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Microwave dielectric resonances in TTF-TCNQ Barry, Charles Patrick 1977

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MICROWAVE DIELECTRIC RESONANCES IN TTF-TCNQ by CHARLES PATRICK BARRY B.Sc,  McGill University, 1974  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required  standard  I  THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 ©  Charles Patrick Barry 1977  In p r e s e n t i n g  this  thesis  in p a r t i a l  fulfilment of  an advanced degree at the U n i v e r s i t y of B r i t i s h the L i b r a r y s h a l l I  f u r t h e r agree  for scholarly by h i s of  this  written  make i t  that permission  for  Columbia,  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  It  gain s h a l l  permission.  Depa rtment  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  that  thesis or  is understood that copying or p u b l i c a t i o n  for financial  The U n i v e r s i t y o f B r i t i s h  for  study.  purposes may be granted by the Head of my Department  representatives. thesis  freely available  the requirements  Columbia  not be allowed without my  ii  ABSTRACT  D i e l e c t r i c resonances i n the microwave r e g i o n 16 t o 40 GHz  have  been observed i n s i n g l e c r y s t a l s of the h i g h l y a n i s o t r o p i c m a t e r i a l TTF-TCNQ below  25°K.  The a p p l i c a t i o n of o p e n - c i r c u i t boundary c o n d i t i o n s  to the d i e l e c t r i c r e s o n a t o r problem  i s shown to be i n a p p r o p r i a t e f o r t h i s  m a t e r i a l and a t r a c t a b l e a l t e r n a t i v e t h e o r y , which takes e x t e r n a l i n t o account, i s p r e s e n t e d .  U s i n g t h i s t h e o r y , the d i e l e c t r i c modes were  4.2°K to o b t a i n the complex d i e l e c t r i c c o n s t a n t  analyzed at  crystallographic b-axis.  fields  a l o n g the  For f o u r samples s t u d i e d i n d e t a i l , the  average  v a l u e s f o r the r e a l and imaginary p a r t s of the d i e l e c t r i c c o n s t a n t a r e  4200 and 16 r e s p e c t i v e l y a t 25 GHz.  found to be i s found;  however, i t i s not c l e a r whether t h i s f r e q u e n c y dependence  i s i n t r i n s i c to the samples or i s due solution.  A f r e q u e n c y dependence of  to the approximate  No f r e q u e n c y dependence of £^ was  observed.  n a t u r e of the  TABLE OF CONTENTS  Abstract Table List  of Contents of T a b l e s  L i s t of F i g u r e s Ac knowled g ement s  CHAPTER I - INTRODUCTION 1.1  Background  1.2  Purpose of t h e Experiment  CHAPTER I I - THEORY OF ANISOTROPIC DIELECTRIC RESONATORS 2.1  Introduction  2.2  The OCB Model  2.3  Solutions f o r the Anisotropic  2.4  S o l u t i o n s f o r t h e D i e l e c t r i c Resonator i n t h e L i m i t b ->  Slab  00  2.5  C o a x i a l Modes  CHAPTER I I I - THE EXPERIMENT 3.1  D e s i g n and Apparatus  3.2  Procedure  3.3  Results  CHAPTER IV - INTERPRETATION OF THE RESULTS 4.1  Mode  Identification  4.2  The R e a l and Imaginary P a r t s of e  I V  Page CHAPTER V - CONCLUSIONS  58  APPENDIX A:  THE ANISOTROPIC DIELECTRIC SLAB  63  APPENDIX B:  THE INFINITE COAXIAL LINE  68  References  77  0  V  LIST OF TABLES  Pag Summary o f n u m e r i c a l r e s u l t s f o r t h e E ^  m  modes  52  vi  LIST OF FIGURES Page 1.  The o r g a n i c  2.  C r y s t a l s t r u c t u r e o f TTF-TCNQ  3.  One-dimensional e l e c t r o n i c bands and c o r r e s p o n d i n g of  4.  donor m o l e c u l e TTF and a c c e p t o r m o l e c u l e TCNQ  3 density  s t a t e s f o r t h e TTF and TCNQ c h a i n s .  5  Microwave c o n d u c t i v i t y o f TTF-TCNQ o b t a i n e d u s i n g  the c o a x i a l  resonator technique. 5.  Conductivity  2  6  from a one-dimensional e l e c t r o n gas w i t h phonon  coupling.  8  6.  Behaviour o f e from a phenomenological theory o f CDW.  8  7.  C o o r d i n a t e system chosen f o r t h e r e c t a n g u l a r  geometry o f  TTF-TCNQ.  14  8.  The lowest o r d e r TE mode f i e l d omn  9.  a)  D i e l e c t r i c slab of thickness  b)  E^^ f i e l d  configuration  c)  field  configuration  10.  Graphical  configurations. t  s o l u t i o n f o r the TE^ d i e l e c t r i c  19  s l a b even (a) and odd  (b) modes. 11.  15  23  H (a) and H (b), as a f u n c t i o n o f d i s t a n c e x y  from t h e c e n t r e  of t h e s l a b .  23  12.  Equations  (2.39) ( 1 ) , and (2.40) ( 2 ) , s o l v e d  13.  D i v i s i o n of the region outside  14.  Waveguide assembly, i n s e r t e d i n t o a dewar.  32  15.  Copper b l o c k .  33  the resonator.  graphically.  27 27  vxx  Page 16.  Waveguide assembly o u t s i d e the dewar and b l o c k diagram of the d e t e c t i o n e l e c t r o n i c s .  17.  (1-the n o r m a l i z e d  35  s p e c t r a ) f o r one  sample a t t h r e e d i f f e r e n t  l e n g t h s b. 18.  38  F i e l d c o n f i g u r a t i o n near a s h o r t e d end of a waveguide e x c i t e d i n the dominant T E ^ Q mode.  End  ( a ) , top and  s i d e (b) views  a r e shown. 19.  41  C o u p l i n g as a f u n c t i o n of p o s i t i o n i n the waveguide f o r modes A to D.  20.  42  C o u p l i n g as a f u n c t i o n of r o t a t i o n i n the waveguide f o r modes A to D.  -43  21.  1/Q  (normalized to the v a l u e a t 24°K) vs 1/T.  22.  Q v a l u e s and resonant  23.  Mode p l o t showing s e v e r a l modes o f which f o u r c o u l d  45  f r e q u e n c i e s f o r the A mode, (sample 14). be  identified. 24.  49  Frequency dependence of e' o b t a i n e d from second-order b  theory  of d i e l e c t r i c r e s o n a t o r s .  51  25.  Q v a l u e s e x t r a p o l a t e d to 1/b  26.  D e n s i t y of s t a t e s w i t h an i m p u r i t y band which has been g i v i n g r i s e to l o c a l i z e d  2  =0.  states  near the  56 split,  Fermi  energy. 27.  60  Temperature dependence of the c o n d u c t i v i t y from a d e n s i t y of states,  28.  46  shown i n f i g u r e 26.  60  C y l i n d r i c a l geometry of a d i e l e c t r i c rod i n a c i r c u l a r waveguide.  69  V l l l  Page 29.  Mode p l o t f o r a u n i a x i a l c y l i n d r i c a l d i e l e c t r i c waveguide.  30.  Mode p l o t f o r the TM^Q c o a x i a l mode f o r v a r i o u s v a l u e s o f R /R 2  31.  •  R  Mode p l o t  Electric section  73  f o r t h e TM^Q c o a x i a l mode f o r v a r i o u s v a l u e s o f  e . z 32.  73  74 field  l i n e s f o r a c o a x i a l mode w i t h a c e n t r e  of f i n i t e  length.  76  ACKNOWLEDGEMENTS  I would l i k e t o acknowledge t h e support o f Dr. W.N. Hardy i n the s u p e r v i s i o n o f t h i s p r o j e c t . numerous d i s c u s s i o n s w i t h Dr.  I have a l s o b e n e f i t e d g r e a t l y from  A.J. Berlinsky.  Most o f t h e samples used  i n t h i s p r o j e c t were s u p p l i e d by Dr. L a r r y W e i l e r F i n a l l y , I would l i k e t o thank t h e o r g a n i c  and h i s co-workers.  c o n d u c t o r s group a t t h e  U n i v e r s i t y of P e n n s y l v a n i a f o r sending us some of t h e i r  samples.  CHAPTER I - INTRODUCTION  1.1  The  organic material  (TTF-TCNQ) c o n t i n u e s  Background  tetrathiafulvalene-tetracyanoquinodimethane  to be a m a t e r i a l o f c o n s i d e r a b l e  t h e o r i s t s and e x p e r i m e n t a l i s t s .  The h i g h a n i s o t r o p y  c o n d u c t i v i t y ^ " ^ , as w e l l as m o l e c u l a r  i n t e r e s t both t o i n the e l e c t r i c a l  o r b i t a l c a l c u l a t i o n s ^ ^ , have  i n d i c a t e d t h a t t h i s m a t e r i a l may be d e s c r i b e d  i n terms of a one-dimensional  band s t r u c t u r e . The TTF with  anisotropy  and TCNQ m o l e c u l e s  r e s u l t s from the s t a c k i n g o f t h e r e l a t i v e l y  ( F i g u r e 1 ) . The TTF and TCNQ s e p a r a t e l y form  stacks  s t r o n g c o u p l i n g between m o l e c u l e s o f t h e same s t a c k r e l a t i v e t o t h a t  of m o l e c u l e s on d i f f e r e n t graphic should has  flat  stacks  ( F i g u r e 2 ) . As a r e s u l t , t h e c r y s t a l l o -  b - a x i s , which i s the d i r e c t i o n of t h e s t r o n g m o l e c u l a r have t h e h i g h e s t  conductivity.  sometimes been a p p l i e d t o d e s c r i b e  intermolecular coupling  The e x p r e s s i o n  coupling,  "quasi-one-dimensional"  the f a c t t h a t the a n i s o t r o p y  i n the  i s l a r g e , but f i n i t e .  (3) Peierls  has shown t h a t a o n e - d i m e n s i o n a l m e t a l w i l l be un-  stable with respect the Fermi l e v e l , effect  t o a l a t t i c e d i s t o r t i o n , which w i l l open up a gap a t  thus l e a d i n g t o a s m a l l gap semiconducting s t a t e .  i s not r e s t r i c t e d  t o s t r i c t l y one-dimensional systems;  however, t h e  d i s t o r t i o n s , n e c e s s a r y to c r e a t e an energy gap over t h e e n t i r e Fermi are i n g e n e r a l more complicated  i n higher  This  surface,  dimensions.  T h i s type of mechanism i s b e l i e v e d t o be r e s p o n s i b l e f o r the l a r g e drop i n t h e b - a x i s c o n d u c t i v i t y i n TTF-TCNQ around 53°K, where a t r a n s i t i o n from a m e t a l l i c t o a semiconducting s t a t e i s observed.  I n terms o f t h e  2  II  •H  \  ||  c=c  H'  H  TTF  N  H  „  c  ,c=c  V,c=c /  / /  N  H  \ /  H  \ /.  C  =  C  x  H  N  c  / c=c \  %  N  TCNQ F i g u r e 1.  The o r g a n i c donor m o l e c u l e TTF a c c e p t o r m o l e c u l e TCNQ  and  (b)  (a) F i g u r e 2.  ia3A  (b)  C r y s t a l s t r u c t u r e of TTF-TCNQ. (a) View down t h e a - a x i s which connects t h e TTF molecule to the TCNQ. The open c i r c l e s r e p r e s e n t atoms on the TTF m o l e c u l e , w h i l e the s o l i d ones r e p r e s e n t those on t h e TCNQ. (b) View down the b - a x i s which i s t h e s t a c k i n g direction. Here, the s o l i d c i r c l e s r e p r e s e n t atoms t i l t e d out o f t h e p l a n e of the paper w h i l e the open c i r c l e s a r e those t i l t e d i n t o i t .  4  one-dimensional bands shown i n figure 3, a l a t t i c e i n s t a b i l i t y of wavevector q = 2kp, where k^ i s the Fermi wavevector, leads to the introduction of a gap 2A i n the energy spectrum. One of the presently disputed questions has to do with the mechanism responsible for the temperature dependence of the conductivity, both above and below 53°K.  The strong temperature dependence, as well as  the large magnitude of the d.c. conductivity i n i t i a l l y observed by Coleman (4> et a l .  , could not be understood by them on the basis of  metallic conduction.  So,- the suggestion  conventional  of viewing the behaviour of  TTF-TCNQ i n terms of c o l l e c t i v e many-body effects associated with the soft phonon P e i e r l s i n s t a b i l i t y was made ^  .  Figure 4 shows the temperature  dependence of the conductivity at a frequency of 30 GHz, the coaxial resonator  technique ^ .  obtained  using  The normalized conductivity increa-  ses as the temperature decreases, attaining a maximum value of V9 at T=53°K. Below t h i s temperature, the conductivity drops sharply. well as subsequent conductivity measurements  These r e s u l t s , as  reveal that, although the  peak conductivity r a t i o s are not as large as i n i t i a l l y reported by Coleman et a l . ^  ( a(53°K) / a(R.T.) > 500 ) , ah understanding of the  conducti-  v i t y mechanism on the basis of single p a r t i c l e scattering alone remains unclear. To study the conductivity i n a one-dimensional system, one (8 s) consider a model proposed by F r o h l i c h  who  can  examined the coupling of  non-interacting electrons to phonons i n a j e l l i u m model.  In a mean f i e l d  theory, one singles out the interactions with phonons of wavevector q=2kp. Frohlich showed that this system can support a current by propagating a  F i g u r e 3.  One^dimensional e l e c t r o n i c bands and c o r r e s p o n d i n g d e n s i t y of s t a t e s f o r the TTF and TCNQ c h a i n s b e f o r e (a) and a f t e r (b) the P e i e r l s t r a n s i t i o n .  10[  f (300°K) = 32.4 GHz o Q (300°K) = 32  8  0  100  0  Figure 4.  T(°K)  2 0 0  Microwave conductivity of TTF-TCNQ obtained using the coaxial resonator technique.  300  7  coupled l a t t i c e and e l e c t r o n i c charge d i s t o r t i o n as a t r a v e l l i n g wave. f u r t h e r argued t h a t f o r s u f f i c i e n t l y unattenuated.  He  s m a l l v e l o c i t i e s , t h e wave should move  A study o f t h e c o n t r i b u t i o n to t h e c o n d u c t i v i t y from such  a c o l l e c t i v e mode was made i n an elegant  paper by Lee, R i c e and  (9) Anderson  .  contributions sitions.  I n i t , they show t h a t , below t h e t r a n s i t i o n , t h e r e a r e no to t h e d.c. c o n d u c t i v i t y from s i n g l e p a r t i c l e i n t e r b a n d  tran-  However, i n c l u s i o n o f t h e F r o h l i c h c o l l e c t i v e mode y i e l d s a c o n -  d u c t i v i t y of t h e form a(io)  =  m  +  6(CJ)  O^(GJ)  (1.1)  m* where 6 (to)  i s the Dirac d e l t a f u n c t i o n .  A p l o t o f a i s shown i n f i g u r e 5.  i s t h e c o n t r i b u t i o n due t o s i n g l e p a r t i c l e i n t e r b a n d a gap 2A, whereas t h e weighted 6 - f u n c t i o n P e i e r l s - F r b h l i c h mode. introduce  For  across  i s t h e c o n t r i b u t i o n due t o t h e  They f u r t h e r argue t h a t v a r i o u s mechanisms  could  an energy gap i n t o t h e e x c i t a t i o n spectrum o f t h i s c o l l e c t i v e  mode and, i n s t e a d o f an i n f i n i t e d.c. c o n d u c t i v i t y , low  transitions  frequency a.c. c o n d u c t i v i t y .  t h e r e w i l l be a l a r g e  The v a r i o u s mechanisms proposed were:  1.  Impurity s c a t t e r i n g  2.  Commensurability w i t h t h e u n d e r l y i n g  3.  I n t e r a c t i o n s r e l a t e d t o 3-dimensional  t h e case CJ < < 2A, they g i v e t h e f o l l o w i n g e x p r e s s i o n  of t h e d i e l e c t r i c  lattice ordering.  f o r the r e a l part  constant:  e<(a))  = 1 + u>  p  2  +u;  p  2  m m*  (1.2) 6A  2  u  2  T  - -u  2  8  F i g u r e 5.  C o n d u c t i v i t y from a one-dimensional e l e c t r o n gas w i t h phonon c o u p l i n g (from R e f e r e n c e 9 ) . The dashed l i n e i s the c o n d u c t i v i t y without i n c l u d i n g the e f f e c t of the F r o h l i c h mode.  0  F i g u r e 6.  Behaviour of e from a phenomenological t h e o r y of  CDW.  9  oi  i s a non-zero f r e q u e n c y due t o some p i n n i n g mechanism, co i s the plasma p  frequency, m t h e e l e c t r o n i c band mass and m* t h e e f f e c t i v e mass o f e l e c t r o n s i n the condensed  state.  The complex d i e l e c t r i c c o n s t a n t , due t o the presence o f charge d e n s i t y waves, can be found w i t h i n a phenomenological t h e o r y o f charge d e n s i t y waves, as f o l l o w s :  l e t m* and N  be the mass and number  e l e c t r o n s i n t h e condensed charge d e n s i t y wave s t a t e .  of  The c l a s s i c a l  equa-  t i o n o f motion g o v e r n i n g t h e d i s p l a c e m e n t X o f t h e condensate from i t s equilibrium position i s N_m*X + yt + kX = N e E  (1.3)  c  Y i s a f r i c t i o n c o n s t a n t , k a harmonic r e s t o r i n g c o n s t a n t and E denotes an I n t r o d u c i n g two new c o n s t a n t s co^ and T:  externally applied e l e c t r i c f i e l d . w  2 T  = • .kN m* g  r =N m* g  and t a k i n g the F o u r i e r t r a n s f o r m o f equation(1.3), we get  X(u>) =  e_ m* E(co)  (1.4)  co - co — i u r T 2  2  X(co) may be i n t e r p r e t e d a s the mean d i s p l a c e m e n t o f teheuphaseuid bif t h e P e i e r l s - F r o h l i c h condensate.  ei-dzt.-  S i n c e the d i p o l e moment P a r i s i n g  from  t h i s d i s p l a c e m e n t i s N eX, then t h e d i e l e c t r i c c o n s t a n t due t o the motion o f a charge d e n s i t y wave i s ! >S E(to) where:  to  2 T  m m*  - co — irto 2  N i s the t o t a l number o f c o n d u c t i o n e l e c t r o n s e  (1.5)  10  and  OJ„2  N f 4ire n 1 , where n = _e m L  =  2  conductor.  , L b e i n g t h e l e n g t h of t h e l i n e a r  T h i s frequency dependence i s drawn i n f i g u r e 6. The  t o t a l d i e l e c t r i c c o n s t a n t e w i l l have a c o n t r i b u t i o n  from  s i n g l e p a r t i c l e a c t i v a t i o n a c r o s s a gap 2A , as w e l l as a c o n t r i b u t i o n from any e x i s t i n g c o l l e c t i v e e(o>)  Tanner thate  ^ ^ a n d Coleman  motion.  = e (a))  +  s p  e  C D W  have argued,  (o))  on t h e b a s i s of o p t i c a l  - 50' and t h a t t h e r e f o r e , another mechanism (such as a c o l l e c t i v e  mode) must be p r o v i d i n g o s c i l l a t o r  s t r e n g t h a t low temperatures  d i e l e c t r i c c o n s t a n t v a l u e s of 3000 have, been r e p o r t e d red  studies,  spectrometry  (15)  where  (12-14)  Far  shows no evidence o f a pinned mode above 7cm  A c c o r d i n g t o a l a t e s t e s t i m a t e by Coleman  infra-  -1  , oi^, i s around 2cm  (or 60 GHz).  1.2 . Purpose of t h e Experiment  The c r y s t a l  s t r u c t u r e of TTF-TCNQ i s m o n o c l i n i c  (16)  and thus,  t h e r e a r e i n g e n e r a l f o u r n o n - v a n i s h i n g components of t h e d i e l e c t r i c tensor:  yv  e  l l  E  12  £  21  e  22  0  0 0  (1.6)  0  S i n g l e c r y s t a l s , grown i n a c e t o n i t r i l e s o l u t i o n s , tend t o have p h y s i c a l dimensions  which c o i n c i d e w i t h t h e a and b c r y s t a l l o g r a p h i c  axes;  the t h i r d c a r t e s i a n a x i s c * makes a 14.46° a n g l e w i t h t h e c r y s t a l l o g r a p h i c c-axis.  Making t h e a p p r o x i m a t i o n t h a t t h e c r y s t a l s t r u c t u r e i s o r t h o -  11  rhombic, t h e r e are now  three non-vanishing  components of £  > coinciding  w i t h the p h y s i c a l axes ( F i g u r e 7 a ) .  s e  Low  x  0  0  e  0  0  0 0  Y  yv  temperature measurements  | .  (1.7)  ez  (12-14)  show t h a t  e _Z_  -  500  and  thus,  this material i s highly anisotropic. The  frequency  r e g i o n , where the d i e l e c t r i c c o n s t a n t  b - a x i s i s b e l i e v e d to be changing  e ralong^the  r a p i d l y , i s p r e s e n t l y j u s t below the  r e a c h of c o n v e n t i o n a l f a r i n f r a r e d  spectrometers,  I t i s a l s o hard to probe  u s i n g a microwave c a v i t y p e r t u r b a t i o n experiment because, even though the a v a i l a b l e c r y s t a l s a r e n e e d l e - l i k e i n shape, d e p o l a r i z a t i o n e f f e c t s be s u b s t a n t i a l because of the extremely  can  h i g h v a l u e of e_. The d e p o l a r i z a > Li Li  t i o n f a c t o r can be estimated revolution.  from the r e s u l t expected  f o r an e l l i p s o i d of  At h i g h e r f r e q u e n c i e s , where s h o r t e r c r y s t a l l e n g t h s a r e  r e q u i r e d , the a c c u r a c y o f t h i s w i l l get worse. P r e v i o u s r e p o r t s of d i e l e c t r i c resonances TTF-TCNQ ^ ^ 1 7  a t 13 GHz,  to h i g h e r f r e q u e n c i e s .  suggested  i n s i n g l e c r y s t a l s of  t h a t the method might be e a s i l y  The d i e l e c t r i c r e s o n a t o r method thus appeared as  a t t r a c t i v e a l t e r n a t i v e f o r p r o b i n g the microwave r e g i o n up to and 60 GHz,  the suggested  extended  v a l u e of u> .  Thus, the purpose of t h i s  an  beyond  experiment  i s to a p p l y the theory of a n i s o t r o p i c d i e l e c t r i c r e s o n a t o r s to TTF-TCNQ i n order to e x t r a c t the frequency  dependence of the complex d i e l e c t r i c  tensor.  12  CHAPTER I I - THEORY OF ANISOTROPIC DIELECTRIC RESONATORS  2.1  I t has  Introduction  been known f o r many y e a r s t h a t a f i n i t e p i e c e of  high  p e r m i t t i v i t y m a t e r i a l i n f r e e space can r e s o n a t e i n v a r i o u s modes. resonators constant  made from l o w - l o s s  - 100),  (19  materials,  have been observed and  such as TiO  High Q  (dielectric  i n v e s t i g a t e d by  s e v e r a l experimen-  21)  ters  '  .  That d i e l e c t r i c r e s o n a t o r s  noting  that electromagnetic  waves w i l l be  can e x i s t may  be e a s i l y seen by  c o m p l e t e l y r e f l e c t e d from  the  i n t e r f a c e between f r e e space and a d i e l e c t r i c , i f the angle of i n c i d e n c e i s g r e a t e r than the c r i t i c a l angle 0 = S i n -1 f l / V e " 1 , where e' i s the r e a l c • X  7  v  p a r t of the d i e l e c t r i c  J  constant.  Exact s o l u t i o n s to the Maxwell e q u a t i o n s are o n l y p o s s i b l e f o r d i e l e c t r i c resonators  w i t h s p h e r i c a l , t o r o i d a l , and  surfaces, while rigorous resonators The  do not  difficulties  exist.  e l l i p s o i d a l boundary  s o l u t i o n s f o r f i n i t e c y l i n d r i c a l and The  first  i n f i n d i n g exact  two  cases were t r e a t e d by  Richtmeyer  s o l u t i o n s stem from the f a c t  f i e l d s or t h e i r d e r i v a t i v e s do n o t . n e c e s s a r i l y v a n i s h a t the the  rectangular  that.the  surfaces  of  resonator. An a p p r o x i m a t i o n t h a t g i v e s a rough p i c t u r e of the modal  i s to c o n s i d e r  the d i e l e c t r i c r e s o n a t o r  " d u a l " of m e t a l w a l l r e s o n a t o r s ,  w i t h an a i r boundary to be  i . e . at the boundary, impose the  n * H  fields the conditions  = o"  (2.1)  n, . 1 = 0 These c o n d i t i o n s contrast  are c a l l e d o p e n - c i r c u i t boundary c o n d i t i o n s  (2.2) (OCB)  in  to the s h o r t - c i r c u i t boundary c o n d i t i o n s of m e t a l l i c c a v i t i e s .  13  The r e a s o n f o r assuming the OCB considering  the i n t e r f a c e between two  t a n t s e'^ and vector  D  e^, r e s p e c t i v e l y .  i s o t r o p i c media w i t h d i e l e c t r i c  The normal  cons-  component o f the d i s p l a c e m e n t  must be continuous a t t h i s i n t e r f a c e , i . e . n  • V  n  • E  = n  Thus,  I f E'^ >>  c o n d i t i o n s can be e a s i l y seen by  e ' J then n • E.^.= 2  (2.l)(often called  0.  1  •  2  K  = '2  n  • E  2  However, the v a l i d i t y of imposing  imposing a magnetic  c o n d i t i o n (2.2) i s not always  D,  w a l l boundary) i n c o n j u n c t i o n w i t h  good even though, e'-j. >>  The OCB modal f i e l d  condition  e'.^.  c o n f i g u r a t i o n s w i l l be the same as those f o r  the m e t a l l i c c a v i t y except t h a t . t h e i r E and H f i e l d s w i l l be i n t e r c h a n g e d . These are shown f o r the lowest modes i n f i g u r e In g e n e r a l , the a c c u r a c y of the OCB poor, p a r t i c u l a r l y  8. assumption  (as w i l l be shown i n s e c t i o n 2.3)  r e s o n a t o r dimensions  are comparable.to  i s expected  when any of the  the e f f e c t i v e w a v e l e n g t h . i n s i d e the  resonator.  Thus, s u b s t a n t i a l f i e l d  resonator.  A more r e a l i s t i c boundary v a l u e problem has been t r e a t e d  Yee  (19)  results.  and Okaya and Barash  (21)  to be,  energy e x i s t s o u t s i d e of the d i e l e c t r i c  and found to y i e l d  substantially  With r e f e r e n c e to f i g u r e , 7 a , they assumed OCB  by  improved  c o n d i t i o n s on the  s u r f a c e s p a r a l l e l to the X - a x i s w h i l e a t the boundary s u r f a c e s p e r p e n d i c u l a r to the X - a x i s , they a p p l i e d exact boundary  conditions.  T h e i r s o l u t i o n s are i n f a c t exact s o l u t i o n s to the i n f i n i t e of t h i c k n e s s c* i l l u s t r a t e d  i n f i g u r e 7b.  slab  An e q u i v a l e n t way,of t h i n k i n g  14  •Figure 7b.  15  fx*  011  • X  x"  • x  031  ,<s  *  (  012  022  F i g u r e 8. &  The lowest order TE  y  032  mode f i e l d c o n f i g u r a t i o n s . omn The s o l i d l i n e s a r e E - f i e l d s w h i l e the H - f i e l d s a r e t r a n s v e r s e to the p l a n e of the paper.  16  of t h e i r assumption of OCB  conditions  a t the boundary s u r f a c e s  to the X - a x i s i s to r e q u i r e a n t i n o d e s of the magnetic f i e l d at y =0,_a. and  z = 0, b  (see f i g u r e 7b) .  parallel  say,  A d i s c u s s i o n of the  to o c c u r  properties  of the s o l u t i o n s f o r the a n i s o t r o p i c d i e l e c t r i c s l a b found by Okaya Barash can be  found in,Appendix A.  c o m p l i c a t e d due i f one  to the a n i s o t r o p y  restricts  The  s p e a k i n g , one Fortunately,  of the s l a b i n the Y-Z  plane.  the problem to the s i t u a t i o n where the f i e l d s  u n i f o r m a l o n g the considerably.  These s o l u t i o n s are i n g e n e r a l  can  Z^axis  (b d i r e c t i o n ) , t h e n the  penalty  paid  f i e l d s can be  however, i t i s p o s s i b l e  corrections f o r f i n i t e resonators  by  very  However, are  simplified  f o r t h i s s i m p l i f i c a t i o n i s that  o n l y a p p l y the r e s u l t s to r e s o n a t o r s  and  strictly  of i n f i n i t e  to determine e x p e r i m e n t a l l y  length.  the  extrapolating results for different  lengths. A f t e r a d i s c u s s i o n of the s o l u t i o n s to the a n i s o t r o p i c r e s o n a t o r w i t h OCB  conditions  i n the f o l l o w i n g s e c t i o n , the TE modes of  the a n i s o t r o p i c d i e l e c t r i c s l a b when the Z-axis,  i.e.  2.:  9 _ 0 w i l l be 9z "  2.4,these s o l u t i o n s w i l l be  found and  f i e l d s are u n i f o r m a l o n g  examined  ( s e c t i o n 2.3).  used to f i n d an e x p r e s s i o n  c i e s of the d i e l e c t r i c r e s o n a t o r of i n f i n i t e l e n g t h and 2.2  The  OCB  f o r the  isotropic material  section  eigenfrequen"a".  Model currents  for  are:  V x E = iuu_ V x H = iw c V • eE = 0 V • H = 0  In  the  f i n i t e width  Maxwell's e q u a t i o n s i n the absence of f r e e charge and a magnetically  dielectric  H  (2.3)  e E  (2.4) (2.5) (2.6)  17  From them, one gets t h e f o l l o w i n g wave e q u a t i o n f o r E: V E - V(V • 1) 2  + io y£_E =  (2.7)  2  S o l u t i o n s f o r a r e c t a n g u l a r body of dimensions L^,  and  which s a t i s f y t h e OCB c o n d i t i o n s a r e : E E E  = A x  x  = A  y  y  = A  z  z  sin k X x  cos k Y y  cos k Z z  cos k X x  sink Y y  cos k Z z  cos k X x  cos k Y y  sink Z z  (2.8)  where k„ x  M_ —  =  k  k,, = mfr y " L y  9  ''£V4>m ' , n are integers; °  Z  '  = nir L~ z  (2.9)  A , A , A a r e c o n s t a n t s t o be x y z  determined.  S u b s t i t u t i n g these s o l u t i o n s i n t o t h e wave e q u a t i o n (2.7), one f i n d s ,  after  a l i t t l e a l g e b r a , t h e f o l l o w i n g s o l u t i o n s to t h e s e c u l a r e q u a t i o n : co  2  1 2y  k  2  x  4.  I y k  z i_  4  x  E  i y £  + +  '  +  y  \ +  k y  £  X  4  £  yj  £  i  i _ E  '+  k<* Z  x  £  ,  yj  1_ E  yj  X  £  X  1_  2  yj  E Z;  1/2  , 2 k k + x z 2  2  Z  i_  2  , 2 k k + y z 2  £  k  Z  2 k k x y 2  +  k  2  (2.10)  i l  (. y  x  The s o l u t i o n s may be c l a s s i f i e d as TE ( E ^  E  0) o r TM ' ( H  x  = 0).  For the  TE modes, one must have m,n ^ 0 , whereas f o r t h e TM modes, £ ^ 0.  I n an  i s o t r o p i c medium, f o r a g i v e n v a l u e of %, m, n, t h e TE and TM modes a r e degenerate (although,the f i e l d  configurations are different).  One can see  18  t h a t an e f f e c t o f the a n i s o t r o p y number o f o b s e r v a b l e  i s t o break t h i s degeneracy and double•the  modes.  S p e c i a l i z i n g fcor the case e = e = e. ( t e t r a g o n a l symmetry), x y the s o l u t i o n s (2.10) reduce to ( f o r y = 1) co  1 = — e  2  —j c  corresponding  z  (k  2  x  + k ) , 1 y H e,  k  2  J  (2.11)  2  z  to TE modes, and co  1  2  T  (k + k + k ) ^ x y z 2  = —  2  (2.12)  2  J  f o r the TM modes. The c* << a, b .  s i n g l e c r y s t a l s o f TTF-TCNQ a v a i l a b l e , h a d t y p i c a l dimensions Because o f t h i s and the f a c t t h a t e  have the lowest  frequencies  f  where m, n f 0. illustrated  i n f i g u r e 8.  should  g i v e n by.  1_ e  = c omn. y  The f i e l d  » . e . , the TE  n. 2  m  -i  1/2 (2.13)  c o n f i g u r a t i o n s f o r the f i r s t  few modes a r e  These f i e l d p a t t e r n s a r e magnetic m u l t i p o l e s  i n c h a r a c t e r except t h a t , w i t h i n the OCB model, the magnetic f i e l d terminate fields  a t the boundary.  In a more r e a l i s t i c model, one expects the  to extend o u t s i d e the r e s o n a t o r  the TEQ^^ mode i n f i g u r e , 9 b .  lines  and form c l o s e d l o o p s ' a s  shown f o r  In terms o f the OCB s o l u t i o n s , t h i s means  an e f f e c t i v e v a l u e of & between 0 and 1.  2.3  S o l u t i o n s f o r the A n i s o t r o p i c D i e l e c t r i c  The.TE (E = 0) mode s o l u t i o n s f o r the d i e l e c t r i c x t illustrated  i n f i g u r e ' 9 a w i l l now be d e r i v e d .  the equations  (2.3) and (2.4) e x p l i c i t l y :  Slab  slab of thickness  We b e g i n by w r i t i n g out  20 •  *  3H . 3H = i . z y [dy k £ 9z J O X ( 9H 9H E = i z X y k e, [3z " 9x J o y E  (2.14)  X  z  =  9H y 9x  i  (2.15)  k E o z  where k  9H 3y  X  (2.16)  j  9E  9E H = - i X y k y [9z o  9E  X  •  E  3E = - i z k y [9y o *•  H  H  9E  = - i z k y o  9x  y  y 9z J  z 9x J 9E  X  9y J  (2.17)  (2.18)  (2.19)  =u)Ui  o  — cc  For TE (E  = 0) modes w i t h no s p a t i a l dependence,along the z - a x i s  (i.-'  ( i . e . 9_ = 0 ) , and assuming a s p a t i a l dependence'- along the y - a x i s -of the 9z form e  l k  y  Y  , the f i e l d s  (2.14)-(2.19) become E  x  H  x  = E = H y z =  K_  ,—  H = i y k y o E  z  will  s a t i s f y equation V E 2  =0  E z  (2.20)  3E  (2.21)  3x  (2.7), i . e . - 9 z . —  (V" • E) + k y c E = 0 o z z 2  dZ  For an a n i s o t r o p i c m a t e r i a l , V • E r 0 i n g e n e r a l s i n c e E T E . y z the e q u a t i o n f o r E  will  c o n t a i n terms, l i n e a r i n the f i e l d  Thus,  derivative.  T h i s i s why the g e n e r a l s o l u t i o n s f o r the s l a b t u r n out to be so c o m p l i c a t e d (See Appendix A ) .  I n the p r e s e n t case however,  = 0 and so the e q u a t i o n 9z  for E  z  becomes, E 9x'  z  + k^yc E = 0 o z z  (2.22)  The boundary v a l u e problem f o r these modes can be s o l v e d e x a c t l y . The treatment  i s s i m p l i f i e d by s e p a r a t i n g them, from the s t a r t , i n t o even  21  and odd modes.  ( I t i s of c o u r s e p o s s i b l e t o s t a r t w i t h g e n e r a l  field  e x p r e s s i o n s and o b t a i n the even and odd modes from t h e r e s u l t i n g problem).  eigenvalue  Even and odd r e f e r t o the symmetry o f E^ about the plane x = 0,  which i s now chosen t o l i e i n the c e n t e r o f t h e s l a b .  (See F i g u r e 9a)  Eyen_Gu_ided_ TE_modesj_ The mode s o l u t i o n f o r even modes i n s i d e the s l a b i s : E  z  = A. cos k .X l xi  where A. i s a c o n s t a n t . l H  (2.23)  , ,  (2.24)  Therefore,  = - i y  Ixl < t ' 2  j  A. k . s i n k .X  —  o  i  xi  X  | |  1  X  < t 2  K  S u b s t i t u t i n g ( 2 . 2 3 ) i n t o ( 2 . 2 2 ) , we g e t : k = k ye — k k xx o -;z • yy 2  The f i e l d  z  i s a constant  o  (2.25)  2  o u t s i d e the s l a b can be w r i t t e n as E  where A  2  H  = A  e"  o  l '  "  X  t  /  2  )  ;  Ixl > t — 2  (2.26)  .  | | > t_  (2.27)  1  1  , and thus: = ^  7  k x o (  k  V  X  A  - xo(l l k  e  x  ~ t/2)  X  °°  2  where k xo 2  y  Q  and C  = k - k y y o o 2  (2.28)  2  e  o  a r e t h e p e r m e a b i l i t y and p e r m i t t i v i t y of t h e surrounding medium  Q  respectively  (without any l o s s o f g e n e r a l i t y , t o be y^ = E  k . and k can be p o s i t i v e q u a n t i t i e s s i n c e E > 1. xi xo z 2  of  2  k  2  Q  = 1).  Both  For p o s i t i v e values  , t h e f i e l d s decay e x p o n e n t i a l l y w i t h i n c r e a s i n g |x| o u t s i d e the s l a b .  Thus, t h e c o n d i t i o n f o r a guided mode i s k > 0. ° xo 2  22  The c o n t i n u i t y of E  z  and H  y  a t the i n t e r f a c e between the two media,  l e a d s to the f o l l o w i n g s e c u l a r e q u a t i o n : k  (k . t l xx  = k . tan xo xi  (2.29)  Odd,Guided TE modes:  In t h i s case, the s o l u t i o n s i n s i d e the s l a b a r e E (X) = B. s i n k .X XX  X  Z  (2.30) H  y  (X) =  ^  i  B. k . cos k .X i xi  2  o while outside E (X) = B e " z o H  7  (X) =  ^  k x  °  "  ( | x |  t  /  2  )  r l l x  - i B  o  k  (2.31)  > 1 2  - k o ( | x | - t/2) e" X k x o  xo  o The c o n t i n u i t y requirements eigenvalue  f o r the E^ and  components  l e a d to the  equation: k  k .t  = - k . cot xo xx  (2.32)  XX  We can get a f e e l i n g f o r the form of the s o l u t i o n s to the e i g e n v a l u e e q u a t i o n s by g r a p h i n g  them.  I t i s convenient  (2.25) and  to combine k .t xx I 2 J  and m u l t i p l y e q u a t i o n k  I  xo 2  k  +  xo  F o r i n s t a n c e , c o n s i d e r the even TE modes. (2.28) i n t o one e q u a t i o n ,  t  k t o  (y = 1) (2.33a)  (2.29) by t/2: t J  k .t  I2  tan  XX  j  k .t  I2 J XX  (2.33b)  (a)  (b)  ^b)  n=1 F i g u r e 11.  H  x  (a) and  n = 2  (b) as a f u n c t i o n of d i s t a n c e from  the c e n t r e of the s l a b .  24  Equation  (2.33b) determines one r e l a t i o n between fk .t] _  may be p l o t t e d on the  1-2 i s a c i r c l e of r a d i u s (e  fk t l xo 2  - l) ^  k t xo' , and 2  (Figure 10a). Equation  (2.33a)  The p o i n t s o f  d e f i n e the r e l a t i o n s h i p o f k . to k xi xo  i s the f a c t t h a t the two graphs w i l l  i n t e r s e c t a t a t l e a s t one p o i n t . have a low frequency  and  2  i n t e r s e c t i o n between the two curves i n t e r e s t i n g property  t xi 2  i n the same p l a n e .  2  Z  An  plane  JI J  X I  k  cutoff.  always  Thus, t h i s f i r s t symmetric mode w i l l n o t  F o r a mode to be guided,  k' must be g r e a t e r xo  than z e r o , and so we need o n l y c o n s i d e r those p o i n t s o f i n t e r s e c t i o n which l i e i n the upper p l a n e  o f the diagram, i.e.: 0 < xl 2 k  f c  < £ , 2  ir < x i 2 k  t  < -31 , ... ~ 2  or i n g e n e r a l , (n - 1) TT < k . < nir - - x i- —  (2.34)  where n i s an odd i n t e g e r . A g r a p h i c a l a n a l y s i s o f the odd modes ( F i g u r e 10b) r e v e a l s t h a t a l l modes have a c u t o f f f r e q u e n c y .  For guided modes ( i . e . k > 0 ) , we xo need o n l y c o n s i d e r p o i n t s o f i n t e r s e c t i o n which l i e i n the i n t e r v a l (n - 1) TT < k . < mr — - xx - — t t  (2.35)  where n i s a non-zero even i n t e g e r . To summarize, the a n i s o t r o p i c d i e l e c t r i c s l a b may s u s t a i n TE  n  (E = 0) modes, where n i s a non-rzero i n t e g e r t h a t can be viewed as t h e x  number o f a n t i n o d e s  i n E (X). z  e x a c t l y and the .eigenvalue  The boundary v a l u e problem can be s o l v e d  equations  a r e found t o be independent o f the  25  v a l u e s o f e and e . x y We w i l l use the s o l u t i o n s found here to see under what c o n d i t i o n s the  a p p l i c a t i o n o f OCB c o n d i t i o n s w i l l be a good a p p r o x i m a t i o n a t the s l a b  boundaries.  T h i s i s e a s i l y done by c o n s i d e r i n g the e q u a t i o n r e s u l t i n g  from combining  e q u a t i o n s (2.33a) and (2.33b):  k .t  2z  -tan  XX  fk t] 2 0 2  'k . t  2  XX  2  2  fk . t l 2  (2  XX  2  When e >> 1, z 2  k .t  I  XX  2  tan  U  k. . t  2  XX  J  T  X  2  z  }2 t  fk .t]  I  XX  2  where A i s the f r e e space wavelength. F o r the lowest mode, ° ' X and s o , i f the t h i c k n e s s t >> o , then  fk .t] XX  tan  2  <k  XX  .t)  »  (2.37)  J k  t x i < jr ; 2 = 2  1  2  x.e. k .t  ir 2  XX  2 Thus, k . = X  t »  1  (2.38)  TT_ ( t h e OCB approximation) w i l l be a good a p p r o x i m a t i o n i f , t  o / / i ~ and the boundary z  s u r f a c e w i l l c o i n c i d e w i t h a node o f the  1  s t a n d i n g wave i n s i d e the s l a b . of  the f i e l d  As the t h i c k n e s s i s reduced however, more  extends o u t s i d e the s l a b  (See f i g u r e 1 1 ) .  In a d d i t i o n ,  when a h i g h e r mode i s e x c i t e d , the f i e l d s w i l l n o t extend o u t s i d e the s l a b as much as f o r lower o r d e r ones, s i n c e a p p r o x i m a t e l y the same f r a c t i o n o f a wavelength  extends o u t s i d e the s l a b .  26  I n s i g h t as to the behaviour limit  ( t << ^ £ f )  c  e  a  n  be gained by examining k  and  of the s o l u t i o n s i n the o p p o s i t e  2  =  (k . + xi  1 ye  2  o  = k  - k . tan xi  2  2  y  (2.39)  (2.28) and  »  z  1, then from  (2.39) , k  - 0;  2  o  k . xx 2  If, i n addition,  k  (2.29):  k .t xx  2  These e q u a t i o n s a r e p l o t t e d as a f u n c t i o n of  e  (2.25)  kO y  z  the e q u a t i o n r e s u l t i n g from combining k  equation  (2.40) k .t XX  i n f i g u r e 12.  If  so (2.40) becomes  tan  2  k .t  (2.41)  XX  t x i << 1, then one can approximate (2.41) f u r t h e r by k .t xx 2  Putting this value f o r k ^  (2.42X  i n t o the e i g e n v a l u e e q u a t i o n  (2.39),  one  gets  2 -  2  y + k  (2.43)  ye The e s s e n t i a l d i f f e r e n c e between these e i g e n f r e q u e n c i e s and those f o r the OCB model when k.^  0 (see e q u a t i o n  found  ( 2 . 1 1 ) ) , i s the l i n e a r r a t h e r  than q u a d r a t i c dependence,on k^ when t i s s m a l l . The samples of TTF-TCNQ a v a i l a b l e f o r t h i s work had t h i c k n e s s e s c* of the o r d e r of 0.03 mm. (X  o  If e  ~ 10  = 1 cm), then X ^ = X //e~ ~ 0.03 eff o z c  3  a t a frequency of 30  GHz  Thus, c* ~ 0.1 X ^ and eff  cm.  c* a p p l i c a t i o n of the OCB  c o n d i t i o n s a l o n g the planes X = ± 7 p  should be  expected to be a.very poor a p p r o x i m a t i o n i n d e e d . The v a l u e s f o r " a " a r e t y p i c a l l y between 10c* and 50c* so t h a t 5 £ ^ f f ~ ^' T h 5 a l o n g the a  e  u s  Figure  13.  D i v i s i o n of the r e g i o n o u t s i d e the r e s o n a t o r  28  boundary planes p e r p e n d i c u l a r to " a " , t h e soundness o f the OCB r e s t r i c t i o n i s at best  marginal.  2.4  S o l u t i o n s f o r t h e D i e l e c t r i c Resonator i n the L i m i t b •+ °°.  The d i e l e c t r i c r e s o n a t o r o f f i n i t e width  " a " may be c o n c e i v e d as  b e i n g formed from the s e c t i o n abed of the s l a b shown i n f i g u r e 9a. I n o r d e r t o t r e a t t h e boundary v a l u e problem a t t h e newly formed s u r f a c e s ab and c d , one can r e q u i r e t h a t these s u r f a c e s c o i n c i d e w i t h of the magnetic f i e l d  (the OCB c o n d i t i o n ) , i . e . k y  where m i s an i n t e g e r ; which has proved  antinodes  = mir "7  however, a treatment  of t h e boundary c o n d i t i o n s  s u c c e s s f u l i n s i m i l a r problems i n t h e theory o f o p t i c a l  (22) waveguides  can be made by c o n s i d e r i n g the f o l l o w i n g :  D i v i d e t h e r e g i o n o u t s i d e the r e s o n a t o r i n t o the e i g h t r e g i o n s shown i n f i g u r e 13.  I f one r e l a x e s t h e requirement  that the f i e l d s  i n these r e g i o n s s a t i s f y exact boundary c o n d i t i o n s along the dashed l i n e s , then s i m p l i f i c a t i o n o f t h e boundary v a l u e problem can be achieved w i t h  the r e s u l t  t h a t f o r the fundamental mode f o r i n s t a n c e ,  E_(X,Y) may be w r i t t e n a s : , cos k .X v  E (X,Y) z  =  ^  cos k X xi e  -k o(|x| x  X  o  v  1  1  - k . ( Y l1 - c*/2) e Ju vo  :  cos k Y yx  - a/2)  |Y| > a 2 ;  ,  cos k .Y yi  „  ;  inside resonator  (2.44)  ,,  X  > c* 2  When these s o l u t i o n s a r e s u b s t i t u t e d i n t o t h e wave e q u a t i o n  (2.22), one  29  gets k k  2  = k . - k yi o  2  (2.45)  = k . - k xi o  (2.46)  2  xo 2  2  yo  2  Matching the f i e l d s at the boundaries of the c r y s t a l gives k  = k . tan x i * k  xo k  yo  X I  (2.47)  C  '  1  = k . tan r i " I yi |-£—  (2.48)  y  Higher order modes may also be solved f o r by writing a sinusoidal dependence i n either the x or y d i r e c t i o n (or both simultaneously) instead of the cosine function.  For these modes, the equations corresponding to  (2.47) and (2.48) w i l l contain the cotangent function. As k . ->• mjr , a l l the ^ a solutions reduce to those f o r the i n f i n i t e slab found i n the previous section. Mod_e_De_s^giiationj_ Unlike metallic c a v i t i e s , the f i e l d patterns of d i e l e c t r i c waveguides and c a v i t i e s are sensitive to £^ > the wavelength and the sample v  dimensions.  This complicates the problem of finding a reasonably  descriptive mode designation scheme. For rectangular metallic waveguides and c a v i t i e s , the accepted approach i s to designate the modes as TE (or H) and TM (or E) and to specify the number of f i e l d maxima i n the X, Y and Z directions with a t r i p l e subscript. When there are no variations, a 0 i s used. Since the rectangular d i e l e c t r i c cavity modes are neither pure TE nor TM, a d i f f e r e n t scheme should be used. here w i l l be based on the following f a c t :  The scheme to be adopted  i f the resonator i s considered  30  as a s e c t i o n of l e n g t h b of a d i e l e c t r i c waveguide, then i n the l i m i t a l a r g e aspect along one electric  ratio  ( i . e . b/c*  >> 1 ) , the e l e c t r i c  of the t r a n s v e r s e axes. field  i s p r i m a r i l y along  According  to t h i s  field  i s primarily  Modes w i l l be d e s i g n a t e d the  of  z E*  i f the  Jim  Z-axis.  scheme, the r e s o n a t o r modes p r e s e n t l y b e i n g 2  c o n s i d e r e d w i l l be denoted E„ where % and m are both non-zero i n t e g e r s . £m z z The  field  c o n f i g u r a t i o n s f o r the E ^  f i g u r e 9b and  f r e e space.  E^  modes are i l l u s t r a t e d  in  9c r e s p e c t i v e l y . 2.5  The  and  C o a x i a l Modes  solutions discussed  thus f a r have been f o r a r e s o n a t o r  in  In p r a c t i c e , the d i e l e c t r i c m a t e r i a l i s p l a c e d i n s i d e a  m e t a l l i c waveguide, so t h a t the e f f e c t of the waveguide w a l l s upon the resonances must be c o n s i d e r e d .  I f the f i e l d s decrease  significantly  w i t h i n a d i s t a n c e l e s s than the d i s t a n c e of the r e s o n a t o r  from the w a l l s ,  then a s m a l l p e r t u r b a t i o n of the f r e e space s o l u t i o n s i s expected; otherwise,  the e f f e c t s of the w a l l s must be c o n s i d e r e d .  One  can get  f e e l i n g f o r the q u a l i t a t i v e f e a t u r e s of the c o a x i a l system by an a n i s o t r o p i c c y l i n d r i c a l problem i s c o n s i d e r e d  rod, i n s i d e a c i r c u l a r  the E  field  lines),  modes which would otherwise The  considering  m e t a l l i c waveguide.  to f i e l d  the c o a x i a l l i n e may  configurations s u s t a i n low  r a d i a t e i f the waveguide were not  frequency  present.  f r e q u e n c i e s of these modes are h i g h l y s e n s i t i v e to the r a d i i of  dielectric  rod and waveguide.  of o b s e r v i n g  This  in'Appendix B where i t i s shown t h a t i n a d d i t i o n to  c o a x i a l h y b r i d modes (which w i l l correspond resembling  a  Hence, one must be aware of the  these modes as w e l l as d i e l e c t r i c modes.  the  possibility  31  CHAPTER I I I - THE  EXPERIMENT  3.1  The to be  experimental  introduced  Design and  Apparatus  apparatus was  designed  to a l l o w s i n g l e samples  i n t o a r e c t a n g u l a r waveguide ( e x c i t e d i n the dominant  mode) i n such a way  t h a t the c o u p l i n g between the r e s o n a t i n g  the waveguide f i e l d  c o u l d be a d j u s t e d  continuously during a run.  i l l u s t r a t e s the segment of the waveguide assembly b u i l t dewar system.  sample  and  Figure  The  14  first  s e c t i o n i n v o l v e d a s h o r t p i e c e of r e c t a n g u l a r waveguide f i t t e d w i t h a T h i s was  ^Q  to f i t i n s i d e a  The waveguide path c o n s i s t e d of t h r e e s e c t i o n s .  brass flange.  t e  small  p l a c e d through a h o l e i n the dewar cover p l a t e , and  the f l a n g e b o l t e d to i t . Next came a s e c t i o n of s t a i n l e s s s t e e l waveguide to reduce thermal c o n d u c t i v e  loss.  The  last  s e c t i o n was  a U-shaped p i e c e  of waveguide w i t h a r a d i u s of c u r v a t u r e chosen so as to permit d u c t i o n of the sample w h i l e The  sample was  a x i s ) a t the end  the waveguide assembly was  by a s t a i n l e s s s t e e l tube.  F i g u r e 15  b l o c k , housing  thermometer, which was  tubing.  The  i n the dewar.  and  of the waveguide.  guided  into position  shows a d e t a i l drawing of the copper  r e c t a n g u l a r base of the b l o c k was  f l a n g e a t the end  intro-  i n s e r t e d i n t o the waveguide (along i t s symmetry  of a s t a i n l e s s s t e e l r o d , which was  a heater  the  f i t t e d a t the end  of  b o l t e d to a r e c t a n g u l a r  A small hole  (dia. =  .044") wide  enough to a l l o w the c r y s t a l to pass through s a f e l y (but s m a l l enough to beyond c u t o f f f o r the frequency the b l o c k , c o n c e n t r i c w i t h the  bands used) was  be  bored through the base of  tube.  The mounting of the samples was through the s m a l l h o l e and  the  extension  conceived  i n t o the guide.  so as to permit The  passage  samples were  32  MICROMETER BLRAATSES P  O-RING  ROD  ST TE AEIN LESS S L WAVEGUD IE COPPER BLOCK 1EATER COILS THERMOMETER F i g u r e 14.  Waveguide subassembly designed to f i t He dewar.  into a  V  33  Figure 15.  Copper block. The metal wedge i s normally not present but can be used to t i l t the waveguide e l e c t r i c f i e l d with respect to the c r y s t a l axis (see section 4.1).  34  a t t a c h e d to a f u s e d q u a r t z f i b r e w i t h a s m a l l amount of was  used  to e l e c t r i c a l l y i s o l a t e the sample from the apparatus  to p r o v i d e the n e c e s s a r y as l i t t l e as p o s s i b l e . passed  epoxy.  Finally,  quartz  as w e l l as  e x t e n s i o n of the sample w h i l e p e r t u r b i n g the A s m a l l p a r t of the t i p of the q u a r t z was  through a m e t a l  position.  The  tube  (a 26 Gauge hypodermic needle) and  the n e e d l e  i t s e l f was  the end of the moveable r o d and was  fields  then  eppxied  into  placed into a hole centered at  h e l d t h e r e by means of a s m a l l screw.  The p o s i t i o n of the sample c o u l d hence be c o n t r o l l e d e x t e r n a l l y by moving the r o d . A Weinschel microwave sweep o s c i l l a t o r  (model 221), w i t h backward  wave o s c i l l a t o r p l u g - i n u n i t s i n the frequency ranges H1826) and  26 to 40 GHz  (model H2640), p r o v i d e d the RF  18 to 26 GHz source.  (model  This  v o l t a g e - c o n t r o l l e d microwave source i s s u i t a b l e f o r both swept and nuous wave measurements. used  A 0 to 20 v o l t s v a r i a b l e ramp generator  to sweep through any frequency range of i n t e r e s t .  w i t h a d i r e c t i o n a l coupler-power 460b) combination,  was  meter  (model 11517A) and  the d.c. output was  I f a v e r y slow v o l t a g e ramp was (Hewlett-Packard  419A  (General Microwave C o r p o r a t i o n model  ( F i g u r e 16).  the s h o r t e d end of the waveguide, was  D.C.  was  The power, l e v e l l e d  coupled i n t o the waveguide c i r c u i t  w i t h a 10 db d i r e c t i o n a l c o u p l e r  conti-  ( d e s c r i b e d above)  The power, r e f l e c t e d a t  d e t e c t e d w i t h a Hewlett-Packard then d i s p l a y e d on an  b e i n g used,  the output was  mixer  oscilloscope. first  amplified  n u l l v o l t m e t e r ) then, s i m u l t a n e o u s l y , d i s p l a y e d  on a s t r i p c h a r t r e c o r d e r and r e c o r d e d d i g i t a l l y on magnetic tape f o r computer  analysis. Most of the measurements were c a r r i e d out a t 4.2°K.  h i g h e r temperatures  were d e s i r e d , a c a l i b r a t e d carbon r e s i s t o r  However, when thermometer  A. D.C.  MULTIPLEX  MAGNETIC TAPE RECORDING UNIT  D.C. AMPLIFIER STRIP CHART RECORDER  0-20VOLTS  RAMP GENERATOR POWER METER  £4 BWO  LmJ  POWER HEAD  lm  nin pij  _  ISOLATOR  f  10 db DIRECTIONAL COUPLERS  Figure 16.  Waveguide assembly outside dewar and block diagrams of e l e c t r o n i c s .  36  and a 590, varnish insulated resistance wire heater were i n s t a l l e d i n the Cu block shown i n figure 15.  Temperature regulation was carried out with a  rate/proportional/integral temperature 3.2  controller.  Procedure  A major inconvenience with the experimental design i s that large standing waves are l i k e l y to occur i n the r e f l e c t e d power spectrum,  especially  at the higher frequencies. The reason for t h i s i s that a large percentage of the incident power i s r e f l e c t e d at the shorted end of the waveguide.  In  the microwave frequency range, the best i s o l a t o r s presently a v a i l a b l e have voltage-standing-wave-ratios i n the neighbourhood  of 1.25.  Even with them,  substantial standing waves w i l l then be set up, both at the source and detector end.  A way around t h i s d i f f i c u l t y i s to take a background scan,  i . e . a reflected power spectrum without the c r y s t a l i n the waveguide.  If  the r e f l e c t i o n curves with the sample i n are now divided by the background, a smooth curve should r e s u l t , with the sample absorption appearing as dips in t h i s normalized spectra.  Since the waveguide section shown i n figure 14  i s approximately 30 inches long, the thermal contraction may  change the RF  path length by a substantial f r a c t i o n of a quarter wavelength. background scan should be performed at each  Hence, the  temperature.  Any power absorption attributable to the presence of the sample i n the waveguide was i n i t i a l l y observed by performing a fast frequency sweep and looking at the r e f l e c t e d power signal on an oscilloscope. absorbtion of varying the coupling were then noted.  The e f f e c t s on the  Lineshape analysis  required knowing the normalized spectra.Doinga slow sweep and recording the  37  amplified mixer signal and sweep voltage d i g i t a l l y on magnetic tape, the data could be stored so as to be l a t e r analyzed by computer to give the normalized spectra. The voltage-frequency c a l i b r a t i o n of the R band (26 - 40 GHz)  unit  under l e v e l l e d conditions was acquired by mixing the signal (at a fixed voltage setting) with a 2 to 4 GHz signal from an a u x i l i a r y o s c i l l a t o r , whose frequency was measured with a frequency counter.  As the frequency  th of the a u x i l i a r y o s c i l l a t o r i s tuned closer to 1/10  .. of that of the output  signal, the lowest frequency harmonic generated by the mixer approaches zero.  This can be observed on an oscilloscope, and so, as long as the th  appropriate harmonic i s c o r r e c t l y i d e n t i f i e d (the 10 the signal frequency can be ascertained.  one i n t h i s case),  This c a l i b r a t i o n was  once and was p e r i o d i c a l l y checked with a frequency meter. dure was followed f o r c a l i b r a t i n g the K-band  performed  The same proce-  (18 to 26 GHz)  unit.  Using the U.B.C. IBM 370 computer, programs were written to read the data o f f the tapes and divide the background from the spectra.  Con-  version of sweep voltages to frequencies was b u i l t into the programs, so p that normalized power _r_ vs. frequency plots could be generated. P  o 3.3  Results i s shown i n figure lt?a for a sample of  A plot of P  o TTF-TCNQ of length b - .55cm at a temperature varying coupling strengths can be seen. peak value i s one.  of 4.2°K.  Four modes with  C r i t i c a l coupling occurs when the  If the modes can be i d e n t i f i e d , then their frequencies  38  SAMPLE 11  B  b = .55 cm  D  .104-  .05;  (a) b = .48 cm  06  B 044-  I  02  D oof  B  (b) b = .44 cm  064  04  D 024-  oo--^KvJ  30 Figure 17.  (c)  FREO.(G3H5z)  40  (1 - the normalized spectra) f o r one sample at three d i f f e r e n t lengths b at 4.2°K; a = .037 cm, c* * .002 cm.  39  can be used t o compute e of p ) •  ( n e g l e c t i n g any  p o s s i b l e frequency  A l t e r n a t i v e l y , a s i n g l e mode can be i s o l a t e d and  £  V  or more of the sample dimensions, a s e r i e s of resonant f u n c t i o n of the dimensions can be o b t a i n e d .  shortened  seen).  As the l e n g t h was  sample f o r t h r e e d i f f e r e n t v a l u e s o f b  by c u t t i n g o f f a segment w i t h a sharp b l a d e ;  the d i e l e c t r i c p r o p e r t i e s , due  one  f r e q u e n c i e s as a  to the i n e v i t a b l y induced  adjustshortened,  f r e q u e n c i e s of a l l the modes were seen to i n c r e a s e .  shows s p e c t r a of one was  by changing  The most c o n v e n i e n t l y  a b l e parameter i n TTF-TCNQ i s the b dimension. the resonant  dependence  Figure  17  (the c r y s t a l no  changes i n  s t r a i n s , were  These modes c o u l d be f o l l o w e d i n d i v i d u a l l y as a f u n c t i o n of l e n g t h ,  because of the d i f f e r e n t c o u p l i n g behaviour d i s p l a y e d by each mode. The  s t r e n g t h of the c o u p l i n g was  a n a l y s i s of the a b s o r p t i o n s .  I t can be  noted by p e r f o r m i n g  by  (18) R(w)  where u  o  i s the resonant  unloaded and guide,  =  jr P  c o u p l i n g Q's.  Then, one  +  '1  1 4  e x t r a c t e d by  - 1 Q,  %  1  0)  2  + 2  %  frequency  t h e r e i s no l o s s due  by  f 1  1 4  -  0)  -, 1  f i n d s from e q u a t i o n  3.1,  I  2 o  ji -  o  (3.1)  2  o  and  Q  0  radiation.  .  the same c o u p l i n g  o  and  Q a r e , r e s p e c t i v e l y , the c  For a d i e l e c t r i c r e s o n a t o r to  lineshape  shown t h a t the r e f l e c t e d power from  a c a v i t y f o r which power i s i n t r o d u c e d and hole i s given  a  +1 that  The  enclosed  observed Q,  i n a wave-  then,  i s given (3.2)  40  where t = 1 - R(m The  +  (-)  o  ) •  s i g n i s used when the mode i s under (over) c o u p l e d . There are u s u a l l y no  difficulties  i n coupling  to the lowest o r d e r  (19) modes  .  T h i s c o u p l i n g was  v a r i e d by  changing the sample p o s i t i o n a l o n g  the waveguide symmetry a x i s as w e l l as by r o t a t i n g the configuration By see  f o r a TE^Q  examining the for instance  field  waveguide near a s h o r t e d  patterns  t h a t the T E ^ ^ z  o r d e r s o l u t i o n modes a r e E ^ where the magnetic f i e l d the c o u p l i n g  f o r the T E  strength  1/Q  O M N  a s  and  a  E^^  i s shown i n f i g u r e  modes ( f i g u r e 8 ) , one  A,  3A/2,  Modes B,  C and  four  of mode A  t i c f e a t u r e of t h i s mode f o r a l l the samples examined was r o t a t i o n a l coupling  have the symmetry of the waveguide f i e l d  A  was characteris-  found  except f o r the A mode, f o r  o f t e n changed d r a s t i c a l l y w i t h s m a l l  angle (See  20).  shows  lowest  that i t did  symmetry was  which the c o u p l i n g  19  D showed a  g e n e r a l l y h i g h l y s e n s i t i v e to s l i g h t changes i n the p o s i t i o n .  figure  18.  can  . . . ) . Figure  f u n c t i o n of p o s i t i o n f o r the  p e r i o d i c b e h a v i o u r w i t h p o s i t i o n , whereas the c o u p l i n g  The  field  ) should c o u p l e s t r o n g l y a t a p o s i t i o n  modes, l a b e l l e d A through D i n f i g u r e 16.  have i t s maximum a t A/2.  The  and TE^,^ modes (the c o r r e s p o n d i n g second z  i s a maximum ( A/2, C  end  sample.  changes i n  not to  41  (c) Figure 18.  F i e l d configuration near the shorted end of a waveguide excited i n the dominant T E ^ Q mode. End (a) top and side (b) views are shown. The absolute value of the H - f i e l d i s shown i n ( c ) .  42  .4  X / 2  8  POSITION (cm) F i g u r e 19.  C o u p l i n g as a f u n c t i o n f o r modes A to D.  of p o s i t i o n i n the waveguide  F i g u r e 20.  C o u p l i n g as a f u n c t i o n f o r modes A to D.  of r o t a t i o n i n the waveguide  44  CHAPTER IV - INTERPRETATION OF THE RESULTS  4.1  Mode I d e n t i f i c a t i o n  For a long time, one of the obstacles preventing the i n t e r p r e tation of the experimental r e s u l t s had been a correct mode assignment to the observed resonances.  A l l d i e l e c t r i c mode assignments to the modes  A to D were inconsistent with the observed coupling symmetries.  As an  example, a glance at f i g u r e 19 reveals that modes A and B (the most strongly coupled) couple weakly at A/2 and A/4 r e s p e c t i v e l y . The r o t a t i o n a l symmetry (Figure 20) indicates that these cannot be TM  resonator  modes (For TM modes, the f i e l d configurations for the E and H f i e l d s are interchanged.  This implies that with respect to the angle 6 defined i n  figure 20, the maximum coupling would occur at 0 = 90° and 270°).  However,  the assignment T E Q ^ for the A mode was not consistent with the observed p o s i t i o n a l coupling, whereas the B mode had a l l the symptoms of being This dilemma was  TE^^.  f i n a l l y resolved when i t was noticed that the  A mode persisted up to room temperature, where TTF-TCNQ i s known to be i n a conducting regime (e*/e^ >> 1); plot of 1/Q,  however, the other modes did not.  A  which i s proportional to the d i e l e c t r i c l o s s , i s shown for  the low temperature regime i n figure 21.  For both modes, the Q decreases  as the temperature increases u n t i l above about 25°K.  Above t h i s temperature.  only the A mode can be seen (although only very f a i n t l y ) , , Q values and resonant frequencies are shown i n figure 22 .  The Q measurements of the  (6)  coaxial resonances reported by Hardy et a l dependence down to about 30°K.  showed a similar temperature  SAMPLE 9 • A mode : f o B mode : f  34.6 GHz; 39.0 GHz;  Q(24°K) = 480 Q(24°K) - 1100  o o o o O  o  OO o  o  .01' .00  .08  .16  T Figure 21.  '•• mom*  .24  46  5000+  500  50+  T(°K)  T(°K) F i g u r e 22.  Q v a l u e s and  resonant  f r e q u e n c i e s f o r the A mode.  47  It thus appeared then that the A mode was a coaxial mode, of the f i n i t e d i e l e c t r i c inside a rectangular waveguide.  As the  temperature i s lowered below the metal-semiconducting t r a n s i t i o n region, the sample goes from a large value of z '/e' 1  b  b  (H  ^' b 7ra  u e  b-axis conductivity) to a small value.  where a, i s the b  —;—  b  The f i e l d configurations of the  coaxial mode for these two cases are shown i n figure 32 of Appendix B. Note that these f i e l d patterns are orthogonal to those of the empty waveguide. Two other facts confirmed that this mode was indeed a coaxial resonance:  1.  The resonant frequency changed substantially as the e f f e c t i v e diameter of the outer conductor was altered.  For instance, as the sample  was withdrawn into the copper block, the frequency rose sharply u n t i l i t was out of the band being examined.  Also, when an oversized wave-  guide was used, the frequency dropped s l i g h t l y .  Only very small  changes i n the frequencies of the other modes, due to perturbations caused by the walls, were seen.  2.  The mode overcoupled strongly when a metallic wedge was  inserted  into the waveguide near the shorted end, as i n figure 15.  This  should be expected, since the wedge w i l l change the waveguide f i e l d so that i t i s no longer orthogonal to the coaxial mode.  48  Once the A mode had been i d e n t i f i e d , assignments consistent with the coupling data could then be made. modes are T E  Q 1 1  ,  TE  021  a n d  TE  031  r e s  P  Within the OCB model, the B, C and D  e c t l v e l  y•  T h e  TE  021  l s  o  n  l  y  e x  P  e c t e d  to couple through asymmetries i n the waveguide f i e l d , and indeed, this coupling i s consistently found to be very weak.  In terms of the second z  order solutions, the respective assignments are  4.2  z  , E-^ and E-^.  The Real and Imaginary Parts of e  A mode plot ( f vs 1/b ) 2  shown i n figure 23.  z  2  t y p i c a l of a l l the samples analyzed i s  Because this particular sample was  i n i t i a l l y quite  long (b = .876 cm), several modes (each represented by a point) can be seen, of which four, l a b e l l e d A - D, showed the same coupling characterist i c s discussed i n previous sections. Longer samples were chosen when i t became apparent that the OCB model was  inappropriate f o r t h i s material  and that the a l t e r n a t i v e theory developed was only v a l i d i n the l i m i t b +  00  . To see that the OCB model i s l i k e l y to be poor, consider what  the theory predicts for the mode p l o t .  According to equation (2.13), the  frequency of the TE modes i s given by ^ omn J  f  = c  2  2  £  b  • -1 2 1 m — a  n b.  (4.1)  Accordingly, the intercepts of the mode plots should follow the quadratic sequence 1, 4, 9 ( i . e . m proportional to nj  2  for m = 1, 2, 3), and the slopes should be  SAMPLE 14  F i g u r e 23.  Mode p l o t  showing s e v e r a l modes o f which f o u r c o u l d be i d e n t i f i e d  50  I t i s c l e a r from the e x p e r i m e n t a l mode p l o t t h a t the do not f o l l o w such a sequence.  I f one uses e q u a t i o n  frequency dependence t h i s would imply f o r e', one b e'(15 GHz) = 390 b e'(18  GHz)  =  1100  e'(21 GHz) b  =  1700  D  (4.1)  intercepts  to see what  finds  Such a huge d i s p e r s i o n would n e c e s s a r i l y i n v o l v e l a r g e v a l u e s of which a r e not seen,and t h e r e f o r e the OCB p r i a t e f o r these samples  (as was  anticipated  assumptions a r e c l e a r l y  i n the p r e v i o u s c h a p t e r ) .  o r i g i n of the n e a r l y l i n e a r r a t h e r than q u a d r a t i c sequence of the mode p l o t  inapproThe  observed  i n t e r c e p t s can be e x p l a i n e d i n terms of the exact s o l u t i o n s f o r  the i n f i n i t e s l a b i n the l i m i t of s m a l l t h i c k n e s s . where c* <<  X^/Ze /, the e i g e n v a l u e e q u a t i o n 7  R e c a l l t h a t i n the  (2.43) i s found  to  limit  be:  When k (4.2)  The  two  terms i n the square b r a c k e t s w i l l be equal when a _ mir c* 2 =  Thus, when a/c* >> will  1 and  f o r s m a l l mode index i n t e g e r m,  the l i n e a r  term  dominate. The f r e q u e n c y dependence of  d i s c u s s e d i n s e c t i o n 2.4  o b t a i n e d by u s i n g the improved model  i s shown i n f i g u r e 24.  S i n c e the t h e o r y a p p l i e s  Figure 24.  Frequency dependence of e£ obtained from the model discussed i n s e c t i o n 2.4. The OCB r e s u l t s f o r sample 14 are a l s o shown f o r comparison.  52  TABLE I - SUMMARY OF NUMERICAL RESULTS FOR THE E? MODES lm  Sample 5: a = .045 cm c*= .0020 cm ^ampl_e_ll_: a = .037 cm c*= .0023 cm  J3ampl_e_12: a = .048 cm c*= .0048 cm  Sample 14: a = .050 cm c*= .0036 cm  i  m  f(GHz)  1  ^x(cm ^  6  19.7  199.0  0.18  57.3  -0.18  2520  3  25.7  349.8  0.31  187.6  -0.10  5440  1  20.3  233.5  0.18  69.3  -0.18  3300  2  26.9  335.5  0.26  147.4  -0.13  4240  3  33.4  410.2  0.32  227.4  -0.11  4480  1  16.9  143.5  0.22  51.3  -0.23  1858  2  20.3  206.2  0.31  110.8  -0.16  3046  3  25.3  251.2  0.38  172.0  -0.13  3301  1  15.5  166.2  0.19  50.9  -0.19  2871  2  18.4  238.8  0.27  108.6  -0.14  4615  3  21.9  291.9  0.33  167.6  -0.11  5380  X  ky(cin]  53  only to the b ->• °° limit, the eigenfrequencies are taken from the extrapolated intercept points in the mode plots (Figure 23).  Table I summarizes  the numerical solutions of equations (2.44) to (2.48) for four samples analyzed in detail.  The degree to which the OCB mode assignment TE , oml (for m = 1, 2, 3) is inappropriate, is indicated by defining parameters  6  x  and 6 : y  k = fm + 6 ) ir y ^ yJ _ (In the OCB model,6 and 6 are zero). x y  The values for 6 and <S , found x y'  from the second order solutions, are also tabulated. Increases  As the .mode: index  6^ -> 0, indicating that the OCB approximation improves  on the planes Y = ± a/2.  However, <S -»• 1, which shows that the assumption  that I = 0 (although never good) gets worse. The second order solutions result in a higher value of e(, as b  well as a milder frequency dependence.  The discrepancy in the results for  the four samples shown can be attributed to: 1.  Non-uniformity of the c* dimension. The solutions are highly sensitive to the value of c* when a/c* >> 1, as can be seen qualitatively from equation (4.2 •).  Since variations in  the thickness along the length of a crystal can be substantial (typically 10 £ a_ ^ 50), only an effective thickness, based on the average along c* any given sample, can be obtained.  (This was calculated by measuring  under a microscope the dimensions of the segments trimmed from a given crystal and averaging the values of c*).  54  2.  Variation in the dielectric properties of the samples.  3.  The approximate nature of the solutions to the dielectric resonator problem.  4.  The assumption that the extrapolated values for the resonant frequencies (obtained from the mode plots) correspond to the actual frequency in the limit b -> «>. There are two reasons to worry about this last point.  One is the  problem of end-effects associated with the finite resonator, the other is the effect of the waveguide walls upon the resonant frequencies.  The f i r s t  point- is considered for the coaxial mode in AppeiidixMB. The second point can be studied quantitatively for the TM coaxial mode (mode A), by applying the solutions derived in Appendix B.  When the fields penetrate the sample,  neglecting the geometrical correction for the noncylindrical shape of the (23) sample  , an effective radius R^ can be defined in such a way that the  cross-sectional area of the effective cylinder equals the cross-sectional area of the samples R  ! = / * * a  /  c  TT  Since the skin depth of the metallic rectangular waveguide i s very small, the effective outer radius R„ w i l l be taken as  where  and  axe the waveguide dimensions.  For sample 14, R^ = .073 mm  while for the R-band waveguide, R„ = 3.4 mm. Hence,  55  Taking e£ = 3000 and ej_= 5, then the numerical s o l u t i o n f o r the value of the intercept i s found to be f = 2 GHz. Experimentally (See f i g u r e 23) the intercept occurs a t f = 11.6 GHz. The discrepancy between these two numbers i s most l i k e l y due to the neglect of geometric c o r r e c t i o n s as w e l l as to end-effects (discussed i n Appendix B ) . These w a l l and end e f f e c t s have not been s t u d i e d f o r t h e  d i e l e c t r i c resonator modes B, C, D.  I t might be suggested that since these  modes have the same slope as mode A i n the mode p l o t s , that they are i n f a c t also hybrid c o a x i a l modes. This i s undoubtedly t r u e , however i t has been shown  (24)  that the resonant frequencies of hybrid c o a x i a l modes are  h i g h l y i n s e n s i t i v e t o t h e v a l u e o f R^.  T h i s i s because one e x p e c t s t h e  f i e l d s t o decay e x p o n e n t i a l l y i n the immediate v i c i n i t y o f t h e r e s o n a t o r and  thus, t h e i n t e r a c t i o n w i t h the waveguide w a l l s w i l l be s m a l l .  observed e x p e r i m e n t a l l y frequency  This i s  i n t h e f a c t t h a t as t h e sample i s withdrawn, t h e  o f t h e d i e l e c t r i c modes changes v e r y l i t t l e  f o r a l l but t h e lowest  z mode. to  Thus, t h e s o l u t i o n s found f o r the E ^  be s t r o n g l y p e r t u r b e d  m  modes should n o t be expected  by the waveguide.  The Q's o f t h e modes g i v e i n f o r m a t i o n on t h e imaginary T h i s i s most e x p e d i e n t l y o b t a i n e d  by making t h e s u b s t i t u t i o n s :  and o  -*• to  i n the eigenfrequency equations.  o  o 2Q  One gets  p a r t o f e^.  57  Figure 25 shows the Q values as a f u n c t i o n of 1/b f o r a B and 2  D mode (The C mode was too weakly coupled to measure i t s Q). From the i n t e r c e p t s one gets: Q(16.9 GHz) = 200 ± 50 Q(25.3 GHz) = 270 ± 50 Using the values of  found f o r the sample (sample 12), one consequently  finds e£(16.9 GHz) = 17 ± 4 e"(25.3 GHz) = 16 ± 4 D  Thus, to w i t h i n the accuracy that the intercept values can be ascertained no frequency dependence can be seen.  58  CHAPTER V - CONCLUSIONS  The complex d i e l e c t r i c constant of TTF-TCNQ a l o n g the b - a x i s between 15 GHz technique.  and 25 GHz,  has been measured u s i n g the d i e l e c t r i c  For f o u r samples a n a l y z e d i n d e t a i l a t 4.2°K, the  v a l u e of the r e a l p a r t  i s found to be 4200;  resonator  average  a f r e q u e n c y dependence  i s i n d i c a t e d , however i t i s not c l e a r whether t h i s i s i n t r i n s i c  to the  samples or a r e s u l t of the approximate n a t u r e of the t h e o r y of d i e l e c t r i c resonators, from 17  A v a l u e of  - 25 GHz.  = 16  ± 4 was  found, w i t h no f r e q u e n c y dependence  T h i s l a t t e r r e s u l t i m p l i e s a f r e q u e n c y dependence of  the c o n d u c t i v i t y s i n c e  =  , hence 03  CT(OJ)  a  u  where  a(16.9 GHz) = .16 ± .04 (fi-cm)""  1  a(25.3 GHz) = .23 ± .06 (ft-cm)" The  temperature  dependence of the Q's  i n d i c a t e s t h a t the c o n d u c t i v i t y i s  t h e r m a l l y a c t i v a t e d f o r T between 15°K and 25°K; temperature  1  f o r i n s t a n c e , the  dependence of the Q shown i n f i g u r e 21 g i v e s an  energy of 79°K.  Two  activation  o t h e r samples a n a l y z e d gave a c t i v a t i o n e n e r g i e s of  90°K and 105°K r e s p e c t i v e l y . These a c t i v a t e d e n e r g i e s a r e c o n s i d e r a b l y lower  230°K found from d.c. m e a s u r e m e n t s ^ . t i v i t y a t 4.2°K i s over two  than the v a l u e of  In a d d i t i o n , the microwave conduc-  o r d e r s of magnitude h i g h e r than the  d.c.  conductivity. These f a c t s can be understood of s t a t e s e x i s t s i n the semiconducting  by p r o p o s i n g t h a t a f i n i t e d e n s i t y gap near the Fermi energy.  Such a  59  s i t u a t i o n might Be r e a l i z e d p h y s i c a l l y by the presence of s t r u c t u r a l disorder  or i m p u r i t i e s .  I f the s t a t e s a r e s t r o n g l y l o c a l i z e d and  electron c o r r e l a t i o n s are strong,  then i n the gap r e g i o n ,  electron-  the d e n s i t y of  s t a t e s f o r a one^-dimensional system would be as i l l u s t r a t e d  i n figure 26.  I f the s t a t e s near the F e r m i energy a r e s t r o n g l y l o c a l i z e d , t h e n t h e r e w i l l be two main c o n t r i b u t i o n s  to the c o n d u c t i v i t y :  1.  Thermal a c t i v a t i o n a c r o s s the semiconducting gap A^ = 2A  2.  T h e r m a l l y induced hopping from an o c c u p i e d l o c a l i z e d s t a t e t o an empty localized state.  The energy r e q u i r e d w i l l be denoted by  The c o n t r i b u t i o n from t h e s e mechanisms can be w r i t t e n  a = Generally  A  1  »  A  2  and  e  -A-i/kT 1  +  e  L  L^.  as  -Ao/kT z  a ^  »  The temperature dependence of t h i s c o n d u c t i v i t y  i s sketched i n f i g u r e 2 7 .  In p r a c t i c e , the a c t u a l d e t a i l s of a(T) w i l l depend on the temperature dependence of A^ and L^.  However, except a t v e r y low temperature, A^ s h o u l d  be independent of temperature The f r e q u e n c y dependence of the c o n d u c t i v i t y due to hopping  has  (29) been shown by Mott  to have the form  g (OJ)  a  =  a(to) - a ,  a |C i  where v ^ i s a parameter dependent interaction.  When the compensation  a  <o  In  Q•C •  upon the s t r e n g t h  10  o f the e l e c t r o n - p h o n o n  i s v e r y low, S = 1.  For doped  0 8 w i t h v e r y lowicompensaf i p n ^ „for instance^taneto -,. 'a / dependence was &  silicon (28) found i n the  f i g u r e 27.  Temperature dependence of the c o n d u c t i v i t y when hopping c o n d u c t i o n i s i n c l u d e d .  61  neighbourhood of 10 KHz, which was attributed to a hopping conduction mechanism. In TTF-TCNQ, a finite density of states in the gap between the valence and conduction bands might be expected to arise from structural (27)  disorder or from impurities.  Eldridge  has recently found evidence of  photoconductivity from small amounts of impurities in TTF-TCNQ. is conceivable that this is also giving rise to the  Thus, i t  observed frequency  and temperature dependence of the conductivity. Although the high values of  remain difficult to understand b  on the basis of conventional single particle scattering alone, the transport properties of many dielectric materials^ 2 ^ SrTiOg (e' ~ 1500 at 60°K)  (e.g.  ~ ^-00)  ,  ) have been successfully explained as being  due to small polaron formation.  Thus, i t is conceivable that alternatives  to the theory of a pinned Frohlich mode could explain the experimental results. When this project was first undertaken, i t was hoped that a viable theory for finite anisotropic resonators could be developed, so that any given mode could be followed as the length b was shortened, thus giving the frequency dependence of e£ over as wide a frequency range as desired.  At present, the theory is only reliable in the limit b •> 0 0 .  Although measurements at several lengths must be taken in order to get the intercept frequency, only one data point per mode can be interpreted reliably.  The frequency of this point is a function of the width "a".  One must be able then to work with samples with variable "a" (as well as "b") dimension in order to extend the present range to higher frequencies.  62  T h i s would be d i f f i c u l t  t o do w i t h TTF-TCNQ because t h e a v a i l a b l e  have a £ .5 mm and one must work w i t h l e n g t h s b ~ 10 mm. t h a t f o r such h i g h l y a n i s o t r o p i c m a t e r i a l s , of t h e d i e l e c t r i c a better  constants using  I t thus appears  further conclusive  d i e l e c t r i c mode a n a l y s i s  t h e o r e t i c a l u n d e r s t a n d i n g o f t h e modes.  samples  studies  should await  63  APPENDIX A :  THE ANISOTROPIC DIELECTRIC SLAB  The nature of the solutions for an anisotropic slab of thickness t and d i e l e c t r i c tensor given by (1.7) w i l l be summarized.  Only TE(E  =0)  modes w i l l be considered, although an analogous treatment can be made f o r the TM(H = 0) modes. x  Assuming a dependence along the Y" and Z d i r e c t i o n s of the form g  i ( k y Y + k Z) ^ ^ z  t  e n  e  i E  e c  x  tric  f i e l d i s written:  = 0  E  - A (X) e y y  E  = A (X) e  i(k Y + y  1 ( k  y  Y  +  k Z) z  (A.l)  k z Z )  where A^(X) and A^(X) are functions to be determined.  Substituting into  (2.7), one gets the coupled d i f f e r e n t i a l equations: A  +  H  A  =  (A. 2)  0_  where the primes indicate d i f f e r e n t i a t i o n with respect to X, A (X) y  (A.3) A (X) z  and k V  o  y  -  k  k k y z  2  z  M k  2  y z k  2  o  E  to —2" 2  If one takes A  (A.4)  k ue - k o z y  k  2  « e^" ^ , then after s u b s t i t u t i o n into a  X  y,z  (A.2), the secular equation for non-zero solutions gives two values f o r a,  64  k y ( e +e ) - (k +k ) ± 2  a  i(±)  =  \  o  y  z  2  2  y  z  k u(e -e ) + ( k - k ) 2  2  o  y  z  y  + 4k k  2  2  z  2  y z  S i n c e R i s symmetric, i t may be d i a g o n a l i z e d by an o r t h o g o n a l t r a n s f o r m a tion, associated with a r o t a t i o n  (by some a n g l e 0^, say) i n t h e Y-Z  plane,i.e. i f cos 9  sin 9  -sin 0  cos 6  then '!(+) R  M R  = 0  'i(-)  One f i n d s t h a t t h e a n g l e 0^ i s g i v e n by  . = 1sin"  0  2  1  k k  1  (A.5)  y z  2 (k u(e -e ) + ( k - k ) ) 2  2  o  Thus, t h e g e n e r a l s o l u t i o n s f o r A  y  and A  y  z  2  y  z  J  are  z  f  •  A (X)  cos  A (X)  -sin  y  sin 0  C  cos 0 ~ " " i  C  e  ( + )  l  a  i( ) +  X  (A.6) z  'i  w  l a i  (-)  e  <-)  X  In an analogous way, the s o l u t i o n s o u t s i d e t h e waveguide a r e E E E  where  x  = 0  = B (X) e y y z  = B (X) e z  1 ( k  1  (  y  Y  V  +  +  k  z k  Z  )  z > Z  (A.7)  65  f  •  B (X)  cos 9 o  y  B (X)  -sin 9 o  z  J  V.  a  k  2  o(--)  k  c  V) V)  cos 0 oj  6  (A. 8) e l a  ° (  ) X  - (k + k ) y z  2  2  0  o(+)  ia (+)x  sin 9 o  (A. 9)  2  (A.10)  2  o  and 9 = 1 sin" ° 2  k k  1  2(k  .y  (A.11)  z  - k ) y z 2  2  For well guided modes, a , < 0. - B . ' o(-) 2  s  However, a 2,,s > 0 and so the o(+)  f i e l d s contain propagating as well as damped contributions.  The r e l a t i v e  strength of these two contributions ( i . e . D^/D^_^) can be found from matching the solutions at the boundary.  After some very tedious  the boundary value problem gives the following secular  a  /,va  / vSin  o(+) o(-)  o(+)  +  f  l i(-) o(+)  g  a  a  S i n  a  K(+) I 2  i(+) i(-) a  t  tan J  f  "  g  2 i(-) o(-) a  a  tan  o(+) 2  C O S  \  a  i(2  f a K-  2  A  -  -A  g  2  - cos  2  = sin  2  {  l i(+) o(-)  8  2 i(+) o(+)  a  J  2  a  C O S  faK+)t] 2  tan  a  a  s i n  tan  J  where ±  tan  tan  g  =  g  equation:  •  V,  J  o(+) 2  c o s  algebra,  (9 ±  eQ)  (6 - 9 ) ±  Q  0  (A.12)  66  The complexity of the solutions arises from the anisotropy in the Y-Z (21) plane. Okaya and Barash show that the solutions are considerably simpler when the system is uniaxial, i . e . e = e . In this case, y z D^/D^  = 0 and so the mode is well guided.  case, this does not occur;  However, in the more general  in fact, i t can be easily shown that D ^ / D ^  gets larger as the anisotropy increases. In the case of high anisotropy (e  >> e ), the secular equation z y  (A.12) can be simplified considerably by noting that, to f i r s t order, (See equation 2.11): k2 + k2 > e v k2 + k2 = e k2 > k2 y z — y z y o o hence, 2  °(+> o(-) Thus, taking c t ^ 2  a  a  i(+)ai(-)tan V  k2  k2.  o „ y  o  V  -(k 2 + k 2 ) * y z  z'  - 0, (A.12) becomes:  i(+) t " tan 2  0  2  J  'o(-)  8  t t lai(+)tan ° i(+)  «  + g2ai(_)tan  0  The problem is now tractable by numerical methods.  (A.13) However, attempting  to solve for the dielectric constants is s t i l l very d i f f i c u l t .  As far as  the author knows, the only published numerical solutions to this equation have been for given values of e and e , where the parameter a . , N was then ° y z l(-)  67  found f o r comparison with a f i r s t order theory When no z-dependence of the f i e l d s i s taken ( i . e . k  -> 0), z then 9 , 8. -> 0, and the f i e l d s reduce to those found i n section 2.3 o i namely equations (2.20), (2.21) and (2.23); to (2.29).  equation (A.13) reduces  68  APPENDIX B  THE INFINITE COAXIAL LINE  The modes o f an i n f i n i t e c y l i n d r i c a l d i e l e c t r i c r o d o f r a d i u s R. i n a c i r c u l a r m e t a l l i c waveguide o f r a d i u s £^ = e  y  can be s o l v e d f o r when  = ej_(See f i g u r e 28). The equations  f o r the f i e l d s i n r e c t a n g u l a r  c o o r d i n a t e s g i v e n by (2.14 ) t o (2.19 ) can e a s i l y be transformed c y l i n d r i c a l coordinates  ( r , <j>, z) by u s i n g the t r a n s f o r m a t i o n s : (j)  s i n cb  F  - s i n <j>  cos <j>  F  3f 3r  cos <j>  s i n <j>  3f_ 3x  3f 3c|>  - r s i n <j>  r c o s <j>  3f 3y  F  COS  r  I  to  J  X  y  and  >  where F and f a r e a v e c t o r and s c a l a r r e s p e c t i v e l y . the  dielectric:  E  r  \ H  =  i  =  k  'c5H  r  T  3r k  i  r  1- c ^  where  CO  The f i e l d s a r e expressed  2  z  z 3<|)  3H toy_ c  COE^  z  3r  r  z 3<t>  (B.2)  )  z 3<|>  3H  (B.3)  J 3E  to c  y _ - k  (B.4)  3r (B.5)  2  £j  and H .  z  z  inside (B.l)  z  3E  X  c  i n terms o f E  to solving f o r E  coordinates i s  toy c  +  3r  z  SH  1 r 1  z  z  'k 3E z z r 3tb  i  = k  =  i s reduced  3E  One f i n d s t h a t  and H , hence, a complete s o l u t i o n  z  Equation  (2.7  ) for E  z  i n cylindrical  F i g u r e 28,  C y l i n d r i c a l geometry o f a d i e l e c t r i c rod i n a c i r c u l a r waveguide.  70  9E  i 1_ r  2  9r'  z  +  3E z 9r  9E 1 z r " g^ 2  2  2  , ik E + z z The equation f o r 9H 2  9r  2  z  +  a)  1 — r  lk z  ye  2  — T  E r 1 ffjfe. r + -— + — — 9r r 9<j> 9  E  - k z 2  z  E  +  =  z  0  (B.6)  i s e a s i l y shown to be . 1  9H z  r  9r  +  1  .  9H z 2  ye  0)  —>r  +  - k z  H  2  z  2  =  0  (B.7)  r " "g^ "" 2  2  Writing the s o l u t i o n s i n the form E ( r , <(,) = F(r) z H (r, z  <fr) =c  e  G(r) e  where n i s an i n t e g e r , then equations (B.6) and (B.7) become 9F 9r  r  9G 9r  1 9G r 9r  2  z  9F . 9r  1  F  =  0  (B.8)  - n G r^  =  0  (B.9)  k? i r  z  and 2  2  where  k  2  2  1  ,2  1 =  e  —•  /  o  (B.10)  ^  The s o l u t i o n s to these Bessel equations which are f i n i t e a t r = 0 are: F(r) = A  i  J (k r ) n l  (B.ll)  2  J (k r)  (B.12)  G(r) = A where A^ and  are constants.  n  ±  Outside the rod, where the medium i s  i s o t r o p i c , the s o l u t i o n s are w r i t t e n as F ( r ) = B. H ^ ( k . r ) + C. H * (k_r) 1 n l 1 n 2  (B.13)  G(r) = B  (B.14)  (  (  H ^(k r) + C 2  where B , B„, C.. and C„ are constants; 1 2 1 2 +  the f i r s t and second k i n d ;  (k r)  (  2  and  n  }  2  2  and  n  are Hankel functions of  71  k  (B.15)  = OJ y e - k Z —y O O Z 2  2  2  c I t i s c l e a r from equations (B.l) to (B.4) that i f the f i e l d s have z  any angular dependence ( i . e . n > 0) , the s o l u t i o n s cannot be separated into the usual TE and TM types.  Such modes, c a l l e d h y b r i d , w i l l not be  solved f o r , here. When n = 0, the TM (H = 0 ) modes have the f o l l o w i n g f i e l d s : z  E, = H = H = 0 <j> r z E  =  k  i  r  55  *  kT  3 E  k e, o J.  z z  - —  3r  where  E (r) z  = ^  5 r<R.  i1 „< o ii>  A  J  k  r  B, H ^(k„r) + C. H ( k . r ) i o z i o z (  ( 2 )  ;R  1<r  <R,  (B.16)  Before proceeding to solve the boundary value problem, some i n s i g h t i n t o the behaviour of the s o l u t i o n s w i l l be sought. In the l i m i t R ->  ( i n which case a c y l i n d r i c a l d i e l e c t r i c  00  2  waveguide i s obtained), a guided wave can only propagate i f k i s because the asymptotic values of the Hankel f u n c t i o n s are  H  (i) o  (  w  )  H >(w) o (2  Therefore i f  i(w - 1/4TT  - AT v irw fl_  e  TTW  = i y where y e Re, then H^dyr) - e^  r  •i(w - 1/4TT  )  )  2  < 0.  This  72  In a d d i t i o n , i n order to s a t i s f y the boundary value problem, one must have k  (and hence k  2  ) > 0.  2  Thus from (B.5) and (B.15), we have the c o n d i t i o n 1  <  OJ  -  —9-  < 1  2  -  where the l e f t and r i g h t hand equal signs apply i n the short and long wavelength l i m i t s r e s p e c t i v e l y . ( S e e f i g u r e 29).  In contrast to m e t a l l i c  c a v i t i e s , the (w/c R^f vs O^R-^f c h a r a c t e r i s t i c s do not terminate at (k R . ) Z  2  = 0 but have a low frequency cutoff when k  J-  ->• 0, i . e . when k  2  A high frequency c u t o f f a l s o occurs when k  -»• 0, i . e . when k  2  to/c. z  Z>  fe^ u/c.  For f i n i t e radius..'Rj,"^ may be p o s i t i v e ; a s . w e l l as negative. no low frequency cutoff, c o n d i t i o n e x i s t s - i n this-case" the s o l u t i o n s J use than  and Y  o  and  o  and C, are that E at r =  2  > 0,  to the Bessel equation are more convenient to ^  o .  o  When k  Thus  The boundary conditions which determine A- , B, 1 1 J  and H  can be continuous at r = R, and that E_ be zero  1  L A  > k  i  l o lV " l o 2 l " l V W J  (k  B  ^i V W B  J  k  ( k  R  (k  f  +  l o 2 2>  J  +  B  :  C  l o Y  ( k  R  C  )  l i< 2 l J  k  2 2 R  )  R  "  )  +  C  l  (B.18)  = 0  Y  l  (  k  2 l R  )  = 0  (B.19)  (B.20)  0  A non-zero s o l u t i o n e x i s t s i f the determinant of the c o e f f i c i e n t matrix vanishes.  This gives r i s e to the equation:  J  £z k  l  l< 2 l> k  R  Y  o  j (kjl^)  ( k  2 2 R  )  " o 2 2> J  ( k  R  VW  (B.21)  = 0  F i g u r e 2 9 . Mode p l o t f o r a u n i a x i a l c y l i n d r i c a l waveguide.  00  .01  .02  03  .04  dielectric  .05  F i g u r e 3 0 . Mode p l o t f o r t h e TM^Q c o a x i a l mode f o r v a r i o u s v a l u e s o f R /R,, when e  *= 1 0 and  0  Z.  L  Z  e,= 2. -L  75  The q u a l i t a t i v e features of these s o l u t i o n s have been discussed by Hardy et a l .  f o r the case of an a n i s o t r o p i c centre conductor.  The features  relevant to the present problem can be seen by s o l v i n g equations (B.15) and (B.21) simultaneously f o r d i f f e r e n t values of 2 ^ l ' ± R  R  e  (B.10), anc  *z" E  Figures 30 and 31 show numerical r e s u l t s f o r k H, << 1 which w i l l be the X  Z  region appropriate f o r the resonances observed i n TTF-TCNQ. An e f f e c t of the outer m e t a l l i c w a l l f o r f i x e d values of  and e ^ i s to modify the slope  and i n t e r c e p t point of the mode p l o t as shown i n f i g u r e 30. When R ^ ^ i and e are kept constant and e i s v a r i e d , again, the slopes and i n t e r c e p t J. z  points are seen to change (Figure 31).  The q u a l i t a t i v e features of the  solutions i n f e r e d from these s o l u t i o n s are: 1.  The slopes f o r  2.  The slopes decrease as £, and £ increase i. z For f i x e d R2/R^> the i n t e r c e p t point decreases w i t h i n c r e a s i n g value of  3.  << 1 are l e s s than 1 but approach u n i t y as R2/R.J-*  00  Further a n a l y s i s has shown that when E /E >> 1, then the zj . intercept point i s not very s e n s i t i v e to e . When the centre d i e l e c t r i c has a f i n i t e length b, then n e g l e c t i n g end e f f e c t s , k = Tr/b. By considering the f i e l d c o n f i g u r a t i o n s i n f i g u r e z 32, one would expect the end e f f e c t s to be comparable t o those f o r the (6)  m e t a l l i c resonator which were found to be of the order of 10%  . A  q u a l i t a t i v e understanding of the consequences of the f i n i t e sample length can be had by examining the f i e l d l i n e s i n f i g u r e 32.  The end e f f e c t s can b  considered as equivalent to the sample having an e f f e c t i v e length b + A where 0 < << 1. Then f « (b + A) or f o r small A/b: b f 2 cc i 2  2  1 - 2 4  b  5  b  F i g u r e 32.  E l e c t r i c f i e l d l i n e s f o r a c o a x i a l mode w i t h a c e n t r e s e c t i o n of f i n i t e l e n g t h when (a) 4™ > >  0>>  W  «  1  !  77  REFERENCES  1.  J.T. Tiedje;  M.Sc. Thesis;  U.B.C.; 1975  2.  A.J. B e r l i n s k y , J.F. Carolan, L. Weiler; (1974)  3.  R.E. P e i e r l s ; Quantum Theory of S o l i d s ; (Oxford U n i v e r s i t y Press, London, 1955)  4.  L.B. Coleman, M.J. Cohen, M.J. Sandman, D.J. Yamagishi, F.G. G a r i t o , A.J. Heeger; S o l i d State Comm.; 12, 1125 (1973)  5.  J . Bardeen;  6.  W.N. Hardy, A.J. B e r l i n s k y , L. Weiler;  7.  Thomas et a l . ;  8.  H. FrOhlich;  9.  P.A. Lee, T.M. Rice, P.W. Anderson;  10.  D.B. Tanner, C S . Jacobsen, A.F. Garito, A.J. Heeger; 13B, 3381 (1976)  Phys. Rev.;  11.  L.B. Coleman, C R . Fincher, A.F. G a r i t o , A.J. Heeger; Sol. (B); 75_, 239 (1976)  Phys. Stat.  12.  S.K. Khanna, E. Ehrenfreund, A.F. G a r i t o , A.J. Heeger; Phys. Rev.; 10B, 2205 (1974)  13.  J.P. Ferraris, T.F. Finnegan; S o l i d State Comm.; 18_, 1169 (1976)  14.  W.J. Gunning, S.K. Khanna, A.F. G a r i t o , A.J. Heeger; 21, 765 (1977)  15.  J.E. E l d r i d g e ;  16.  T.J. Kistenmacher, T.E. P h i l l i p s , D.O. Cowan; A l t a Cryst.; B30, 763 (1974)  17.  S.K. Khanna, A.F. G a r i t o , A.J. Heeger, R.C J a c k l e v i c ; Comm.; 16, 667 (1975)  18.  C M . Hiddy et a l . ; North American Rockwell Science Centre, Thousand Oaks, C a l i f o r n i a ; Report NASA CR-1960; page 54  S o l i d State Comm.; 15, 795  S o l i d State Comm.; 13, 357 (1973)  Phys. Rev.;  Phys. Rev.;  14B, 3356 (1976)  13B, 5105 (1976)  Proc. Royal Soc; A223, 296 (1954) Solid State Comm.; 14_, 703 (1974)  B u l l . Amer. Phys. S o c ;  S o l i d State Comm.;  20, 495 (1975)  S o l i d State  78  19.  H.Y. Yee; Hansen Labs., S t a n f o r d Univ., S t a n f o r d , M.L. Report 1065; J u l y 1963  20.  R.D. Richtmyer;  21.  A. Okaya, L.F.  22.  E.A.J. M a r c a t i l i ;  23.  These c o r r e c t i o n s f o r the TM mode a r e d i s c u s s e d i n R e f e r e n c e 6 where i t i s found t h a t t h e c o r r e c t i o n i s o f t h e o r d e r o f 5%.  24.  M. J a w o r s k i ;  25.  R.O. B e l l , G. Rupprecht;  26.  N. Mott; M e t a l - I n s u l a t o r T r a n s i t i o n s ; London, 1 9 7 4 ) ; Chapter I  27.  J.E. E l d r i d g e ;  28.  M. P o l l a c k , T.H. G e b a l l e ;  29.  N,F, Mott, E.A. D a v i s ; E l e c t r o n i c Processes i n N o n - C r y s t a l l i n e Materials; (Clarendon P r e s s , London, 1971); Chapter 4  J . A p p l . Phys.; Barash;  California;  10, 391 (1939)  I.R.E. P r o c ;  50, 2081 (1962)  B e l l S y s t . Tech. J . ;  48, 2071 (1969)  P r i v a t e communication. I.R.E., Trans,  S o l i d S t a t e Coram;  on M.T.T.;  9, 239 (1961)  (Taylor & Francis L t d . ,  21, 737 (1977)  Phys. Rev.;  122, 1742 (1961)  

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