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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 presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of this thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i ABSTRACT D i e l e c t r i c resonances i n the microwave region 16 to 40 GHz have been observed i n si n g l e c r y s t a l s of the highly anisotropic material TTF-TCNQ below 25°K. The a p p l i c a t i o n of open-circuit boundary conditions to the d i e l e c t r i c resonator problem i s shown to be inappropriate for t h i s material and a t r a c t a b l e a l t e r n a t i v e theory, which takes external f i e l d s into account, i s presented. Using t h i s theory, the d i e l e c t r i c modes were analyzed at 4.2°K to obtain the complex d i e l e c t r i c constant along the c r y s t a l l o g r a p h i c b-axis. For four samples studied i n d e t a i l , the average values f o r the r e a l and imaginary parts of the d i e l e c t r i c constant are found to be 4200 and 16 r e s p e c t i v e l y at 25 GHz. A frequency dependence of i s found; however, i t i s not c l e a r whether t h i s frequency dependence i s i n t r i n s i c to the samples or i s due to the approximate nature of the s o l u t i o n . No frequency dependence of £^ was observed. TABLE OF CONTENTS Abstract Table of Contents L i s t of Tables L i s t of Figures Ac knowled g ement s CHAPTER I - INTRODUCTION 1.1 Background 1.2 Purpose of the Experiment CHAPTER II - THEORY OF ANISOTROPIC DIELECTRIC RESONATORS 2.1 Introduction 2.2 The OCB Model 2.3 Solutions f o r the Anisotropic Slab 2.4 Solutions for the D i e l e c t r i c Resonator i n the Limit b -> 0 0 2.5 Coaxial Modes CHAPTER I I I - THE EXPERIMENT 3.1 Design and Apparatus 3.2 Procedure 3.3 Results CHAPTER IV - INTERPRETATION OF THE RESULTS 4.1 Mode I d e n t i f i c a t i o n 4.2 The Real and Imaginary Parts 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 of numerical r e s u l t s for the E ^ m modes 52 v i LIST OF FIGURES Page 1. The organic donor molecule TTF and acceptor molecule TCNQ 2 2. C r y s t a l structure of TTF-TCNQ 3 3. One-dimensional e l e c t r o n i c bands and corresponding density of states f o r the TTF and TCNQ chains. 5 4. Microwave conductivity of TTF-TCNQ obtained using the coaxial resonator technique. 6 5. Conductivity from a one-dimensional electron gas with phonon coupling. 8 6. Behaviour of e from a phenomenological theory of CDW. 8 7. Coordinate system chosen for the rectangular geometry of TTF-TCNQ. 14 8. The lowest order TE mode f i e l d configurations. 15 omn 9. a) D i e l e c t r i c slab of thickness t 19 b) E^^ f i e l d configuration c) f i e l d configuration 10. Graphical s o l u t i o n for the TE^ d i e l e c t r i c slab even (a) and odd (b) modes. 23 11. H (a) and H (b), as a function of distance from the centre x y of the slab. 23 12. Equations (2.39) (1), and (2.40) (2), solved g r a p h i c a l l y . 27 13. D i v i s i o n of the region outside the resonator. 27 14. Waveguide assembly, inserted into a dewar. 32 15. Copper block. 33 vxx Page 16. Waveguide assembly outside the dewar and block diagram of the detection e l e c t r o n i c s . 35 17. (1-the normalized spectra) for one sample at three d i f f e r e n t lengths b. 38 18. F i e l d configuration near a shorted end of a waveguide excited i n the dominant T E ^ Q mode. End (a), top and side (b) views are shown. 41 19. Coupling as a function of p o s i t i o n i n the waveguide for modes A to D. 42 20. Coupling as a function of r o t a t i o n i n the waveguide for modes A to D. -43 21. 1/Q (normalized to the value at 24°K) vs 1/T. 45 22. Q values and resonant frequencies for the A mode, (sample 14). 46 23. Mode plot showing several modes of which four could be i d e n t i f i e d . 49 24. Frequency dependence of e' obtained from second-order theory b of d i e l e c t r i c resonators. 51 25. Q values extrapolated to 1/b 2 =0. 56 26. Density of states with an impurity band which has been s p l i t , g iving r i s e to l o c a l i z e d states near the Fermi energy. 60 27. Temperature dependence of the conductivity from a density of states, shown i n f i g u r e 26. 60 28. 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 wave-guide. 69 V l l l Page 29. Mode plot 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. 73 30. Mode plot f o r the TM^Q coa x i a l mode for various values of R 2 / R R • 73 31. Mode plo t for the TM^Q coaxial mode for various values of e . 74 z 32. E l e c t r i c f i e l d l i n e s f o r a coaxial mode with a centre section of f i n i t e length. 76 ACKNOWLEDGEMENTS I would l i k e to acknowledge the support of Dr. W.N. Hardy i n the supervision of t h i s p r o ject. I have also benefited g r e a t l y from numerous discussions with Dr. A.J. Berlinsky. Most of the samples used i n t h i s project were supplied by Dr. Larry Weiler and h i s co-workers. F i n a l l y , I would l i k e to thank the organic conductors group at the Unive r s i t y of Pennsylvania for sending us some of t h e i r samples. CHAPTER I - INTRODUCTION 1.1 Background The organic material tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) continues to be a material of considerable i n t e r e s t both to th e o r i s t s and experimentalists. The high anisotropy i n the e l e c t r i c a l conductivity^"^, as well as molecular o r b i t a l c a l c u l a t i o n s ^ ^ , have indicated that t h i s material may be described i n terms of a one-dimensional band structure. The anisotropy r e s u l t s from the stacking of the r e l a t i v e l y f l a t TTF and TCNQ molecules (Figure 1). The TTF and TCNQ separately form stacks with strong coupling between molecules of the same stack r e l a t i v e to that of molecules on d i f f e r e n t stacks (Figure 2). As a r e s u l t , the c r y s t a l l o -graphic b-axis, which i s the d i r e c t i o n of the strong molecular coupling, should have the highest conductivity. The expression "quasi-one-dimensional" has sometimes been applied to describe the fac t that the anisotropy i n the intermolecular coupling i s large, but f i n i t e . (3) P e i e r l s has shown that a one-dimensional metal w i l l be un-stable with respect to 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 at the Fermi l e v e l , thus leading to a small gap semiconducting state. This e f f e c t i s not r e s t r i c t e d to s t r i c t l y one-dimensional systems; however, the d i s t o r t i o n s , necessary to create an energy gap over the entire Fermi surface, are i n general more complicated i n higher dimensions. This type of mechanism i s believed to be responsible f o r the large drop i n the b-axis conductivity 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 to a semiconducting state i s observed. In terms of the 2 H' II \ c=c || •H H TTF N „ H H N c ,c=c c V / \ / ,c=c c=c / \ / . \ / / C = C x % N H H N TCNQ Figure 1. The organic donor molecule TTF and acceptor molecule TCNQ (b) ia3A (a) (b) Figure 2. Cr y s t a l structure of TTF-TCNQ. (a) View down the a-axis which connects the TTF molecule to the TCNQ. The open c i r c l e s represent atoms on the TTF molecule, while the s o l i d ones represent those on the TCNQ. (b) View down the b-axis which i s the stacking d i r e c t i o n . Here, the s o l i d c i r c l e s represent atoms t i l t e d out of the plane of the paper while the open c i r c l e s are those t i l t e d into i t . 4 one-dimensional bands shown in figure 3, a l a t t i c e i n s t a b i lity of wave-vector q = 2kp, where k^ is the Fermi wavevector, leads to the introduction of a gap 2A in 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 conventional metallic conduction. So,- the suggestion of viewing the behaviour of TTF-TCNQ in terms of collective many-body effects associated with the soft phonon Peierls instability was made ^ . Figure 4 shows the temperature dependence of the conductivity at a frequency of 30 GHz, obtained using the coaxial resonator technique ^ . The normalized conductivity increa-ses as the temperature decreases, attaining a maximum value of V9 at T=53°K. Below this temperature, the conductivity drops sharply. These results, as well as subsequent conductivity measurements reveal that, although the peak conductivity ratios 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-vity mechanism on the basis of single particle scattering alone remains un-clear. To study the conductivity in a one-dimensional system, one can (8 s) consider a model proposed by Frohlich who examined the coupling of non-interacting electrons to phonons in a jellium 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 Figure 3. One^dimensional e l e c t r o n i c bands and corresponding density of states for the TTF and TCNQ chains before (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 . 1 0 [ 8 f (300°K) = 32.4 GHz o Q (300°K) = 32 0 0 1 0 0 T(°K) 2 0 0 3 0 0 Figure 4. Microwave conductivity of TTF-TCNQ obtained using the coaxial resonator technique. 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. He further argued that f o r s u f f i c i e n t l y small v e l o c i t i e s , the wave should move unattenuated. A study of the contribution to the conductivity from such a c o l l e c t i v e mode was made i n an elegant paper by Lee, Rice and (9) Anderson . In i t , they show that, below the t r a n s i t i o n , there are no contributions to the d.c. conductivity from sin g l e p a r t i c l e interband t r a n -s i t i o n s . However, i n c l u s i o n of the 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 con-d u c t i v i t y of the form a(io) = m 6(CJ) + O^(GJ) (1.1) m* where 6 (to) i s the Dirac delta function. A plo t of a i s shown i n f i g u r e 5. i s the contribution due to single p a r t i c l e interband t r a n s i t i o n s across a gap 2A, whereas the weighted 6-function i s the contribution due to the P e i e r l s - F r b h l i c h mode. They further argue that various mechanisms could introduce an energy gap into the e x c i t a t i o n spectrum of t h i s c o l l e c t i v e mode and, instead of an i n f i n i t e d.c. conductivity, there w i l l be a large low frequency a.c. conductivity. The various mechanisms proposed were: 1. Impurity s c a t t e r i n g 2. Commensurability with the underlying l a t t i c e 3. Interactions r e l a t e d to 3-dimensional ordering. For the case CJ < < 2A, they give the following expression f o r the r e a l part of the d i e l e c t r i c constant: e<(a)) = 1 + u> p 2 + u ; p 2 m m* (1.2) 6A 2 u 2 - -u2 T 8 Figure 5. Conductivity from a one-dimensional electron gas with phonon coupling (from Reference 9). The dashed l i n e i s the conductivity without including the e f f e c t of the F r o h l i c h mode. 0 Figure 6. Behaviour of e from a phenomenological theory of CDW. 9 oi i s a non-zero frequency due to some pinning mechanism, cop i s the plasma frequency, m the e l e c t r o n i c band mass and m* the e f f e c t i v e mass of electrons i n the condensed state. The complex d i e l e c t r i c constant, due to the presence of charge density waves, can be found within a phenomenological theory of charge density waves, as follows: l e t m* and N be the mass and number of electrons i n the condensed charge density wave state. The c l a s s i c a l equa-t i o n of motion governing the displacement X of the condensate from i t s equilibrium p o s i t i o n i s N_m*X + yt + kX = N ceE (1.3) Y i s a f r i c t i o n constant, k a harmonic r e s t o r i n g constant and E denotes an externally applied e l e c t r i c f i e l d . Introducing two new constants co^  and T: w T 2 = • .k-Ngm* r = Ngm* and taking the Fourier transform of equation(1.3), we get e_ X(u>) = m* E(co) (1.4) co 2 - co2 — i u r T X(co) may be interpreted as the mean displacement of teheuphaseuid ei-dzt.-bif the P e i e r l s - F r o h l i c h condensate. Since the dipole moment P a r i s i n g from t h i s displacement i s N eX, then the d i e l e c t r i c constant due to the motion of a charge density wave i s !> S m m* (1.5) E(to) to T 2 - co2 — irto where: N i s the t o t a l number of conduction electrons e 10 and O J „ 2 = f 4ire 2n 1 m N , where n = _e , L being the length of the l i n e a r L conductor. This frequency dependence i s drawn i n fi g u r e 6. The t o t a l d i e l e c t r i c constant e w i l l have a contribution from single p a r t i c l e a c t i v a t i o n across a gap 2A , as well as a contribution from any e x i s t i n g c o l l e c t i v e motion. e(o>) = e s p ( a ) ) + e C D W ( o ) ) Tanner ^ ^ a n d Coleman have argued, on the basis of o p t i c a l studies, thate - 50' and that therefore, another mechanism (such as a c o l l e c t i v e mode) must be providing o s c i l l a t o r strength at low temperatures where (12-14) d i e l e c t r i c constant values of 3000 have, been reported (15) red spectrometry Far i n f r a --1 shows no evidence of a pinned mode above 7cm According to a l a t e s t estimate by Coleman , oi^ , i s around 2cm (or 60 GHz). 1.2 . Purpose of the Experiment (16) The c r y s t a l structure of TTF-TCNQ i s monoclinic and thus, there are i n general four non-vanishing components of the d i e l e c t r i c tensor: 0 0 0 0 yv e l l E12 £21 e22 (1.6) Single c r y s t a l s , grown i n a c e t o n i t r i l e solutions, tend to have physical dimensions which coincide with the a and b cry 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 axis c* makes a 14.46° angle with the cr y s t a l l o g r a p h i c c-axis. Making the approximation that the c r y s t a l structure i s ortho-11 rhombic, there are now three non-vanishing components of £ > coinciding with the physical axes (Figure 7 a). s x 0 0 e yv 0 e Y 0 | . (1.7) 0 0 ez (12-14) e Low temperature measurements show that _Z_ - 500 and thus, t h i s material i s highly a n i s o t r o p i c . The frequency region, where the d i e l e c t r i c constant e ralong^the b-axis i s believed to be changing r a p i d l y , i s presently j u s t below the reach of conventional f a r i n f r a r e d spectrometers, I t i s also hard to probe using a microwave c a v i t y perturbation experiment because, even though the a v a i l a b l e c r y s t a l s are needle-like i n shape, depolarization e f f e c t s can be s u b s t a n t i a l because of the extremely high value of e_. The depolariza-> Li Li t i o n f a c t o r can be estimated from the r e s u l t expected for an ellipsoid of re v o l u t i o n . At higher frequencies, where shorter c r y s t a l lengths are required, the accuracy of t h i s w i l l get worse. Previous reports of d i e l e c t r i c resonances i n si n g l e c r y s t a l s of TTF-TCNQ ^ 1 7 ^ at 13 GHz, suggested that the method might be e a s i l y extended to higher frequencies. The d i e l e c t r i c resonator method thus appeared as an a t t r a c t i v e a l t e r n a t i v e for probing the microwave region up to and beyond 60 GHz, the suggested value of u> . Thus, the purpose of t h i s experiment i s to apply the theory of anisotropic d i e l e c t r i c resonators to TTF-TCNQ i n order to extract the frequency dependence of the complex d i e l e c t r i c tensor. 12 CHAPTER II - THEORY OF ANISOTROPIC DIELECTRIC RESONATORS 2.1 Introduction It has been known for many years that a f i n i t e piece of high p e r m i t t i v i t y material i n free space can resonate i n various modes. High Q resonators made from low-loss materials, such as TiO ( d i e l e c t r i c constant - 100), have been observed and investigated by several experimen-(19 21) ters ' . That d i e l e c t r i c resonators can e x i s t may be e a s i l y seen by noting that electromagnetic waves w i l l be completely r e f l e c t e d from the i n t e r f a c e between free space and a d i e l e c t r i c , i f the angle of incidence -1 i s greater than the c r i t i c a l angle 0 = Sin X f l / V e 7 " 1 , where e' i s the r e a l c v • J part of the d i e l e c t r i c constant. Exact solutions to the Maxwell equations are only possible for d i e l e c t r i c resonators with s p h e r i c a l , t o r o i d a l , and e l l i p s o i d a l boundary surfaces, while rigorous solutions for f i n i t e c y l i n d r i c a l and rectangular resonators do not e x i s t . The f i r s t two cases were treated by Richtmeyer The d i f f i c u l t i e s i n f i n d i n g exact solutions stem from the f a c t that.the f i e l d s or t h e i r d e r ivatives do not.necessarily vanish at the surfaces of the resonator. An approximation that gives a rough picture of the modal f i e l d s i s to consider the d i e l e c t r i c resonator with an a i r boundary to be the "dual" of metal wall resonators, i . e . at the boundary, impose the conditions n * H = o" (2.1) n, . 1 = 0 (2.2) These conditions are c a l l e d open-circuit boundary conditions (OCB) i n contrast to the s h o r t - c i r c u i t boundary conditions of m e t a l l i c c a v i t i e s . 13 The reason for assuming the OCB conditions can be e a s i l y seen by considering the i n t e r f a c e between two i s o t r o p i c media with d i e l e c t r i c cons-tants e'^  and e^, r e s p e c t i v e l y . The normal component of the displacement vector D must be continuous at t h i s i n t e r f a c e , i . e . n • V = n • D 2, Thus, n • E 1 = '2 n • E 2 K If E'^ >> e ' 2 J then n • E.^ .= 0. However, the v a l i d i t y of imposing condition ( 2 . l ) ( o f t e n c a l l e d imposing a magnetic wall boundary) i n conjunction with condition (2.2) i s not always good even though, e'-j. >> e'.^. The OCB modal f i e l d configurations w i l l be the same as those for 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 interchanged. These are shown for the lowest modes i n figure 8. In general, the accuracy of the OCB assumption i s expected to be, poor, p a r t i c u l a r l y (as w i l l be shown i n section 2.3) when any of the resonator dimensions are comparable.to the e f f e c t i v e wavelength.inside the resonator. Thus, su b s t a n t i a l f i e l d energy e x i s t s outside of the d i e l e c t r i c resonator. A more r e a l i s t i c boundary value problem has been treated by (19) (21) Yee and Okaya and Barash and found to y i e l d s u b s t a n t i a l l y improved r e s u l t s . With reference to figure,7a, they assumed OCB conditions on the surfaces p a r a l l e l to the X-axis while at the boundary surfaces perpendicular to the X-axis, they applied exact boundary conditions. Their solutions are i n fa c t exact solutions to the i n f i n i t e slab of thickness c* i l l u s t r a t e d i n figure 7b. An equivalent way,of thinking 14 •Figure 7b. 15 0 1 1 0 1 2 0 2 2 fx* • X x " • x 031 0 3 2 *( y ,<s Figure 8. The lowest order TE mode f i e l d configurations. & omn The s o l i d l i n e s are E - f i e l d s while the H - f i e l d s are transverse to the plane of the paper. 16 of t h e i r assumption of OCB conditions at the boundary surfaces p a r a l l e l to the X-axis i s to require antinodes of the magnetic f i e l d say, to occur at y =0,_a. and z = 0, b (see f i g u r e 7b) . A discussion of the properties of the solutions for the anisotropic d i e l e c t r i c slab found by Okaya and Barash can be found in,Appendix A. These solutions are i n general very complicated due to the anisotropy of the slab i n the Y-Z plane. However, i f one r e s t r i c t s the problem to the s i t u a t i o n where the f i e l d s are uniform along the Z^axis (b d i r e c t i o n ) , t h e n the f i e l d s can be s i m p l i f i e d considerably. The penalty paid for t h i s s i m p l i f i c a t i o n i s that s t r i c t l y speaking, one can only apply the r e s u l t s to resonators of i n f i n i t e length. Fortunately, however, i t i s possible to determine experimentally the corrections for f i n i t e resonators by extrapolating r e s u l t s for d i f f e r e n t lengths. After a discussion of the solutions to the anisotropic d i e l e c t r i c resonator with OCB conditions i n the following section, the TE modes of the anisotropic d i e l e c t r i c slab when the f i e l d s are uniform along the Z-axis, i . e . 9 _ 0 w i l l be found and examined (section 2.3). In section 9z " 2 . : 2.4,these solutions w i l l be used to f i n d an expression for the eigenfrequen-cies of the d i e l e c t r i c resonator of i n f i n i t e length and f i n i t e width "a". 2.2 The OCB Model Maxwell's equations i n the absence of free charge and currents for a magnetically i s o t r o p i c material are: V x E = iuu_ H (2.3) V x H = iw e E (2.4) c V • eE = 0 (2.5) V • H = 0 (2.6) 17 From them, one gets the following wave equation f o r E: V 2E - V(V • 1) + io 2y£_E = (2.7) Solutions f o r a rectangular body of dimensions L^, and which s a t i s f y the OCB conditions are: E = A s i n k X cos k Y cos k Z x x x y z E = A cos k X s i n k Y cos k Z (2.8) y y x y z E = A cos k X cos k Y s i n k Z z z x y z where k„ = M_ 9 k,, = mfr " L y x — y k = nir Z ' L~ z (2.9) ''£V4>'m, n are integers; A , A , A are constants to be determined. ° x y z Substituting these solutions into the wave equation (2.7), one f i n d s , a f t e r a l i t t l e algebra, the following solutions to the secular equation: co 2 1 2y k 2 x I y z 4. k ' + y £ £ X i + k 2 + Z £ £ , x yj k 4 x i £ y i _ E 2 k 2 k 2 + x y , 2 k 2 k 2 + y z , 2 k 2 k 2 + x z \ k 4 + y i _ £ yj yj (. y x i _ E Z £ X 1_ E Z; i l E '- k<* + Z 1_ E X yj 1/2 (2.10) 0) or TM ' (H x = 0). For the The solutions may be c l a s s i f i e d as TE (E^ TE modes, one must have m,n ^ 0 , whereas f or the TM modes, £ ^ 0. In an i s o t r o p i c medium, for a given value of %, m, n, the TE and TM modes are degenerate (although,the f i e l d configurations are d i f f e r e n t ) . One can see 18 that an e f f e c t of the anisotropy i s to break t h i s degeneracy and double•the number of observable modes. S p e c i a l i z i n g fcor the case e = e = e. (tetragonal symmetry), x y the solutions (2.10) reduce to (for y = 1) co2 1 ( k 2 + k 2) , 1 k 2 (2.11) —j = — x y H z c z e J e, corresponding to TE modes, and co2 1 ( k 2 + k 2 + k 2 ) (2.12) T = — ^ x y z J for the TM modes. The s i n g l e c r y s t a l s of TTF-TCNQ available,had t y p i c a l dimensions c* << a, b. Because of t h i s and the fac t that e » . e . , the TE should have the lowest frequencies given by. f = c omn. y 1_ e m n. 2 - i 1/2 (2.13) where m, n f 0. The f i e l d configurations f o r the f i r s t few modes are i l l u s t r a t e d i n fig u r e 8. These f i e l d patterns are magnetic multipoles i n character except that, within the OCB model, the magnetic f i e l d l i n e s terminate at the boundary. In a more r e a l i s t i c model, one expects the f i e l d s to extend outside the resonator and form closed loops'as shown for the TEQ^^ mode i n figure,9b. In terms of the OCB solutions, t h i s means an e f f e c t i v e value of & between 0 and 1. 2.3 Solutions for the Anisotropic D i e l e c t r i c Slab The.TE (E = 0) mode solutions f o r the d i e l e c t r i c slab of thickness x t i l l u s t r a t e d i n figure'9a w i l l now be derived. We begin by wr i t i n g out the equations (2.3) and (2.4) e x p l i c i t l y : 20 • * E = i . 3H z . 3H y (2.14) H = - i 3E z 9E y (2.17) X k £ O X [dy ( 9H X 9z J X k y o [9y *• 9z J E = i 9H z (2.15) H = - i 9E X 9E z (2.18) y k e, o y [3z " 9x J • y k y o [9z 9x J E = i 9H y 9H X (2.16) H = - i 9E y 9E X (2.19) z k E o z 9x 3y j z k y o 9x 9y J where k =u)Ui o — cc For TE (E = 0) modes with no s p a t i a l dependence,along the z-axis ( i . - ' ( i . e . 9_ = 0), and assuming a s p a t i a l dependence'- along the y-axis -of the 9z form e l k y Y , the f i e l d s (2.14)-(2.19) become E = E = H = 0 x y z H = K_ E x , — z H = i y 3E k y 3x o (2.20) (2.21) E z w i l l s a t i s f y equation (2.7), i . e . V 2E - 9 (V" • E) + k 2yc E = 0 z . — o z z d Z For an anisotropic material, V • E r 0 i n general since E T E . Thus, y z the equation f o r E w i l l contain terms, l i n e a r i n the f i e l d d e r i v a t i v e . This i s why the general solutions f o r the slab turn out to be so complicated (See Appendix A). In the present case however, = 0 and so the equation 9z for E becomes, z 9x' E + k^yc E = 0 z o z z (2.22) The boundary value problem f o r these modes can be solved exactly. The treatment i s s i m p l i f i e d by separating them, from the s t a r t , into even 21 and odd modes. (It i s of course possible to s t a r t with general f i e l d expressions and obtain the even and odd modes from the r e s u l t i n g eigenvalue problem). Even and odd r e f e r to the symmetry of E^ about the plane x = 0, which i s now chosen to l i e i n the center of the slab. (See Figure 9a) Eyen_Gu_ided_ TE_modesj_ The mode solu t i o n f o r even modes insi d e the slab i s : E = A. cos k .X Ixl < t (2.23) z l x i ' 2 where A. i s a constant. Therefore, l H = - i A. k . s i n k .X , , (2.24) y j — i x i X 1 | X| < t o K 2 Substituting(2.23)into(2.22), we get: k 2 = k 2ye — k k 2 (2.25) xx o -;z • yy The f i e l d outside the slab can be written as E = A e " k x o ( l X ' " t / 2 ) ; Ixl > t (2.26) z o 1 1 — 2 where A i s a constant , and thus: o H = ^ k A e - k x o ( l x l ~ t/2) . | X| > t_ (2.27) 7 V X ° ° 2 where k 2 = k 2 - k 2 y e (2.28) xo y o o o y Q and CQ are the permeability and p e r m i t t i v i t y of the surrounding medium res p e c t i v e l y (without any l o s s of generality, to be y^ = E Q = 1). Both k 2. and k 2 can be p o s i t i v e quantities since E > 1. For p o s i t i v e values x i xo z of k 2 , the f i e l d s decay exponentially with increasing |x| outside the slab. Thus, the condition for a guided mode i s k 2 > 0. ° xo 22 The continuity of E and H at the in t e r f a c e between the two media, z y leads to the following secular equation: k = k . tan xo x i (k . t l xx (2.29) Odd,Guided TE modes: In t h i s case, the solutions inside the slab are E (X) = B. s i n k .X Z X X X H (X) = i B. k . cos k .X y ^ i x i o 2 (2.30) while outside E (X) = B e " k x ° ( | x | " t / 2 ) z o H (X) = - i B k e " k x o 7 ^ o xo o -k Xo(|x| - t/2) r l x l > 1 2 (2.31) The con t i n u i t y requirements for the E^ and components lead to the eigenvalue equation: k = - k . cot xo xx k .t X X (2.32) We can get a f e e l i n g for the form of the solutions to the eigen-value equations by graphing them. For instance, consider the even TE modes. It i s convenient to combine (2.25) and (2.28) into one equation, (y = 1) k .t xx I 2 J + k t xo k t o (2.33a) and multiply equation (2.29) by t/2: k t xo k .t X X tan k .t X X I 2 J I 2 j I 2 J (2.33b) (a) (b) n=1 ^b) n = 2 Figure 11. H x (a) and (b) as a function of distance from the centre of the slab. 24 k t k t Equation (2.33b) determines one r e l a t i o n between x i and xo' , and 2 2 plane (Figure 10a). Equation (2.33a) may be plotted on the fk .t] X I _ fk tl xo 1-2 J I 2 J i s a c i r c l e of radius (e - l) ^ 2 i n the same plane. The points of Z 2 in t e r s e c t i o n between the two curves define the r e l a t i o n s h i p of k . to k x i xo An i n t e r e s t i n g property i s the fac t that the two graphs w i l l always i n t e r s e c t at at l e a s t one point. Thus, t h i s f i r s t symmetric mode w i l l not have a low frequency c u t o f f . For a mode to be guided, k' must be greater xo than zero, and so we need only consider those points of i n t e r s e c t i o n which l i e i n the upper plane of the diagram, i.e.: 0 < k x l f c < £ , ir < k x i t < -31 , ... 2 2 2 ~ 2 or i n general, (n - 1) TT < k . < nir (2.34) - - x i - — where n i s an odd integer. A graphical analysis of the odd modes (Figure 10b) reveals that a l l modes have a cutoff frequency. For guided modes ( i . e . k > 0), we xo need only consider points of 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 (2.35) — - xx - — t t where n i s a non-zero even integer. To summarize, the anisotropic d i e l e c t r i c slab may sustain TE (E = 0) modes, where n i s a non-rzero integer that can be viewed as the n x number of antinodes i n E z ( X ) . The boundary value problem can be solved exactly and the .eigenvalue equations are found to be independent of the 25 values of e and e . x y We w i l l use the solutions found here to see under what conditions the a p p l i c a t i o n of OCB conditions w i l l be a good approximation at the slab boundaries. This i s e a s i l y done by considering the equation r e s u l t i n g from combining equations (2.33a) and (2.33b): k .t X X 2 2 -tan z 'k .t X X 2 2 fk t] 0 2 fk . t l X X 2 (2 2 2 When e >> 1, z k .t X X 2 t a n 2 k. .t X X U T } z t 2 fk .t] X X I 2 J 2 X I 2 J (2.37) k t where A i s the free space wavelength. For the lowest mode, x i < jr ; ° ' X 2 = 2 and so, i f the thickness t >> o , then x.e. fk .t] X X tan <k . t ) X X 2 2 k .t X X ir 2 2 » 1 (2.38) Thus, k . = TT_ (the OCB approximation) w i l l be a good approximation i f , X 1 t t » o / / i ~ and the boundary surface w i l l coincide with a node of the z 1 standing wave insi d e the slab. As the thickness i s reduced however, more of the f i e l d extends outside the slab (See figu r e 11). In addition, when a higher mode i s excited, the f i e l d s w i l l not extend outside the slab as much as for lower order ones, since approximately the same f r a c t i o n of a wavelength extends outside the slab. 26 Insight as to the behaviour of the solutions i n the opposite l i m i t (t << ^ e£f) c a n be gained by examining equation (2.25) k 2 = 1 ye (k z . + k O x i y (2.39) and the equation r e s u l t i n g from combining (2.28) and (2.29): k 2 = k 2 - k 2. t a n 2 o y x i These equations are plotted as a function of k .t xx k .t X X (2.40) i n figure 12. If e » 1, then from (2.39) , k 2 - 0; so (2.40) becomes z o k 2. t a n 2 xx k .t X X (2.41) k t I f , i n addition, x i << 1, then one can approximate (2.41) further by k 2 . t xx (2.42X Putting t h i s value f or k ^ into the eigenvalue equation (2.39), one gets 2 - 2 ye y + k (2.43) The e s s e n t i a l d i f f e r e n c e between these eigenfrequencies and those found for the OCB model when k.^ 0 (see equation (2.11)), i s the l i n e a r rather than quadratic dependence,on k^ when t i s small. The samples of TTF-TCNQ a v a i l a b l e for t h i s work had thicknesses c* of the order of 0.03 mm. I f e ~ 10 3 at a frequency of 30 GHz (X = 1 cm), then X ^ c = X //e~ ~ 0.03 cm. Thus, c* ~ 0.1 X ^ and o e f f o z e f f c* a p p l i c a t i o n of the OCB conditions along the planes X = ± 7 p should be expected to be a.very poor approximation indeed. The values f or "a" are t y p i c a l l y between 10c* and 50c* so that 5 £ a ^ e f f ~ ^' T h u s 5 along the Figure 13. D i v i s i o n of the region outside the resonator 28 boundary planes perpendicular to "a", the soundness of the OCB r e s t r i c t i o n i s at best marginal. 2.4 Solutions for the D i e l e c t r i c Resonator i n the Limit b •+ °°. The d i e l e c t r i c resonator of f i n i t e width "a" may be conceived as being formed from the section abed of the slab shown i n figu r e 9a. In order to trea t the boundary value problem at the newly formed surfaces ab and cd, one can require that these surfaces coincide with antinodes of the magnetic f i e l d (the OCB condition), i . e . k = mir y "7 where m i s an integer; however, a treatment of the boundary conditions which has proved successful i n s i m i l a r problems i n the theory of o p t i c a l (22) waveguides can be made by considering the following: Divide the region outside the resonator into the eight regions shown i n figu r e 13. If one relaxes the requirement that the f i e l d s i n these regions s a t i s f y exact boundary conditions along the dashed l i n e s , then s i m p l i f i c a t i o n of the boundary value problem can be achieved with the r e s u l t that for the fundamental mode for instance, E_(X,Y) may be written as: , v -k v o.( Yl - c*/2) cos k .X e Ju 1 : |Y| > a 2 E z(X,Y) = ^ cos k X cos k Y ; inside resonator (2.44) x i y x -k Xo ( | x | - a/2) , „ ,, e x o v 1 1 cos k .Y ; X > c* y i 2 When these solutions are substituted into the wave equation (2.22), one 29 gets k 2 = k 2. - k 2 (2.45) xo y i o k 2 = k 2. - k 2 (2.46) yo x i o Matching the fields at the boundaries of the crystal gives k = k . tan k x i C * (2.47) xo X I ' 1 k = k . tan r y i " I (2.48) yo y i |-£— Higher order modes may also be solved for by writing a sinusoidal dependence in either the x or y direction (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 for the in f i n i t e slab found in the previous section. Mod_e_De_s^giiationj_ Unlike metallic cavities, the fi e l d patterns of dielectric wave-guides and cavities are sensitive to £^v> the wavelength and the sample dimensions. This complicates the problem of finding a reasonably descriptive mode designation scheme. For rectangular metallic waveguides and cavities, 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 in the X, Y and Z directions with a triple subscript. When there are no variations, a 0 is used. Since the rectangular dielectric cavity modes are neither pure TE nor TM, a different scheme should be used. The scheme to be adopted here w i l l be based on the following fact: i f the resonator is considered 30 as a section of length b of a d i e l e c t r i c waveguide, then i n the l i m i t of a large aspect r a t i o ( i . e . b/c* >> 1), the e l e c t r i c f i e l d i s p r i m a r i l y z along one of the transverse axes. Modes w i l l be designated E* i f the Jim e l e c t r i c f i e l d i s p r i m a r i l y along the Z-axis. According to t h i s scheme, the resonator modes presently being 2 considered w i l l be denoted E„ where % and m are both non-zero integers. £m z z The f i e l d configurations for the E ^ and E ^ modes are i l l u s t r a t e d i n figure 9b and 9c r e s p e c t i v e l y . 2.5 Coaxial Modes The solutions discussed thus f a r have been for a resonator i n free space. In p r a c t i c e , the d i e l e c t r i c material i s placed i n s i d e a m e t a l l i c waveguide, so that the e f f e c t of the waveguide walls upon the resonances must be considered. If the f i e l d s decrease s i g n i f i c a n t l y within a distance l e s s than the distance of the resonator from the walls, then a small perturbation of the free space solutions i s expected; otherwise, the e f f e c t s of the walls must be considered. One can get a f e e l i n g for the q u a l i t a t i v e features of the coaxial system by considering an anisotropic c y l i n d r i c a l rod, i n s i d e a c i r c u l a r m e t a l l i c waveguide. This problem i s considered in'Appendix B where i t i s shown that i n addition to coaxial hybrid modes (which w i l l correspond to f i e l d configurations resembling the E f i e l d l i n e s ) , the coaxial l i n e may sustain low frequency modes which would otherwise radiate i f the waveguide were not present. The frequencies of these modes are highly s e n s i t i v e to the r a d i i of the d i e l e c t r i c rod and waveguide. Hence, one must be aware of the p o s s i b i l i t y of observing these modes as well as d i e l e c t r i c modes. 31 CHAPTER I I I - THE EXPERIMENT 3.1 Design and Apparatus The experimental apparatus was designed to allow s i n g l e samples to be introduced into a rectangular waveguide (excited i n the dominant t e^Q mode) i n such a way that the coupling between the resonating sample and the waveguide f i e l d could be adjusted continuously during a run. Figure 14 i l l u s t r a t e s the segment of the waveguide assembly b u i l t to f i t i n s i d e a dewar system. The waveguide path consisted of three sections. The f i r s t section involved a short piece of rectangular waveguide f i t t e d with a small brass flange. This was placed through a hole i n the dewar cover p l a t e , and the flange bolted to i t . Next came a section of s t a i n l e s s s t e e l waveguide to reduce thermal conductive l o s s . The l a s t section was a U-shaped piece of waveguide with a radius of curvature chosen so as to permit the i n t r o -duction of the sample while the waveguide assembly was i n the dewar. The sample was inserted into the waveguide (along i t s symmetry axis) at the end of a s t a i n l e s s s t e e l rod, which was guided into p o s i t i o n by a s t a i n l e s s s t e e l tube. Figure 15 shows a d e t a i l drawing of the copper block, housing a heater and thermometer, which was f i t t e d at the end of the tubing. The rectangular base of the block was bolted to a rectangular flange at the end of the waveguide. A small hole (dia. = .044") wide enough to allow the c r y s t a l to pass through sa f e l y (but small enough to be beyond cutoff for the frequency bands used) was bored through the base of the block, concentric with the tube. The mounting of the samples was conceived so as to permit passage through the small hole and extension into the guide. The samples were 32 BRASS PLATE STAINLESS STEEL WAVEGUIDE MICROMETER O-RING ROD COPPER BLOCK 1EATER COILS THERMOMETER Figure 14. Waveguide subassembly designed to f i t into a He dewar. V 33 Figure 15. Copper block. The metal wedge is normally not present but can be used to t i l t the waveguide electric f i e l d with respect to the crystal axis (see section 4.1). 34 attached to a fused quartz f i b r e with a small amount of epoxy. The quartz 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 as well as to provide the necessary extension of the sample while perturbing the f i e l d s as l i t t l e as po s s i b l e . A small part of the t i p of the quartz was then passed through a metal tube (a 26 Gauge hypodermic needle) and eppxied into p o s i t i o n . F i n a l l y , the needle i t s e l f was placed into a hole centered at the end of the moveable rod and was held there by means of a small screw. The p o s i t i o n of the sample could 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 rod. A Weinschel microwave sweep o s c i l l a t o r (model 221), with backward wave o s c i l l a t o r p lug-in units i n the frequency ranges 18 to 26 GHz (model H1826) and 26 to 40 GHz (model H2640), provided the RF source. This voltage-controlled microwave source i s s u i t a b l e for both swept and c o n t i -nuous wave measurements. A 0 to 20 v o l t s v a r i a b l e ramp generator was used to sweep through any frequency range of i n t e r e s t . The power, l e v e l l e d with a d i r e c t i o n a l coupler-power meter (General Microwave Corporation model 460b) combination, was coupled into the waveguide c i r c u i t (described above) with a 10 db d i r e c t i o n a l coupler (Figure 16). The power, r e f l e c t e d at the shorted end of the waveguide, was detected with a Hewlett-Packard mixer (model 11517A) and the d.c. output was then displayed on an o s c i l l o s c o p e . If a very slow voltage ramp was being used, the output was f i r s t amplified (Hewlett-Packard 419A D.C. n u l l voltmeter) then, simultaneously, displayed on a s t r i p chart recorder and recorded d i g i t a l l y on magnetic tape f o r computer a n a l y s i s . Most of the measurements were c a r r i e d out at 4.2°K. However, when higher temperatures were desired, a c a l i b r a t e d carbon r e s i s t o r thermometer MULTIPLEX STRIP CHART RECORDER A. D.C. MAGNETIC TAPE RECORDING UNIT D.C. AMPLIFIER POWER METER 0 - 2 0 V O L T S RAMP GENERATOR POWER HEAD £4 LmJ lm B W O _ ISOLATOR f nin pij 10 db DIRECTIONAL COUPLERS Figure 16. Waveguide assembly outside dewar and block diagrams of electronics. 36 and a 590, varnish insulated resistance wire heater were installed in the Cu block shown in figure 15. Temperature regulation was carried out with a rate/proportional/integral temperature controller. 3.2 Procedure A major inconvenience with the experimental design i s that large standing waves are l i k e l y to occur in the reflected power spectrum, especially at the higher frequencies. The reason for this i s that a large percentage of the incident power is reflected at the shorted end of the waveguide. In the microwave frequency range, the best isolators presently available 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 this 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 crystal in the waveguide. If the reflection curves with the sample in are now divided by the background, a smooth curve should result, with the sample absorption appearing as dips in this normalized spectra. Since the waveguide section shown in figure 14 is approximately 30 inches long, the thermal contraction may change the RF path length by a substantial fraction of a quarter wavelength. Hence, the background scan should be performed at each temperature. Any power absorption attributable to the presence of the sample in the waveguide was i n i t i a l l y observed by performing a fast frequency sweep and looking at the reflected power signal on an oscilloscope. The effects on the absorbtion of varying the coupling were then noted. 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 later analyzed by computer to give the normalized spectra. The voltage-frequency calibration of the R band (26 - 40 GHz) unit under levelled conditions was acquired by mixing the signal (at a fixed voltage setting) with a 2 to 4 GHz signal from an auxiliary oscillator, whose frequency was measured with a frequency counter. As the frequency th of the auxiliary oscillator 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 is correctly identified (the 10 one in this case), the signal frequency can be ascertained. This calibration was performed once and was periodically checked with a frequency meter. The same proce-dure was followed for calibrating the K-band (18 to 26 GHz) unit. Using the U.B.C. IBM 370 computer, programs were written to read the data off the tapes and divide the background from the spectra. Con-version of sweep voltages to frequencies was built into the programs, so p that normalized power _r_ vs. frequency plots could be generated. P o 3.3 Results A plot of P o is shown in figure lt?a for a sample of TTF-TCNQ of length b - .55cm at a temperature of 4.2°K. Four modes with varying coupling strengths can be seen. C r i t i c a l coupling occurs when the peak value is one. If the modes can be identified, then their frequencies 38 .104-.05; 06 044-I 02 oof 064 04 024-B SAMPLE 11 b = .55 cm D (a) b = .48 cm B D B o o - - ^ K v J (b) b = .44 cm D (c) 30 35 FREO.(GHz) 40 Figure 17. (1 - the normalized spectra) for one sample at three different lengths b at 4.2°K; a = .037 cm, c* * .002 cm. 39 can be used to compute e (neglecting any possible frequency dependence of £ p V ) • 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 by changing one or more of the sample dimensions, a series of resonant frequencies as a function of the dimensions can be obtained. The most conveniently adjust-able parameter i n TTF-TCNQ i s the b dimension. As the length was shortened, the resonant frequencies of a l l the modes were seen to increase. Figure 17 shows spectra of one sample for three d i f f e r e n t values of b (the c r y s t a l was shortened by c u t t i n g o f f a segment with a sharp blade; no changes i n the d i e l e c t r i c properties, due to the i n e v i t a b l y induced s t r a i n s , were seen). These modes could be followed i n d i v i d u a l l y as a function of length, because of the d i f f e r e n t coupling behaviour displayed by each mode. The strength of the coupling was noted by performing a lineshape analysis of the absorptions. I t can be shown that the r e f l e c t e d power from a c a v i t y for which power i s introduced and extracted by the same coupling hole i s given by (18) R(w) = j r -P 1 f 1 - 1 2 + 0) - 0) o 2 4 % Q, o 1 '1 + 1 2 ji - o 2 4 % o (3.1) where u i s the resonant frequency and Q and Q are, r e s p e c t i v e l y , the o 0 c unloaded and coupling Q's. For a d i e l e c t r i c resonator enclosed i n a wave-guide, there i s no l o s s due to r a d i a t i o n . The observed Q, then, i s given by -, 1 . I +1 (3.2) Then, one f i n d s from equation 3.1, that 40 where t = 1 - R(m ) • o The + (-) sign i s used when the mode i s under (over) coupled. There are usually no d i f f i c u l t i e s i n coupling to the lowest order (19) modes . This coupling was varied by changing the sample p o s i t i o n along the waveguide symmetry axis as well as by r o t a t i n g the sample. The f i e l d configuration for a TE^Q waveguide near a shorted end i s shown i n f i g u r e 18. By examining the f i e l d patterns for the T E O M N modes (figure 8), one can see for instance that the T E ^ ^ and TE^,^ modes (the corresponding second z z order so l u t i o n modes are E ^ and E^^ ) should couple strongly at a p o s i t i o n where the magnetic f i e l d i s a maximum ( A/2, A, 3A/2, . . . ) . Figure 19 shows the coupling strength 1/QC a s a function of p o s i t i o n for the four lowest modes, l a b e l l e d A through D i n f i g u r e 16. Modes B, C and D showed a periodic behaviour with p o s i t i o n , whereas the coupling of mode A was generally highly 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 . A c h a r a c t e r i s -t i c feature of t h i s mode for a l l the samples examined was that i t did not have i t s maximum at A/2. The r o t a t i o n a l coupling symmetry was found to have the symmetry of the waveguide f i e l d except for the A mode, for which the coupling often changed d r a s t i c a l l y with small changes i n angle (See f i g u r e 20). 41 (c) Figure 18. Field configuration near the shorted end of a waveguide excited in the dominant T E ^ Q mode. End (a) top and side (b) views are shown. The absolute value of the H-field i s shown in (c). 42 .4 X / 2 8 POSITION (cm) Figure 19. Coupling as a function of p o s i t i o n i n the waveguide fo r modes A to D. Figure 20. Coupling as a function of r o t a t i o n i n the waveguide fo r modes A to D. 44 CHAPTER IV - INTERPRETATION OF THE RESULTS 4.1 Mode Identification For a long time, one of the obstacles preventing the interpre-tation of the experimental results had been a correct mode assignment to the observed resonances. A l l dielectric mode assignments to the modes A to D were inconsistent with the observed coupling symmetries. As an example, a glance at figure 19 reveals that modes A and B (the most strongly coupled) couple weakly at A/2 and A/4 respectively. The rota-tional 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 fields are interchanged. This implies that with respect to the angle 6 defined in 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 positional coupling, whereas the B mode had a l l the symptoms of being TE^^. This dilemma was 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 is known to be in a conducting regime (e*/e^ >> 1); however, the other modes did not. A plot of 1/Q, which is proportional to the dielectric loss, i s shown for the low temperature regime in figure 21. For both modes, the Q decreases as the temperature increases u n t i l above about 25°K. Above this temperature. only the A mode can be seen (although only very faintly),,Q values and resonant frequencies are shown in figure 22 . The Q measurements of the (6) coaxial resonances reported by Hardy et a l showed a similar temperature dependence down to about 30°K. SAMPLE 9 • A mode : f o B mode : f 34.6 GHz; Q(24°K) = 480 39.0 GHz; Q(24°K) - 1100 O O O o o o o o o o '•• mom* . 0 1 ' . 0 0 . 0 8 .16 .24 T Figure 21. 46 5000+ 500 50+ T ( ° K ) T ( ° K ) Figure 22. Q values and resonant frequencies for the A mode. 47 It thus appeared then that the A mode was a coaxial mode, of the f i n i t e dielectric inside a rectangular waveguide. As the temperature is lowered below the metal-semiconducting transition region, the sample goes from a large value of z1'/e' (H ^'7rab where a, i s the b b — ; — b u e b b-axis conductivity) to a small value. The f i e l d configurations of the coaxial mode for these two cases are shown in 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 effective 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 slightly. Only very small changes in 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 in 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 is no longer orthogonal to the coaxial mode. 48 Once the A mode had been identified, assignments consistent with the coupling data could then be made. Within the OCB model, the B, C and D modes are T E Q 1 1 , T E021 a n d T E031 r e s P e c t l v e l y • T h e T E021 l s o n l y e x P e c t e d to couple through asymmetries in the waveguide f i e l d , and indeed, this coupling is consistently found to be very weak. In terms of the second z z z order solutions, the respective assignments are , E-^ and E-^. 4.2 The Real and Imaginary Parts of e A mode plot ( f 2 vs 1/b2) typical of a l l the samples analyzed is shown in figure 23. 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, labelled A - D, showed the same coupling characteris-tics discussed in previous sections. Longer samples were chosen when i t became apparent that the OCB model was inappropriate for this material and that the alternative theory developed was only valid in the limit b + 00. To see that the OCB model is l i k e l y to be poor, consider what the theory predicts for the mode plot. According to equation (2.13), the frequency of the TE modes is given by ^ J omn f 2 = c 2 £b • -1 m 2 — 1 n a b. (4.1) Accordingly, the intercepts of the mode plots should follow the quadratic sequence 1, 4, 9 (i.e. m2 for m = 1, 2, 3), and the slopes should be proportional to nj SAMPLE 14 Figure 23. Mode p l o t showing several modes of which four could be i d e n t i f i e d 50 It i s clear from the experimental mode p l o t that the intercepts do not follow such a sequence. If one uses equation (4.1) to see what frequency dependence t h i s would imply for e', one f i n d s b e'(15 GHz) = 390 b e'(18 GHz) = 1100 D e'(21 GHz) = 1700 b Such a huge dis p e r s i o n would ne c e s s a r i l y involve large values of which are not seen,and therefore the OCB assumptions are c l e a r l y inappro-p r i a t e f or these samples (as was anticipated i n the previous chapter). The o r i g i n of the nearly l i n e a r rather than quadratic sequence of the observed mode plot intercepts can be explained i n terms of the exact solutions f or the i n f i n i t e slab i n the l i m i t of small thickness. R e c a l l that i n the l i m i t where c* << X^/Ze7/, the eigenvalue equation (2.43) i s found to be: When k (4.2) The two terms i n the square brackets w i l l be equal when a _ mir c* = 2 Thus, when a/c* >> 1 and for small mode index integer m, the l i n e a r term w i l l dominate. The frequency dependence of obtained by using the improved model discussed i n section 2.4 i s shown i n f i g u r e 24. Since the theory a p p l i e s Figure 24. Frequency dependence of e£ obtained from the model discussed i n section 2.4. The OCB r e s u l t s for sample 14 are also shown for comparison. 52 TABLE I - SUMMARY OF NUMERICAL RESULTS FOR THE E? MODES lm m f(GHz) ^x(cm ^ 6 X ky(cin ]-Sample 5: 1 19.7 199.0 0.18 57.3 -0.18 2520 a = .045 cm c*= .0020 cm 3 25.7 349.8 0.31 187.6 -0.10 5440 ^ampl_e_ll_: 1 20.3 233.5 0.18 69.3 -0.18 3300 a = .037 cm c*= .0023 cm 2 3 26.9 33.4 335.5 410.2 0.26 0.32 147.4 227.4 -0.13 -0.11 4240 4480 J3ampl_e_12: 1 16.9 143.5 0.22 51.3 -0.23 1858 a = .048 cm c*= .0048 cm 2 3 20.3 25.3 206.2 251.2 0.31 0.38 110.8 172.0 -0.16 -0.13 3046 3301 Sample 14: 1 15.5 166.2 0.19 50.9 -0.19 2871 a = .050 cm c*= .0036 cm 2 3 18.4 21.9 238.8 291.9 0.27 0.33 108.6 167.6 -0.14 -0.11 4615 5380 i 53 only to the b ->• °° limit, the eigenfrequencies are taken from the extra-polated 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 and 6 : x y k = f m + 6 ) ir y ^ yJ _ (In the OCB model,6 and 6 are zero). The values for 6 and <S , found x y x y' from the second order solutions, are also tabulated. As the .mode: index Increases 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 first 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 is very small, the effective outer radius R„ wil 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 solution for the value of the intercept i s found to be f = 2 GHz. Experimentally (See figure 23) the intercept occurs at 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 corrections as well as to end-effects (discussed i n Appendix B). d i e l e c t r i c resonator modes B, C, D. It might be suggested that since these modes have the same slope as mode A i n the mode pl o t s , that they are i n fact also hybrid coaxial modes. This i s undoubtedly true, however i t has highly i n s e n s i t i v e to the value of R^. This i s because one expects the f i e l d s to decay exponentially i n the immediate v i c i n i t y of the resonator and thus, the i n t e r a c t i o n with the waveguide walls w i l l be small. This i s observed experimentally i n the f a c t that as the sample i s withdrawn, the frequency of the d i e l e c t r i c modes changes very l i t t l e f o r a l l but the lowest z mode. Thus, the solutions found f o r the E ^ m modes should not be expected to be strongly perturbed by the waveguide. The Q's of the modes give information on the imaginary part of e^. This i s most expediently obtained by making the s u b s t i t u t i o n s : These wall and end e f f e c t s have not been studied f o r the been shown (24) that the resonant frequencies of hybrid coaxial modes are and -*• to o o o 2Q i n the eigenfrequency equations. One gets 57 Figure 25 shows the Q values as a function of 1/b2 for a B and D mode (The C mode was too weakly coupled to measure i t s Q). From the intercepts one gets: Q(16.9 GHz) = 200 ± 50 Q(25.3 GHz) = 270 ± 50 Using the values of found for the sample (sample 12), one consequently finds e£(16.9 GHz) = 17 ± 4 e"(25.3 GHz) = 16 ± 4 D Thus, to within 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 along the b-axis between 15 GHz and 25 GHz, has been measured using the d i e l e c t r i c resonator technique. For four samples analyzed i n d e t a i l at 4.2°K, the average value of the r e a l part i s found to be 4200; a frequency dependence i s indicated, however i t i s not clear 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 nature of the theory of d i e l e c t r i c resonators, A value of = 16 ± 4 was found, with no frequency dependence from 17 - 25 GHz. This l a t t e r r e s u l t implies a frequency dependence of the conductivity since = , hence 03 CT(OJ) a u where a(16.9 GHz) = .16 ± .04 (fi-cm)""1 a(25.3 GHz) = .23 ± .06 (ft-cm)" 1 The temperature dependence of the Q's indicates that the conductivity i s thermally activated f or T between 15°K and 25°K; f o r instance, the temperature dependence of the Q shown i n f i g u r e 21 gives an a c t i v a t i o n energy of 79°K. Two other samples analyzed gave a c t i v a t i o n energies of 90°K and 105°K r e s p e c t i v e l y . These activated energies are considerably lower than the value of 230°K found from d.c. measurements^. In a d d i t i o n , the microwave conduc-t i v i t y at 4.2°K i s over two orders of magnitude higher than the d.c. conductivity. These f a c t s can be understood by proposing that a f i n i t e density of states e x i s t s i n the semiconducting 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 impurities. If the states are strongly l o c a l i z e d and e l e c t r o n -electron c o r r e l a t i o n s are strong, then i n the gap region, the density of states for a one^-dimensional system would be as i l l u s t r a t e d i n f i g u r e 26. If the states near the Fermi energy are strongly l o c a l i z e d , then there w i l l be two main contributions to the conductivity: 1. Thermal a c t i v a t i o n across the semiconducting gap A^ = 2A 2. Thermally induced hopping from an occupied l o c a l i z e d state to an empty l o c a l i z e d s tate. The energy required w i l l be denoted by L^. The contribution from these mechanisms can be written as -A-i/kT L -Ao/kT a = e 1 + e z Generally A 1 » A 2 and » a ^ The temperature dependence of t h i s conductivity i s sketched i n f i g u r e 27. In p r a c t i c e , the ac 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 at very low temperature, A^ should be independent of temperature The frequency dependence of the conductivity due to hopping has (29) been shown by Mott to have the form g a (OJ) = a(to) - a , a <o a | C i Q • C • In 10 where v ^ i s a parameter dependent upon the strength of the electron-phonon i n t e r a c t i o n . When the compensation i s very low, S = 1. For doped s i l i c o n 0 8 -& (28) with very lowicompensaf ipn^ „for instance^taneto -,. 'a / dependence was found i n the f i g u r e 27. Temperature dependence of the conductivity when hopping conduction i s included. 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. Thus, i t is conceivable that this is also giving rise to the 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^ (e.g. ~ -^00) , SrTiOg (e' ~ 1500 at 60°K) ) 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 This would be d i f f i c u l t to do with TTF-TCNQ because the a v a i l a b l e samples have a £ .5 mm and one must work with lengths b ~ 10 mm. I t thus appears that f o r such highly a n i s o t r o p i c materials, further conclusive studies of the d i e l e c t r i c constants using d i e l e c t r i c mode an a l y s i s should await a better t h e o r e t i c a l understanding of the modes. 63 APPENDIX A : THE ANISOTROPIC DIELECTRIC SLAB The nature of the solutions for an anisotropic slab of thickness t and dielectric 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 for the TM(Hx = 0) modes. Assuming a dependence along the Y" and Z directions of the form gi(kyY + k zZ) ^ t ^ e n e i e c t r i c f i e l d i s written: E = 0 x i(k yY + k zZ) E = A (X) e 1 ( k y Y + k z Z ) E - A (X) e y y (A.l) where A^(X) and A^(X) are functions to be determined. Substituting into (2.7), one gets the coupled differential equations: A + H A = 0_ where the primes indicate differentiation with respect to X, A y(X) A z(X) (A. 2) (A.3) and M k V - k 2 o y z k k y z k k y z k 2ue - k 2 o z y (A.4) k 2 E to2 o —2" If one takes A « e^"a^X, then after substitution into y,z (A.2), the secular equation for non-zero solutions gives two values for a, 64 ai(±) = \ k2 y(e +e ) - (k 2+k 2) ± o y z y z k2u(e -e ) + ( k 2 - k 2 ) o y z y z + 4k2 k 2 y z Since R i s symmetric, i t may be diagonalized by an orthogonal transforma-t i o n , associated with a r o t a t i o n (by some angle 0^, say) i n the Y-Z plane,i.e. i f cos 9 - s i n 0 si n 9 cos 6 then R M R = '!(+) 0 'i ( - ) One finds that the angle 0^ i s given by 0 . = 1 s i n " 1 1 2 k k y z 2 (k 2u(e -e ) + ( k 2 - k 2 ) ) o y z y z J (A.5) Thus, the general solutions f o r A and A are y z s i n 0 • f A y(X) cos A z(X) - s i n cos 0 C ( + ) e l a i ( + ) X C e l a i < - ) X (-) (A.6) ' i ~ w " " i In an analogous way, the solutions outside the waveguide are E = 0 x E = B (X) e 1 ( k y Y + k z Z ) y y E = B (X) e 1 ( V + k z Z > z z (A.7) where 65 and B y(X) f cos • 9 sin 9 o o B z(X) V. J -sin 9 cos 0 o oj a 2 o(--) k2 -0 (k 2 + k 2) y z o(+) ia c(+)x V ) 6 V ) e l a ° ( - ) X k 2 o 9 = 1 s i n " 1 ° 2 k k .y z 2(k 2 - k 2) y z (A. 8) (A. 9) (A.10) (A.11) For well guided modes, a 2 , s < 0. However, a2,,s > 0 and so the - - B . ' o(-) o(+) fields contain propagating as well as damped contributions. The relative strength of these two contributions (i.e. D^/D^_^) can be found from matching the solutions at the boundary. After some very tedious algebra, the boundary value problem gives the following secular equation: a /,va / v S i n o(+) o(-) o(+) + a i ( + ) a i ( - ) c o s o(+) tan tan faK+)t] 2 { 2 J 2 g l a i ( - ) a o ( + ) S i n " g 2 a i ( - ) a o ( - ) C O S f K(+)t tan a i ( -A I 2 J 2 • f o(+) tan f aK- -A 2 \ J 2 V, J - g l a i ( + ) a o ( - ) C O S 8 2 a i ( + ) a o ( + ) s i n tan tan = 0 (A.12) where g± - cos 2 (9 ± - e Q ) g 2 = s i n 2 (6± - 9Q) 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. However, in the more general case, this does not occur; 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 first order, (See equation 2.11): k2 + k2 > e v k2 + k2 = e k2 > k2 y z — y z y o o hence, 2 k2. ° (+> - o „ o(-) o y z' k2 V 0 -(k2 + k2) * y z Thus, taking c t 2 ^ - 0, (A.12) becomes: a i ( + ) a i ( - ) t a n ai(+) t" tan 2 V J 2 'o(-) 8 l a i ( + ) t a n °t i ( + ) t + g 2 a i ( _ ) tan « 0 (A.13) The problem is now tractable by numerical methods. However, attempting to solve for the dielectric constants is s t i l l very diff icult . 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 for comparison with a f i r s t order theory When no z-dependence of the fields i s taken (i.e. k -> 0), z then 9 , 8. -> 0, and the fields reduce to those found in section 2.3 o i namely equations (2.20), (2.21) and (2.23); equation (A.13) reduces to (2.29). 68 APPENDIX B THE INFINITE COAXIAL LINE The modes of 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 rod of radius R. i n a c i r c u l a r m e t a l l i c waveguide of radius can be solved f o r when £^ = e y = ej_(See f i g u r e 28). The equations f o r the f i e l d s i n rectangular coordinates given by (2.14 ) to (2.19 ) can e a s i l y be transformed to c y l i n d r i c a l coordinates ( r , <j>, z) by using the transformations: and F r I J 3f 3r 3f 3c|> COS (j) - s i n <j> s i n cb cos <j> cos <j> -rs i n <j> s i n <j> rcos <j> F X F y 3f_ 3x 3f 3y > where F and f are a vector and scalar r e s p e c t i v e l y . One f i n d s that i n s i d e the d i e l e c t r i c : E r = i k z 3E z + toy 3r c 1 SH 1 z r 3<|) (B.l) \ = i 'k z r 3E z toy_ 3tb c 3H z 3r (B.2) H r = kT 'c5H z 3r COE^ X c r 3E ) z 3<|> J (B.3) = i k z r 3H z to 3<t> c 3E 3r (B.4) where 1- CO 2 c ^ y £ j_ - k 2 (B.5) The f i e l d s are expressed i n terms of E and H , hence, a complete s o l u t i o n z z i s reduced to solving f o r E and H . Equation (2.7 ) f o r E i n c y l i n d r i c a l z z z coordinates i s Figure 28, 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. 70 92E i 3E 92E z + 1_ z 1 z l k r 9r r 2" g^ 2 z 9r' 1 E + 9 E r 1 ffjfe. — r + - — + — — r 9r r 9<j> , i k E + z z a)2 ye - k 2 — T z z E = 0 z The equation for i s easily shown to be 92H . 9H . 92H z + 1 z + 1 z + 9 r 2 r 9r r 2" "g^2"" Writing the solutions i n the form 0) ye - k 2 —>r z z H 2 = 0 E ( r , <(,) = F(r) e z H z ( r , <fr) =c G(r) e (B.6) (B.7) where n i s an integer, then equations (B.6) and (B.7) become 9 2F 1 9F . 9 r z r 9r k? i r z F = 0 and (B.8) 92G 1 9G 9 r 2 r 9r k 2 - n 2 -1 r ^ G = 0 where ,2 e / o 1 = —• ^ (B.9) (B.10) The solutions to these Bessel equations which are f i n i t e at r = 0 are: F(r) = A J (k r) i n l G(r) = A 2 J n ( k ± r ) ( B . l l ) (B.12) where A^ and are constants. Outside the rod, where the medium i s is o t r o p i c , the solutions are written as F(r) = B. H ( ^ ( k . r ) + C. H (* } (k_r) 1 n l 1 n 2 G(r) = B 2 H ( ^ ( k 2 r ) + C 2 ( k 2 r ) (B.13) (B.14) where B +, B„, C.. and C„ are constants; and are Hankel functions of 1 2 1 2 n n the f i r s t and second kind; and 71 k 2 = O J 2 y e - k 2 Z —y O O Z c z (B.15) I t i s clear from equations (B.l) to (B.4) that i f the f i e l d s have any angular dependence ( i . e . n > 0) , the solutions cannot be separated into the usual TE and TM types. Such modes, called hybrid, w i l l not be solved f o r , here. When n = 0, the TM (H =0) modes have the following f i e l d s : z where E, = H = H =0 <j> r z E = i r 5 5 * kT k 3 E z k e , z o J. - — 3r A i J„< ki r> 1 o i 5 r<R. E z ( r ) = ^ B, H (^(k„r) + C. H ( 2 ) ( k . r ) ; R 1 < r<R, i o z i o z (B.16) Before proceeding to solve the boundary value problem, some insight into the behaviour of the solutions w i l l be sought. In the l i m i t R 2 -> 0 0 ( 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 waveguide i s obtained), a guided wave can only propagate i f k 2 < 0. This i s because the asymptotic values of the Hankel functions are H ( i ) ( w ) - A T o v irw i(w - 1/4TT ) H(2>(w) o fl_ e TTW •i(w - 1/4TT ) Therefore i f = i y where y e Re, then H ^ d y r ) - e ^ r 72 In addition, i n order to s a t i s f y the boundary value problem, one must have k 2 (and hence k 2 ) > 0. Thus from (B.5) and (B.15), we have the condition 1 < O J 2 < 1 - —9- -where the l e f t and rig h t hand equal signs apply i n the short and long wave-length l i m i t s respectively.(See figure 29). In contrast to metallic c a v i t i e s , the (w/c R^f vs O^R-^f characteristics do not terminate at (k R.) 2 = 0 but have a low frequency cutoff when k 2 ->• 0, i . e . when k to/c. Z J- Z> z A high frequency cutoff also occurs when k 2 -»• 0, i . e . when k fe^ u/c. For f i n i t e radius..'Rj,"^ may be positive; as.well as negative. Thus no low frequency cutoff, condition e x i s t s - i n this-case" When k 2 > 0, the solutions J and Y to the Bessel equation are more convenient to o o ^ use than and . The boundary conditions which determine A- , B, o o J 1 1 and C, are that E and H1 can be continuous at r = R, and that E_ be zero at r = L A l Jo ( klV " B l Jo ( k2 Rl ) " C l V W = 0 > ^i V W + f k i k: B l J i < k 2 R l ) + C l Y l ( k 2 R l ) = 0 B l Jo ( k2 R2> + C l Y o ( k 2 R 2 ) " 0 (B.18) (B.19) (B.20) A non-zero solution exists 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: Jl<k2Rl> Y o ( k 2 R 2 ) " Jo ( k2 R2> V W (B.21) £z j ( k j l ^ ) k l = 0 Figure 29. Mode plo t for 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. 0 0 .01 .02 03 .04 .05 Figure 30. Mode plo t for the TM^Q c o a x i a l mode f o r various values of R0/R,, when e *= 10 and e,= 2. Z. L Z -L 75 The q u a l i t a t i v e features of these solutions have been discussed by Hardy et a l . for the case of an anisotropic centre conductor. The features relevant to the present problem can be seen by solving equations (B.10), (B.15) and (B.21) simultaneously for different values of R2^ Rl' e± a n c* E z " Figures 30 and 31 show numerical results for k H, << 1 which w i l l be the Z X region appropriate for the resonances observed i n TTF-TCNQ. An effect of the outer metallic w a l l for fixed values of and e^is to modify the slope and intercept point of the mode plot as shown i n figure 30. When R ^ ^ i and e are kept constant and e i s varied, again, the slopes and intercept J. z points are seen to change (Figure 31). The qu a l i t a t i v e features of the solutions infered from these solutions are: 1. The slopes for << 1 are less than 1 but approach unity as R2/R.J-* 0 0 2. The slopes decrease as £, and £ increase i . z 3. For fixed R2/R^ > the intercept point decreases with increasing value of Further analysis has shown that when E /E >> 1, then the z j . intercept point i s not very sensitive 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 neglecting end e f f e c t s , k = Tr/b. By considering the f i e l d configurations i n figure z 32, one would expect the end effects to be comparable to those for the (6) metallic resonator which were found to be of the order of 10% . A qualit 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 figure 32. The end effects can b considered as equivalent to the sample having an eff e c t i v e length b + A where 0 < << 1. Then f 2 « (b + A) 2 or for small A/b: b f 2 cc i b 5 1 - 2 4 b Figure 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 with a centre section of f i n i t e length when (a) 4 ™ > > 1 0>> W « ! 77 REFERENCES 1. J.T. Tiedje; M.Sc. Thesis; U.B.C.; 1975 2. A.J. Berlinsky, J.F. Carolan, L. Weiler; Solid State Comm.; 15, 795 (1974) 3. R.E. P e i e r l s ; Quantum Theory of Solids; (Oxford University Press, London, 1955) 4. L.B. Coleman, M.J. Cohen, M.J. Sandman, D.J. Yamagishi, F.G. Garito, A.J. Heeger; Solid State Comm.; 12, 1125 (1973) 5. J. Bardeen; Solid State Comm.; 13, 357 (1973) 6. W.N. Hardy, A.J. Berlinsky, L. Weiler; Phys. Rev.; 14B, 3356 (1976) 7. Thomas et a l . ; Phys. Rev.; 13B, 5105 (1976) 8. H. FrOhlich; Proc. Royal Soc; A223, 296 (1954) 9. P.A. Lee, T.M. Rice, P.W. Anderson; Solid State Comm.; 14_, 703 (1974) 10. D.B. Tanner, CS. Jacobsen, A.F. Garito, A.J. Heeger; Phys. Rev.; 13B, 3381 (1976) 11. L.B. Coleman, CR. Fincher, A.F. Garito, A.J. Heeger; Phys. Stat. Sol. (B); 75_, 239 (1976) 12. S.K. Khanna, E. Ehrenfreund, A.F. Garito, A.J. Heeger; Phys. Rev.; 10B, 2205 (1974) 13. J.P. Ferraris, T.F. Finnegan; Solid State Comm.; 18_, 1169 (1976) 14. W.J. Gunning, S.K. Khanna, A.F. Garito, A.J. Heeger; Solid State Comm.; 21, 765 (1977) 15. J.E. Eldridge; B u l l . Amer. Phys. Soc; 20, 495 (1975) 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. Garito, A.J. Heeger, R.C Jacklevic; Solid State Comm.; 16, 667 (1975) 18. CM. 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 78 19. H.Y. Yee; Hansen Labs., Stanford Univ., Stanford, C a l i f o r n i a ; M.L. Report 1065; July 1963 20. R.D. Richtmyer; J . Appl. Phys.; 10, 391 (1939) 21. A. Okaya, L.F. Barash; I.R.E. P r o c ; 50, 2081 (1962) 22. E.A.J. M a r c a t i l i ; B e l l Syst. Tech. J . ; 48, 2071 (1969) 23. These corrections for the TM mode are discussed i n Reference 6 where i t i s found that the cor r e c t i o n i s of the order of 5%. 24. M. Jaworski; P r i v a t e communication. 25. R.O. B e l l , G. Rupprecht; I.R.E., Trans, on M.T.T.; 9, 239 (1961) 26. N. Mott; Metal-Insulator Transitions; (Taylor & Francis Ltd., London, 1974); Chapter I 27. J.E. Eldridge; S o l i d State Coram; 21, 737 (1977) 28. M. Pollack, T.H. Geballe; Phys. Rev.; 122, 1742 (1961) 29. N,F, Mott, E.A. Davis; E l e c t r o n i c Processes i n Non-Crystalline  Materials; (Clarendon Press, London, 1971); Chapter 4 

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