes e x p o n e n t i a l l y t o z e r o i n the s e m i c o n d u c t i n g phase a t low t e m p e r a t u r e s . From [12] one would e x p e c t a f r a c t i o n a l i n c r e a s e i n the v e l o c i t y o f sound o f N(e^.) C2/(2BS) i n c o o l i n g TTF- TCNQ. t h r o u g h i t s meta1 - i n s u l a t o r t r a n s i t i o n . T h i s i s q u a l i t a t i v e l y t h e e f f e c t w h i c h i s e x p e r i m e n t a l l y o b s e r v e d i n the e x t e n s i o n a l modes. To see how the s i z e o f t h i s e f f e c t compares w i t h e x p e r i m e n t we sub- s t i t u t e an a v e r a g e d e n s i t y o f s t a t e s f o r t h e TTF-TCNQ bands ( B e r l i n s k y e t a l 1974) o f 5.8 e v - 1 and use B s = 2 X 1 0 1 1 dynes/cm 2 as d i s c u s s e d e a r l i e r . For a f i t t o the o b s e r v e d 1.5% v e l o c i t y anomaly C = O.38 ev. T h i s number i s c e r t a i n l y r e a s o n a b l e s i n c e the a v e r a g e bandwidth from t h e m o l e c u l a r o r b i t a l c a l c u l a t i o n s ( B e r l i n s k y e t a l 1974) i s 0.32 e v , and t i g h t - b i n d i n g bands a r e e x p e c t e d t o have d e f o r m a t i o n p o t e n t i a l s o f the o r d e r o f the bandw i d t h (M i t r a 1969). The much s m a l l e r a n o m a l i e s o b s e r v e d i n t h e shear v o l o c i t i e s a r e a l s o e x p l a i n e d by t h i s model because the d e f o r m a t i o n p o t e n t i a l f o r s h e a r waves i s e x p e c t e d to be s m a l l f o r m e t a l s i n w h i c h a l l o f the Fermi s u r f a c e i s i n one B r i l l o u i n zone ( K i t t e l 1963)• There i s no re a s o n t o e x p e c t any o f the TTF-TCNQ Fermi s u r f a c e t o l i e o u t s i d e t h e f i r s t zone. I t i s i n t e r e s t i n g t o e x t r a p o l a t e the e x p e r i m e n t a l l y o b s e r v e d s o f t e n i n g i n the sound v e l o c i t y up t o q - 2kf w i t h t h e h e l p o f the 187 F i g . 5k - Zero t e m p e r a t u r e l o n g i t u d i n a l c a l c u l a t e d from [8] u s i n g the c o u p l i n g c o n s t a n t a c o u s t i c phonon d i s p e r s i o n e x p e r i m e n t a l e l e c t r o n - p h o n o n 188 F i g . 55 Z e r o t e m p e r a t u r e l o n g i t u d i n a l a c o u s t i c phonon d i s p e r s i o n c a l c u l a t e d f r om [8] f o r an 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 w h i c h y i e l d s a T c o f 5̂ K q dependent p o l a r i z a b i 1 i t y e x p r e s s i o n [11]. I f one assumes a q independent c o u p l i n g c o n s t a n t g, then the p e r t u r b e d l o n g i t u d i n a l a c o u s t i c phonon s p e c t r u m i s g i v e n by the s o l i d l i n e i n Fig.54, f o r ( h y p o t h e t i c a l ) m e t a l l i c TTF-TCNQ a t z e r o t e m p e r a t u r e . The dashed l i n e i s the u n p e r t u r b e d phonon f r e q u e n c y . The s m a l l Kohn anomaly i n Fig.5** i s much t o o s m a l l t o a c c o u n t , i n t h e mean f i e l d t h e o r y ( R i c e and S t r a s s l e r 1973) f o r t h e o b s e r v e d m e t a l - i n s u l a t o r t r a n s i t i o n t e m p e r a t u r e . T h i s d i s c r e p a n c y may n o t be u n r e a s o n a b l e s i n c e the mean f i e l d model i s not e x p e c t e d t o be v e r y a c c u r a t e . N e v e r t h e l e s s , i t i s i n t e r e s t i n g t o pu r s u e some o f the i m p l i c a t i o n s o f t h e s m a l l Kohn anomaly i n Fig.54, i n the framework o f t h e mean f i e l d t h e o r y . 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 g would have t o be about a f a c t o r o f f i v e b i g g e r , i n o r d e r t h a t t h e Kohn anomaly i n the a c o u s t i c phonon be l a r g e enough t o a c c o u n t f o r t h e o b s e r v e d m e t a l - i n s u l a t o r t r a n s i t i o n t e m p e r a t u r e . I f the l a r g e r v a l u e o f g i s sub- s t i t u t e d i n t o [11] then t he l a r g e z e r o t e m p e r a t u r e s o f t e n i n g o f the a c o u s t i c phonon mode shown i n Fig.55 r e s u l t s . Such a d r a s t i c s o f t e n i n g i s n ot o b s e r v e d i n the i n e l a s t i c n e u t r o n s c a t t e r i n g measurements o f S h a p i r o e t a l (1977). In f a c t t h e Kohn anomaly o b s e r v e d by n e u t r o n s c a t t e r i n g r e s e mbles the much s m a l l e r anomaly i n Fig.54. These ob- s e r v a t i o n s s u g g e s t t h a t the m e t a l - i n s u l a t o r t r a n s i t i o n i n TTF-TCNQ i s caused by a s t a t i c d i s t o r t i o n i n a c o m b i n a t i o n o f i n t r a m o l e c u l a r modes as proposed by R i c e and L i p a r i (1977). I f a m e t a l - i n s u l a t o r t r a n s i t i o n can be produced by c o u p l i n g o f the c o n d u c t i o n e l e c t r o n s t o i n t r a m o l e c u l modes i n a d d i t i o n t o t h e i n t e r m o l e c u l a r a c o u s t i c modes, then i t w i l l be more d i f f i c u l t t o s t a b i l i z e t he low t e m p e r a t u r e m e t a l l i c s t a t e o f an o r g a n i c s o l i d , e s p e c i a l l y i f the m a t e r i a l i s composed o f e x t e n d e d m o l e c u l e s . Even though the model d e s c r i b e d above seems t o be i n e x c e l l e n t agreement w i t h t h e low f r e q u e n c y sound v e l o c i t y measurements below the m e t a l - i n s u l a t o r t r a n s i t i o n , t h e c a l c u l a t i o n s were made i n the quantum l i m i t f o r w e l l d e f i n e d phonon and e l e c t r o n s t a t e s . That i s , the l i f e t i m e b r o a d e n i n g o f t h e e l e c t r o n and phonon s t a t e s has been i m p l i c i t l y assumed t o be n e g l i g i b l e . In r e a l i t y f o r the f r e q u e n c i e s used i n the e x p e r i m e n t s ql « 1 ( r e c a l l I i s the e l e c t r o n mean f r e e p a t h ) , and t h e r e f o r e t he l i f e t i m e b r o a d e n i n g o f the e l e c t r o n i c ex- c i t a t i o n s i s enormous compared t o Tito. Th i s b roaden i ng has a d r a s t i c e f f e c t on the u l t r a s o n i c a t t e n u a t i o n where t r a n s i t i o n r a t e s f o r energy c o n s e r v i n g t r a n s i t i o n s must be c a l c u l a t e d . However, i n the p r e s e n t c a s e we a r e o n l y i n t e r e s t e d i n v i r t u a l t r a n s i t i o n s t o a con t i n u u m o f e l e c t r o n s t a t e s . The f a c t t h a t t he e l e c t r o n s t a t e s a r e much b r o a d e r than the energy change i n v o l v e d i n an e l e c t r o n - p h o n o n s c a t t e r i n g p r o - c e s s i s not i m p o r t a n t s i n c e energy does not need t o be c o n s e r v e d i n v i r t u a l t r a n s i t i o n s anyway. On the av e r a g e we e x p e c t the v i r t u a l t r a n s i t i o n s t o c o i n c i d e w i t h t he w e l l d e f i n e d quantum t r a n s i t i o n , and th e quantum r e s u l t s h o u l d s t i l l be v a l i d a t low f r e q u e n c i e s . A n o t h e r e q u a l l y n o n - r i g o r o u s argument can be made f o r the v a l i d i t y o f t he quantum r e s u l t a t low f r e q u e n c i e s . In n e a r l y f r e e e l e c t r o n models the u n p e r t u r b e d phonon f r e q u e n c y i s t h e l a t t i c e plasma f r e q u e n - cy ttn = (4frne2/M)^ where M i s the i o n mass. The e f f e c t o f a d d i n g e l e c t r o n s , w h i c h w i l l i n t e r a c t w i t h the i o n s i n such a way so as t o s c r e e n the i o n s , i s t o reduce the plasma f r e q u e n c y t o ^p/*/^q~ • The new phonon b r a n c h now has a l i n e a r d i s p e r s i o n down t o z e r o f r e q u e n c y ( P i n e s and N o z i e r e s 1965)• The d i e l e c t r i c c o n s t a n t tq can be c a l c u - l a t e d from the p o l a r i z a b i 1 i t y u s i n g the same quantum l i m i t a p p r o a c h and the same a p p r o x i m a t i o n t h a t we have used i n c a l c u l a t i n g the change i n phonon f r e q u e n c y . The d i e l e c t r i c c o n s t a n t a p p r o a c h l e a d s t o t h e c o r r e c t low f r e q u e n c y sound v e l o c i t y . By a n a l o g y , o u r c a l c u l a t i o n o f the phonon f r e q u e n c i e s s h o u l d a l s o l e a d t o the c o r r e c t low f r e q u e n c y v e l o c i t y . ( i i ) Thermodynamic L i m i t These d i f f i c u l t i e s w i t h the a p p l i c a b i l i t y o f the quantum f o r m a l i s m i n t he low f r e q u e n c y l i m i t a l l v a n i s h i f i n s t e a d we a p p r o a c h t h e pr o b l e m from t h e p o i n t o f view o f c a l c u l a t i n g a s t a t i c e l a s t i c c o n s t a n t . The e l a s t i c c o n s t a n t s f o r a m a t e r i a l i n thermodynamic e q u i l i b r i u m a r e d e t e r m i n e d by t a k i n g second o r d e r s t r a i n d e r i v a t i v e s o f an a p p r o p r i a t e thermodynamic p o t e n t i a l , as d i s c u s s e d i n S e c t i o n 3-1 above. For e l e c t r o n s the thermodynamic l i m i t i s v a l i d p r o v i d e d t h a t qt « 1 (see P a r t A, C h a p t e r I) w h i c h i s always w e l l s a t i s f i e d i n the e x p e r i m e n t d e s c r i b e d h e r e . As p o i n t e d out i n the p r e v i o u s s e c t i o n i t i s p r i m a r i l y t h e h i g h e s t o c c u p i e d e l e c t r o n energy bands w h i c h d e t e r m i n e the e l a s t i c c o n s t a n t s . In o r d e r t o c a l c u l a t e the t o t a l c o n t r i b u t i o n o f the e l e c t r o n s t o t h e e l a s t i c c o n s t a n t s we need t o know the dependence o f the e l e c t r o n e n e r g y bands on s t r a i n , t o second o r d e r i n the s t r a i n . However, i f we 192 F i g . 56 - Band s t r u c t u r e o f TTF-TCNQ d i s c u s s e d i n the t e x t . The dashed t i n e shows the e f f e c t o f an e x a g g e r a t e d b a x i s compress i o n 193 F i g . 57 - E f f e c t o f a b a x i s c o m p r e s s i o n on the d e n s i t y o f o c c u p i e d s t a t e s f o r the band s t r u c t u r e shown i n F i g . 56 can be s a t i s f i e d w i t h c a l c u l a t i n g the s m a l l change i n e l a s t i c con- s t a n t s b r ought about by a m e t a l - i n s u l a t o r t r a n s i t i o n , then the pr o b l e m i s e a s i e r . In t h i s c a s e i t i s not n e c e s s a r y t o know the s t r a i n depen- dence o f a l l o f the band p a r a m e t e r s t o second o r d e r , p r o v i d e d the f e a t u r e s w h i c h l e a d t o a change i n the e l a s t i c c o n s t a n t a t the m e t a l - i n s u l a t o r t r a n s i t i o n a r e c o r r e c t l y d e s c r i b e d . We c o n s i d e r t h e f o l l o w i n g s i m p l e model f o r the s t r a i n dependence o f t he band s t r u c t u r e . The en e r g y bands f o r the TTF and TCNQ c h a i n s a r e t a k e n t o be one d i m e n s i o n a l t i g h t - b i n d i n g bands w i t h band p a r a - meters as c a l c u l a t e d by B e r l i n s k y e t a l (1974). In t h i s band s t r u c t u r e the TTF(TCNQ) band has a maximum (minimum) a t k = 0 (see F i g . 56). The Fermi l e v e l f o r t h e two bands i s d e t e r m i n e d by assuming a c h a r g e t r a n s f e r p o f .59. In the s p i r i t o f the r i g h t - b i n d i n g a p p r o x i m a t i o n we assume i d e n t i c a l s t r a i n dependences f o r the two bandwidths and s t r a i n independent c e n t r e s o f g r a v i t y . That i s , a one d i m e n s i o n a l s t r a i n i n the c o n d u c t i n g d i r e c t i o n i s assumed t o s c a l e t he w i d t h o f b o t h t h e TTF and the TCNQ bands and t h e i r energy gaps i n the i n s u l a t i n g s t a t e , by the same f a c t o r exp(-Bc), where £ i s the b a x i s s t r a i n and B i s a d i m e n s i o n l e s s p a r a m e t e r . In t h i s model a s t r a i n w i l l have no e f f e c t on the Fermi e n e r g y but wi 11 change the Fermi w a v e v e c t o r k̂ . and hence the ch a r g e t r a n s f e r . The e f f e c t o f a b a x i s c o m p r e s s i o n on the d e n s i t y o f o c c u p i e d s t a t e s i s i l l u s t r a t e d i n Fig.57 f o r the t i g h t - b i n d i n g band s t r u c t u r e shown i n Fig.56. Note the a d d i t i o n a l c h a r g e t r a n s f e r f r o m t h e TTF t o the TCNQ band. We now i n v e s t i g a t e the e f f e c t o f t h i s s m a l l s t r a i n dependent c h a r g e t r a n s f e r on the e l a s t i c modulus c 2 2 . As d i s c u s s e d i n the p r e v i o u s c h a p t e r the e x p e r i m e n t measures t h e a d i a b a t i c m o d u l i , and the t e m p e r a t u r e o f the sample w i l l be s t r a i n dependent. However, the l a t t i c e s p e c i f i c h eat i s v e r y much l a r g e r than the e l e c t r o n i c s p e c i f i c h e a t . For t h i s r eason the t e m p e r a t u r e o f the sample w i l l be d e t e r m i n e d by t h e r e q u i r e m e n t t h a t the l a t t i c e e n t r o p y be a c o n s t a n t and not the e l e c t r o n i c e n t r o p y . S i n c e the e l e c t r o n s a r e t h e r m a l l y c o u p l e d t o the l a t t i c e , t h e i r t e m p e r a t u r e w i l l be de- t e r m i n e d by the phonon heat b a t h . A c c o r d i n g l y the c o n t r i b u t i o n o f the c o n d u c t i o n e l e c t r o n s w i l l be somewhere between a d i a b a t i c and i s o t h e r m a l . S i n c e the d i f f e r e n c e between t h e two moduli i s n o t e x p e c t e d t c be v e r y s i g n i f i c a n t we w i l l t a k e the e a s i e s t a p p r c a c and c a l c u l a t e the i s o t h e r m a l modulus. The i s o t h e r m a l c o n t r i b u t i o n t o the modulus c 2 2 ' s d e t e r m i n e d from the b a x i s s t r a i n dependence o f the f r e e e nergy d e n s i t y [13] F = i c 2 2 r, 2 - kT ln(\ + e x p ( - -^jj N F(e,c) + N Q ( e , ? ) ] d E where a l l c o n t r i b u t i o n s t o the e l a s t i c modulus not c o n n e c t e d w i t h e l e c t r o n t r a n s f e r have been i n c o r p o r a t e d i n t o t he f i r s t term. We have taken the Fermi energy t o be z e r o f o r c o n v e n i e n c e . The TTF and TCNQ bands have s t r a i n dependent d e n s i t y o f s t a t e s f u n c t i o n s g i v e n by Np(e,r.) and N^(e,c) r e s p e c t i v e l y . The B r i l l o u i n zone boundary f o l l o w s the s t r a i n s e l f c o n s i s t e n t l y . The e l a s t i c c o n s t a n t c 2 2 i s o b t a i n e d by d i f f e r e n t i a t i n g the f r e e energy [13] t w i c e w i t h r e s p e c t t o £. The r e s u l t i s [14] c 2 2 = c°22 - B 2 j [ ( c + n p ) 2 N F ( e ) + (e + n Q ) 2 NQto](- I f . ) de + B 2 | [ N f ( E ) + N Q ( e ) J e f (e) de where rip and HQ a r e t h e e n e r g i e s o f t h e c e n t r e o f the TTF and TCNQ bands r e s p e c t i v e l y . S i n c e the r e l a t i v e p o s i t i o n o f the two bands i s d e t e r m i n e d by t h e c h a r g e t r a n s f e r p, t h e p o s i t i o n s o f the band c e n t r e s a r e g i v e n by n p ^ = 2 t p ^ j cos ("^jr) as shown i n F i g . 57 where 4 | t p | and J»|tJ a r e t h e a p p r o p r i a t e b a n d w i d t h s . We now examine the be- h a v i o u r o f the l a s t two terms i n [14] when energy gap opens up i n t he d e n s i t y o f s t a t e s and the m a t e r i a l becomes an i n s u l a t o r . The l a s t term d e c r e a s e s by an amount p r o p o r t i o n a l t o B 2 i^A 2 Np(o) + A 2 NQ(O)J where Ap and A^ a r e the energy gaps on the two c h a i n s . S i m i l a r l y the second l a s t term d e c r e a s e s i n magnitude from 3 2 n2- N (o) NQ(O)J t o z e r o a t z e r o t e m p e r a t u r e . C l e a r l y t h i s change i s much b i g g e r than the change i n t h e l a s t term i n [14] s i nee (^Ap/rip^ 2, ^AQ/T1Q^2 « 1. I f the l a s t term i n [14] i s n e g l e c t e d the e l a s t i c modulus w i l l i n c r e a s e by A C_22. = 2 ^ 2 2 . & v°>+ -a vo)] c 2 2 v 2 2 i n c o o l i n g from the m e t a l l i c phase, t h r o u g h the meta1 - i n s u l a t o r t r a n - s i t i o n towards z e r o t e m p e r a t u r e . We now a p p r o x i m a t e the l o n g i t u d i n a l b u l k wave v e l o c i t y i n the b d i r e c t i o n , v 2 2 by t h e b a x i s e l o n g a t i o n a l v e l o c i t y a t T =0, and use tp = .05 ev and t ^ = -.11 ev f o r the o v e r l a p p a r a m e t e r s ( B e r l i n s k y e t a l 1974). In t h i s c a s e 3 = 4.1 g i v e s agreement w i t h t he o b s e r v e d low t e m p e r a t u r e v e l o c i t y anomaly. T h i s v a l u e f o r 8 may be compared . w i t h 3 ~ 6 i n f e r r e d from the m o l e c u l a r o r b i t a l c a l c u l a t i o n o f the o v e r l a p p a r a m e t e r s . The agreement between the o b s e r v e d and c a l c u l a t e d 8 i s c o n s i d e r e d t o be e x c e l l e n t . The much s m a l l e r anomaly i n the modes i n v o l v i n g o n l y s h e a r moduli can now be e x p l a i n e d as f o l l o w s . S i n c e t h e r e i s no volume change a s s o c i a t e d w i t h s h e a r modes, t h e r e i s no f i r s t o r d e r change i n the l a t t i c e c o n s t a n t s due t o s t r a i n and hence a c o r r e s p o n d i n g l y s m a l l e r s t r a i n dependence t o the b a n d w i d t h s . The c h a r g e t r a n s f e r mechanism w i l l then p l a y o n l y a minor r o l e i n d e t e r m i n i n g the s h e a r m o d u l i . There a r e a t l e a s t two p o s s i b l e e x p l a n a t i o n s f o r t h e low t e m p e r a t u r e anomaly i n t h e t e m p e r a t u r e dependence o f the a a x i s e l o n g a t i o n a l mode. The f i r s t p o s s i b i l i t y i s t h a t t h e anomaly i s due t o c o u p l i n g t o the b a x i s m o l e c u l a r o v e r l a p . T h i s c o u p l i n g c o u l d r e s u l t from a l i b r a t i o n o f t h e m o l e c u l e s a b o u t , f o r example, t h e a a x i s caused by a s t r a i n a l o n g the a d i r e c t i o n . A second p o s s i b i l i t y i s t h a t a s t r a i n i n the a d i r e c t i o n a l s o tends t o f a v o u r a change i n the charg e t r a n s f e r be- tween c h a i n s , perhaps by a l t e r i n g t he e l e c t r o s t a t i c Madelung energy ( T o r r a n c e and S i l v e r m a n 1977). T h i s change i n ch a r g e t r a n s f e r c o u l d be m o d e l l e d by p o s t u l a t i n g t h a t an a a x i s s t r a i n p r o d u c es a r i g i d s h i f t o f the TTF and TCNQ. bands r e l a t i v e t o one a n o t h e r . As i n the b axis case the strain dependent charge transfer would disappear in the insulating phase, leading to a s t i f f e r lattice, and a low temperature velocity anomaly. It is instructive to consider some of the implications of the above model of the low temperature elastic anomaly. The dominant term in the expression for the electronic softening of the elastic modulus in [14] is practically identical to the expression for the Paul i spin susceptibility except that u is replaced by gn and Bn-.. D r Q This similarity is more general than the particular model used here o suggests, and is due to the dependence of both phenomena on the den- sity of thermally accessible states near the Fermi level. The apparent lack of an anomaly in the sound velocity near the phase transition at 38K is consistent with the corresponding spin susceptibility data (Scott et al 1974). Also the small positive curvature in the tempe- rature dependence of the b axis elongational velocity in the range of 200K-300K (see Fig.46) is consistent with the temperature depen- dence of the spin susceptibility over the same temperature range. Another feature of the charge transfer model proposed here is that it predicts a pressure dependence of the charge transfer and of the associated Fermi wavevector. If we neglect the pressure depen- dence of the bulk modulus, use the experimentally determined value of 3 and consider only the b axis strain contribution to the charge transfer, then the pressure dependence of kp is 3*10~3 /Kbar corresponding to an additional charge transfer of 1%/Kbar. In this model one would also expect a change in charge transfer to result from thermal e x p a n s i o n . The e x p e c t e d change i n kp between OK and 300K i s c o n s i s t e n t w i t h t h e t e m p e r a t u r e dependence o f t h e k kp anomaly o b s e r v e d i n X-ray measurements by Kagoshima e t a l (1976) and Pouget e t a l (1976) . In c o n c l u s i o n , the e l a s t i c measurements show a s m a l l s t i f f e n i n g a t t e m p e r a t u r e s below 52K f o r modes o f v i b r a t i o n w h i c h a r e a s s o c i a t e d w i t h a volume change. The s t i f f e n i n g i s i n t e r p r e t e d as a r i s i n g from a s t r a i n i n d u c e d c h a r g e t r a n s f e r between the TTF-TCNQ. c o n d u c t i o n bands. T h i s c h a r g e t r a n s f e r s o f t e n s t h e l a t t i c e i n the m e t a l l i c phase above 52 K but i s i n h i b i t e d i n the low t e m p e r a t u r e i n s u l a t i n g phase by t h e app e a r a n c e o f an energy gap i n the e l e c t r o n e nergy bands. ( i i i ) Comment on A c o u s t i c A b s o r p t i o n The c h a r g e t r a n s f e r model s u g g e s t s an a d d i t i o n a l a c o u s t i c l o s s mechanism f o r the l o n g i t u d i n a l modes. The i d e a i s t h a t i t t a k e s t i m e f o r the c h a r g e t r a n s f e r between c h a i n s t o o c c u r and d u r i n g t h i s time the e l a s t i c modulus r e l a x e s from c 2 2 b e f o r e t h e c h a r g e t r a n s f e r can t a k e p l a c e , t o c 2 2 (see [14]) a f t e r e q u i l i b r i u m i s r e a c h e d . The c o n t r i b u t i o n o f t h i s r e l a x a t i o n p r o c e s s t o the damping i s (Zener 19^8) [15] •1 _  c22 ~ C22 COT C 2 2 1 + CO^T 2 T2 T h i s e x p r e s s i o n i s a n a l o g o u s t o [13] i n C h a p t e r I I , the e x p r e s s i o n f o r t h e r m o e l a s t i c damping. The r e l a x a t i o n time x i s the time r e - q u i r e d f o r t h e c h a r g e t r a n s f e r between c h a i n s t o t a k e p l a c e . I n t u i t i v e l y one would e x p e c t the c h a r g e t r a n s f e r r e l a x a t i o n t i m e t o be comparable w i t h the a a x i s d i e l e c t r i c r e l a x a t i o n time ( 4 i r e a / a a ) where e a = h ( B a r r y and Hardy 1 977 ) i s the a a x i s d i - e l e c t r i c c o n s t a n t and a a i s the a a x i s c o n d u c t i v i t y . From d.c. c o n d u c t i v i t y measurements ( T i e d j e 1 9 7 5 ) o a - 3 - 8 ( f i - c m ) " 1 a t 60K. The c o r r e s p o n d i n g d i e l e c t r i c r e l a x a t i o n time i s 1 . 5 x 1 0 _ l l s . At 60K we e s t i m a t e from F i g . kS t h a t A c 2 2 / c 2 2 ' s 3%- W i t h t h e s e numbers, and the 261 khz l o n g i t u d i n a l r e s o n a n c e f r e q u e n c y o f Sample #23 we can use [15] t o c a l c u l a t e the a b s o r p t i o n . We f i n d t h a t 0 . " 1 = 0 . 8 x 1 0 ~ 5 a t 60K whereas the o b s e r v e d a b s o r p t i o n i s about I O - 4 . We c o n c l u d e from t h i s r e s u l t t h a t the c o n t r i b u t i o n o f t h e charge t r a n s f e r mechanism t o the damping i s p r o b a b l y not i m p o r t a n t i n the t e m p e r a t u r e regime i n w h i c h the sample i s m e t a l l i c . A t f i r s t g l a n c e one might e x p e c t t h i s c o n t r i b u t i o n t o the ab- s o r p t i o n t o be independent o f t e m p e r a t u r e below t h e m e t a l - i n s u l a t o r t r a n s i t i o n because A c 2 2 / c 2 2 d e c r e a s e s e x p o n e n t i a l l y as the tem- p e r a t u r e i s l o w e r e d and the d i e l e c t r i c r e l a x a t i o n t i m e i n c r e a s e s e x p o n e n t i a l l y , w h i l e the damping depends on the p r o d u c t . However, as d i s c u s s e d i n the p r e c e e d i n g s e c t i o n A c 2 2 / c 2 2 i s e x p e c t e d t o be p r o p o r t i o n a l t o the s p i n s u s c e p t i b i l i t y whereas the d i e l e c t r i c r e - l a x a t i o n time w i l l depend on the c o n d u c t i v i t y . S i n c e the a c t i v a t i o n e n ergy f o r the s u s c e p t i b i l i t y ( T o r r a n c e e t a l 1 9 77 ) i s lower (90K) than t h e a c t i v a t i o n e nergy (180K) f o r the c o n d u c t i v i t y ( E l d r i d g e 1 9 77 ) one would e x p e c t t h e t e m p e r a t u r e dependence o f the r e l a x a t i o n time t o 7 6 h 5 O 4 r 3 0 0 10 2 0 3 0 5 0 6 0 7 0 TOO F i g . 58 - Low t e m p e r a t u r e l o s s peak f o r a l o n g i t u d i n a l mode. dominate and the damping t o i n c r e a s e below the m e t a l - i n s u l a t o r t r a n s i t i o n . E v e n t u a l l y a t v e r y low t e m p e r a t u r e s when WT > 1 o r t h e t e m p e r a t u r e dependence o f the c o n d u c t i v i t y becomes dominated by i m p u r i t i e s the damping s h o u l d e i t h e r d e c r e a s e o r become tem- p e r a t u r e i n d e p e n d e n t . The t e m p e r a t u r e dependence o f the damping i n the i n s u l a t i n g phase i s shown i n F i g . 58 f o r the fundamental l o n g i t u d i n a l mode o f Sample #10 a t low t e m p e r a t u r e s . I f we p o s t u l a t e t h a t the e x c e s s low tem- p e r a t u r e a b s o r p t i o n above the a b s o r p t i o n minimum near 60K i s due t o the c h a r g e t r a n s f e r c o n t r i b u t i o n then we can e s t i m a t e the r e l a x a t i o n t i m e T. For c o n v e n i e n c e c o n s i d e r a t e m p e r a t u r e o f 35K where Sample #10 r e s o n a t e s a t 384 khz. A t t h i s t e m p e r a t u r e we e s t i m a t e from F i g . 49 t h a t A c 2 2 / C 2 2 ' s 0.6% and f r o m F i g . 58 t h a t the a d d i t i o n a l a b s o r p t i o n i s 5 x 1 0 _ l t . W i t h t h e s e numbers we can use [15] t o c a l - c u l a t e T. We f i n d T = 3*10" 8s. From d.c. c o n d u c t i v i t y measurements ( T i e d j e 1975) oa - 0.03 (^-cm)" 1 a t 35K, and the c o r r e s p o n d i n g d i e l e c t r i c r e l a x a t i o n time i s 2 x 1 0 ~ 9 s . T h i s r e l a x a t i o n time i s an o r d e r o f magnitude s h o r t e r than the t i m e i n f e r r e d from the a b s o r p t i o n d a t a . However, one c o u l d a r g u e t h a t t h i s d i e l e c t r i c r e l a x a t i o n t i m e i s an u n d e r e s t i m a t e o f the t r u e r e l a x a t i o n time because the c o r r e c t con- d u c t i v i t y t o use i s s m a l l e r than the a a x i s c o n d u c t i v i t y . The reason i s t h a t the a a x i s c o n d u c t i v i t y a r i s e s from a s e r i e s c o n n e c t i o n o f an e x t e n d e d m o l e c u l e w h i c h has l i t t l e o r no r e s i s t a n c e and a h i g h r e - s i s t a n c e (low c o n d u c t i v i t y ) element w h i c h d e t e r m i n e s the c h a r g e t r a n s - f e r r e l a x a t i o n t i m e . A l t h o u g h t h e above argument i s c e r t a i n l y not c o n c l u s i v e i t can n o t be r u l e d o u t as a p o s s i b l e e x p l a n a t i o n o f the anomalous low t e m p e r a t u r e sound a b s o r p t i o n i n TTF-TCNQ_. SUMMARY 1. The Main R e s u l t s o f t h i s Work The e l e c t r o n i c c o n t r i b u t i o n to the a t t e n u a t i o n o f u l t r a s o n i c waves i n one and two d i m e n s i o n a l m e t a l s has been c a l c u l a t e d f o r a r b i t r a r y qZ u s i n g a t r a n s p o r t e q u a t i o n a p p r o a c h . The a t t e n u a t i o n i s found t o be a n o m a l o u s l y low and s t r o n g l y t e m p e r a t u r e dependent i n one d i m e n s i o n a l m e t a l s . In the quantum l i m i t , when the e l e c t r o n mean f r e e p a t h i s l o n g compared t o an a c o u s t i c w a v e l e n g t h , the a t t e n u a t i o n i n one and two d i m e n s i o n a l m e t a l s depends s t r o n g l y on the d i r e c t i o n o f p r o p a g a t i o n o f the a c o u s t i c wave. On the o t h e r hand the a t t e n u a t i o n o f sound i n non - d e g e n e r a t e e l e c t r o n gases ( s e m i c o n d u c t o r s ) i s independent o f t h e d i m e n s i o n a l i t y o f the e l e c t r o n gas. A much s i m p l e r method o f s o l v i n g the Boltzmann t r a n s p o r t e q u a t i o n , t o o b t a i n the a m p l i f i c a t i o n o f sound i n the p r e s e n c e o f a d.c. e l e c t r i c f i e l d , has been d i s c o v e r e d . A c a p a c i t i v e t e c h n i q u e has been d e v e l o p e d f o r making sound v e l o c i t y and a t t e n u a t i o n measurements on s m a l l samples. In t h i s t e c h n i q u e an r f c a r r i e r s i g n a l i s used as a probe t o d e t e c t s m a l l d i s p l a c e m e n t s o f the sample. The r f c a r r i e r method i s shown t o be s u p e r i o r t o the c o n v e n t i o n a l v i b r a t i o n p i c k u p w h i c h i s based on a p p l y i n g a d.c. b i a s v o l t a g e t o the p i c k u p c a p a c i t o r . Three d i f f e r e n t e l a s t i c c o n s t a n t s f o r TTF-TCNQ have been measured as a f u n c t i o n o f t e m p e r a t u r e . TTF-TCNQ i s found t o be s l i g h t l y s t i f f e r p e r p e n d i c u l a r t o the c o n d u c t i n g d i r e c t i o n than p a r a l l e l t o the con- d u c t i n g d i r e c t i o n . A l t h o u g h the m a t e r i a l i s v e r y a n i s o t r o p i c e l e c t r i c a l l y , e l a s t i c a l l y i t i s not f a r from b e i n g i s o t r o p i c . The s t r o n g t e m p e r a t u r e dependence o f t h e e l a s t i c c o n s t a n t s i s a t t r i b u t e d t o the imp o r t a n c e o f m o l e c u l a r l i b r a t i o n s and i n t r a m o l e c u l a r modes i n the l a t t i c e e n t r o p y o f TTF-TCNQ above 2 0 K . The b u l k modulus a t z e r o t e m p e r a t u r e , the p r e s s u r e dependence o f the b u l k modulus, the G r u n e i s e n c o n s t a n t and the room t e m p e r a t u r e s p e c i f i c h eat a r e e s t i - mated. A s m a l l anomaly i n the t e m p e r a t u r e dependence o f the Young's modul a t low temperatures i s i n t e r p r e t e d as b e i n g due t o the f r e e z i n g o u t o f the c o n d u c t i o n e l e c t r o n s below the meta1 - i n s u l a t o r t r a n s i t i o n . The low t e m p e r a t u r e anomaly g i v e s a d i r e c t measure o f t h e q -> 0 e l e c t r o n - phonon c o u p l i n g c o n s t a n t . The i n t e r p r e t a t i o n o f the e l a s t i c anomaly s u g g e s t s an e x p l a n a t i o n f o r the t e m p e r a t u r e dependence o f the 4kp s p o t s o b s e r v e d by X-ray s c a t t e r i n g . W i t h i n the r e s o l u t i o n o f the measurements, no s m a l l d i s c o n t i n u i t i e s i n the t e m p e r a t u r e dependence o f t h e v e l o c i t i e s , o f t h e type e n v i s a g e d by P h i l l i p s (1977). were o b s e r v e d . A method i s p r o p o s e d f o r d e t e r m i n i n g the c a x i s t h e r m a l c o n d u c t i v i o f TTF-TCNQ by m e a s u r i n g the f r e q u e n c y dependence o f t h e damping o f low f r e q u e n c y f l e x u r a l v i b r a t i o n s . For a summary o f the e x p e r i m e n t a l r e s u l t s on TTF-TCNQ see T a b l e X below. " TABLE X Summary o f the E x p e r i m e n t a l R e s u l t s Modulus 1 Room Temperature E l a s t i c Moduli 2 ( I 0 n d y n e s / c m 2 ) P e r c e n t I n c r e a s e From R.T. t o OK 3 Ea C 6 6 3.1 ± 0.7 1.27 ± 0.09 0.5 ± 0.1 37 58.0 37 G r u n e i s e n c o n s t a n t 4 2.6 Room t e m p e r a t u r e heat c a p a c i t y (C ) 35R P r e s s u r e d e r i v a t i v e o f b u l k modulus ( 3 B / 3 P ) 5 16±2 Average d e f o r m a t i o n p o t e n t i a l G 0.38ev P r e s s u r e dependence o f kp 7 0.5%/kbar 1 For an e x p l a n a t i o n o f t h e s e l a b e l s see P a r t B S e c t i o n 2.2. 2 These a r e a d i a b a t i c m o d u l i . For the d i f f e r e n c e between the a d i a b a t i c and i s o t h e r m a l Young's moduli see p.1 4 8 . 3 See F i g s . 46 and 4 8 . 4 See p.174. 5 See p.175 6 See p.186 7 See p.198 2. S u g g e s t i o n s f o r F u r t h e r Work I n the c a l c u l a t i o n o f the a t t e n u a t i o n o f sound i n a one d i m e n s i o n a l metal we have assumed a p e r f e c t l y one d i m e n s i o n a l m e t a l , w i t h n o n - z e r o c o n d u c t i v i t y i n o n l y one d i r e c t i o n . However, i n r e a l m a t e r i a l s t h e r e i s a lways some c o n d u c t i v i t y i n a l l d i r e c t i o n s . I t would be w o r t h w h i l e t o check t h a t the v e r y s m a l l a t t e n u a t i o n i n the at « 1 l i m i t i s n o t an a r t i f a c t o f t h e p e r f e c t l y one d i m e n s i o n a l l i m i t t h a t we have c o n s i d e r e d h e r e . T h i s c a l c u l a t i o n c o u l d be done by a l l o w i n g the e l e c t r o n i c band s t r u c t u r e t o have some t h r e e d i m e n s i o n a l c h a r a c t e r , and then c a l c u - l a t i n g the a t t e n u a t i o n u s i n g the method o f R i c e and Sham (1970) f o r example. Methods f o r m e a s u r i n g s m a l l d i s p l a c e m e n t s have r e c e i v e d a c o n s i d e r - a b l e amount o f a t t e n t i o n r e c e n t l y i n c o n n e c t i o n w i t h t h e d e t e c t i o n o f g r a v i t a t i o n a l r a d i a t i o n . From the d i s c u s s i o n i n S e c t i o n 1.2 o f P a r t B, i t a ppears t h a t a d i s p l a c e m e n t s e n s i t i v i t y o f 1 0 " 1 6 - 1 0 - 1 7 /Kv cm c o u l d be o b t a i n e d , u s i n g the c a p a c i t i v e t e c h n i q u e d e s c r i b e d i n t h i s t h e s i s , s i m p l y by u s i n g a l a r g e r p i c k u p c a p a c i t a n c e and s u p e r c o n d u c t i n g , LC r e s o n a n t c i r c u i t s . T h i s s e n s i t i v i t y i s comparable w i t h the h i g h e s t s e n s i t i v i t y o b t a i n e d t o d a t e i n c u r r e n t g r a v i t a t i o n a l r a d i a t i o n de- t e c t o r s . More e x p e r i m e n t s a r e r e q u i r e d t o d e t e r m i n e the r e a l l i m i t s t o the s e n s i t i v i t y o f the c a p a c i t i v e v i b r a t i o n p i c k u p . I n S e c t i o n 3.2 o f P a r t B, we have p r o p o s e d two d i f f e r e n t i n t e r p r e - t a t i o n s o f the low t e m p e r a t u r e anomaly i n the Young's m o d u l i . One i s based on a quantum m e c h a n i c a l c a l c u l a t i o n o f the v e l o c i t y o f sound i n t h e p r e s e n c e o f an a r b i t r a r y d e f o r m a t i o n p o t e n t i a l 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 . The o t h e r i n t e r p r e t a t i o n i s based on an e l e c t r o n - phonon i n t e r a c t i o n w h i c h i s p e c u l i a r t o the band s t r u c t u r e o f TTF- TCNQ. I f i t i s l e g i t i m a t e t o e x t r a p o l a t e the quantum c a l c u l a t i o n t o q -*.0, then one would e x p e c t a s i m i l a r v e l o c i t y anomaly more o r l e s s independent o f the e x a c t n a t u r e o f the band s t r u c t u r e , i n o t h e r m a t e r i a l s w h i c h undergo P e i e r l s t r a n s i t i o n s . On the o t h e r hand i f the second mechanism (change t r a n s f e r ) r e a l l y i s n e c e s s a r y t o ex- p l a i n the low t e m p e r a t u r e anomaly then one would o n l y e x p e c t t o see s i m i l a r a n o m a l i e s i n m a t e r i a l s i n w h i c h t h e r e i s a p o s s i b i l i t y o f e l e c t r o n t r a n s f e r from one p a r t o f the energy band t o a n o t h e r . The a m b i g u i t y might be r e s o l v e d by l o o k i n g f o r . v e l o c i t y a n o m a l i e s i n o t h e r m a t e r i a l s w h i c h undergo P e i e r l s t r a n s i t i o n s , but have a d i f f e r e n t band s t r u c t u r e , such as T a S 3 (Sambongi e t a l 1 9 7 7 ) . TSeF-TCNQ would a l s o be an i n t e r e s t i n g m a t e r i a l t o l o o k a t . More measurements s h o u l d be made o f the damping o f l o n g i t u d i n a l modes to d e t e r m i n e the n a t u r e o f the a b s o r p t i o n a n o m a l i e s near 52K and 150K. A l s o c a r e f u l measurements o f the damping o f the low f r e - quency f l e x u r a l modes c o u l d be used t o o b t a i n the t r a n s v e r s e t h e r m a l d i f f u s i v i t y . A t t e m p e r a t u r e s w e l l below the metal i n s u l a t o r t r a n s i t i o n the microwave c o n d u c t i v i t y o f TTF-TCNQ i s known to be s e v e r a l o r d e r s o f magnitude l a r g e r than the d.c. c o n d u c t i v i t y . A s t r o n g f r e q u e n c y dependence might be e x p e c t e d i f t h e low t e m p e r a t u r e c o n d u c t i v i t y were dominated by l o c a l i z e d c a r r i e r s h o p p i n g between p i n n i n g s i t e s . A measurement o f the f r e q u e n c y dependence o f the c o n d u c t i v i t y i n the range 0-1 Ghz c o u l d g i v e some i n f o r m a t i o n about the hopping t i m e . The measurement might be done w i t h s u b s t a n t i a l l y the same a p p a r a t u s as t h a t used i n the a c o u s t i c resonance e x p e r i m e n t s . APPENDIX § 1. Diode Temperature Sensor Calibration Diode #D2755 Current 10yA T(K) Voltage T(K) Voltage 4.2 2.1956 60 1.0273 5 2.1581 65 1.0143 6 2.1123 70 1.0011 7 2.0688 75 • 9877 8 2.0286 77 .9814 9 1.9917 80 • 9742 10 1.9585 90 .9468 11 1.9290 100 .9192 12 1.9023 110 .8916 13 1.8772 120 .8639 14 1.8514 130 .8361 15 1.8226 140 .8083 16 1.7503 150 .7803 17 1.7543 160 .7522 18 1.7099 170 .7241 19 1.6516 180 .6959 20 1.5791 190 .6677 21 1.4951 200 • 6393 22 1.4063 210 .6108 23 1.3228 220 .5824 24 1.2549 230 • 5539 25 1.2076 240 .5259 26 1.1735 250 .4985 27 1.1496 260 .4710 28 1.1335 270 .4434 29 1.1223 280 .4155 30 1.1140 290 .3869 32 1.1022 300 • 3578 34 1.0939 36 1.0873 38 1.0819 40 1.0765 45 1.0644 50 1.0523 55 1.0400 211 § 2. C i r c u i t Diagrams (ii) Diode D e t e c t o r C i r c u i t TWO FERROXCUBES IN ^ l O p f 4h 500pf 3-9K FERROXCUBE TOROIDAL INDUCTOR * OUT 33K DIODE: HP 5082-2800 S c h o t t k y B a r r i e r Diode ( i i ) MOSFET P r e a m p l i f i e r -12V OUT 0.082 15K 100K 68 0.082 0.082 15K 100K 100 ( a l l c a p a c i t a n c e s i n m i c r o f a r a d s ) 212 P r e a m p l i f i e r S p e c i f i c a t i o n s MOSFET T l 3N211 3db Bandwidth .05-12 Mhz Gain 20db Output Impedance 50ft Output Power ~ -8dbm ( i i i ) Phase S h i f t e r ( d e s i g n e d by S. Knotek) S p e c i f i c a t i ons MOSFET T e t r o d e s 3N128 Bandwidth .05-15 Mhz Input l8-30dbm Output Impedance 500. Output Power 7dbm ( C i r c u i t d i a g r a m , n e x t page) 2 1 3 214 ( i v ) L o r e n t z i a n G e n e r a t o r 10K(ten t u r n ) 68K S i g n a l A v e r a g e r A/VV-- » Sweep Output 1 0 K i — W r DIODE: HP 5082-2800 r VCG .0015 2.1uH CRT ^ . 0 0 5 ? 10K i ( f 0 = 2.8Mhz Q = 66 ) (v) S i g n a l A v e r a g e r Input A m p l i f i e r 12V low n o i s e r e s i s t o r s BJT's 2N4403 Gain 180 Bandwidth 0.03-50 OOOhz 215 § 3« Thermal Expansion Correction (Jericho et al 1977) b axis a a r ,d c axis expansion: T tJUliX) 300 0 290 .10 280 .21 270 • 31 260 .42 250 .52 240 .63 230 .73 220 .83 210 • 93 200 1.03 190 1.13 180 1.22 170 1.32 160 1.41 150 1.51 140 1.60 130 1.70 120 1 --78 110 1.87 100 1.95 90 2.02 80 2.09 70 2.16 60 2.21 50 2.25 40 2.29 30 2.31 20 2.33 10 2.34 0 2.34 «a ~ a c ~ -3 a D BIBLIOGRAPHY Abragam, A. 1961. P r i n c i p l e s o f N u c l e a r Magnetism, O x f o r d U n i v e r s i t y P r e s s , London. A b r i k o s o v , A.A., Gorkov, L.P. and D z y a l o s h i n s k i , I.E. 1963- Methods o f Quantum F i e l d Theory i n S t a t i s t i c a l P h y s i c s , P r e n t i c e H a l l , T o r o n t o . A k h i e z e r , A . I . , Kaganov, M.I. and L i u b a r s k i i , G . l a . 1975, Sov. Phys. JETP 5, 685. Andre\ J . - J . , B i e b e r , A. and G a u t i e r , F. 1976. Ann. Phys. J , 145. A u l d , B.A. 1973- A c o u s t i c F i e l d s and Waves i n S o l i d s , V o l . I and I I , J . W i l e y & Sons, T o r o n t o . B a r i S i c , S. 1972. Ann. Phys. ]_, 23- Barmatz, M. and Chen, H.S. 1974. Phys. Rev. _B_9, 4073. Barmatz, M. T e s t a r d i , L.R.. G a r i t o , A.F. and Heeger, A . J . 1974. S o l i d S t a t e Comrnun. J_5, 1299. Barmatz, M. T e s t a r d i , L.R. and D i S a l v o , F . J . 1975- Phys. Rev. B12, 4367- B a r r y , C P . and Hardy, W.N. 1977. P r i v a t e comrnun i c a t i o n . B e h r e n s , E. 1968. T e x t i l e Res. J . 38, 1075- B e r l i n s k y , A . J . , C a r o l a n , J . F . and W e i l e r , L. 1974. S o l i d S t a t e Comrnun. J 4 . , 347- B h a t i a , A.B. and Moore, R.A. I960. Phys. Rev. 121, 1075- B h a t i a , A.B. 1967- U l t r a s o n i c A b s o r p t i o n , O x f o r d U n i v e r s i t y P r e s s , London. B l e s s i n g , R.H. and Coppens, P. 1974. S o l i d S t a t e Comrnun. \5_, 215- B l o u n t , E . I . 1959. Phys. Rev. 114, 418. B r a g i n s k i i , V.B. and Manukin, A.B. 1977- Measurement o f Weak F o r c e s i n P h y s i c s E x p e r i m e n t s , U n i v e r s i t y o f Ch i c a g o P r e s s , C h i c a g o . B r a g i n s k i i , V.G., M i t r o f a n o v , Rudenko, V.N. and Khorev, A.A. 1971. I n s t r u m e n t s and E x p e r i m e n t a l T e c h n i q u e s 14,-1239. B u l a e v s k i i , L.N. 1975- Sov. Phys. Usp. J_8, 131. C a n t r e l l , J.H. and B r e a z e a l e , M.A. 1974. P r o c . IEEE U l t r a s o n . Symp. 537- C a n t r e l l , J.A. and B r e a z e a l e , M.A. 1977- J . A c o u s t . Soc. Am. 6j_, 403- Cohen, M o r r e l H., H a r r i s o n , M.J. and H a r r i s o n , W.A. 1960. Phys.Rev. 117, 937- Cohen, M.J., Coleman, L.B., G a r i t o , A.F. and Heeger, A . J . 1976. Phys. Rev. Bl3, 5111. Coleman, L.B., Cohen, M.J., Sandman, D.J., Y a m a g i s h i , F.G., G a r i t o , A.F. and Heeger, A . J . 1973- S o l i d S t a t e Commun. J_3, 943- C r a v e n , R.A., Salomon, M.B., D e P a s q u a l i , G. and Herman, R.M. 1974. Phys. Rev. L e t t . 3_2, 769- D a n i e l s , W.B. 1963. L a t t i c e Dynamics, P r o c . o f Copenhage-i C o n f e r e n c e , ed. R.F. V.'al 1 i o , Pergamon, New Yor k . Debray, D., M i l l e t , R., Jerome, D., B a r i s i c , S. F a b r e , J.M. and G i r a l , L. 1977. J . P h y s i q u e L e t t , ( t o be p u b l i s h e d ) . D j u r e k , D., F r a n n l o v i c , K., P r e s t e r , M. and T o n i c , S. 1977- Phys. Rev. L e t t . 38, 715. D o n i a c h , S. and Sondheimer, E.H. 1974. Green's F u n c t i o n s f o r S o l i d S t a t e P h y s i c i s t s , W.A. Benjamin I n c . , R e a d i n g , Mass. E l d r i d g e , J.E. 1977- S o l i d S t a t e Commun. 2J_, 737. F e t t e r , A.L. and W a l e c k a , J.D. 1971. Quantum Theory o f Many P a r t i c l e Systems, M c G r a w - H i l l , New York. Gewurtz, S., K i e f t e , H., Landheer, D., McLaren, R.A. and S t o i c h e f f , B.P. 1972. Phys. Rev. L e t t . 2J_, 1454. G i r i f a l c o , L.A. 1973- S t a t i s t i c a l P h y s i c s o f M a t e r i a l s , J . W i l e y £ Sons, T o r o n t o . G l a s e r , W. 1974. F e s t k o r p e r p r o b l e m e 14, 205. 218 Goens, E. 1931- Ann. Phys. J J _ , 649. Hardy, W.N., B e r l i n s k y , A . J . and W e i l e r , L. 1976. Phys. Rev. B l 4 , 3356. H a r r i s o n , M.J. I960. Phys. Rev. 119, 1260. H a r t w i g , W.H. 1973- P r o c . IEEE 6l_, 58. Hearmon, R.F.S. 1961. An I n t r o d u c t i o n t o A p p l i e d A n i s o t r o p i c E l a s t i c i t y , O x f o r d U n i v e r s i t y P r e s s , London. Henry, N.F.M. and L o n s d a l e , K. 1952. I n t e r n a t i o n a l T a b l e s f o r X-ray C r y s t a l l o g r a p h y , V o l . I , Symmetry Groups, The Kynoch P r e s s , Birmingham, E n g l a n d . H o l s t e i n , T. 1959- Phys. Rev. 113, 479- Hughes, R.C. 1975. Ph.D. T h e s i s , Simon F r a s e r U n i v e r s i t y , Burnaby. H u n t i n g t o n , H.B. 1958. Sol i d S t a t e P h y s i c s 7, 213. H u t s o n , A.R., McFee, J . l i . and W h i t e , D.L. 1961. Phys. R-;v. L e t t . ]_. 237- I s h i g u r o , T., Kagoshima, S. and A n z a i , H. 1977- J - Phys. Soc. Japan 42, 365. J e r i c h o , M.H., Roger, W.A. and Simpson, A. 1977- P r i v a t e c o m m u n i c a t i o n . Kagoshima, S., I s h i g u r o , T. and A n z a i , H. 1976. J . Phys. Soc. Japan 4_1_, 2061. K e l l e r , H.J. 1977- (ed.) C h e m i s t r y and P h y s i c s o f One D i m e n s i o n a l Meta1s, NATO Advanced Study I n s t i t u t e S e r i e s , V o l . 25, Plenum, New Yor k . Khanna, S.K., Pouget, J.P., Comes, R., G a r i t o , A.F. and Heeger, A . J . 1977- To be p u b l i shed. K i t t e l , C. 1963. Quantum Theory o f S o l i d s , J . W i l e y £ Sons, T o r o n t o . K i t t e l , C. 1971- I n t r o d u c t i o n t o S o l i d S t a t e P h y s i c s , J . W i l e y & Sons, T o r o n t o . Landau, L.D. and L i f s h i t z , E.M. 1959- F l u i d M e c h a n i c s , Addison-Wes1ey, Re a d i n g , Mass. Landau, L.D. and L i f s h i t z , E.M. 1969- S t a t i s t i c a l P h y s i c s , A d d i s o n - Wesley, R e a d i n g , Mass. Landau, L.D. and L i f s h i t z , E.M. 1970. Theory o f E l a s t i c i t y , A d d i s o n - Wesley, R e a d i n g , Mass. L e i s s a , A.W. 1969- V i b r a t i o n o f P l a t e s , N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n , W a s h i n g t o n , D.C. L e k h n i t s k i i , S.G. 1963- Theory o f E l a s t i c i t y o f an A n i s o t r o p i c E l a s t i c Body, Holden-Day, San F r a n c i s c o . L i n d h o l m , U.S., Kana, D.D., Chu, W.-H. and Abramson, H.N. 1965- J . S h i p R e s e a r c h 9_, 11. L o r d J r , R.C. 1941. J . Chem. Phys. 9_, 693- Love, A.E.H. 1944. A T r e a t i s e on t h e M a t h e m a t i c a l Theory o f E l a s t i c i t y , 4th e d . , Dover, New York. McGuigan, D.F., Lam, C C , Gram, R.O.., Hoffman, A.W., D o u g l a s s , D.H. and Gutche, H. 1977. J . Low Temp. Phys. ( t o be p u b l i s h e d ) . M i k o s h i b a , N. 1959- J . Phys. Soc. Japan J_4, 1C31. M i t r a , T.K. 1969. J . Phys. C, 2, 52. Motchenbacher, C D . and F i t c h e n , F . C 1973- Low N o i s e E l e c t r o n i c D e s i g n , J . W i l e y & Sons, T o r o n t o . P a u l , W. and Warschauer, D.M. 1963- (ed.) S o l i d s Under P r e s s u r e , McGraw- H i l l , New Y o r k . P h i l l i p s , J . C 1977- J . Phys. Soc. Japan 42, 1051. P i n e s , D. and N o z i e r e s , P. 1966. The Theory o f Quantum L i q u i d s , W.A. Benjamin I n c . , R e a d i n g , Mass. P i p p a r d , A.B. 1955. P h i l . Mag. 46, 1104. P i p p a r d , A.B. 1963. P h i l . Mag. 8_, 161. Pouget, J.P., Khanna, S.K., Denoyer, F., Comes, R., G a r i t o , A.F. and Heeger, A . J . 1976. Phys. Rev. L e t t . 37_, 437- R i c e , M.J. and L i p a r i , N.O. 1977- Phys. Rev. L e t t . 38, 437- R i c e , M.J. and S t r a s s l e r , S. 1973- S o l i d S t a t e Comrnun. J_3, 125. R i c e , T.M. and Sham, L . J . 1970. Phys. Rev. BJ_, 4546. Salarnon, M.B., B r a y , J.W., D e P a s q u a l i , G. Cr a v e n , R.A., S t u c k y , G. and S c h u l t z , A. 1975- Phys. Rev. B11, 619- Sambongi, T., T s u t s u m i , K., S h i o z a k i , Y., Yamamoto, M. Yamaya, K. and Abe, Y. 1977- S o l i d S t a t e Comrnun. 22, 729. S c h a f e r , D.E., Thomas, G.A. and Wudl, F. 1975- Phys. Rev. BJ_2, 5532. S c o t t , J.C. G a r i t o , A.F. and Heeger, A.J. 1974. Phys. Rev. B10, 3131- S h a p i r a , Y. and Lax, B. 1965- Phys. Rev. 138, A1191. S h a p i r o , S.M., S h i r a n e , G., G a r i t o , A.F. and Heeger, A.J. 1977- Phys. Rev. EM5, 2413. S o u t h w e l l , R.V. 1922. P r o c . Roy. Soc. A101, 133- S p e c t o r , H.N. 1962. Phys. Rev. 127, 1084. . . S p e c t o r , H.N. 1966. S o l i d S t a t e P h y s i c s 19, 291. S p e c t o r , H.N. 1968- Phys. Rev. 165, 562. T i e d j e , J.T. 1975- M.Sc. T h e s i s , 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 , Vancouver. Timoshenko, S. and G o o d i e r , J.N. 1951- Theory o f E l a s t i c i t y , 2nd e d . , McGraw-Hi11, New Yor k . Timoshenko, S., Young, D.H. and Weaver, W. 1974. V i b r a t i o n Problems i n E n g i n e e r i n g , J . W i l e y & Sons, T o r o n t o . T o r r a n c e , J.B. and S i l v e r m a n , B.D. 1977- Phys. Rev. B15, 788. T o r r a n c e , J.B., T o m k i e w i c z , Y. and S i l v e r m a n , B.D. 1977- Phys. Rev. B15, 4738. T u c k e r , J.W. and Rampton, V.W. 1972. Microwave U l t r a o n i c s i n S o l i d S t a t e P h y s i c s , E l s e v i e r , New York. V r b a , J . and H a e r i n g , R.R. 1973- Can. J . Phys. 5J_, 1350. W a l l e r , M.D. 1961. C h l a d n i F i g u r e s , G. B e l l and Sons, London. W e i l e r , L. 1977- P r i v a t e c o m m u n i c a t i o n . W e i n r e i c h , G. 1956. Phys. Rev. 104, 321. W h i t e , D.L. 1962. J . A p p l . Phys. 33., 2547. W i l s o n , J.A., D i S a l v o , F . J . , Mahajan, S. 1975. Adv. Phys. 24_, 117. Z e n e r , C. 1948. E l a s t i c i t y and A n e l a s t i c i t y o f M e t a l s , U n i v e r s i t y o f C h i c a g o P r e s s , C h i c a g o . Ziman, J.M. 1972. Theory o f S o l i d s , 2nd. e d . , Cambridge U n i v e r s i t y P r e s s , London.

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