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Acoustic wave propagation in TTF-TCNQ Tiedje, J. Thomas 1977

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ACOUSTIC WAVE PROPAGATION IN TTF-TCNQ by J. THOMAS TIEDJE B.A.Sc, University of Toronto, 1973 M. Sc. , University of British Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1977 (c) J. Thomas Tiedje In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 7/77 ABSTRACT Detailed measurements have been made of the temperature dependence of the velocity of three different modes of sound propagation in TTF-TCNQ crystals, in the range 7~300K. Values for the a and b axis Young's moduli and the shear modulus C55 are inferred from the sound velocities. TTF-TCNQ. is found to be stiffer perpendicular to the conducting direction than parallel to it. The elastic anisotropy is typical of crystalline solids even though the anisotropy of the electrical conductivity is unusually large'. A small (1.5%) increase in the velocity of extensicnal waves below the meta1 -insulator transition is interpreted as being due to the disappearance of the conduction electrons. A quantitative theory of the low temperature velocity anomaly leads to an accurate estimate of the q -»• 0 e 1 ectron-phonon coupling constant. The sound velocity measurements were made using an acoustic resonance technique. Resonant modes of vibration of single crystals of TTF-TCNQ were excited electrostatically and detected capacitively using a UHF carrier signal. The detection scheme is shown to be more sensitive than conventional d.c. biased capacitive pickups. A theoretical study of the electronic contribution to the attenuation of sound in one and two dimensional metals and semiconductors is pre sented. The attenuation in one dimensional metals is shown to be anomalously small. In both one and two dimensional metals, in the quantum limit the attenuation depends strongly on the direction of propagation of the wave. A transport equation solution to the problem of calculating the amplification of sound waves in a solid in the presence of a d.c. electric field is described. The treatment is much less complex than any that is currently available. i v TABLE OF CONTENTS Page ABSTRACT •• TABLE OF CONTENTS iv LIST OF FIGURES vii LIST OF TABLES * ACKNOWLEDGEMENTS xi INTRODUCTION 1. The Organic Conductor TTF-TCNQ 1 2. Organization of the Thesis 7 PART A: Electronic Contribution to Attenuation and Amplification of Acoustic Waves 9 I ATTENUATION IN METALS 10 1.1 Introduction1.2 Transport Equation Approach to Ultrasonic Attenuation 12 1.3 The Quantum Limit 2\.k Metals 2? (i) Three Dimensional Metals 27 (ii) Two Dimensional Metals 31 (iii) One Dimensional Metals 6 1.5 Anisotropy of Attenuation ^ (i) One Dimensional Metals ^5 (ii) Two Dimensional Metals 50 1.6 Summary 5II ATTENUATION IN SEMICONDUCTORS 57 2.1 Quantum Limit 52.2 Transport Equation Approach 59 2.3 Metal Semiconductor Transition 6A III AMPLIFICATION 71 3.1 Introduction3.2 Transport Equation 72 3.3 Energy Transfer 9 3.k Conclusion ^5 V PART B: Measurements on TTF-TCNQ 87 I EXPERIMENTAL METHOD 88 1.1 Capacitive Measurement Technique 88 (i) Electronics 9° (ii) Sample Support 100 1.2 Sensitivity of the Measurement Technique. 104 (i) Minimum Detectable Length Change .... (ii) A.C. Method 106 (iii) D.C. Method 1Q9 II THE MODES OF VIBRATION OF TTF-TCNQ, CRYSTALS.. 1T» 2.1 Low Frequency Modes of an Elongated Plate 11** (i) Flexural Modes 115 (ii) Torsional Modes 119 (iii) Comments on a Short Plate 121 (iv) Elongational Modes 123 (v) Crystallographic Symmetry 125 2.2 Interpretation of Experimental Mode Spectrum 130 (i) Vibrating Reed Support 132 (ii) Central Pin Support 9 (iii) Mode Coupling ]j*3 (iv) Support Modes .. 2.3 Vibration Damping 1^5 (i) Q Measurement 1~*-> (ii) Thermoelastic Damping 1^' (iii) Elongational Modes 152 (iv) Effect of Air on Resonance Fre quency and Q 155 III INTERPRETATION OF TEMPERATURE DEPENDENCE OF SOUND VELOCITY 162 3.1 Overall Temperature Dependence 162 3-2 Low Temperature Anomaly 17° (i) Quantum Limit 1°(ii) Thermodynamic Limit 191 (iii) Comment on Acoustic Absorption .... 199 SUMMARY 20lt 1. The Main Results of this Work 204 2. Suggestions for Further Work 7 vi APPENDIX Pa9e 1. Diode Temperature Sensor Calibration 210 2. Circuit Diagrams 211 (i) Diode Detector Circuit 21(ii) MOSFET Preamplifier 21(iii) Phase Shifter 212 (iv) Lorentzian Generator 21M (v) Signal Averager Input Amplifier 21H 3- Thermal Expansion Correction 215 BIBLIOGRAPHY 216 vi i LIST OF FIGURES Figure Page 1. TTF and TCNQ Molecules 2 2. . TTF-TCNQ Crystal Structure 3 3. TTF-TCNQ b Axis Conductivity 5 k. Temperature Dependence of Electronic Energy Gap 6 5- Crossection of the Fermi Sphere Showing the Inter action Surface 25 6. Acoustic Attenuation as a Function of q£ for 2 and 3D Metals 35 7. Fermi Surface and Interaction Surface in a 1D Metal..37 8. Angular Dependence of Attenuation in a 1D Metal in the Quantum Limit ^6 9. Angular Dependence of Attenuation in 1D "eta' for A = 8 *»9 10. Angular Dependence of Attenuation in a 2D Metal in the Quantum Limit 52 11. Angular Dependence of Attenuation in a 2D Metal for A = 8 53 12. Interaction Surfaces - Metallic Band 66 13. Interaction Surfaces - Semiconductor Band 67 1A. Attenuation Near a Meta1-Semi conductor Transition in a 1D Metal in the Quantum Limit 68 15. Drive and Detector Circuit 89 16. Drive and Bridge Detector Circuit 89 17. Block Diagram of Electronic Equipment 93 18. Sample Mounting Configurations 95 19. Photographs of Mounted Samples 6 20. Cut-Away View of Sample Box 97 vi i i Figure Page 21. Photograph of the Outside of the Sample Holder 98 22. Photograph of the Inside of the Sample Holder 99 23. Shielded Electrode 101 2k. Special Notched Electrode 1025- Cryostat for Low Temperature Measurements 102 26. Capacitive Vibration Detector - A.C. Method 106 21. Capacitive Vibration Detector - D.C. Method 109 28. Noise Equivalent Circuit 110 29. Contour Map of Sensitivity of D.C. Method 112 30. Flexural Mode Shapes 117 31. Modes of a Square Cantilever Plate ....122 32. Modes of a Rectangular Free Plate 122 33- Arrangement of TTF and TCNQ. Molecules in the a c Plane 124 3**. Experimental Flexural Mode Spectrum 131 35. Low Frequency Flexural and Torsional Mode Dispersion Diagram 133 36. Effect of D.C. Bias Voltage on Flexural Resonance Frequency 135 37. Longitudinal Mode Dispersion Diagram 138 38. Flexural and Torsional Mode Crossing 1*2 39. Interference with Fac Flexural Mode I*4** i«0. Wiggles 1*7 k\. Flexural Mode Damping 1*9 k2. Longitudinal Mode Dampi ng 151 ix Figure Page 43- Sound Velocity and Attenuation Near the Metal-Semiconductor Transition 153 44. Air Entrained by a Flexural Mode 156 45. Air Entrained by a Torsional Mode 157 46. Temperature Dependence of b Axis Elongational and Torsional Mode Velocities 163 47. Temperature Dependence of b Axis Flexural Mode Velocity 164 48. Temperature Dependence of a Axis Flexural and Torsional Mode Velocities 165 49. Low Temperature Anomaly in the Young's Modulus Velocity 177 50. Excitation Spectrum for a Non-Interacting 1D Tight-Binding Band 179 51. Dyson's Equation 1d2 52. Electron Gas Polarization Diagram 183 53. Hartree Polarizabi1ity for a 1D Tight-Binding Band...184 54. Acoustic Phonon Dispersion for Experimental Electron-Phonon Coupling Constant 187 55- Acoustic Phonon Dispersion for an Electron-Phonon Coupling Constant which gives Tc = 54K 188 56. Band Structure of TTF-TCNQ. 192 57- Strain Dependence of the Density of Occupied States..193 58. Low Temperature Longitudinal Mode Dampi ng 201 LIST OF TABLES Table Page I TTF-TCNQ. Material Parameters *7 II Attenuation for 1, 2, 3D Metals in a D.C. Electric Field 81 III Flexural Mode Parameters 116 IV Flexural Mode Elastic Constants ..128 V Room Temperature Sound Velocities for TTF-TCNQ, \h\ VI Effect of Air on Flexural and Torsional Mode Frequencies 158 VII Young's Moduli for Various Materials 162 VIII Griineisen Constant- for Various Materials 17* IX Pressure Derivative of Bulk Modulus for Various Materials 176 X Summary of the Experimental Results 206 ACKNOWLEDGEMENTS I am grateful to my supervisor Rudi Haering for the considerable amount of time and effort he has spent helping me with this project. His physical intuition and common sense were invaluable, to say the least. The capacitive measurement technique was suggested by Walter Hardy, and his help was essential in making it work. The quantum approach to the interpretation of the low temperature velocity anomaly was suggested by E. Tosatti. All aspects of the interpretation of the temperature dependence of the sound velocities were worked out in collaboration with Manfred Jericho. 1 benefited from many useful discussions with B. Bergersen and W. I. Friesen. I would like to thank L. Weiler for supplying me with TTF-TCNQ samples and for explanations of various chemistry problems. The TTF-TCNQ crystals used in the experiment were synthesized and grown by Y. Hoyano. It is a pleasure to thank Susan Haering for taking, developing and printing the photographs in this thesis. Lore Hoffmann simplified the final production process with her efficient typing and thoughtful approach to the organization of the thes i s. I am grateful to the National Research Council for financial support in the form of a Science Scholarship. 1 INTRODUCTION 1. The Organic Conductor TTF-TCNQ Tetrathiofulvalinium (TTF) tetracyanoquinodimethanide (TCNQ) is an electrically conducting organic solid composed of the TTF and TCNQ molecules shown in Fig. 1. The material has several unusual proper ties. First it is a better conductor than almost any other organic material known. In fact, at 60K its conductivity (lO4 (fi-cm)"1) is comparable with mercury at room temperature. The availability of good organic conductors opens up the possibility of making materials with desirable electronic properties by chemical modification of the constituent molecules. Potential applications include new or improved electronic devices and higher temperature superconductors. A second unusual feature of TTF-TCNQ is that its conductivity is very anisotropic. The anisotropy arises from the nature of the crystal structure; as shown in Fig. 2, the large flat TTF and TCNQ molecules are arranged in segregated stacks. The molecular orbitals for neigh bouring molecules on the same stack overlap much more strongly than the molecular orbitals for molecules on different stacks. The result is that electrons are able to move more freely along the chains than perpendicular to the chains. The corresponding anisotropy in the con ductivity is large enough that the material may be regarded as a nearly one dimensional metal. Theory predicts a number of unique properties for one dimensional conductors. One of these characteristics, namely the Peierls insta-Fig. 1 TTF and TCNQ Molecules 3 Fig. 2 - Photograph of a model of the TTF-TCNQ. crystal structure bility, causes a structural phase transition in which the one dimen sional metal changes into a semiconductor. Electrical conductivity measurements reveal that TTF-TCNQ does undergo a phase transition of this type at low temperatures (see Fig. 3). X-ray and neutron scattering experiments seem to confirm that the phase transition is a Peierls transition. However, below the metal-semiconductor transition there is at least one and possibly as many as three (Djurek et al 1977) additional phase transitions. The two most prominent phase transitions, near 38K and 5^K, are best illustrated by the temperature dependence of the electronic energy gap in the semiconducting phase as shown in Fig. 4 (Tiedje 1975). The phase transitions are reflected in sharp increases in the electronic energy gap 2A, as a function of temperature. The nature of these phase transitions is not well understood. Most of the initial work on TTF-TCNQ was stimulated by the ob servation in a few samples of TTF-TCNQ of an anomalously high con ductivity peak (Coleman et al 1973) just above the meta1 -insulator transition. The anomalous conductivity was interpreted at the time as due to superconducting fluctuations enhanced by the onset of the Peierls distortion. No one has succeeded in duplicating these measure ments, although there have been many attempts. For reviews of recent work on TTF-TCNQ and related materials see Bulaevskii (1975), Andre et al (1976) and Keller (1977). 5 Fig. 3 - Temperature dependence of the TTF-TCNQ b axis conductivity. Sample §2k came from the same batch as many of the samples on which acoustic measurements were made. 6 O o ro o H 7? 4^ O ACT) (K) B O O cn O ro o o 1 1 1 X © CO > > r~ r~ X V m m s\ X X @ x — — © X e x © X @ X © X © © X @ X • © u. © X c S X © X © X © X © X © X ® X — ©X GX © < T 1 1 1 Fig. k - Temperature dependence of the electronic energy gap (2A) in the insulating phase of TTF-TCNQ. Notice the change in slope near 52K and 38K. 7 2. Organization of the Thesis The remainder of this thesis is divided into two parts. In Part A we investigate the effect of the dimensionality of the electron gas on the electronic contribution to the attenuation of sound in metals and semiconductors. Both electromagnetic and de formation potential coupling between the electrons and the sound wave are considered. The dependence of the attenuation of high frequency acoustic waves on the direction of propagation of the wave is calculated for one and two dimensional metals. An im proved transport theory of the amplification of sound in the presence of a d.c. electric field is also presented. In Part B we discuss some experimental measurements on the pro pagation of sound in crystals of TTF-TCNQ.. The large anisotropy in the electrical properties of TTF-TCNQ are illustrated by electrical conductivity measurements (Hardy et al 1976) dielectric constant measurements (Cohen et al 1976) and molecular orbital calculations (Berlinsky et al 197*). Similarly the temperature dependence of the lattice constants (Blessing and Coppens 197*) and the nature of the bonding in the crystal suggest that the lattice may be elastically highly anisotropic as well. In order to measure the anisotropy in the elastic properties and to help clarify the nature of the low tem perature phase transitions detailed measurements have been made of the temperature dependence of the velocity and attenuation of sound in TTF-TCNQ. 8 Part B is divided into three chapters. In the first chapter the capacitive technique that was used to excite and detect acoustic re sonances in single crystals of TTF-TCNQ is described in detail. An analysis of the sensitivity of capacitive displacement detectors is included. The second chapter explains how the vibration spectrum can be used to determine a number of different elastic constants for TTF-TCNQ. The principal damping mechanisms for samples vibrating in air and in a vacuum are discussed. The final chapter contains an interpretation of the temperature dependence of the velocity of sound. A small low temperature anomaly in the velocity is interpreted as being due to an electronic con tribution to the elastic moduli. A possible explanation ror an in crease in damping of some of the acoustic modes at low temperatures is proposed. PART A ELECTRONIC CONTRIBUTION TO ATTENUATION AND AMPLIFICATION OF ACOUSTIC WAVES 10 CHAPTER I Attenuation 1.1 Introduction The electronic contribution to ultrasonic attenuation has been studied extensively in three dimensional metals (Pippard 1955. Cohen et al I960, Rice and Sham 1970). The recent discovery of highly anisotropic quasi one and two dimensional metals has stimulated Interest In the properties of electronic systems of reduced dimen sionality (Glaser 197*, Wilson et al 1975). In this chapter we extend the theory of ultrasonic attenuation so that it applies to such systems We present general expressions for the attenuation constant of longi-tudinal and transverse waves, and we obtain limiting expressions valid In the hydrodynamical and in the quantum mechanical limits. We show that the ultrasonic attenuation in one and two dimensional systems differs significantly from the attenuation in three dimensional structures. The difference is particularly significant in the case of one dimensional systems where energy and momentum selection rules are difficult to satisfy. It Is well known that the absorption of sound in metals depends on the relation between the mean free path of an electron at the 2jL q In Fermi surface, £, and the wave length of the sound wave, X = this chapter, the Boltzmann transport equation is used to calculate the attenuation as a function of frequency for arbitrary values of ql. Quantum mechanical perturbation theory is also used to derive the attenuation in the quantum limit. Although other contributions to the attenuation will be present In real metals, only the electronic contribution will be considered here. With this limitation the ultrasonic attenuation problem is formulated for a three dimensional electron gas using the Boltzmann equation in the relaxation time approximation. The electrons are assumed to interact with the lattice through "collisions", self-consistent electromagnetic fields, and a scalar deformation potential. Then with the deformation potential interaction alone, the high frequency (q£ » 1) limit is rederived using quantum mechanical perturbation theory. The two approaches mentioned above are then specialized to the three, two, and one dimensional electronic systems. In each case the attenuation is calculated explicitly for free electrons. in all cases the electrons are assumed to be contained in a three dimensional crystal lattice. 1.2 Transport Equation Approach to Ultrasonic Attenuation The Boltzmann equation, in the relaxation time approximation, js given by (see e.g. Ziman 1972) sound wave, and f is the e'ectron distribution function. De formation potential coupling is included at a later stage. The electrons tend to relax towards the local equilibrium distribution assume that the electrons relax to an equilibrium distribution function centered at the local lattice velocity. Also, since the scattering processes are local, the electron density is not affected. For this reason the local equilibrium chemical potential must be de termined self-consistently from the as yet unknown local electron density. The velocity of an electron In an arbitrary band structure 's» — = ^ Z-k*-^)' Hence the energy of an electron in a frame of reference moving with the local lattice velocity u_ is given by [1] where E Is the self-consistent electric field generated by the function f, through scattering. Following Holstein (1959) we e'(k) = e(k) -fik-u Adding the term 7ik;u_ to the electron energy corresponds to tilting the band structure In space. To first order thts corresponds to a Fermi surface shifted in k_space by . Thus, the local equilibrium distribution function Is T (k, x, t) -f. (e'(k); v(x, t))« f. - (*k- u + n ^ where u(x, t) is the self-consistent chemical potential mentioned above, f0 is the equilibrium electron distribution in the absence of e sound wave, and rsi is the .oscillating component .of the electron number density. The Boltzmann equation may now be solved to first order by linearizing. Let f = f + fi where f\ « ^X is the o oscillating component of the distribution function that is induced by the sound wave. Equation [1] can now be written ~ »"fl + |i ZkE OO-afj - e|-V ke(k) §{° This solution to first order In E, n^, u, and fj, all of which are proportional to the amplitude of the sound wave, Is [2] fx « 3u T e v-E - ti k-u - •—• nx 3n 1,1 3f, 1 - lux + I ajvT 3e The electronic current density is defined by J —c 2e (2^) I 3 I d3k Vfi [3] mu a- E — E • — - ni e S R where S => ^ is the phase velocity of the sound wave, and 2e2x [4a] a - 7^3 J d3k 1 - iojx ^ f.3£.\ iojx + ig/vr \ 9e / 2e2r ti [4b] I = -^73 ~ d3k  1 - iuT + igjyj I 3e J ^ a-fir TT£P d3k 1 - i OJX + iqjvr ^ 3c y The total current associated with the sound wave consists of the sum of the ion current and the electronic current which Is Induced 15 by the self-consistent electric field. The total current density Is J - J n e u — —e — If we restrict our attention to monovalent metals, the electron number density n will equal the ion number density. The continuity equation for electrons Is to nj e + q_*J.e = 0 or [S] Je„ - " nie'S where Je|, Is the component of Je parallel to g_. The self-consistent electric field is determined from the total current density by the Maxwell equations (Kittel 1963). The result is [6a] and [6b] E „ - hlL (j „ + n e u„ J L The next step Is to obtain expressions for the self-consistent electric field and current In terms of the lattice velocity. The problem can be simplified by assuming that the sound wavevector q_ Is In the direction of a principal axis of the electron constant energy surfaces. In this case the conductivity tensors are all diagonal, and the components can be treated separately. in the case of longitudinal waves (qjru), [5] can be used to eliminate n^ from [3] to give. mu G E XX XX where a' = =— , E' - -z -— , and the wavevector q Is In XX I "" K XX 1 — K ~LM X X the x direction. For longitudinal waves, only the x components of _J , E_ and u_ are non-zero. Now if [6a] is substituted for E , then E1 knio xx xx [8] J = - n e u 1 J ex x Oo 0) ^TTIO' 1 + XX. CO n e^T where a0 ° . Notice that J a - n e u .when co «a' ~ E 1 . m x xx xx In this (perfect screening) limit the Ion current Is exactly matched by the electron current. An expression for E ts obtained by sub-X stltutlng [8] Into [7]: [9] neu xx E« xx XX 1 + 0) XX bl E' xx Oo In the transverse wave case (q_1 u), there is no electron density oscillation hence. mu MOl J - o E - E —^-LAUJ ey yy y yy ex where.a is in-thex direction, and the latti.ce-wave is polarized In the y direction. Substitute [6b] into [10] then J = - n e u ey y y _ U) \c M. (o \c/ yy and E » -y neu yy yy 0O «»> W yy yy i - yy The screening of the transverse waves Is less efficient than the screening of the longitudinal wave because the charge carriers Inter-18 act through magnetic forces rather than through the stronger electric forces. The factors of ( —| reflect this difference. The expressions may be written more compactly by observing that the contribution of the conduction electrons to the dielectric constant of a metal with conductivity cr, Is 4TTI Thus, for longitudinal waves [11] x n e u xx CO XX - 1 and [12] J =• - n e u ex x 1 + r*1 -"xx XX where e' = 1 + a* xx u xx Also for transverse waves n e u .. /_\2 y HIT i / SY t u (cl Eyy w [13] yy - l and [14] 19 where «„. =» 1 - Q ayy The work done on the electrons per unit time by the sound wave Is (Blount 1959) where the electronic energy band has been assumed to be parabolic. The first term In [15] Is the ohmlc loss due to the presence of the self-consistent field. The second term is a viscous drag effect, which results from the electron-lattice collisions where by the electrons reach local equilibrium with the lattice. This effect does not depend on the charge of the electrons, and Is present even In the absence of the self-consistent field. However, since internal electric fields force the electrons to follow the lattice wave closely, the inclusion of a self-consistent field greatly reduces the collision drag term. As a result the first term In [15] dominates, except at very high frequencies. The attenuation of the sound wave Is found by dividing the power 1 i 2 absorbed per unit volume, P, by the energy flux Tr P |u_| S v/here p Is the density of the lattice. Thus P The attenuation of longitudinal and transverse waves Is found by substituting [11], [12] and [13], [14] respectively, Into [15], For longitudinal waves we find [16a] a pSx nm Re whereas for transverse waves [16b] where .€ = 1 + for? 0) \cj " The last two results are new expressions for ultrasonic attenuation. So far, the coupling between the sound wave and the electrons has been assumed to be due to collisions and to a self-consistent electromagnetic field. A deformation potential tensor can also couple the electrons to the wave. If the deformation potential is a scalar, it affects only the interaction of electrons with longitudinal waves, and does not change the interaction with transverse waves. If a scalar deformation potential, C, Is Introduced into the Boltzmann equation (see Harrison I960, or Tucker and Rampton 1972), and the expression for the power dissipation is suitably modified, It can be shown that the attenuation of acoustic waves Is given by c where D = ,, ? determines the relative Importance of electromagnetic and deformation potential coupling. For example, If C » —jj*r- , the deformation potential will dominate. The three terms in [17] may be inter preted as follows. The first term is due to the self-consistent field plus the screened collision coupling. The last term is the screened deformation potential alone, and the second term is a cross term which . I n.c uudas „con tr 5 bu ti ons-. from .-all - »three-<mechan«sras. We now develop an alternative method which is valid in the quantum limit, and includes only the deformation potential inter action. 1.3 The Quantum Limit If the energy tiii) of the externally impressed phonon is larger than the uncertainty in the electronic energy ^ due to the collisions, then one is said to be working in the quantum limit. In this limit quantum mechanical perturbation theory can be used to evaluate the attenuation of a sound wave. The following additional assumptions are made: 1. The electrons are free, except for dimensionality constraints. 2. The phonon dispersion is linear. 3. The phonon energy fica is much smaller than the Fermi energy. 4.. The .elect ron-phonon, i n terac tion. is ..adequately .des cr i bed by the standard interaction Hamiltonian where C is a scalar deformation potential, a^ is a phonon creation operator, and c* is an operator which creates an electron with wavevector k_ and spin a. Most of the remainder of this section is material which has been described in detail elsewhere (Kittel 1963, Tucker & Rampton 1972). It is included here to provide a framework for dealing with the one and two dimensional systems. The probability per unit time that a phonon of wavevector c|_ Is absorbed by. an electron Is and the probability that a phonon q Is emitted Is - 1>u - e (k-gjj where the rates are normalized to unit volume, and fo(k), n(q) are the equiIibrium electron and phonon distribution functions respectively. Acoustic attenuation can be defined as the net rate of absorption of phonons In the mode q_ divided by the phonon flux n(q_)S. Thus where we have assumed that kT » tiw. This expression may be evaluated by converting the summation to an Integral. Then riot 1 c2(TI d3k ^f0(k). ~ f0(k+ou)^fi ^e(k) + fiu> - e(k + ojj It Is of Interest to Investigate which electronic states contribute to the attenuation. Wavevector ("momentum") must be conserved In any phonon emission or absorption process, otherwise the matrix element of the electron-phonon Interaction is zero. In an absorption process, for example, k_l=k.+CL. Furthermore, in the golden rule approximation used here, the transition rate Is zero unless energy Is also conserved. Thus e(k,') = e(k) + TiSq For free electrons the energy conservation requirement reduces to TiSq m 2m * The energy and momentum conservation requirements define a set of electronic states which can Interact with the sound wave. These states lie In the vicinity of a surface in k space, across which an electron scatters in any phonon absorption or emission process (PIppard 1963). A crossectlon of this Interaction surface Is shown In Fig. 5» for a three dimensional Fermi sphere. Clearly the major .25-' INTERACTION SURFACE Fig. 5 - Crossection of the Fermi sphere showing the interaction surface contribution to the scattering rate will come from the neighbourhood of the Fermi surface where there are empty electronic states for the electrons to scatter Into. As will be shown In a later section, this situation Is drastically altered In one dimensional metals. 27 1.4 Metals (i) Three Dimensional Metals Acoustic attenuation in three dimensional metals has been studied extensively. The results will be presented here for comparison with the one and two dimensional cases. The trans port equation method will be used first. For free electrons, and kT « Cp , the conductivity tensor £ ='JL, is given by xx 1 - i COT a3 - j jtan"1 (A - cox) + tan"1 (A + COT)J . I , h + (A - (,vr)2\ I + T l09\l + (A + MTJaj J _ = _J> 3_ yy zz 1 - i COT 2a 3 (^)[ tan"1 (A-COT) + tan"1 (A+COT) log p + (A - COT)2] |_1 + (A + COT)2J where a = —, A=q-£, and ^v^r is the mean free path of an electron 1" I COT T » p at the Fermi surface. The other components of the conductivity tensor are zero. The sound wave is assumed to propagate in the x direction. The only non-zero component of R_ is [19] A2 I COT ( 1 - i COT ) 28 Furthermore, If S «Vp, then the conductlvfty expression reduces to [20a] a - r—v2— ^T  1 1 xx 1-IOJT a 3 j a - tan"1 A - I A(i)T 1+A2 [20b] a °o = v-?2- r^r (a2 + 1) ftan"1  1 J yy zz 1 — icur 2a3 J ' V . T+A^y " a The attenuation of longitudinal and transverse waves is found by substituting [19], [20a] and [20b] into [16a] and [16b]. In the low frequency limit (A « 1) where the screening is perfect {to « a„, a' for longitudinal waves, cu « (l)2°-'(c)2°yy for transverse wavesj the attenuation In the absence of deformation potential coupling Is h nm , ? al= 15 PST A „ 1 nm A2 t 5 pSx for longitudinal and transverse waves respectively. For higher frequencies in which A > 1, but the screening is still perfect, the following more general expressions may be used: [21] nm *l PST A2 tan"1 A 3 (A - tan"l AJ" - 1 nm 't pSx 2A3 3 [(1+A2)tan"l A - A] -] In the high frequency limit (A » 0 where the screening Is perfect, the attenuation Is al" o" pT qVF h nm t 3ir pS M F For comparison with the result of the quantum calculation, the ex pression for the attenuation, which includes the deformation potential may also be evaluated in the quantum limit. In this limit the re laxation time is allowed to become large so that A » 1, and u 2-»- » u, where to is the plasma frequency. "When"these conditions are P * P satisfied, the expression [17] for the attenuation in the presence of d .deformation potential, reduces to nm r, Substituting the appropriate limit of the conductivity [20a], into the last equation, we obtain 2 where R3 \6iu\e2- / /Z Is the three dimensional Fermi-Thomas screening length. It Is Interesting to note that (\ + (q Ra)"2] Is the mvp Fermi-Thomas dielectric constant valid for q « —— . ti In the quantum limit, the attenuation may also be found by Inte grating [18]. The result Is r__, ir nm # v. . W al= 6" pT I IT) qvF A comparison of the last result with [22] reveals two things. First the quantum calculation does not take into account the screening of the sound wave by the conduction electrons. Secondly, the electro magnetic ccupl.ing .mechanism...Is.equlvalen.t...to.a deformation potential of strength —. This is just the potential arising from a charge density oscillation of the form ne . It follows from [22] that when qR3 « 1 the self consistent field coupling is equivalent 2 e to a deformation potential of strength 3* F . The electromagnetic electron-lattice coupling normally dominates in metals at all reasonable frequencies. To summarize, the quantum approach gives the transport result If an electromagnetic coupling energy of -^2— Is added to the deformation potential and the entire Inter action potential is screened by dividing by the Fermi-Thomas dielectric constant. (ii) Two Dimensional Metals By two dimensional metals we mean metals In which the electronic energy depends on k and k but not on k . For example, the Fermi x y z surface of a two dimensional free electron gas is a cylinder, centered on the k axis. Associated with the two dimensional nature z of the electron gas, there are three modes of sound propagation In addition to the longitudinal and transverse waves propagating in the conducting plane and polarized in the conducting plane. First there is a transverse wave travelling in the conducting plane and polarized perpendicular to the conducting plane. Secondly there are two modes, one longitudinal and one transverse, propagating perpendicular to the conducting plane. Waves polarized in the non-conducting direction cannot deliver energy to the electrons in the linear approximation considered here. Hence these waves are not attenuated. The remaining special wave is the transverse wave propagating in the non-conducting direction. Referring back to the expression for the conductivity tensor [4a], we see that for this mode q*v = o, hence a = a - -r^r— and a = -1— xx yy 1-IIOT zz This value for the conductivity substituted Into [16b] yields the attenuation [24] at = nm pSx (cox): 1 + (COT)2 f -1 which Is much smaller than the corresponding three dimensional result In al1 1imlts. tn calculating the attenuation of .the two modes travelling tn the conducting plane and polarized in the conducting plane, we use the same procedure as tn the three dimensional case. The non-zero components of the conductivity tensor for a two dimensional free electron system with kT « Cp are given by [25a] a - -r-^- ~ { 1 - \ [25b] o-„ = — //i2+f -1 1 yy 1-IUT a2 \ ) where the symbols have the same meaning as in the three dimensional problem. The only non-zero component of Is R c A2 xx x itox (1-icut) 2oa which differs only in the factor of 2 from the corresponding three dimensional result. The attenuation of the longitudinal and transverse modes ts found by substituting the last three results into [16a] and [16b] respec tively. In the low frequency (A « 1) perfect screening limit, the attenuation is 1 nm . 2 ala at "h~ psT ' For arbitrary A, and perfect screening A2 nm l26] °c = °t ~ p!7 2(/T+A^-l) ] In the large A limit the last result simplifies to 1 nm al~\= 2 pT qVF For purposes of comparison with the result of the quantum calculation, the attenuation of longitudinal waves In the long relaxation time, or quantum 1Imlt, Is / C + /|TRNE2 \ V ro-7i _ * nm I q 1 qVF  W al~2 pT I ^ j (l + (qR2)"2^ where R2 =(l(7Tne2J 's tne two dimensional Fermi-Thomas screening^ length. The attenuation in this limit may also be found by evaluating [18] for a two dimensional electron gas, which gives W al = 2 PT fe) qVF ' As In the three dimensional case the quantum result [28] Is Identical 3* with the Boltzmann equation result [27], except that the latter in cludes electromagnetic coupling and screening. A graph of the attenuation of longitudinal acoustic waves in two and three dimensional metals, calculated from eqns. [21] and [26] as a function of A (= at) is shown in Fig. 6. The similarity of the two curves in Fig. 6 suggests the following interpolation formula for the inverse tangent function: This formula is asymptotically exact for large and small x and de viates from the exact value of tan_1x by a maximum of abcut 1.1% for x = 2. 35 Fig. 6 - Acoustic attenuation as a function of at for 3D and 2D metals in units of nmVr as calculated from [21] and [26] PS (iii) One Dimensional Metal We now investigate acoustic attenuation in one dimensional metals. By one dimensional, we mean that the electronic energy depends only on k and not on k or k . As a result the Fermi surface consists x y z of two parallel planes, perpendicular to the axis. As in the two dimensional metal, there are five distinct cases depending on the relative orientation of the conducting axis, the sound propagation direction and the polarization vector. Only two of the waves interact with the electrons. One is a transverse wave polarized along the conducting axis, and the other is a longitudinal wave travelling aiong the conducting axis. The attenuation of the first of these is identical to the attenuation of the similar mode in the two dimensional metal; hence it is also given by [24]. In the case of the longitudinal wave propagating along the con ducting axis, it is instructive to consider the quantum limit first. The calculation is similar to the two dimensional problem except that the electronic energy e(k) is a function of k^ only. The result, however, is quite different. The reason for the difference is most easily understood by examining the interaction surface. As has been described above, the interaction surface defines the electron j< states, which are allowed by momentum and energy conservation to 37 Fig. 7 - Fermi surface and interaction surface in a one dimensional metal interact with a sound wave of wavevector q «kp. Unlike the inter action surface for the two and three dimensional cases, the inter action surface for one dimension does not intersect the Fermi sur face (see Fig.7). Although this feature does not restrict electrons from scattering with phonons of wavevector ~ 2kp in the present case where q « kp , we expect a much reduced attenuation, which approaches zero at low temperatures. In fact, this conclusion is borne out by evaluating [18] in the S « vp limit: This expression differs from the corresponding two and three dimen-/£F\ -EF sional results by the presence of the factor! j e r/j" . Clearly ct£ approaches zero as T goes to zero. The quantum limit may also be extracted from the transport inte grals. However, the A,tor ->• » limit must be taken with care. Since the acoustic attenuation expression contains a real part operator, a small real part may be significant even when the imaginary part is much larger. The non-zero part of the conductivity tensor for longi tudinal sound waves in a one dimensional metal may be written [29] This integral may be evaluated in the tor •> °° limit by using the functional relation Urn \ ^= p -L- + i IT 6 (v - S) WT » V-SM + —) where P means principal part. The real part of the conductivity (imaginary part of the integral) may now be found exactly, and the imaginary part can be approximated by treating ^- |^ as a delta function at ± Vp. The result is - °° 0 - -7-xx A kT - i — Once again there is a simple relation between R_ and g^. The only non-zero component of R_ is A* °-R = - xx x iiox (1 - icox) Oo When the last two results are substituted into the attenuation equation, [17], we obtain x eF (c + ^f\ qvF nm / F\ _ - r=- / q2 , _L [30] «£= Zir^lj^Je kT . 2e F where R, =(-^—i—ol is the one dimensional Fermi-Thomas screening l2iTne2| length. The transport method once again gives the same result [30], as the quantum method [29]» barring the expected failure of the quantum calculation to include electromagnetic coupling and screening. The extreme quantum limit that is considered above (cox >> 1) is unlikely to be attainable in practice. In this limit the delta function in [18] implies strict energy conservation in electron phonon collision processes, and the attenuation arises from the thermal broadening of the Fermi surface. If the condition cox >> 1 is re laxed to the weaker condition A >> 1, the finite collision time may be taken into account by replacing the delta function in [18] by the Lorentzian TL irx (e(k) + fuo - e(k + q^ +^ -1 The ultrasonic attenuation, then arises from the intersection of the Fermi surface with the tail of the Lorentzian and is given by , , n m /c \2 ^ al= pTT fa) ' Aside from a factor of two, this expression differs from the corresponding two dimensional result [28] by a factor of A"1. This factor reflects the non-intersection of the interaction and Fermi surfaces. The attenuation [31] may also be obtained using the transport equation method, by neglecting the self-consistent field and collision drag, in favour of the deformation potential 41 As pointed out earlier, the self-consistent field, election sound wave coupling, is expected to dominate in metals at least in the low frequency limit. We first try to obtain an expression analogous to the two dimensional result [26], using the transport equation method. However, if the same approximations are made as in the two (or three) dimensional cases, the attenuation in a one dimensional electron gas is identically zero. To obtain a non-zero result we relax the zero temperature approximation and calculate the transport tensors to order [ — J . The transport integrals (f depend on temperature through the temperature dependence of the Fermi function. As the temperature rises from zero, the unit step in the Fermi function broadens to a width of order kT and the chemical potential increases from the Fermi energy to y = e. Her) Provided that both of these effects are included, the non-zero component of the conductivity tensor is found to be TT2 /kT\2 to first order in 6 where 0 = —/ — j « 1. Similarly A2 ° [33] R = . • r— (1 - 9) L J x IU)T(I-IIOT) cr Substituting [32] and [33] into [17] we obtain [34] a 3 PST n m 1 + A2 for a free electron gas. This expression is valid for arbitrary A, provided that the perfect screening limit holds « a0, and that A is not so large that the extreme quantum limit applies. The transport equation method, in the relaxation time approximation, may also be used to calculate the ultrasonic attenuation for an arbitrary band structure, although the general expression which results, is much more complicated than [17]. However, in the perfect screening limit the expression for the attenuation is once again quite simple. We then find that the attenuation of longitudinal waves travelling in the x direction is given by [35] where terms of order have been ignored, and the transport tensors £ , ^ and are given by [4a], [4b] and [4c] respectively. Similarly a = ne^T where m f (k)d3k is the total number of electrons in the conduction band. Notice that the m dependence of [35] is fictitious since both aG and £ are defined to be proportional to m"1. The one dimensional transport integrals are sufficiently simple that the temperature dependent part of the attenuation may be evaluated for an arbitrary band structure, in the perfect screening limit. If the band structure in the vicinity of the Fermi surface is given by e(k) = ep + f,(k - kp) vp + 2m* F where 1 i>£ VF f. 8k k = kr and (m*)"1 = fp" k = kr then the temperature dependence of the Fermi level is V = e F 12 ep where cp = j m* vp2. The ultrasonic attenuation is given by ir2 nm"' [36] ^--g-^r / *kF \/kT\2 A5 where A = qvpx as before and the total number of carriers in the band is n = DpfikpVp where Dp is the density of states at the Fermi surface. The last result closely resembles the free electron result [34]. The non-intersection of the Fermi surface with the interaction surface is reflected in the fact that the attenuation [34] becomes frequency independent for A » 1, instead of approaching a linear frequency dependence. The same behaviour has already been noted in [31] for deformation potential coupling. On the other hand, the absence of the leading term in the ultrasonic attenuation in the A « 1 limit cannot be explained on the basis of the energy and wavenumber selection rules. In fact it is easy to show, using the collision broadened energy conservation requirement that for A << 1 electrons at the Fermi surface are just as likely to interact with the sound wave as the electrons on the interaction surface. The reason for the small attenuation in the one dimensional metal for A << 1 lies in the absence of the higher order relaxation times Tj j = 2, 3» • • •» discussed by Bhatia and Moore (i960)'. Further more the expression [34] (or [36]) should be taken only as an order of magnitude estimate of the attenuation since another term of /kT\2 order J — \ has been omitted by assuming the sound wave to be \eF/ isothermal [Akhiezer, Kaganov and Liubarskii (1957)]-1.5 Anisotropy of Attenuation In one and two dimensional metals it is possible to adjust the position of the interaction surface simply by changing the angle of propagation of the acoustic wave relative to the conducting direction (or plane). In the quantum limit, when the interaction surface intersects an extremum of the Fermi surface there is a large enhancement in the sound absorption. For this reason the attenuation of acoustic waves will be very anisotropic in one and two dimensional metals, in the quantum limit. In this section we assume for con venience that the acoustic wave is coupled to the electrons only through a deformation potential C. (i) One Dimensional Metal The interaction surface has been defined above as the surface in k space for which e (k+q_) - e (k_) = fiqS with q« kp. In a one dimen sional metal the electronic energies depend only on kx so that the definition of the interaction surface reduces to e(kx + q cos 6) - e(kx) = fiqS for a one dimensional metal in which the acoustic wave propagates at an angle 0 relative to the conducting direction. Note that changing the direction of q_ relative to the conducting axis is mathematically equivalent to replacing the speed of sound S by S/cos 0. This di rection dependence of the apparent velocity of sound has a drastic effect on the ultrasonic attenuation in the quantum limit. First we consider the col 1 i s ionless quantum regime in which COT » 1. As was pointed out in the previous section, in this limit the absorp tion processes arise from the intersection of the exponential tail of the Fermi-Dirac distribution function with the energy conserving 6 46 e o Fig. 8 - Angular dependence of ultrasonic attenuation in 1D metal C ^ calculated from [37] for T = 300K normalized to aQ - j ~j (^~J VF The inset is an enlargement of the peak attenuation near 9 = 90°. function. Evaluating [18] we find rvm tD\ * nm / C \2 eF qvF . 2 [37] a„(6) = — — h—) -,-= sechz L 1 I 2 pS UEp J kT cos 6 This result reduces to [29] as 8 + 0 and goes to zero as 6 ->- TT/2. At intermediate values of 6 the attenuation goes through a large maximum when the interaction surface touches the Fermi surface. The 0 dependence of the absorption is shown in Fig. 8. In this figure, TTF-TCNQ material parameters taken from Table I have been used. TABLE I - TTF-TCNQ Material Parameters . (T^t - «F) / » 2.8x1021 cm"3 1.62 g/cm3 4x105 cm/S m VF eF TTF Band 8 mQ 0.5x107 0.5x103 b axis scattering time (60K) TCNQ Band 107 cm/S 103 K 5x10' •lU Berl insky et al (1972*) 48 The absorption peak occurs at 9max = cos" (S/vp) and has an angular width of order S/vp radians. The peak absorption is a JL /C \2 11 11 for kT > fito and a JL JDJE / C \2 EF for kT < tito. The first result is a factor of order (ep/kT)(vp/S) larger than the corresponding three dimensional result [23] reflecting the fact that in a one dimensional metal the interaction surface can intersect all of one sheet of the Fermi surface at once, rather than just a narrow ring as in a three dimensional metal. When the condition cox » 1 is relaxed to the weaker condition A» 1, the attenuation in a one dimensional metal comes from the intersection of the Fermi surface with the tails of the broadened energy conserving "6" function. If we replace the 6 function in [18] by a Lorentzian, as discussed in the preceeding section, then in the low temperature limit where the Fermi surface is sharply defined, a trivial integration leads to [38] 49 Fig. 9 " Angular dependence of ultrasonic attenuation irfa ID metal calculated from [58] with A = 8. 50 for the dependence of the attenuation on the propagation direction. This expression reduces to the earlier results at 8= 0 ([31]) and TT/2, and has a peak at 6max = COS_1(1/A). The peak value of the attenuation is much smaller than in the col 1 i s ion less (tor » 1) regime. The peak attenuation is a nm -cmax pSx (2e which is a factor of order A larger than the corresponding three dimensional result. A plot of a^(^) f°r A = 8 is shown in Fig. 9-(ii) Two Dimensional Metals The ultrasonic attenuation in two dimensional metals is also ani sotropic in the quantum limit. In two dimensional metals there is a large peak in the attenuation when the interaction surface moves out to touch the surface of the Fermi cylinder tangentia 11y. This peak occurs when the acoustic wave propagation direction is nearly perpendicular to the conducting plane. We now use [17] to calculate the attenuation of an acoustic wave propagating at an angle 6 to the conducting plane in a two dimensional metal. In the extreme quantum limit in which OJT » 1 and energy is strictly conserved in electron-phonon collision processes /n\ _ 1 nm a£(9) = 2pTsu, F 1 -Vp2COS20 £p 1 " Vp2cos28 Ei qvF COS0 for kT > tico. In this expression when the argument of the square root is negative the result is taken to be zero. With this proviso the attenuation is zero at 9 = TT/2 and matches the previous result [28] at 6 = 0. For cose > S/vF 2 \ ~i [39] at(Q) - (cos29 - j o£(0) where ct^(O) is given by [28]. For 9 near 9max = cos_1(S/vp) the attenuation reaches a peak of VF a£max = -S-\^/ a£(0) " A non-zero temperature vkT > tico) broadens the sharp Fermi surface and reduces the peak attenuation to %iax~~ IkT al{0)' When kT > tico the attenuation as a function of 9 is given by 'er\i £4°1 a£(9) = c^(kT) a£(0) J x^ sech2(x-e)dx wi th 6 I Vp2COs20) kT The integral in [40] must be done numerically. However, for cos9 > S/ 52 50 3000 T = I00K 40 2000 30 a, (0) 1000 pni 0 89.82 89.84 89.86 89.88 6 (°) 10 0 J L 0 Fig. 10 - Angular dependence of ultrasonic attenuation in a 2D metal calculated from [40]. The inset shows the peak attenuation with an expanded horizontal scale. 53 Fig. 11 - Angular dependence of ultrasonic attenuation in a 2D metal calculated from [41] with A = 8. and at temperatures low compared to the Fermi temperature the last result ([40]) is equal to [39] to a good approximation. A plot of the attenuation as a function of angle is shown in Fig. 10, using eF = lO^K and S/vp = 10"3. It is unlikely that the extreme quantum limit (COT » 1) can be achieved in practice. If the condition COT » 1 is relaxed to the more realistic condition q£ » 1 then there is no longer strict energy conservation in electron-phonon scattering processes. In this case the 6 function in [18] is replaced by a Lorentzian as described above. In the low temperature limit the two dimensional integration in [18] may be done exactly for arbitrary direction of sound propagation. .The result is r,.., /„x 2 tan-1(AcosO) 1 j . ,nS [41] a£(6) = - [—J^ T + A2cos2ej A «£«>) • This expression reduces to the col 1 isionless result [40] at 8 = 0,TT/2; however, just as in the one dimensional case collisions drastically reduce the acoustic absorption peak near 6 = TT/2. A plot of a^(Q) as a function of 0 is shown in Fig. 11 for A = 8. The peak attenuation is a£max=0-226Aa£(0) which occurs at an angle 8max = cos-1(1.825/A) with respect to the conducting plane. Just as in the one dimensional case the peak attenuation at 6 = 0m=v, in the two dimensional metal is a factor of order A larger than the corresponding isotropic attenuation in a three dimensional metal. 1.6 Summary We have extended the theory of ultrasonic attenuation in metals so that this theory may be applied to metals whose electronic band structures are one or two dimensional. Our results are valid for arbitrary values of A and include electron-phonon coupling via a scalar deformation potential as well as coupling via collisions and via the self-consistent electric field which arises from the response of the electrons to the sound wave. We have shown that the ultra sonic attenuation in one and two dimensional systems differ signifi cantly from we 11-known results for three dimensional systems. In particular the attenuation is shown to be anomalously small and strongly temperature dependent for metallic one dimensional systems. In addition the attenuation is shown to be highly anisotropic in one and two dimensional metals in the quantum limit. There are other applications of the theory in addition to the application to quasi one and two dimensional metals. For example, it may be applied to the attenuation resulting from an accumulation layer in an MOS junction, or to the attenuation associated with layered metal-insulator heterostructures. The theory also applies to a three dimensional electron gas in a strong magnetic field. In this case, one dimensional behavior results from the quantization of the electron moti.on in the plane perpendicular to the applied field. The Fermi surface for a particular Landau level may be "tuned" to match the interaction surface by varying the magnetic field. In the limit A» 1 this result in large peaks in the acoustic attenuation as a function of magnetic field. These "giant quantum oscillations" have been observed in gallium by Shapira and Lax (1965). This magnetic field tuning of the Fermi surface is analogous to changing the position of the interaction surface in a one dimensional metal by varying the direction of propagation of the acoustic wave. The principal experimental obstacle to observing the anisotropy of the attenuation in one and two dimensional metals is in obtaining a long enough electron mean free path and high enough frequency to achieve the A > 1 limit. CHAPTER I I Attenuation in Semiconductors 2.1 Quantum Limit In this chapter the techniques that have already been applied to metals will be used to calculate the acoustic attenuation due to electrons in n-type semiconductors. Both deformation potential and electromagnetic coupling will be considered, and an energy independent scattering time is assumed. This assumption was unnecessary in the previous chapter because in metals only the scattering time for Fermi energy electrons is important. The first step is to use perturbation theory to calculate the net phonon absorption rate in the quantum regime where A» 1. The only difference from the metallic approach is in the definition of the electron distribution function f(k). For metals f (k_) = ^exp j^e (k_) -Cp^/kT^ + 1^_1 whereas for semiconductors f (k) = exp J^u-e (k_) ^/kT^ where u is the chemical potential. In the metal case £p » kT while for semiconductors u<0. If we assume the energy bands to be parabolic, the electron energy may be decomposed into a sum of three parts where , n\2 and 1TI3 are effective masses. This decomposition makes it possible to write the attenuation a = as a product of sums [1] a TrC2q 7^ X where q_ || kx. This ability to factorize the electron distribution function means that the attenuation along the principal directions is independent of the dimensionality of the electron gas. The dimen sionality only affects the form of the expression for the charge carrier density. The expression in [1] may be evaluated by converting the summations into integrals in the standard way. In two limiting cases the inte grations may be performed explicitly. For S« v , and for S » v th where v , = (2kT/m)^ is the thermal velocity of the electrons. This th thermal velocity takes the place in semiconductors of the Fermi ve locity in metals. Similarly the important mean free path in semi conductors is £ = Vj-p1- At room temperature and using the free electron mass v , - 107cm/S, so that S/v . - 30. In order for the sound velocity th th to be of the same order as the thermal electron velocity the temperature must be lowered to about 0.3K- In cases in which the thermal velocity of the electrons is comparable to the speed of sound a numerical inte gration is required to evaluate the attenuation. 2.2 Transport Equation Approach In the same way as for metals the 3oitzmann transport equation can be used to calculate the acoustic attenuation for arbitrary values of qZ. In what follows the sound wave is assumed to propagate along x which is assumed to be a conducting direction for the one and two dimensional semiconductors. The outstanding difference between the semiconductor and metal results is the lack of dependence on dimen sionality in the semiconductor case because of the absence of a Fermi surface. The first step is to evaluate the transport tensors [4a] and [4c] of Chapter I. The frequency dependent conductivity tensor has two distinct non-zero components for two and three dimensional semicon ductors. Only one of these components is zero in one dimension. They are where a = A(1 - itox)-1 and $(z) is the error function defined by 2 'z are valid for COT < 1 only. Also exp (-t2)dt. The expressions for the conductivity o R = A2 pxx x icoT (1 - JCOT) 2a if the sound wave couples to the electrons primarily through a self-consistent electromagnetic field the attenuation of longitudinal and transverse waves is found by substituting the above transport tensors into [16a] and [16b] of Chapter I. In the low frequency (A « 1) perfect screening limit the attenuation is [6] a£ = pT7 for longitudinal waves and 1 nm , p for transverse waves. The transport equation approach may also be applied in the quantum limit in which A» 1. Just as for the one dimensional metal the A -> 00 limit must be taken with care in order to avoid losing a small real or imaginary part. The integration is most easily done in rectangular cc ordinates as follows: Since (- 3f/9e) can be factored the ky and kz integrations are straight forward. The kx integration is done using lim 1 n 1 ,•!•/ c\ —> rt 7-7— = P —7 + nr 6(v„ - S) T-*- 00 q(v - S) - I/T qCvy " s' The principal part integral can be done analytically in two limits. For S » v. th # exp (-41)* I ^] xx 1 - icoT a , vth v v-h and for S « v , th a L v2 v. J XX 1 - icox a L The Oyy component of the conductivity tensor is needed to calculate the attenuation of transverse waves. It is calculated in a similar way. For S « v . th a = a 'o 1 yy zz 1 - iojx a >, . 2S " and for S » v th a = a yy zz q° i ex / _si\ 1 - itoT a [u exp \ v2hy + i 'th These expressions may be substituted directly into [16a] and [16b] in Chapter I, to find the acoustic attenuation. In the perfect screening limit in the absence of a deformation potential, the attenuation for the different cases is given below. For S« v4, /r7 nm th t TT L and for S » v th [8] at 2^^ S" >S Vth exp 'th/ a. - — 1 V?h 2~S2~ A£ A comparison of these results ([7] and [8]) with [2] and [3] reveals that in the quantum limit, electromagnetic coupling is equivalent to a deformation potential equal to kT. This relationship is analogous to the equivalence in metals of electromagnetic coupling to a de formation potential equal to the Fermi energy, as discussed in Chapter I. In semiconductors the carrier density may be small enough that the screening is incomplete and the electromagnetic coupling (4irne2/q2) is small compared to deformation potential or piezoelectric coupling. When the deformation potential is the dominant coupling mechanism, the attenuation of longitudinal waves is determined by the last term in eqn. [17] of Chapter I. We consider the quantum limit first. Substituting the appropriate transport tensors calculated above we find for S« v , th and for S » v , th where R = (kT/4-nne2)^ is the Debye-Huckel screening length. The re sults [9] and ]10] above are identical to the expressions [2] and [3] obtained using quantum perturbation theory except that the deformation potential C is replaced by a screened deformation potential. The factors (1 + (qk)"2) and (l - iop2/u,2) are dielectric constants for a classical (non-degenerate) electron gas in the low frequency and high 64 frequency (S » v^) limits respectively (Kittel 1963) - The expression [9] has been obtained by Spector (1966). The transport equation method can also be used to calculate the attenuation in the low frequency limit (A « 1) when the deformation potential is the dominant coupling mechanism. After expanding the conductivity expression [4] to lowest order in A and substituting the expansion into the last term in eqn. [17] of Chapter I we find that The results [9] and [11] above will be rederived in Chapter III below in connection with acoustic amplification in the presence of a d.c. electric field. The relationship between the attenuation of longitu dinal waves and transverse waves is discussed in that chapter, along with a description of how the deformation potential results are modified if there is a piezoelectric interaction. 2.3 Meta1-Semiconductor Transition We have just shown that the electronic contribution to the attenuation of a sound wave propagating along a conducting direction in a semicon ductor is independent of the dimensionality of the semiconductor. On the other hand in Chapter I above, the attenuation of sound in a one dimensional metal was found to be anomalously small. These results suggest that when a one dimensional conductor undergoes, a transition [11] from a metallic to a semiconducting state (TTF-TCNQ for example) the electronic contribution to the ultrasonic attenuation may in crease rather than decrease as might be intuitively expected. In order to treat the problem of a meta1-semiconductor transition it is necessary to drop the assumption of free electron energy bands. In its place we assume a one dimensional band structure consisting of a single half-filled tight-binding band given by [12] e(k) = - ep cos k b where b is the lattice constant along the conducting direction and 2ep is the bandwidth. The effect of a meta1-semiconductor transition is to open a gap in the middle of the band so that in the semiconduc ting phase the energy band is given by [13] e(k) = ± / cosz kb + AZ(T) where 2A(T) is a temperature dependent electronic energy gap. The + sign applies for (k| > kp and the - sign for |k( < kp. It is very difficult to calculate the attenuation in this situation for arbitrary values of A. The quantum limit on the other hand is more tractable. We will consider this limit only. The deformation potential coupling C is assumed to be a constant, independent of k and the size of the energy gap, eventhough it is not clear how good this assumption is, particularly for electronic states close to the gap. As discussed in the previous chapter, in the extreme quantum limit sound wave attenuation results from electron-phonon scattering processes in which both energy and momentum are strictly conserved. The electronic states allowed by energy and momentum conservation to participate in electron-phonon scattering processes define a surface in k space known as the interaction surface. In a one dimensional metal, when the sound wave propagation is in the conducting direction, the inter action surface is well separated from the Fermi surface. As a result the attenuation is anomalously small. In a metallic tight-binding band there are two distinct interaction surfaces reflecting the fact that the band contains both positive and negative curvature portions. One of the interaction surfaces is near the origin in k_ space and the other is near the zone boundary, as illustrated in Fig. 12. e(k) *- k Fig. 12 - Positions of Interaction Surfaces - Metallic Band If the contribution to the attenuation from both parts of the band are included then [14] a _ ? nm- / C V 'I PS \eF) (" iS)Q VF according to eqn. [18] of Chapter I. In this expression m" = "h2/Epb>2 is the effective mass for an electron near the bottom of the band and vc = erb/"h is the velocity of a Fermi electron. The above expression r r is similar to the corresponding result [29] given in the previous chapter for a free electron gas. When an energy gap opens up at the Fermi surface two additional places in the band become available where the energy and momentum selection rules can be satisfied. As illustrated in Fig. 13 these new interaction surfaces are close to the Fermi level (or chemical e(k) *» k _7T b TT b Fig. 13 - Positions of Interaction Surfaces - Semiconducting Band 68 Fig. 14 - Ultrasonic attenuation as a function of temperature near a metal-semiconductor transition in a ID conductor. The attenuation was calculated from [15] and is normalized to aQ 2ntrr pS Vep, (i)2 q vf • potential) if the energy gap is not too large, and hence can provide a significant contribution to the attenuation of sound waves. The attenuation in the semiconducting phase can be calculated by evaluating [18] of Chapter I using [13] for the electronic energy band. The result is 2 assuming q« kp. This time the effective mass m" = A graph of the temperature dependence of the attenuation predicted by [15] is shown in Fig. 14 for a one dimensional metal which undergoes a metal-semiconductor transition at 50K. The energy gap is assumed to have a BCS-like temperature dependence below the transition temperature. The attenuation in the metallic phase is extremely small in the col 1ision less UT » 1 limit. If the condition WT » 1 is relaxed to A» 1, then the attenuation in the metallic phase will be comparable to [31]. Even though the metallic attenuation will now be much larger than in the col 1ision less regime it will still be small compared to the peak attenuation in the semiconducting phase. On the other hand, the semiconductor phase attenuation is not significantly affected by relaxing the WT » 1 condition to A » 1, since the interaction surface is already close to the Fermi level, without any collision broadening. In practice the low frequency A« 1 limit is much more likely to be physically realizeable than the quantum limit, particularly in a material such as TTF-TCNQ. where the electron mean free path is only a few lattice constants. From earlier work in Chapter I we expect the electro magnetic coupling mechanism to be far larger than the deformation potential coupling at low frequencies. In this case the metallic attenuation is very low so that one might also expect to see an increase in the low frequency attenuation, to something approaching the three dimensional value, in going from the metal to the semi conductor. CHAPTER I I I Amplification 3.1 Introduction It is well known that under certain circumstances ultrasonic waves may be amplified in metals or semiconductors if the conduction electrons have a d.c. drift velocity (Hutson, McFee and White 1961> Vrba and Haering 1973). The amplification of sound waves may be regarded as a negative attenuation that can occur in the presence of a d.c. electric field. Accordingly the methods described in Chapter I for calculating the attenuation may be applied to the amplification problem. In general the attenuation and hence the amplification of sound waves by electrons depends on the relation between the electron mean free path t an<H the sound wavelength 2-rr/q. The theory of acoustic amplification in the presence of a d.c. electric field has been worked out by Weinreich (1956) and White (1962) for the low frequency {ql « 1) limit and by Pippard (1963) for the high frequency limit [qt « 1). Spector (1962) has used the Boltzmann equation to produce a theory that is valid for arbitrary qt. The Spector (1962) theory is very complex for the following reason. When a d.c. electric field is introduced into the transport treatment of ultrasonic attenuation a large number of additional terms of equal order are generated, none of which can be neglected. In this chapter we outline a theory where the d.c. eletric field is taken into account by shifting the distribution function in k space. This approach eliminates the need to deal with a large number of new terms arising 72 from the d.c. field, and vastly simplifies the problem. Using the new approach, we are able to confirm Spector's results which have never been verified previously. In addition our approach is valid for strong d.c. electric fields in the same way that the method of Spector (19&8) is valid for strong fields. In this chapter we calculate the attenuation (amplification) of sound waves in n-type semiconductors and three dimensional nearly-free-electron metals, in the presence of a d.c. electric field. The calculation follows the transport equation approach developed in Chapter I. In both the metal and semiconductor the conduction electrons are modelled by a free electron gas and non-electronic contributions to the attenuation are ignored. In the metal a self-consistent electric field is used to couple the sound wave to the conduction electrons. In the semiconductor a deformation potential tensor is assumed to be the dominant coupling mechanism. First, the problem will be set up for arbitrary electron statistics and both self-con sistent field and deformation potential coupling. 3.2 Transport Equation In the. presence of a sound wave described by a local lattice velocity u^ « exp [ i (qx-iot) ] , the Boltzmann transport equation for electrons is [1] 73 where f is the electron distribution function, is a self-con sistent electric field and £ is a deformation potential tensor. In the relaxation time approximation used here scattering processes cause the electron distribution function to relax to the local equilibrium distribution function f. As in Chapter I, Section 1.2, f can be approximated by [2] 7(v,r_,t) = f0(v-u; y(r,t)) 9fo / 2eF \ where f is the equilibrium electron distribution function in the absence of a sound wave, p is a self-consistent chemical potential, Cp is the electron Fermi energy, n is the equilibrium electron number density and n1 is a small oscillatory component of the electron dens i ty. A d.c. electric field will be introduced into eqn.[l] by postulating that the sole effect of the field is to shift the equilibrium electron distribution by the average drift velocity = - —• JEQ where EQ is the d.c. field. This assumption is correct to first order in the electric field,. In this approximation, the d.c. field may be introduced by re defining the local equilibrium distribution function f by 9fd [3] f(v,r,t) * ^-^(.(v-^-u+l^n,) for a three dimensional metal where is a doppler shifted equilibrium distribution function. The Boltzmann equation [1] with f defined by [3] may now be solved to first order by substituting where f° is the d.c. part of the perturbation from equilibrium, and fj1 tt exp [ i (qx-cot) ] is the a.c. part of the distribution function. We find that f? - fd " f 1 o o and [4] f} 9e qq-C.-u mu\ / \ 2 eF ex ET + " T = ' V"L" 7Tnl .—1 eico ex/ \ dy 3 n \ ^1-iiox+iqvx^-1 . The electronic contribution to the a.c. current is given by [5] J = 2e e " ~ f2lrT3 J v f} d3k. Now define a doppler shifted a.c. component to the distribution function by exlEi + qq/£*u_ mu e i to et (^1 - i (to-io^Jx + iqvx^ 2 eF 8e where wd = a.-Then the current expression [5] may be rewritten as and J = J - n i e v —e —e 1 —< where [6] Jd —e eiio mu nieSdRd d3kfi =T2fy3 j d3 kf^ d 2e2r £ =J2^ J d3 k v v 1 - i (to-Wj)x + iqvx V de - ~ 6Tr3n Sd dd k 3f 1 - Ru-co^x + iqvx \ 3 (- ° and S = (co-to^j)/q is the doppler shifted sound phase velocity. The total a.c. current density due to the sound wave is the sum of the electronic current and the background ion current. Thus [7] J = J + n e u — —e — where is the total current, and n is the ion number density. in a monovalent metal (n-type semiconductor) the ion (ionized impurity) number density will equal the equilibrium electron number density. In the presence of a sound wave the ion number density will contain a small oscillatory component given by nu/S where u/S is the strain induced by the sound wave. The oscillatory term in the electron number density may be found by using the continuity equation for electrons tonje + g.*^ = 0. This equation can also be written in terms of the doppler shifted quanti ties (to-io,)nie + q'J^ = 0 d — e or [9] nieSd = - /u where J^.. is the component of parallel to q_. Also the electric field which is consistent with the total current may be found from Maxwell's equations to be [10a] EN-^L /je|| + neuH) and [10b] The next step in obtaining the net power flow from the sound wave to the electron gas is to obtain expressions for the self-consistent electric field and current in terms of the local lattice velocity. For a free electron gas, and a sound wavevector q_ in the x direction, the conductivity tensor is diagonal. In this case the vector notation may be dropped and the components can be treated independently. For longitudinal waves (q_ II u) eqn. [9] can be used to eliminate n^ from [6]. Thus q2C u [11] ex = old Elx + XX X eito mu ex where ald = I (1 - Rd ).. Only the x components of Ej and u_ are XX X non-zero for longitudinal waves. Substitute [10a] for E then X [12] where t ex neu T c^d x _x 4d L °o fori old (l + D S X \ XX)_ id X 1 + •^LL ala is a dielectric constant, y ~ 1 -toy x . • q_'v to n?.r and D = 1* "Xo . Also if [12] is substituted into [11] 'o m xx ^Trne2 we find the following expression for the total driving field felt by the electrons: [13] Elx + q2C u XX X e i to neu dd ,ld 1 + D xx ya oJ The self-consistent electric field and current associated with a transverse sound wave may be found in exactly the same way. When a transverse wave is polarized in the y direction and propagates in the x direction there are electric currents in both the x and y directions, Using the electron continuity equation and [6], we find that the doppler shifted current in the x direction is, q2cwttu ex xy y e i to If the current and electric field are required to satisfy the Max well equation [10a] which in this case reduces to [15] E AiTi lx co ex then the total current in the x direction is [16] ex ,ld <xd (Q2Z U xy y eicoy Similarly, the self-consistent current in the y direction is, [17] J = a Elv ey y \ lv ex mu \ v, S ex where is the y component of the electron drift velocity v^, and J is given by [16]. By using the self-consistency requirement [10b] 6X we find that [18] J ey n e u /i, • /C\2 J °D \ V^ J y_ / *t7T i /S\^ d _ y dy ex et I co \c) % o0J s t t 4in /S\2 d where e = 1 - — 1 a . The electric field is found by substituting y co \c J y [18] back into [17]. Thus i»iri /S\2 [19] Ely co et \c a \ v, ne u (1 - J S ex We are now ready to evaluate the energy transfer between the sound wave and the electron gas. 3.3 Energy Transfer The power transferred from the sound wave to the electrons is The first term is the work done per unit time by the self-consistent electric field and deformation potential gradient on the electrons. The second term in [20] is due to the interaction of the lattice wave with the electrons through collisions (see Chapter I) and is important only for high frequencies (co ~ o^). Notice that all d.c. fields and currents have been omitted from [20]. In the linear approximation considered here, d.c. quantities contribute terms in the power expression which either have a zero time average or re present the ohmic losses associated with the drifting electron distribution. The attenuation (amplification) factor for ultrasonic waves is obtained from [20] by dividing by the acoustic energy flux. Thus where a is the attenuation. Rather than evaluating the attenuation with both self-consistent field and deformation potential coupling, we consider two limiting cases. In the first limit, the deformation potential is assumed to be negligible compared with the comparable electromagnetic coupling energy —. This assumption is valid up to high frequencies [20] a = P / (Jrp|u|2) (- 10 Ghz) in metals. In the second limiting case the deformation 4ime2 potential is assumed to be much greater than and hence is the dominant coupling mechanism. This is a good assumption at high frequency in semiconductors. First we consider longitudinal waves in metals and neglect the deformation potential. The attenuation of longitudinal waves is found by substituting [10a] and [13] into [20]. If the deformation Ani potential is set equal to zero and e = 1 + r o to aQ, then [21] nm al " PST Re rid' ITd" This expression should be compared with [16a] in Section 1.2, the analogous expression in the absence of a d.c. field. The attenuation may be evaluated by substituting the appropriate transport tensors given in Section 1.4, into [21]. For a three dimensional metal to the lowest order in S/vp, for to « ox and A = at, the result is r„„n nm T A2 tan"1 A ,.. s t W al = JST [3 (A - tan-l-AT ° " y) " \ for arbitrary A, where u = cpv, /to. For A« 1 eqn. [22] reduces to nm -u + A2 (1 and for A» 1 nm IT . /. \ Similarly, by substituting the transport tensors for the one and two dimensional metals from Section 1.4 into eqn.[21], one obtains the attenuation for the lower dimensionality metals in a d.c. electric field. The results are summarized in Table II for arbitrary A in the co « oJ, limit. The three dimensional result [22] is identical with the expressions obtained by Spector (1962), in the appropriate 1imi ts. TABLE I I Attenuation in 1, 2 and 3 Dimensional Metals in a D.C. Electric Field / nm . . Y 1D 2D 3D A2 (1 - u) A2 tan-1 A (1 - u) ~V 2(/I + Az - 1) 3(A - tan"1 A In the absence of a deformation potential, the d.c. electric field has no significant effect on the attenuation of transverse waves. To show this, substitute [17] and [18] into [20], then a. = nm pSx Re ik . 4TT i / S \2 where tz =1 [—J a . To lowest order in S/v,. the attenuation is o co \c/ o F the same as in the zero field case discussed in Section 1.4. In conventional metals u must be very small compared with one in order to avoid unrealistically large ohmic power dissipation. There-82 fore, it follows from [22] that unless u > 1 can be achieved, acoustic amplification through interaction with a d.c. field is impractical in metals, except possibly at low frequencies (A « 1). However, in this case there is an additional complication. In the low frequency limit in the presence of a large d.c. field the attenuation becomes sensitive to a second order term in J such as —e the acoustoelectric current (Mikoshiba, 1959). For example if u - 1 then the self-consistent field ET_ - (u/S) A2 .E . Since the strain u/S < 10~5 and A« 1, the self-consistent electric field will be much smaller than the d.c. electric field Eq. The large d.c. field coupled with a small second order d.c. term in could give rise to a significant contribution to the attenuation, not included in [22]. For maximum acoustic gain one would like a material with a large electron-sound wave coupling, high mobility carriers, and a relatively low carrier density to limit the ohmic losses. These characteristics are available in some semiconductors. In non-piezoelectric semiconductors we expect the deformation potential to be the most important coupling mechanism between the electrons and the sound wave. For deformation potential coupling the attenuation of longitudinal waves is found by substituting [13] into [20] and assuming D » 1. Thus xx ^Trne2 [23] a pSt nm D 2 xx Re In order to evaluate this expression one needs to know the transport tensor components ad and Rd, which are defined by the integrals following [6]. After doing the integrations with Boltzmann statistics for the electrons as in Section 2.2, we find that in the limit coxy < 1 [24] 1 • ~3 i a I - iwyt az - /ir" exp where a = q£/(l - icoyt) and the mean free path £ is defined in terms of a thermal velocity as £ = (2kT/m)^x, and *(x) = 7r exp ( -t2) dt In the quantum limit in which q£, cox » 1, the conductivity is [25] d 2°o °x = "qT /rf th 'th where v , = (kT/M)2and v , » S has been assumed. In most semiconductors th th the thermal velocity of the electrons v ^ is much larger than the speed of sound S for temperatures above liquid helium temperature. The x com ponent of the diffusion vector Rd is related to the conductivity by [26] R = L J x q£2 iwyx (1 - toyx) 2aQ and the other components of R_ are zero. The attenuation of longitudinal waves can now be evaluated in the low frequency limit (q£ « 1) and in the quantum limit (q£ » 1) by substituting [24] and [25] respectively into [23] and [26]. In the q£ « 1 limit 84 /C \2 (qR)1* (1 -y) W al " pSx \~kT (l + M)2)2 +(^)2(1-M): where R = [kT/(4-rme2)]2 is the Debye-H'uckel screening length and (4irne2/m)^ is the plasma frequency. Similarly in the q£» 1 0) P 1 imi t t28^ a£ ~ T P~S7 VTTJ (, + (qR)-2)2 for S « vlL. th The above procedure will now be applied to transverse waves ir. semiconductors. The attenuation of transverse waves is found by substituting the current expressions [16] and [18] and the electric field expressions [15] and [19] into the power equation [20]. In the case D » 1, the current and field parallel to the propagation direction give the important contribution to the attenuation. The attenuation is given by [29] cr = D2 _ L J t pSx xy Re A comparison of this result with the corresponding result for longi tudinal waves, [23], reveals that the two are identical except that the diagonal component C of the deformation potential tensor is xx replaced by the off diagonal component C . Accordingly the ql « 1 xy and qt » 1 limits given in [27] and [28] respectively, also apply to transverse waves provided that C is replaced by C xx xy Although the preceeding attenuation expressions have been derived assuming deformation potential electron-sound wave coupling, they may be readily modified to include a piezoelectric coupling constant. In piezoelectric materials such as CdS where the piezoelectric inter action is much larger than the deformation potential interaction at all attainable frequencies, the deformation potential factor qC can be replaced by ed , where d is the piezoelectric constant. When xy xy this replacement is made the expression for the attenuation in the low frequency limit [27] reduces to that obtained by White (1962) using a different method. Note that in the regime in which the deformation potential (or piezoelectric) coupling is dominant, ultrasonic waves are amplified when the electron drift velocity in the direction of the sound propagation is greater than the sound phase velocity. This amplification has been observed in CdS (Vrba and Haering, 1973) and the measured acoustic gain is consistent with eqn.[27] (Hughes, 1975). 3-4 Conclusion To summarize, we have outlined a transport equation approach to the problem of acoustic amplification in a d.c. electric field, that is much less complex than any that is currently available (Spector 1962, 1968). The new treatment duplicates the results of the earlier work and is simple enough to be readily generalized to apply to one and two dimensional metals. Furthermore the method is not restricted to 86 metals and can be easily applied to semiconductors, unlike the earlier treatment. In our calculation there is no need to make any special assumptions about the direction of the d.c. drift field. PART B MEASUREMENTS ON TTF-TCNQ CHAPTER I Experimental Method 1.1 Capacitive Measurement Technique Capacitive transducers have been widely used for exciting and detecting small amplitude vibrations (Barmatz and Chen 1974, Cantrell and Breazeale 1977, McGuigan et al 1977). The two main reasons for the popularity of capacitive transducers are their practical simplicity and high sensitivity to small displacements. In the conventional ca pacitive displacement detector, a large d.c. bias voltage is applied between the test object and a nearby electrode. Oscillations in the position of the test object will modulate the capacitance between the test object and the electrode and cause a current to flow in a large series resistor. The voltage signal on the resistor becomes progressively smaller and harder to measure as the frequency of oscillation of the test object is lowered and the pickup capacitance decreases. In this paper we describe an alternative capacitive detection scheme which does not significantly lose sensitivity at low frequencies and low capacitance values. The new approach to the capacitive vibration pickup was designed to make accurate sound velocity and absorption measurements in single crystals of TTF-TCNQ. A capacitive technique was chosen to avoid having to make good low loss acoustic bonds to the small and somewhat irregular TTF-TCNQ crystals. In order to be able to see longitudinal modes.in the elongated TTF-TCNQ platelets, the pickup transducer must 89 Fig. 16 - Drive and bridge detector circuit operate effectively with a total capacitance of less than 0.1 pf. In the remainder of this section we describe the new capacitive displacement measuring technique, and the electrostatic vibration excitation scheme. (i) Electronics Acoustic resonances were excited in the samples electrostatically by relying on the force between the plates of a charged capacitor. As shown schematically in Fig. 15 the left hand end of the sample forms one side of the capacitor and a nearby electrode forms the other side. When this capacitor is driven by the oscillator shown as e^ in Fig. 15, there is a periodic force on the sample. This oscillator - a frequency synthesizer - is operated in the frequency range 0-10 MHz. To increase the force on the sample the synthesizer output voltage is stepped up by a factor of four to a maximum of 110 V peak-to-peak by a Vari-L,LF-452 wideband transformer. To further increase the driving force and to reduce the relative importance of the second harmonic component of the driving force, a d.c. bias of up to hOO V can be superimposed on the a.c. signal across the drive capacitor. The vibration of the sample is detected by using an rf carrier signal to measure changes in a pickup capacitance. In the simplest case of a parallel plate capacitor, the capacitance will be inversely proportional to the distance between the sample and the pickup electrode. When the sample moves the detector capacitance will change leading to a corresponding change in its impedance. This impedance variation may be detected using the circuit shown schematically in Fig. 15-The carrier signal generator marked ec in Fig. 15 operates in the frequency band 300-1000 MHz, and its output is connected to the sample. When the sample vibrates, the varying impedance of the pickup capacitor, amplitude modulates the rf signal flowing through it. The amplitude modulation on the carrier is recovered by a Schottky barrier diode detector and low pass filter (see Appendix § 1.(i) for circuit diagram). Hence any displacement of the sample is reflected in the diode output voltage. Although this simple detection scheme is adequate for many purposes, the sensitivity can be improved by using a bridge circuit as shown schematically in Fig. 16. In this circuit the rf signal transmitted through the sample pickup capacitor is compared .with the signal through a stationary dummy capacitor. If the sample and dummy capacitances are equal the output from the balanced/unbalanced wideband transformer (Vari-L, HF-122) will be zero. Of course the inductive and resistive components of the impedance in the two arms must also be equal for a null output. In principle a tuned transformer would give better sensitivity, however, a wideband transformer was used in the TTF-TCNQ measurements because it is more convenient when dealing with a variety of samples of different size and shape. The output of the transformer is followed by a low noise UHF ampl i f ier.. (Avantek, AMM-1010) and then by a diode detector. The bridge improves the sensitivity of the detector for two main reasons. The first reason is that the rf signal generator noise is cancelled out when the bridge is balanced, since this noise is the same in both arms. Secondly the signal is a small deviation from a null whereas without the bridge the signal is a small ripple on top of a large rf carrier. Since the bridge output signal is small it may be amplified in a low noise rf amplifier before detection. This preliminary amplification reduces the importance of diode noise. The close proximity of the detector electrode to the drive electrode, particularly for small samples, can lead to problems with electrical pickup. Even though both electrodes are shielded there is a substantial direct capacitive pickup from the drive electrode to the detector electrode. This spurious pickup can cause problems; however, it may be filtered out with a shorted section of coaxial cable as shown schematically in Figs. 15 and 16. The length of the coaxial cable is chosen so that it is a quarter wavelength long at the frequency of the rf carrier. The quarter wave short acts as an open circuit at the carrier frequency and as a short circuit at the lower drive frequency. If a quarter wave short is connected to the bridge output, the direct pickup is eliminated before it can cause problems. A similar coaxial short at the output of the rf generator prevents any backflow of the drive signal into the rf oscillator. As described above, when the drive oscillator is turned on the sample will vibrate and generate a signal that is proportional to the vibration amplitude at the output of the diode detector. This signal is detected using a heterodyne detection scheme, outlined in the block diagram in D.C. BIAS VOLTAGE RF SIGNAL GENERATOR HP 8640B STEP UP TRANSFORMER FREQUENCY SYNTHESIZER HP 3330 A ,4. PHASE SHIFTER SAMPLE AND BRIDGE ASSEMBLY LOW NOISE RF AMP. V DIODE DETECTOR AMPLIFIER MIXER LP F AMPL ILTER IFIER i SIGNAL - AVERAGER -0 Fig. 17 Block Diagram of Electronic Equipment Fig. 17, which we now describe in detail. The first step is to amplify the diode output by about 40 db using a PAR 114 (plug-in 119) low noise preamp. The PAR preamp is used when the drive frequency is in the range 100 Hz to 1 MHz. For drive frequencies between 1 MHz and 10 MHz a MOSFET amplifier (see Appendix § 1.(ii) for circuit diagram) was used, followed by Avantek wideband amplifiers UA 105 and UA 106. The total gain for this combination was also about 40 db. The amplified signal then goes into a double balanced mixer (Minicircuits ZAD-6) together with a reference signal from the drive oscillator. Since the reference signal and the signal coming from the vibrating sample are normally at the same frequency, the mixer will have a d.c. output whose amplitude will depend on the relative phase of the two input signals. The mixer output will be a maximum for zero phase difference and zero for 90° phase difference. Clearly the detector diode and amplifiers will introduce various unspecified phase shifts into the signal coming from the sample. To compensate, the phase of the reference signal can be adjusted by a phase shifter (see Appendix § 1.. (iii) for circuit diagram). The mixer output is amplified (see Appendix § 1. (v) for the amplifier circuit diagram) and fed into a Nicolet 535 signal averager. In operation the frequency synthesizer steps automatically through a preselected frequency interval in 100 or 1000 steps at a rate of 1, 3 or 10 ms per step. The signal averager stores the mixer output at each step so that successive frequency sweeps may be accumulated to improve the signal to noise ratio. 95 SCALE i—-} 1 mm TOP VIEW 9mm Fig. 18 - Sample mounting configurations 96 Fig. 19 - Photographs of mounted samples Scale: 8 x actual size 97 WORM GEAR DRIVESHAFT DIODE TEMPERATURE SENSOR COPPER BOX HEATER POST SPUR GEAR TRANSFORMER SAMPLE SHIELDED DETECTOR ELECTRODE SCALE: } 1 i cm Fig. 20 - Cut-away view of sample box 98 Fig. 21 - Photograph of the outside of the sample holder with the door plate removed. Fig. 22 - Photograph of the inside of the sample holder. Note the a axis sample between the electrodes. In practice one observes a series of peaks in the output as a function of the drive frequency. These peaks correspond to acoustic resonances in the sample or its support. The peak width, amplitude and centre frequency of these resonances are the quantities of inter est. (i i) Sample Support In this section we describe the sample support mechanism and the ancillary low temperature apparatus. In all cases the TTF-TCNQ. samples were glued to a conducting support with Dupont 4929 silver paint which was diluted with 2-butoxyethy1 acetate to lengthen the drying time. Two basic mounting configurations were used as shown in Fig.1B and the photographs ; n F!g.1S- in the "vibrating reed" configuration (Fig.18a) one end of the sample is glued to a brass support. In the other configuration (Fig.l8b) the sample is attached to a pointed tungsten wire at the centre of its broad crystallographic a b face. In this case the glue contact spot is typically about .15 mm in diameter. A variety of different size tungsten wires were used from .003" to .015" in diameter. A point was etched on the wire electro-lytically, in a IM NaOH solution. The mounted sample is held in a copper box between moveable shielded electrodes. A cut-away view of a sample mounted in the box is shown in Fig.20, and photographs of the outside and interior of the box are shown in Figs.21 and 22. Its main upper part was machined out of a single piece of copper to reduce rf leakage and improve the mechanical integrity 101 0-80 BRASS SCREW NYLON INSULATION GUIDE TAB BRASS SHIELD Fig. 23 - Shielded electrode Scale: 8 x actual size Fig. 2k - Special notched electrode for smal1 samples 102 JLXSL ©t at SUPPORT FLANGE STAINLESS STEEL VACUUM CAN LIQUID NITROGEN COAXIAL CABLES AND WORM GEAR DRIVE RODS HELIUM EXCHANGE GAS SAMPLE HOLDER LIQUID HELIU Fig. 25 - Cryostat for low temperature measurements of the electrode drive mechanism. The electrodes can be removed from the box, so that different shape electrodes can be installed to match the size of the sample and the type of mode to be excited. Two of the electrodes used are shown in Figs.23 and 2k. The notched electrode in Fig.2k was used with very small samples. The spacing between the electrode and the sample is adjusted to .01 - .10 mm by turning nylon spur gears threaded onto the 0-80 threaded brass shaft of the electrode. The spur gears in turn are driven by worms. One rotation of the worm gear moves the electrode by .013 rnm, so that fine adjustments can be made to the spacing between the sample and the electrodes. In addition to the electrode drive mechanism the copper box also contains the transformer and dummy sample needed for the bridge de tection scheme described above (see Fig.20). The entire assembly is suspended by three 3 mm semi-rigid coaxial cables inside an evacuated stainless steel can, as shown in Fig.25. In addition, there are three pieces of thin walled stainless steel tubing connected to the worm gears. The three pieces of tubing protrude out of the top of the vacuum can through 0 ring seals, allowing adjustment of the positions of the electrodes when the sample chamber is evacuated. For low tem perature measurements helium exchange gas is introduced into the can and allowed to circulate inside the copper box which contains the sample. As long as the pressure is £ 0.5Torr, the exchange gas does not appreciably damp the sample vibration. During a low temperature run the vacuum can is normally suspended above the liquid level, in a glass dewar containing liquid helium (Fig.25). In this mode of operation four litres of liquid helium will last about 2k hours. The diode temperature sensor and heater on the top of the copper sample box (Fig.20) are used in conjunction with a Lakeshore Cryo-tronics diode temperature controller to maintain the copper box temperature constant to < 10 mK. The diode (#D2755) was calibrated commercially and this calibration is given in the Appendix § 1. It was checked at liquid helium, liquid nitrogen and roughly at room temperature. In addition the diode was used to make four probe d.c. conductivity measurements as a function of temperature on a TTF-TCNQ crystal. The measured phase transition temperatures were consistent with earlier measurements (Tiedje 1975). 1.2 Sensitivity of the Measurement Technique (i) Minimum Detectable Length Change The sensitivity of the apparatus to small displacements of the sample was determined by driving an a axis sample in its fundamental flexural mode with a known driving voltage. When a force F is applied to the free end of a cantilever beam of length £, according to elementary beam theory the resonant displacement of the free end is AH - °- F£2 Ad  - — where E is the Young's modulus, I is the area moment of inertia of the crossection of the beam, and Q. is the quality factor of the mechanical resonance. The minimum detectable displacement is deter mined from the signal to noise ratio (SNR) for a known driving force F by Admin = Ad/SNR. The force on the end of the beam can be estimated from the force on the plates of a charged parallel plate capacitor which is _ _ eoA V2 where A is the area of the plates, d is the plate separation and V is the voltage on the capacitor. . We consider Sample #22 with dimensions 0.335x0.204x0.018 mm as an example. This sample had its fundamental flexural mode at 118 khz, with an air damped 0. of 100 at room temperature. The drive and de tector electrodes covered the bottom third of the sample and were separated from the sample by approximately 0.08 and 0.02 mm re spectively. Using the room temperature a axis Young's modulus of 3-lxlO11 dynes/cm2 (see Chapter II below), the measured driving voltage (10V) and signal to noise ratio (176), we calculate that the minimum detectable displacement is Adm;n - 8x10-11 cm. For the SNR of 176 quoted above, the signal averager output noise bandwidth was 1.2 Hz. According to the parallel plate formula, the pickup capacitor had a capacitance of 9x10~3 pf. (ii) A.C. Method In this section we examine some of the practical limitations to the sensitivity of the capacitive displacement measuring technique. We consider the simplified model of the detector circuit shown in the schematic in Fig.26 below. Fig. 26 - Capacitive Vibration Detector - A.C. Method The noise source en is assumed to be due to Johnson noise in the re sistive element in the LC resonant circuit, in which C is the pickup capacitor. We have ignored the generator noise, because in principle it can be cancelled out by a suitable bridge detection scheme. We also assume that the effective impedance of both the generator and the amplifier input can be adjusted to any desired value by impedance matching using transformers, for example. The matching circuitry has been omitted in Fig. 26 for clarity. For optimum noise performance most rf amplifiers require a source impedance about equal to their input impedance. This condition will be satisfied at resonance provided Ra = Rn + Rq. Furthermore, we assume that the amplitude of the rf carrier signal is limited by the power dissipation in the sample, or equivalently by the power dis sipation in the resistor Rn. In this situation the optimum transformed generator impedance is zero. Normally one detects changes in the pickup capacitance by monitoring the amplitude of the trans mitted rf signal. In this case the circuit is most sensitive when frequency of the LC circuit and Qg is the quality factor for the electrical resonance. When all of the above conditions are met the minimum detectable change in capacitance is where T is the temperature of the resistor Rn, Ta is the noise temperature of the amplifier (Motchenbacher and Fitchen 1973) and Av is the output noise bandwidth of the detector. In eqn.[l] ^ is a factor which sets the desired detection threshold for a capacitance change. For example when ? = 1, a signal is considered to be detected if it has an amplitude equal to the rms noise level. In this section we use r, = 2. P is the power dissipated ih Rn. An expression similar to [1] has been obtained by Braginskii and Manukin (1977)-the generator frequency where coe is the resonant [1] AC _2_ [k(T + Ta)Av] C C Qe pl In practice the power dissipation is not usually a limitation. The limiting factor is more likely to be the capacitor breakdown voltage. In this case, when the breakdown voltage is Vmax, the minimum detectable capacitance change is C (ioe C Qe)i Vn d 'max where Ad/d is the corresponding fractional change in the spacing of the capacitor plates. An order of magnitude estimate for the maximum Q_e of an LC resonant circuit at room temperature is 100, taking into account coupling losses. Much higher Q.e's could be obtained using superconducting circuits. For example, Q_e > 106 can be achieved for superconducting LC circuits and even higher Qe's are possible in superconducting cavity resonators (Hartwig 1973). However, we will consider room temperature circuits only. It is interesting to compare the experimental sensitivity to the theoretical limit [2]. If the following values are used: T = 300K, Ta = 170K (Avantek AMM-1010 amplifier), toe = 4 x 109 s"1, C = 9x10-3 pf, Qe = 100 and Vmax/d = 10s V/cm (Cantrell and Breazeale 1974) then the limiting displacement is Ad - 2x10"13 /Av cm. Experimentally the rf field on the detector capacitor is likely to be at least an order of magnitude less than 105 V/cm. Accordingly the experimental sensitivity of 8x10-11 ZA\7 cm is probably only about an order of magnitude less than the limiting value predicted from [2], using the actual rf field. By judicious selection of the resonance frequency we, (coe C Qe)-1 ~ 10 can be achieved at room temperature for a wide range of pickup capa citance. For (we C Qe)-1 = 10, Ad = 10"llf /A~V cm. Braginskii et al (1971) have built a detection system capable of measuring Ad = 3x10"ll+ /Kv cm with C - 2000 pf and oje = 3xl07 s"1. (iii) D.C. Method For comparison purposes we now consider the d.c. analogue of the capacitive displacement measuring technique described above. In the d.c. approach, a constant bias voltage rather than an a.c. signal, is used as a probe to measure changes in the pickup capacitance. A schematic of the detector < ircuit is shown in Fig. 27 below. Changes in the pickup capacitance C alter the stored charge and induce currents in the bias resistor RD. Accordingly any vibration of the sample is reflected in a voltage on the bias resistor. C Fig. 27 - Capacitive Vibration Detector - D.C. Method 110 There are two main noise sources in the circuit shown in Fig. 27-They are the Johnson noise in the bias resistor and the amplifier noise. In order to estimate the sensitivity of the detector some information about the noise performance of the amplifier input stage is required. For vibration frequencies up to 10 MHz at least, the best available device for the input stage is a junction field effect transistor (JFET). To be specific, we will consider a Siliconix 2N4867 transistor which is a good low noise commercially available devi ce. A noise equivalent circuit for the capacitive detector using an amplifier with a JFET input stage, is shown in Fig. 28 below. jioVVC C Rb ib Cj R; in Fig. 28 - Noise Equivalent Circuit The Norton equivalent circuit has been used for the bias resistor noise generator. The mean-square resistor noise current T^2 is 4kTAv/RD-The current signal generated by the oscillating capacitance is repre sented by an equivalent current source jcoACV, where co is the mechanical vibration angular frequency. At midband the rms voltage noise en for the 2N4867 JFET is 2x10-9 V and the current noise in is 3x10-15 A /Kv. Below 30 hz en is dominated by 1/f noise and above 10 khz in increases 1 i nearly wi th co. From the equivalent circuit in Fig. 28 we find that the minimum detectable capacitance change is for a 1 hz bandwidth. The highest sensitivity is achieved by using the largest possible bias resistor up to a resistance of [co(C + C;)]"1 or where the current noise in dominates, at which time the sensitivity becomes independent of R^. A reasonable upper limit for R^ is 108 fi. We assume the input resistance of the JFET is larger than this and that Cj = 5 pf- If we substitute the circuit parameters given above into [3] and use C = 9x10-3 pf and co = 7-4x105 s-1 corresponding to the first flexural mode of Sample #22 then the minimum detectable displacement is Ad - 10"10 cm when the bias field on the capacitor is 105 V/cm. A contour map in the (co,C) plane of the displacement measuring sen sitivity of the d.c. capacitive technique is shown in Fig. 29. From the contour map we conclude that the d.c. method works best for high vibration frequencies and large pickup capacitances. In this regime Fig. 29 - Minimum detectable displacement Ad frequency f and pickup capacitance method. The contour labels are in as a function of C using the d.c. centimeters. its sensitivity is comparable with the a.c. method where Ad - 10~1 ^*^ cm. However, the d.c. method is less satisfactory for low vibration frequencies and small samples for which the pickup capacitance is necessarily smal1. CHAPTER I I The Modes of Vibration of TTF-TCNQ Crystals 2.1 Low Frequency Modes of an elongated plate The acoustic mode spectrum of an elastic body is exceedingly complex even for objects with simple shapes. It is feasible to calculate the acoustic resonance frequencies for only a few special cases. Provided that the material is elastically isotropic, the infinite medium, the infinite thin plate, the infinite cylinder, and the sphere are soluble. For arbitrary crystallographic symmetry only the infinite medium is soluble, although the other cases are soluble for certain types of elastic ani sotropy. It is interesting to note that the acoustic resonator problem is much more complex than the comparable electromagnetic resonator problem. An intuitive explanation is that there are only two electro magnetic waves possible in an infinite medium, namely two linearly polarized transverse waves, whereas in the acoustic case there are three waves possible - two transverse waves and one longitudinal wave. The acoustic resonator we are interested in here is an elongated, approximately rectangular parallelopiped with monoclinic symmetry. Although it is not possible to calculate all of the resonance frequencies of a TTF-TCNQ crystal, it is possible to obtain some very good approxi mations for the low frequency modes. To begin with let us assume that TTF-TCNQ is elastically isotropic, in order to simplify the discussion. The complications arising from elastic anisotropy will be discussed later, although it will turn out that TTF-TCNQ. is not far from being isotropic, elastically. (i) Flexural Modes The lowest frequency mode of the elongated platelet is a bending ("flexural") mode with displacements perpendicular to the broad face of the platelet. Since the TTF-TCNQ platelets are geometrically similar to an ordinary plastic ruler, the low frequency modes can be readily visualized with the help of a ruler. The ruler has three different types of modes which could be classified as flexural. The lowest frequency type (labelled F^ ) corresponds to bending along the length of the ruler with displacements perpendicular to the broad face. The second type (F^ ) is the same as the first except that the displacement is perpendicular to the edge of the ruler. The third type (F ) is a bend across the width of the ruler with displacements 3 C perpendicular to the broad face. The rationale behind the mode labelling scheme will become clear later. The propagation of a flexural wave along a beam whose long axis is parallel to y, is described by the wave equation in the limit that the wavelength is long compared to the beam thick ness. In equation [1] u is the displacement of the beam centre line from the equilibrium position, A is the crossectional area of the [1] EI 9% = 0 pA ay1* beam, and E is its Young's modulus. The area moment of inertia I is the moment of inertia of the beam crossection about a line through the beam centre line which is perpendicular to both the direction of the displacement u and the long axis of the beam. Consider the foot long ruler again as an example. If t is its thickness and w its width then I = t3w/12 for the soft Fbc modes and I = w3t/12 for the stiff F^g modes. Equation [1] also applies to flexural modes in thin plates. For an isotropic plate with Poisson's ratio v and a width much larger than a flexural wavelength, the Young's modulus E in [1] is replaced by the plate modulus E/(l - v2). The flexural resonance frequencies are determined by looking for the solutions of [1] which satisfy the boundary conditions or the ends of the beam. The resonance frequencies are given by (Timoshenko 1974) [2] f = _L /H fe-\2 LZJ Tn 2TT / pA \l) where £ is the length of the beam and mn is the ntn root of a transceden-tal equation which is specified by the boundary conditions. The boundary conditions of interest here are "c1amped-free" for the vibrating reed (cantilever beam) configuration and "free-free" for the central pin support. For these boundary conditions the first four mn values are given in Table III TABLE I I I Flexural Mode Parameters iriQ mi rri2 m3 Clamped-Free 1.875 4.694 7.855 10.996 Free-Free 0 4.730 7-853 10.996 (n + JT)TT 1.571 4.712 7.854 10.996 and the corresponding mode shapes are shown in Fig. 30. n Clamped-Free n Free-Free Fig. 30 - Flexural Mode Shapes For large n, mn asymptotically approaches (n + i)ir. As pointed out earlier the one dimensional wave equation [1] is only a good approximation if the flexural wavelength is long compared to the beam thickness. Furthermore the approximation will break down if the shear modulus is very small. The importance of the shear modulus is best illustrated by considering the extreme case of a pad of paper where the shear force between sheets is nearly zero. In this case the flexural rigidity of the pad of paper is dominated by the shear modulus between sheets and is therefore close to zero. In less extreme cases the contribution of the shear modulus is negligible, and the beam rigidity is determined entirely by the compression of the concave side of the beam and the extension of the convex side. Nevertheless, because of the chainlike nature of the crystal structure of TTF-TCNQ it has been suggested that the interchain shear moduli may be unusually small (Barmatz et al 197**). The shear modulus and finite beam thickness can be taken into account by adding some more terms to [1] (Timoshenko 1974). If the modified differential equation is solved for the resonance fre quencies, one finds that the additional terms in the equation reduce the effective Young's modulus for the ntn flexural mode in [2], by the factor (Goens 1931) [3] 1 + K (I)' m* | where K ~ 1 and G is a shear modulus. The beam thickness t is measured in the direction of the beam displacement during bending. In order to obtain an exact expression for the resonance frequency of even the lowest frequency flexural mode one would have to solve a three dimen sional differential equation and look for solutions which match the boundary conditions over the entire surface of the beam rather than just at its ends. In the samples which are supported by the central pin, the post at the centre provides an additional constraint which is not easily dealt with 119 using simple beam theory. However, for the lowest flexural mode a semiquantitative estimate of the effect of the central support can be obtained from the numerical calculations of Southwell (1922) for centrally pinned discs. The effect of even a very small pin contact area is to raise the zero frequency n = 0 mode of a free-free beam up to about 80% of the n = 1 mode frequency. The flexural modes with a mode at the centre and the higher frequency flexural modes are not affected very much by the pin. (ii) Torsional Modes We now consider the low frequency torsional (T) modes of the elongated platelet (or ruler). The displacement of the platelet for the torsional modes we are interested in here, is a volume con serving strain in which neighbouring crossections perpendicular to the long axis of the sample are twisted relative to one another. The propagation of torsional waves along a beam which is long com pared to its lateral dimensions is described by the wave equation (Landau and Lifshitz 1970) m 924> = C 92<j> 1 J 9t2 pi 3y2 where the long axis of the beam is along y as before, and <j> is the angle of rotation of a crossection. Ip is the area moment of inertia of a crossection about its centre, and C and p are the torsional rigidity and density of the beam respectively. The torsional resonance frequencies for a beam of length t are given by l5' fn - M; M For the clamped-free boundary conditions n is an odd integer and for the free-free boundary conditions n is an even integer. Unless the crossection of the beam is circular, plane crossections become warped under torsion. Because of this feature, the torsional rigidity is a complicated function of the shape of the crossection (Timoshenko 1951). The torsional rigidities for some simple shapes are given by Landau and Lifshitz (1970). For a beam with a rectangular crossection of width w and thickness t < 0.2 w. C is given by (Timoshenko 1951) [«] "'t('^;)' In the t« w limit, the torsional resonance frequencies in [5] reduce to f = — M~ — n w / p 21 where G is the shear modulus, as before. In theory, if the lateral dimensions of the beam are comparable to a torsional wavelength then a correction factor analogous to [3] is needed in the expression for the torsional resonance frequencies. In practice the correction factor does not change the frequency much even when the lateral dimension is equal to a half wavelength. Furthermore an approximate theoretical calculation of the correction factor is apparently in disagreement with experimental results (Behrens 1968). This correction will be ignored here. (iii) Comments on a Short Plate We now describe qualitatively the low frequency modes of a plate in which the width of the plate is comparable to its length. This situation is of interest because it illustrates what happens to the modes of an elongated plate when the wavelength is comparable to its width. In addition some measurements were made on nearly square platelets cut off the end of standard elongated TTF-TCNQ crystals. The low frequency modes of a plate are usually illustrated by Chladni figures, which are the patterns of nodal lines of the modes (Waller 1961, Leissa 1969)• The experimentally determined patterns of nodal lines for the first few modes of a square cantilever plate and a rectangular free plate are shown in Fig.31 and 32 respectively. Observe that the first mode in Fig.31 appears to be purely flexural, the second one torsional and the third one flexural again. The fourth mode looks like a trans verse flexural mode, but it is not purely flexural because of the edge clamp. Although the frequency of this mode cannot be calculated accurately using the simple beam formulas, the frequency of the first two modes can be calculated with reasonable accuracy from [2] and [5], since the vibration wavelength for both modes is at least four times the width of the "beam". In addition the first two modes are well separated from 2.63 6.27 8.14 9.22 Relative Frequency Fig. 31 " Modes of a Square Cantilever Plate 0 1 2 Fig. 32 -Modes of a Rectangular (2:1) Free Plate their neighbours in frequency, making them easy to identify experi mental ly. An examination of the nodal patterns in Fig.32 shows that just as for the cantilever plate, the higher frequency modes are likely to be a complex mixture of torsion and flex. It is difficult to obtain information about the elastic moduli from these modes because of their complicated nature. We now temporarily leave the subject of torsional and flexural vibrations and investigate another simple type of vibration of a long plate. (iv) Elongational Modes The final type of mode which we will consider is the longitudinal stretch or elongational (L) mode. The propagation of an extensional wave along a rod oriented parallel to the y axis, is described by the wave equation m liiL = £ iiy. L/J 8t2 p 9y2 where u is the displacement of the rod along its axis. The Young's modulus E appears in [7] rather than a bulk wave longitudinal modulus because the rod is free to expand or contract laterally depending on whether it is being compressed or extended longitudinally. As before, the resonance frequencies are obtained by looking for solutions of the wave equation which satisfy the boundary conditions Fig. 33 - Arrangement of TTF and TCNQ molecules in the a c plane. The solid dots are tipped up above the plane. on the ends of the beam. For free-free boundary conditions the re sonance frequencies are For short beams and high frequencies the longitudinal acoustic wave length may be comparable with the transverse dimensions of the beam. In this situation, the one dimensional wave equation [7] is no longer a good approximation just as the one dimensional equations for the flexural and torsional waves are no longer a good approximation in the same limit. Provided the wavelength is not too short, the elon gational resonance frequencies may be corrected by dividing the Young's modulus E by the factor (Love 19^4) for a beam with Poisson's ratio v and a rectangular crossection. (v) Crystallographic Symmetry Up to now we have assumed the vibrating plate or beam to be made of an isotropic material. However, the TTF-TCNQ samples on which the measurements were made are not isotropic; rather, they are monoclinic crystals (space group P2j/C ). The positions of the TTF and TCNQ molecules in the crystallographic a c plane are shown in Fig. 33 This notation is explained by Henry and Lonsdale (1951). (Blessing and Coppens 1974). Fig. 2 shows the segregated stacking arrangement of the molecules in the b direction which is largely re sponsible for the unusual electrical properties of TTF-TCNQ. All pf the UBC grown TTF-TCNQ crystals have the unique ( b ) axis parallel to the long axis of the crystal and the a axis parallel to the broad transverse dimensions. The c axis is about 14.5° away from being perpendicular to the other two axes; hence it is not quite per pendicular to the broad a b face of the crystal. This feature allows a two fold ambiguity in the direction of the c axis in a real crystal and opens up the possibility that what appears to be a single crystal may actually be twinned. For convenience we will use the reciprocal lattice vector c" which is defined to be perpendicular to a and b, instead of c when discussing the vibration modes in relation to the crystallographic symmetry axes. Monoclinic acoustic resonators are not easy to deal with theoretically, because in general 13 independent elastic constants need to be considered (Auld 1973). In a beam made of a monoclinic material, the flexural modes are coupled to the torsional modes in a complex way, by certain elastic constants. However, if the TTF-TCNQ c axis were perpendicular to a (or the crystal were suitably twinned), the flexural modes and torsional modes would be uncoupled. To make the vibration problem manageable we will assume that c is perpendicular to a. This amounts to assuming that the samples have orthorhombic symmetry and that the crystallographic symmetry axes are aligned along the symmetry axes of the sample crystal. The approximation is probably not unreasonable since the c axis is only 14.5° away from being perpendicular to the a axis. In general an orthorhombic material has nine elastic constants, which may be broken down into three Young's moduli, three Poisson's ratios and three shear moduli. The elastic moduli are most con veniently defined in terms of the 6X6 compliance matrix s j j (Auld 1973). The Young's moduli Ea, and Ec for the three crystallo-graphic symmetry directions are equal to s^"1, s^1 and Sg"1 respectively when the a b and c" axes are aligned parallel to the x y and z axes. Similarly the three shear moduli C44, C55 and 055 are equal to s^"1, s^1 and Sg"1 respectively (Lekhnitskii 1963). The expressions for the resonance frequencies of isotropic beams can be readily generalized to apply to orthorhombic beams. Let us consider the flexural modes first. The Young's modulus in the frequency expression [2] should be the modulus along the direction which the beam is compressed and extended during vibration. For example, the relevant Young's modulus for the Fbc and Fb modes is Eb. The shear modulus which goes into the correction factor [3] is best described with the help of the pad of paper analogy. If the beam were a stack of weakly interacting sheets, bent in its softest direction, the appropriate shear modulus to put into the correction factor would be the modulus against sliding of the sheets on top of one another in a direction parallel to the long axis of the beam. For the F^c modes, c^ is the appropriate shear modulus. The elast constants which apply to the three experimentally observed types of flexural modes are summarized in Table IV. TABLE IV Flexural Mode Elastic Constants Mode f Young's Modulus E Shear Modulus ^ G Fbc Eb c66 Ea c55 . refer to eqn. [2] and [3] The torsional rigidity has been calculated for an orthorhombic beam with a rectangular crossection in the books by Hearmon (1961) and Lekhnitskii (1963). The results may be readily adapted to the two different types of torsional modes which were studied experimentally. In the first type of mode (Ta mode) the torsion axis is parallel to the a axis. In this case the torsional rigidity [6] should be replaced by r _ r t3w / 192 t /c66\*\ [10] C " c66 — M ^^(c^J J for t < 0.2 w and c6£ ~ c^^. The thickness t is to be measured in the c" (thin) direction, and the width is measured in the b direction. Similarly,when the torsion axis is parallel to b (T^ mode) r -, r t3w /1 192 t /c66\*\ W C = C66 — ^ "TT w(ci?) ) where t is measured in the c" as before, and the width w is measured in the a di rection. It is trivially easy to generalize the expression for the elongational mode frequencies to an orthorhombic beam. One need only replace the isotropic Young's modulus E in [8] by the b axis Young's modulus E^. Although only the b axis modes were measured experimentally, the fre quencies of elongational modes for beams with long axes in the a or c direction could be calculated in an exactly analogous way. Similarly, the correction factor [9] for the b axis mode should be replaced by (Behrens 1968) r,^i 1 j. 1 /niiA2 / 2 2 . 2 .2\ [12] 1 + J (2IJ (V12 w + v23 t j for an orthorhombic beam where v12 and v23 are Poisson's ratios defined in terms of the elements of the compliance matrix by v^/lia = - Si2 and V23/Ec = " s23-Before going on to describe some of the experimental results we first discuss a more drastic approximation to the elastic symmetry of TTF-TCNQ. This approximation will be useful later on in obtaining an estimate for the bulk modulus from the experimental data. The salient feature of the TTF-TCNQ structure is the linear stacking of the molecules along the b direction. Accordingly one might expect the elastic properties to be different depending on whether the strain is parallel or per pendicular to the molecular stacks. The highest symmetry crystal system for which this distinction is possible is the hexagonal system, in which there are five independent elastic constants. A crystal belonging to the orthorhombic system will have hexagonal symmetry if two of the orthorhombic symmetry directions are equivalent This relationship implies that when b is the preferred direction in a hexagonal material, the a c plane is elastically isotropic. A necessary condition for the hexagonal symmetry to be a good approxi mation is that the difference between the a and c axis Young's moduli be small, at least compared to the difference between the a and b axis Young's moduli. The idea that the a and c directions are approxi rnately equal by comparison, is supported by recent room temperature compressibility measurements (Debray et al 1977) and thermal expansion data (Blessing and Coppens 1974). In conclusion a hexagonal model is the simplest approximation to the structure of TTF-TCNQ that still includes the essential anisotropy of the material. 2.2 Interpretation of Experimental Mode Spectrum The expressions derived in the previous section for the vibration frequencies of elongated plates will be used to interpret the experi mental mode spectrum of TTF-TCNQ crystals. As we have already pointed out there are severe mathematical difficulties in calculating the vibration frequencies of a rectangular resonator made of a monoclinic 10 khz 30 khz 50 khz Experimental flexural mode spectrum. The numbered marks below the experimental trace indicate the frequencies of the flexural modes computed from [2] using the measured sample dimensions and a sound velocity to fit the second flexural mode. The noise level is comparable to the thickness of the line. material. However, a more serious practical limitation to the accurate calculation of the resonant frequencies is the somewhat irregular geometry of the available TTF-TCNQ crystals. For example, the sample thickness (c* dimension) which is normally the least uniform dimension, typically tapers off substantially near the ends. If the ends are cut off with a razor blade a good crystal will not vary in thickness by more than about 10% over its length. In addition, the silver paint clamp on the end of the sample in the vibrating reed configuration, is neither perfectly rigid nor perfectly uniform. Similarly the central support point for the longitudinally mounted samples perturbs the free-free boundary conditions. For all of these reasons we expect to see deviations from the idealized resonance fre quencies given in Section 2.1 above. (i) Vibrating Reed Support The vibrating reed support configuration was used to study flexural and torsional modes. An experimental flexural mode spectrum is given in Fig.34, which shows the first four flexural modes along with a theoretical spectrum obtained from [2] by adjusting the Young's modulus to fit the second experimental flexural mode. The first resonance in Fig.34 appears to be weaker than the second one because it has a lower Q (see Section 2.3 below) and the amplifier gain is smaller at low frequencies. The resonance lines are antisymmetric because the phase of the reference input to the mixer has been set to detect the component of the sample response which is out of phase with the driving force. 133 400 300-f (khz) 200-I00h 3 4 5 ( ir/2 units) Fig. 35 - Low frequency flexural and torsional modes of a vibrating reed. By increasing the system gain, it was possible to see the next five harmonics in the series shown in Fig.3^. In addition, six low fre quency 1^ torsional and flexural modes were identified. By ana logy with the continuous dispersion curves for acoustic waves pro pagating along an infinitely long sample, dispersion curves can also be plotted for the modes of a finite length sample as a series of discrete points. The low frequency mode dispersion diagram for a sample in the vibrating reed configuration is shown in Fig.35. In addition to the modes plotted in Fig.35, a large number of unidenti fied higher frequency modes are observed experimentally up to about 1 MHz. Three pieces of information are helpful in identifying the various flexural and torsional modes. First, the torsional modes can be se parated from the flexural modes by their temperature dependences. The torsional mode frequencies depend on a shear velocity which has a weaker temperature dependence than the Young's modulus velocity which determines the flexural frequencies (see Chapter III below). Secondly, the two different types of flexural modes (F^a and Fbc) may be distinguished by the way they couple to the drive and detector electrodes. For example the F^a flexural modes are preferentially excited if the axis of the drive and detector electrodes is aligned parallel to the a axis. Needless to say, no matter which type of mode is preferentially excited it is almost impossible to avoid a slight excitation of all the other types of modes through non-ideal sample and electrode geometry. Finally, once the modes have been 135 BIAS VOLTAGE Fig. 36 ~ Effect of d.c. bias voltage on fundamental Fbc flexural mode resonance frequency. identified as Fba, Fbc or Tb, the frequency equations [2] and [5] can be used to assign harmonic numbers and as a check on the Fba, Fbc and T"b identification. The resonance frequencies of a thin vibrating reed can be artifi cially reduced if a large d.c. bias voltage is applied to the drive or detector electrodes (Barmatz and Chen 197*0. Because of the d~2 dependence of the force on the plates of a charged parallel plate capacitor, the spring constant for a bent reed will have an electrical component as well as an elastic component. In the technique described here no d.c. bias field is required on the detector capacitor. However, a d.c. bias is used at the drive electrode to increase the driving force on the sample. The effect of this bias voltage on the frequency of the. fundamental Fbc flexural mode is shown in Fig.36. The d.c. field has a much smaller effect on the torsional and higher harmonic flexural modes. As long as the long axis of the reed is aligned parallel with the crysta1lographic b axis, the low frequency modes give no information about the Young's moduli perpendicular to the b axis. In order to measure the a axis Young's modulus, thin slices were cut off the end of normal TTF-TCNQ. crystals perpendicular to the b axis. The shape of the four slices studied experimentally ranged from nearly square to rectangular with the a dimension twice as long as the b dimension. One end (be* plane) of the sample was glued to a support to produce a small vibrating reed with its long axis in the a direction. Only the first three or four modes could be confidently interpreted in terms of the mode patterns shown in Fig.31 of Section 2.1 (iii) above. The first and third resonances are Fac flexural modes, whose frequencies are determined by the a axis Young's modulus Ea as outlined in Table IV. Section 2.1 (v) above. Similarly, the second mode is a Ta torsional mode whose frequency is determined by the shear modulus c^g as indicated in eqn. [11]. The measured sound velocities are summarized in Table IV. The signal to noise ratio for the fundamental flexural mode was greater than 100 with a 1 Hz noise bandwidth even for the smallest sample studied, which was a 2ug, 0.3 mm long slice. Since the a axis samples were significantly less uniform than the bigger b axis samples the experimental value for Ea is not as accurate as the experimental value for E^. Another experimental difficulty arose on cooling the a axis samples. Unless the length (a dimension) of the sample was significantly bigger than its width (b dimension), differential thermal contraction of the bond at the end would split the sample in half along its length. Thermal cracking of the sample shows up as a large irreproducib1e dis continuity in the temperature dependence of the resonance frequency, and in a splitting of a single resonance line into a doublet or multiplet. The thermally induced cracks are visible if the crystal is examined under a microscope. Of the four a axis vibrating reeds measured only one would cycle down to helium temperature and back to room temperature without breaking. It is possible that some small fractures were also produced near the bond in the longer b axis vi brating reed samples during thermal cycling. 2.0 1.5 1.0 0 / SAMPLE #16 . / L MODE A y © / / / / / / / / / / / / / / [/_ 0 1 2 3 4 5 6 7 WAVENUMBER <jrU units) Fig. 37 - Longitudinal mode dispersion (ii) Central Pin Support The problems with differential thermal contraction may be largely avoided if the sample is supported by a very small silver paint contact on the end of a pointed wire, as described in Chapter I, Section 1.1 (ii). This support configuration is a favourable one for exciting elongational acoustic modes in the sample. For a long sample with a uniform cross-section, the elongational resonance frequencies are given by [8] pro vided one ignores the small pin contact at the centre of the broad a b face of the sample. This approximation is expected to be best for the odd numbered modes since these modes have a node at the support pin. If the experimental elongational mode frequencies are plotted as a function of the wavenumber 2-nn/Z, where L is the length of the sample, then one obtains the discrete dispersion curve shown in Fig.37. Eqn. [8] predicts a linear dependence of the resonance frequency on wavenumber. A close examination of Fig.37 revea1s that the even harmonics are slightly below the straight line and the odd harmonics are slightly above the straight line. In addition, the even harmonics tend to be more heavily damped than the odd harmonics. We attribute these diffe rences between the even and odd harmonics to the effect of the central pin support. The effect of the finite length correction factor [9] (or [12]) is to reduce the frequency of the seventh harmonic by 1-2%. Beyond the fourth harmonic, more than one frequency is plotted in Fig.37 for each wavenumber. The reason for the multiplicity is that there are several modes of nearly equal strength near the frequency where a longitudinal resonance should be. Presumeably, as the wave-length of the longitudinal mode becomes shorter, the non-uniformities in the sample dimensions become progressively more important and the longitudinal modes are coupled more strongly to other acoustic cavity modes. The other acoustic modes which could be coupled to the longi tudinal modes are high frequency harmonics of the b axis flexural and torsional modes and transverse Fac type flexural modes. Moreover, the sample whose longitudinal resonance frequencies are shown in Fig. 37 begins to support transverse shear wave resonances at frequencies corresponding to n > 7. Above 2 MHz the acoustic mode spectrum de generates into a closely spaced set of heavily damped resonances. Of course all of these modes are affected in some complicated way by the central support. This complexity of the high frequency acoustic mode spectrum provides a practical upper limit to the frequency range for which the acoustic resonance method is useful. The measurements could probably be extended to higher frequencies by studying acoustic pulse propagation along the sample rather than looking at resonances. A comparison of eqns. [2] and [8] in light of the discussion on the crystalline anisotropy in Section 2.1 (v) above, reveals that the elongational and b axis flexural resonance frequencies are determined by the same elastic constant to first approximation - namely Eb. The Young's modulus velocity (ED/p)5 has been measured for eight TTF-TCNQ. samples by substituting the experimentally determined flexural and longitudinal resonance frequencies into [2] and [8] respectively. Within our experimental error there is no systematic softening of the Fbc type flexural modes as would be expected from [3], if the shear modulus CL^ were anomalously small as has been suggested by Barmatz et al (197*0 and Ishiguro et al (1977)- The shear softening of the higher frequency type modes is consistent with the shear modulus C55 that is determined from the torsional modes. Even though the flexural modes can be used to determine the same extensional velocity (E^/p)^ as the longitudinal mode, the longitudinal one generally gives a more accurate estimate of this velocity. The reason is that the longitudinal resonance is less sensitive to variations in the sample thickness (c* dimension) which is the least uniform dimen sion. The room temperature sound velocities as determined from the acoustic resonance frequencies of fifteen different samples, are summarized in Table V. The b axis extensional velocity (E^/p)^ and the shear velocity (cgg/p)^ are consistent with inelastic neutron scattering measurements of Shapiro et al (1977). The a axis exten-1 sional velocity (Ea/p)? is consistent with recent compressibility measurements of Debray et al (1977). The detailed temperature depen dence of these velocities is disccused in the next chapter. TABLE V Room Temperature Sound Velocities in TTF-TCNQ Mode Velocity (10s cm/s) b axis extensional 2.8 ± .1 a axis extensional k.k ± .5 egg shear 1.7 ± .2 100 140 180 220 T (K) Fig. 38 - Flexural and torsional mode crossing (iii) Mode Coupli ng Before going on to discuss the temperature dependence of the sound velocities in detail, we first outline some of the effects of extraneous mode coupling on the temperature dependence of the resonance frequencies. As pointed out in the next chapter, the b axis Young's modulus velocity has a stronger temperature dependence than the shear velocity. For this reason, if a mode which depends on the Young's modulus is at nearly the same frequency as a torsional mode at one temperature the two modes may cross as the temperature is changed. A mode crossing of this type is shown in Fig. 38. The torsional mode starts off just above a nearby flexural mode at high temperatures. As the temperature is lowered the frequency of the torsional mode moves below the flexural mode because of the difference in their temperature dependences. Coupling between the two modes prevents them from actually intersecting. The coupling could be caused by the off diagonal component S26 'n the monoclinic compliance matrix (Hearmon 1961) or by asymmetry in the sample or support. The torsional mode in Fig.38 is the fundamental torsional mode and the flexural mode is probably a symmetric Fbc type flexural mode with three nodes. These resonances were observed with the sample supported in the longitudinal mode configuration. Mode coupling can be a problem in making accurate measurements of the temperature dependence of the sound velocity. The problem seems to be particularly severe for elongational modes in the temperature range between 20K and 52K, where the Young's modulus has an anomalously strong temperature dependence compared with the shear modulus (see next chapter). 93.0 T Fig. 39 - Fac flexural mode with interference from an unidentified mode of the support The strong temperature dependence makes it more likely that the re sonance of interest will cross some other mode with a weaker tem perature dependence. Fig. 39 shows a typical mode crossing of this type. Here an Fac type mode is crossed near 39K by an unidentified mode, that is probably related to the support. (iv) Support Modes It is virtually impossible to avoid some interference with the sample resonances from the modes of the support. However, it is possible to avoid support modes over a limited frequency range. For example no support modes were observed among the low frequency flexural and torsional modes of b axis crystals clamped in the vibrating reed configuration. On the other hand, as illustrated in Fig.39 there were some problems with interference with the flexural modes of the a axis slices probably because these modes were at higher frequencies. Inter ference from the modes of the tungsten support wire for the longitudinal mounted samples, was alleviated by using a thick support post (0.015" diameter) so that the support modes were widely spaced and at relatively high frequencies. In any case the sample modes and support modes can be easily distinguished at high temperatures, by the dramatic difference in their temperature dependences. 2.3 Vibration Damping (i) Q_ Measurement The acoustic resonances can be used to determine the absorption of sound as well as the velocity of sound. The absorption is proportional to the width of the resonance line. A convenient measure of the ab sorption is the quality factor Q_ defined by Q"1 = Af/f where f is the resonance frequency and Af is the "full width at half maximum" of the symmetric (in phase) amplitude response. Equivalently Af is the separation between extrema in the antisymmetric (out of phase) resonant response. The Q_ is related to the intensity attenuation factor a discussed in Part A above by a = Q._1q where q is the sound wavenumber. The Q_ was measured experimentally by comparing the symmetric re sponse of the sample with a synthetic lorentzian on a dual beam oscilloscope. The synthetic lorentzian was generated by sweeping a voltage controlled oscillator through the resonance frequency of a tuned circuit (Q_ ~ 70) as shown in the circuit diagram in Appendix § 1.(v). By adjusting the amplitude of the frequency sweep of the voltage controlled oscillator, the apparent width of the synthetic line could be adjusted to match the width of the sample resonance. After calibration the synthetic lorentzian provided a convenient means for measuring linewidth to a relative accuracy of 1%. In order to make accurate linewidth measurements one must be careful to avoid distorting the sample resonance lines. For example if the drive oscillator is swept too quickly through the acoustic resonance, a ringing phenomena known in NMR as "wiggles" (Abragam 1961) will occur. An extreme example of wiggles is shown in Fig. kO. Horizontal scale: 1 Hz/cm Sweep rate: 3-3 Hz/s Fig. 40 - Wiggles (ii) Thermoelastic Damping Heat conduction is an important loss mechanism for the low fre quency flexural modes of TTF-TCNQ. crystals. As discussed by Zener (1948) and Bhatia (1967), whenever the isothermal and adiabatic elastic moduli (see Chapter III) are not equal, thermal conduction between compressions and rarefactions will cause acoustic damping. In conventional metals thermal conduction does not cause significant damping of longitudinal waves at frequencies below about 10 Ghz. However, the damping can be substantial for flexural modes of thin plates where the compressed part of the plate is close to the ex-. panded part. The thermoelastic damping a rectangular crossection is of a flexural resonance of a reed with given by (Bhatia 1967) [13] Q. ^— f2 + f2 • fo-2"rT where Es and Ey are the adiabatic and isothermal Young's moduli re spectively, f is the resonance frequency and t is the reed thickness, measured along c" for the FDC modes. The thermal diffusivity D is the ratio of the thermal conductivity K to the specific heat at constant volume, Cy. To obtain a numerical estimate for the ab sorption we approximate (Es - Ej)/Es by (Bs - Bj)/Bs where Bg (By) is the adiabatic (isothermal) bulk modulus, and use By/Bs = Cv/Cp derived by Bhatia (1567) (see also eqn. [2] in Chapter III below). With the expression for the difference between Cp and Cv given by Landau and Lifshitz (1969) and using the Gr'uneisen approximation discussed in Section 3-1 of Chapter III below, we obtain ~ Ey C - Cv [14] %s = PCp = YaT where y ~ 2.6 is a Gr'uneisen constant and a is the volume expansion coefficient. From eqns [13] and [14] we can estimate Q"1 for the flexural modes using published thermal expansion (Schafer et al 1975, Blessing and Coppens 197*0, b axis thermal conductivity (Salamon et al 1975) and specific heat data (Craven et al 197*0. For example, consider Sample #13 which was 0.038 mm thick and had its fundamental flexural resonance 25 20 15 I (IO"4) 10 0 T T SAMPLE 13 MODE NUMBERS 12 3 4 oo oo o 4 A x * o o 0°° * e© o O O © © ©o e © ° © o o o _L 1 20 40 T (K) 60 Fig. 41 - Damping of first four Fbc flexural modes at 2.1 khz. For this mode we estimate Q-1 = 8x10"^ using a thermal relaxation frequency fQ = 21 khz. The experimentally determined absorption is shown in Fig. 1*1 as a function of temperature for the first four flexural modes. Although the estimated value of Q"1 at 55K of 8x10-lt is close to the measured value of 13x10-l+ for the fundamental flexural mode at 55K, the dependence of the damping on the flexural mode frequency is wrong. With fQ = 21 khz in [13] the absorption will be larger for the second flexural mode whereas experimentally it is observed to be smaller. In addition, above 55K the damping of the second, third and fourth harmonics is only weakly temperature dependent (see Fig. 41). To help explain these features of the experimental da"a we make the following observations. First the thermal relaxation rate fD will increase with decreasing temperature at least as fast as T-1. This temperature dependence should be valid down to 30K, at least. Secondly YaTfQ will be a weak function of temperature. The ex perimental data in Fig.41 may now be accounted for qualitatively by postulating that fQ satisfies fi < fQ < f2 where fi(f2) are the first (second) flexural resonance frequencies. Furthermore, if we observe that the damping is nearly equal for the first two modes at 40K, we can estimate fQ at this temperature. This estimate for fG (5 khz) implies that the thermal conductivity in the c" direction is 0.02 W/cm-K. This value compares with 0.12 wa/cm-K for the b axis thermal conductivity measured by Salamon et al (1375) at the same temperature. Similarly fc = 5 khz implies that Q"1 = 12x10-tt 6 4 I 4) 2 0 1 1 1 1 1 SAMPLE ' #23 • 4 r i P A W © y»\ .® © 1 I / ii / o f » ^ o © "V* © *• 1 1 1 1 1 0 50 100 150 200 250 300 T (K) hi - Damping of fundamental longitudinal mode. The discontinuity at 205K arose when a measurement was repeated after allowing the sample to remain at low temperatures overnight. at kOK, which is about a factor of two bigger than the observed absorption. Clearly, all of these calculations are very rough. Nevertheless, one can conclude that thermal conduction will have a significant effect on the damping of the flexural modes. It should be possible to determine the transverse thermal diffusivity, by careful measurements of the damping of flexural modes. Finally, we note that thermoelastic damping may account for the anomalous frequency dependence observed by Barmatz et al (1975) in the damping of flexural modes in 2H-TaSe2. (iii) Elongational Modes The room temperature damping of the fundamental elongational mode in TTF-TCNQ. is typically an order of magnitude, smaller than the room temperature damping of the fundamental flexural mode. In addition the absorption for the elongational modes increases with harmonic number unlike the first few flexural modes. The temperature dependence of the acoustic absorption for the fundamental longitudinal mode of Sample #23 is shown in Fig. kl. Although the magnitude and detailed temperature dependence of the experimentally observed longitudinal mode damping are not completely reproducible between different samples and different runs with the same sample, certain gross features are always present. There is a broad minimum in the vicinity of 60K where Q ~ IO4. As the sample is cooled through the metal insulator transition the absorption begins to increase, reaching a peak in the range 3O-40K where it is a factor of 3"6 times larger than the absorption near 60K. As the temperature is lowered still further the absorption decreases again. 153 — © U if) O o o E^23 2 8 o ° 8 8 o o o o o o o o o o o o o o © Vc •0 46 48 50 52 54 56 Fig. A3 - Sound velocity and attenuation near the metal-insulator trans i t i on In Sample #23 there is also a 10K wide absorption peak near 150K and a very narrow (< 0.5K wide) peak in the damping just below the metal-insulator transition. The details of the temperature dependence of the damping and the velocity of sound near the metal-insulator transition are shown in Fig. 43. The temperature dependence of the velocity of sound is discussed in detail in the next chapter. Al though neither the narrow absorption peak near the transition nor the wider maximum near 150K were observed in any other sample, it is quite possible that they were missed by not taking measurements at fine enough temperature intervals. The absorption peak near the metal-insulator transition is reminiscent of a similar feature observed near the incommensurate charge density wave transition in 2H-TaSe2 by Barmatz et al (1975). Although the peak near 150K may be due to some extraneous effect, it is tempting to try to relate it to the dis appearance of the 2kp scattering of diffuse X-rays, observed by Khanna et al (1977) near 150K. An upper limit to the conventional electronic contribution to the absorption can be obtained by using the expressions derived in Part A, Chapter I, for the attenuation of sound in three dimensional metals, or the expression for the peak attenuation just below the metal-semiconductor transition, discussed in Part A, Section 2.3-From the bandstructure (Berlinsky et al 1974), d.c. conductivity, and crystal structure we estimate the electronic effective mass nr = 6 in-,, the Fermi velocity Vp - 107 cm/s, the electronic scattering time x ~ 5x10~ll*s at 60K and the carrier density n = 2.8x1021 cm"3. With these values for the material parameters, the electronic con tribution to the damping of a 300 khz longitudinal mode is calculated to be Q-1 - 10~9. Since the measured absorption is of order 10-1*, we conclude that the observed damping is not due to the conduction electron loss mechanism discussed in Part A. If TTF-TCNQ. remained metallic at low temperatures it is possible that this electronic loss mechanism would eventually become important at low temperatures, as it does in ordinary metals. Although thermal conduction is probably the dominant loss mechanism for the low frequency flexural modes, its contribution to the damping of the longitudinal modes is completely negligible. Dislocation damping (Bhatia 1967) and coupling to low Q support modes are probably important sources of loss for the longitudinal modes. An additional damping mechanism is suggested by the model used in the next chapter to explain the temperature dependence of the sound velocity. We comment on this loss mechanism at the end of the next chapter. (iv) Effect of Air on Resonance Frequency and Q Before going on to discuss the temperature dependence of the sound velocity in TTF-TCNQ we briefly outline the effect of air at one at mosphere on the resonant frequencies and Q of the vibrating sample. The shift in the resonant frequency is due to the mass of entrained air that accompanies the vibrating sample. When the sample vibration frequency is low enough that the corresponding wavelength of sound in air is long compared to the transverse ( a ) dimension of the sample, the surrounding air may be treated as an incompressible non-viscous fluid. For typical TTF-TCNQ crystals this condition is well satisfied up to about 50 khz. In this limit the air entrained by an F|3C flexural mode may be approximated by a cylinder with its axis along b and its diameter equal to the width of the crystal, as shown in Fig. 44. c" 3. M^taftT. 1" III n entra i ned a i r a c" sample crossection Fig.44 -Air Entrained by a Flexural Mode It is easy to show that the effective increase in the mass of the sample per unit length leads to a reduction in the Flexural (FDC) resonant frequency by the factor [15] 1 + TT W p a i r 4 t p where p •„ and p are the densities of air and the sample respectively. The same result has been obtained by Lindholm et al (1965) using a more sophisticated approach. Lindholm et al (1965) also calculate the corresponding mass loading for torsional modes and obtain the correction factor This factor may be approximated by calculating the additional axial moment of inertia contributed by two cylinders of air parallel to the long axis of the sample ( b ) and with diameters equal to half the width of the sample as shown in Fig. 45. The resulting correction factor is o entra i ned a i r a c" sample crossection Fig.45 - Air Entrained by a Torsional Mode the same as [16] except that the numerical factor 3^/32 is replaced by 9TT/64. The frequency shifts caused by air loading can be measured by looking for a change in resonance frequency when the sample chamber a -a-Is evacuated. For example consider Sample #14. A comparison be tween the observed effect of air on the resonance frequency and the predictions of [15] and [16] is shown in Table VI for the second and fifth Fbc flexural modes and the second and third Tb torsional modes. In Table VI the air is assumed to be at one atmosphere with density 0.00129 g/cm3. The observed frequency shifts are in reasonable agreement with the predicted values. TABLE VI Effect of Air at One Atmosphere on Flexural and Torsional Mode Frequenci Mode Frequency (khz) Frequency Theory Shift (hz) Expt. Fbc2 5-3 19 25 Fbc5 44.6 162 120 \ 2 47-2 65 100 Tb 3 73-5 101 100 The elongational modes are at too high a frequency for the air to be treated as an incompressible non-viscous fluid. The primary source of air entrainment in a longitudinal mode is the viscous boundary layer which attaches itself to the broad ab surface of the crystal because of the non-zero viscosity of the air. The effective thickness of the boundary layer is 6/2 (Landau and Lifshitz 1959) where 6 = (^n/ojp^; is a skin depth for shear waves in a fluid with viscosity n. I f we take n = 1.8x10-l+ poise (g/cm2-s) at room tem perature then 6/2 - 2x10-t*cm for a 300 khz mode. This layer of air will reduce the resonant frequency of a 300 khz elongational mode by 12 Hz. Experimentally the change in resonance frequency is usually observed to be larger than 12 Hz. Furthermore the shift does not always have the same sign for different samples. This behavious is interpreted as being due to a change in coupling be tween the elongational mode and other nearby modes caused by the increase in Q when the air is removed. The Q of a lossless sample vibrating in air is determined by power loss due to viscous heating of the air and radiated acoustic energy. The air damping may be calculated approximately for standard size samples above 50 khz. In this case the viscous loss is relatively small compared to the radiation loss and the interior dimensions of the sample container are large compared to the sound wavelength, so that walls can be ignored. First we consider the high frequency limit in which the wavelength of sound in air is small compared to both the a and b dimensions of the sample. This limit is applicable at typical longitudinal mode frequencies. The radiated sound power is given by AI = Pair vs AA where vs is the velocity of sound in air and un is the normal component of the velocity of the surface element AA. If we assume that the en tire surface area generates sound, partly because of surface roughness and partly because there may be coupled lateral motions of the sample, then un = u and [17] P tot 2 p ai r where Q_/2TT is defined as the energy stored divided by the energy dissipated per cycle. The actual air damped Q. of Sample #10 was 176 whereas [17] predicts a Q of 1*»3. In a vacuum the measured Q. was 3100. In the intermediate frequency range in which the sound wavelength in air is longer than the a dimension of the sample but still shorter than the b dimension, the sample may be modelled by an infinite cylinder with diameter w. The sound power radiated by an inter mediate frequency flexural mode may be approximated by the power radiated by a transversely oscillating cylinder, given by (Landau and Lifshitz 1959) per unit length where u is the velocity of the cylinder. The corres ponding expression for the Q is [18] P a 1 r P 1 If we substitute values appropriate to the fifth Fbc flexural mode (44.6 khz) of Sample #14, then [18] yields Q = 993- In the absence of air the observed Q is 1500. Adding this measured intrinsic loss to the calculated air loss leads to a net air damped Q of 597- This compares with an experimental value of 500. Although our calculations of the effect of air on the damping and resonance frequencies of vibrating TTF-TCNQ samples are only approximate, the physical origin of the observed effects appear to be well understood. CHAPTER I I I Interpretation of Temperature Dependence of Sound Velocity 3.1 Overall Temperature Dependence Before discussing the temperature dependence of the elastic moduli, we first compare the measured elastic constants (see Table V, in the previous chapter) and anisotropy of TTF-TCNQ with some common material Table Vll contains a list of Young's moduli for covalent (Si), metalli (Au, Pb, Na), ionic (NaCl) and van der Waals (Ar) solids. TABLE Vll Young's Moduli For Various Materials Material Young's Modulus T(K) (1011 dynes/cm2) Si 13-1 300 Au 4.65 0 NaCl 4.37 300 TTF-TCNQ a 4.3 0 b 2.0 0 Pb 2.05 0 Na 0.241 90 Ar 0.117 82 Huntington (1958), Ki ttel (1971), Gewurtz et al (1972) 2.6 TTF-TCNQ b AXIS VELOCITIES 2.5 2.4 ELONGATIONAL MODE 1 _S_ample 23 2.or G © O O 2.2^ 2.1 2.0 TORSIONAL MODE • Sample 21 c c 0 50 100 150 T (K) 200 250 300 g. hG - Temperature dependence of b axis elongational and torsional mode velocities 164 2.85H 2.80 r-E U lO O > 2.75 h 2.70 0 20 40 60 80 100 120 Fig. 47 - Temperature dependence of b axis Young's modulus velocity obtained from an Fbc mode 165 2.6 TTF-TCNQ a AXIS VELOCITIES 5.3 5.2 E o tn O 2.5 2.4 >-O O £ 2.3f < o oo 2.2f-or o 2.1 ELONGATIONAL VELOCITY TORSIONAL MODE e e 9 Sample 22 5.1 e o If) O 15.0 t i— o o 4.9 LU > < O 14.8 < o UJ 14.7 2.0F 4.6 0 50 100 150 T (K) 200 250 300 Fig. 48 - Temperature dependence of a axis Young's modulus velocity and shear velocity obtained from Fac and Ta modes respectively The bonding in the a direction in TTF-TCNQ is expected to be at least partly ionic and the corresponding Young's modulus is comparable with the ionic solid NaCl. Similarly in the b direction where we ex pect some metallic component to the bonding, the TTF-TCNQ Young's modulus is comparable to lead, a soft metal. Certainly within the category of metallic solids and to a lesser extent the ionic solids there is a wide range of elastic constants. Nevertheless, it is clear that the elastic constants of TTF-TCNQ are comparable to those for other common materials. An anomalously soft a axis modulus might be expected if the conducting molecular stacks were weakly coupled. However, both the a and b axis moduli are significantly bigger than the modulus for a weakly bound solid such as Argon at 82K (see Table Vll). Some idea of the elastic anisotropy is obtained by taking the ratio of the Young's modulus along a to the Young's modulus along b. This ratio is about 2.2 for TTF-TCNQ, whereas for zinc (hexagonal) the ratio of the Young's moduli parallel and perpendicular to the hexagonal symmetry axis, is 3-37. By this measure zinc is more anisotropic elastically than TTF-TCNQ. A com parison with the elastic constants of other crystalline materials confirms that the elastic anisotropy of TTF-TCNQ is more or less typical of non-cubic crystalline materials. Now let us consider the temperature dependence of the velocity of sound in TTF-TCNQ. The temperature dependence of the b axis Young's modulus velocity (E^/p)^ is shown in Figs.46 and 47- The data in Figs.46 and47were obtained from an elongational mode and an F[3C flexural mode respectively. Fig.48 shows the temperature dependence of the a axis Young's modulus velocity (Eg/p)^ as determined from an Fac flexural mode. As discussed in Chapter II, Section 2.1 two different torsional modes can be used to measure the shear velocity (c6€/p)^, to a good approximation. The temperature dependence of this shear velocity is shown in Figs.46 and 48. The shear velocity data in Fig.46 was obtained from a Tu, torsional mode and the data in Fig.48 from a Ta mode. All of the above sound velocity measurements have been corrected for thermal expansion using the correction of Jericho et al (1977) given in the Appendix § 3. This correction was inferred from the X-ray data of Blessing and Coppens (1974) and the b axis thermal expansion measure ments of Schafer et al (1975). The temperature dependence of the elastic constants fol lows directiy from the temperature dependence of the sound velocity and the density. The main features in the velocity results are the kink in the Young's modulus velocities near 52K, and the strong temperature dependence in the higher temperature region. Before discussing the anomaly near 52K, we first consider the large overall temperature dependence of the velocities. The velocity results imply that the elastic moduli de crease by ~ 40% between OK and room temperature. Although this may seem like a large temperature dependence when compared with conventional metals where the elastic constants typically change by only a few per cent over this range, a large temperature dependence would be expected from the relatively low solidification temperature for TTF-TCNQ of 498K (Weiler 1977). In fact, if one compares the temperature dependence of the elastic moduli for a variety of solids from OK up to just below their melting point a large reduction in modulus (~ 50%) is normally observed. We now consider the bulk modulus since the temperature dependence of this modulus is the easiest to calculate. If we make the hexagonal approximation to the crystallographic symmetry of TTF-TCNQ, described in Chapter II, Section 2.1, then the experimental Young's moduli may be used to obtain a rough estimate of the bulk modulus. In the hexago nal approximation the bulk modulus is given by Poisson's ratios. If the Poisson's ratios are restricted to lie between 0 and i (Landau and Lifshitz 1970) the bulk modulus must be between ^fT" + ar,d infinity. Obviously this is not a very precise estimate. Although v and v' have not been measured experimentally, nevertheless, if we arbitrarily set v = v1 = 0.2 then the room temperature values for the elastic constants given in Table V of Chapter II combined with the temperature dependence presented earlier in this chapter suggest that B - 2 x 1011 dynes/cm2 at OK. This estimate for the zero temperature bulk modulus compares with a room temperature bulk modulus of 0.94 x 1011 dynes/cm2 measured by Debray et al (1977). Even though it is not possible to arrive at a very accurate estimate for B from the experimental data on Ea and Eb, since all of the measured elastic moduli have similar overall temperature dependences it is reasonable to assume the bulk modulus follows where Ea and Eb are the a and b axis Young's moduli and v and v' are the other moduli as a function of temperature. In general it is extremely difficult to calculate the bulk modulus of a solid, as it means deriving an expression for the pressure of a solid as a function of its volume, in other words an equation of state. In the conventional adiabatic approximation the equation of motion of the lattice, and hence the sound velocity, is determined by the depen dence of the electronic energy eigenvalues on the position of the nuclei and by the direct electrostatic interaction between the nuclei. The direct interaction between different nuclei is probably small compared to the interaction between nuclei and core electrons and between core electrons on different nuclei. Accordingly if one knew how the electron energy eigenvalues depended on the positions of the ions one could cal culate all of the elastic constants. In one special case, namely the alkali metals, this can be done with a reasonable degree of accuracy. In this case the outer electrons can be closely approximated by free electrons and the nuclei and core electrons can be ignored because they are thoroughly screened between different sites. The free electron nature of the electron energy states does not change when the lattice is strained. The only effect of the strain is to cause a change in the Fermi level necessary to maintain local charge neutrality. Thus, the dependence of the electronic energy states on the position of the ions is known and the elastic moduli can be calculated (Kittel 1971). However, for almost any material other than the alkali metals more than one electron energy band needs to be considered and the bands change in complicated ways with strain. Although the contribution of particular bands to the elastic moduli can still be calculated, it is no longer reasonable to try to calculate the moduli from first principles. Instead the normal procedure is to start with a para meterized intermolecular potential. We will follow this procedure here. In general the isothermal bulk modulus for any system is given by (Landau and Lifshitz 1969) where F is the free energy of the system, and V is its volume. Simi larly the adiabatic bulk modulus is given by where U is the internal energy, and the volume derivatives are taken at constant entropy S. In order to calculate the temperature dependence of B.j. and Bs we need F and U as functions of temperature and volume. The standard expressions are (Girifalco 1973) [la] F = F0 + [lb] U = UQ + where the intermolecular potential has been expanded to third order in the volume strain. The third order term is necessary because of thermal expansion, as will be clear later. The thermal phonon fre quencies u)q will depend on volume in some complicated way. We approximate their volume dependence by to. (V) = % (VD) (VG/V)" where y 's a volume independent Gruneisen constant. In this approxi mation the volume expansion coefficient a(T) is given by (Callen 196O) m m YCP(T) YCv(T) [2] A(T) = V(TTB7TTT~ V(T)Bt(T) where Cp and Cy are the. specific heats at constant pressure and volume respectively. By integrating [2] we find [3] V(T) « VoT^SVoT^ where U is the internal energy of the lattice. fitoq /fWq\ U o I -JL coth (^j q In eqn.[3] we have assumed that Cp - Cv and that the temperature depen dence of the volume and bulk modulus is small compared to the temperature dependence of the specific heat. These approximations improve at low temperatures. Using the definition of the isothermal bulk modulus and the ex pression for the free energy [la], one can now write down the iso thermal bulk modulus as a function of temperature in terms of the parameters y and A. Both of these parameters are manifestations of the anharmonic part of the intermolecular force constants. It is useful to express A in terms of the pressure derivative of the bulk modulus since this quantity can be measured directly. The pressure derivative of By is obtained from by dividing by By. The second term in [4] is expected to be at most comparable with the total tempera Lure dependent part of By whereas A will be shown to be > 10B for TTF-TCNQ. To a good approximation the second term in [4] can be neglected and 8By/3P = A/By. In terms of 9By/3P the isothermal bulk modulus is [5a] B (T) = BQ + 1 vo The first term in the square brackets in [5a] tends to make the lattice stiffer at high temperatures. This term arises from the phonon pressure. The negative 8By/8P term comes from the thermal expansion combined with the softening of the intermolecular potential with increasing volume. Similarly, theadiabatic bulk modulus is found by taking partial de rivatives of the internal energy with respect to volume, maintaining (y+1) 8 By TP~ U -Y T the entropy constant. In the Gr'uneisen approximation, the entropy may be kept fixed by allowing the temperature to be a function of volume in such a way that Wq/T is independent of volume. Since the entropy is a function of Wq/T only, it will be constant if w^/T is constant. A procedure similar to that used in obtaining the iso thermal modulus then leads to [5b] BS(T) = BQ + Normally ultrasonic techniques measure adiabatic moduli. Whether it is the adiabatic or isothermal modulus which is appropriate depends on the relation between the period of the sound wave and the time for thermal relaxation between a rarefaction and compression in the wave. In the TTF-TCNQ. experiment the adiabatic moduli determine the reso nance frequency of all of the modes studied with the possible exception of the fundamental flexural mode and its first few harmonics, depending on the geometry of the sample. In a flexural mode the compressed and expanded parts of the sample are separated by a distance equal to the sample thickness only, even though the mode frequency may be relatively low. In this.case the isothermal modulus applies when the resonant frequency is small compared to the thermal relaxation frequency fD discussed in Section 2.3 (ii) and the adiabatic modulus applies at higher frequencies. Y+1 8P Eqn. [2] combined with our estimate of Bs(0) and the published ther mal expansion and specific heat data, enable us to estimate y. From the X-ray structure data (Blessing and Coppens 1974) we estimate ct(T) = 1.6 a^Cl) where %(!") is the b axis linear expansion coefficient measured by Schafer et al (1975). Using this estimate for a(T) we obtain y = 2.56, which is not an unreasonable value. The Gruneisen constants for a variety of different materials are given in Table VIM. TABLE VIII GrUneisen Constants for Various Materials Material GrUneisen y T(K) Si 0.4': 300 Au 3.0 0 NaCl 1.55 300 TTF-TCNQ 2.6 0 Na 1.14 90 Ar 2.7 82 t Daniels (1963), Gewurtz et al (1972) Now that we have an estimate for y [2] can be used to predict the heat capacity beyond the 12K range near 55K measured by Craven et al (1974). The heat capacity which results is typical of molecular solids (Lord 1941). Near room temperature the predicted heat capacity is 35R and increasing approximately linearly at 0.05 R/K. R is the gas con stant per mole of TTF-TCNQ formula units. The large value of the heat 175 capacity clearly indicates the importance of intramolecular and librational degrees of freedom. These low frequency Einstein modes of the TTF and TCNQ. molecules dominate Cp above 20K. It is also possible to make a direct comparison between the tem perature dependence of the bulk modulus and the thermal expansion co efficient. Differentiating [5b] with respect to volume and neglecting the temperature dependence of 9BS/9P we get [6] J_ lis „ B,. dT 9 BQ Y+1 a(T) It may not be a very good approximation to neglect the temperature de pendence of 9BS/9P (Daniels 1963), however, there is neither experimental data nor a reasonable model available to describe its temperature depen dence. If we make the previously stated assumption that the temperature dependence of the bulk modulus follows the temperature dependence of the measured elastic moduli, then in conjunction with the experimental velocity data, [6] may be used to estimate 9BS/9P. Using the b axis velocity data, together with a = 1.6 ab and y = 2.56 we find 9BS/9P ~ 15~17 - In light of the approximations we have made, this number compares well with 9B-J-/9P ~ 12 inferred from the pressure measurements of Debray et al (1977). The pressure derivative of the bulk modulus is given in Table IX for a number of materials. Between 52 K and about 200K for the b axis extensional modes and over a wider temperature range for the other modes, the shape of the velocity TABLE IX t Pressure Dependence of Bulk Modulus t Material 9BT IF Si 5.3 Au 6.1 NaCl 5.7 TTF-TCNQ 15-17 Na 3-3 Ar 8.5 Daniels (1963), Paul and Warschauer (1963) curves is consistent with [6]. (Pvecall that in our approximation v-1 9v/9T = (2B)"1 9B/9T, where v is a sound velocity.) We there fore suggest that if TTF-TCNQ remained metallic down to OK, the temperature dependence of the velocities would be of the general form shown by the dashed line in Fig. 48. For this reason we inter pret the anomaly near 52K in the extensional mode velocities as a stiffening in the modulus above a background which represents the contribution to the elastic constants from anharmonic effects. The other possible interpretation is to regard the anomaly as a broad softening between 40K and 120K. This interpretation is rejected be cause the resulting temperature dependence does not match the thermal expansion data quite as well, and secondly there is no other experi mental evidence for such a broad transition temperature region (>80K). 177 T (K) Fig. kS - Enlargement of low temperature anomaly in the Young's modulus velocity. The dashed line is obtained from eqn.[6] in the text. The break in the curve near k2K results from a splitting of the first longitudinal mode caused by interference from another mode, probably a harmonic of the fundamental flexural or torsional modes. Summarizing, we conclude from the large value for the heat capacity and from the interdependence of the heat capacity, thermal expansion and bulk modulus that the temperature dependence of all three quantities is dominated by librations and intramolecular modes above ~ 20K. 3-2 Low Temperature Anomaly We now discuss the low temperature anomaly in the sound velocity, taking the point of view that it is a stiffening in the elastic modulus above the background temperature dependence. The velocity anomaly is shown in Fig. kS. The low temperature velocity data can be summarized as follows. Below the metal-insulator transition there is an anomalous increase in velocity for modes which involve a volume change. This increase reaches a maximum of about 1.5% above the extrapolated back ground at OK. The fractional increase is about-the same for the Young's modulus modes in the a and b directions and much smaller or even absent in the shear modes. In the remainder of this chapter we show how these experimental results may be interpreted in terms of the contribution of the conduction electrons to the sound velocity. This section is divided into two parts. In the first part we show that an electron-phonon interaction in the high frequency quantum limit leads to a softening of the sound velocity in the metallic phase. In the second part we consider the tight-binding band structure of TTF-TCNQ in detail and show that the electron-phonon coupling can also lead to a softening in the metallic phase at zero frequency. Fig. 50 - Excitation spectrum for a non-interacting tight-binding electron band (TCNQ. band) and an uncoupled acoustic phonon branch. (i) Quantum Limit In the quantum limit (see Part A, Chapter I, Section 1.3) the effect of the conduction electrons on the sound velocity may be calculated by treating the electron-phonon interaction as a perturbation on the un coupled electron and phonon system. The unperturbed energy levels for the electrons are described by a single particle tight-binding band. Consistent with the tight-binding approximation (Barisic 1972), the unperturbed phonons exist in a lattice of neutral 'ions'. Thus there is a phonon mode (acoustic phonon) which propagates down to zero fre quency with a linear dispersion in the unperturbed system. This be haviour contrasts with nearly free electron models where the unper turbed lattice consists of charged ions, and the corresponding un perturbed phonon frequency is an ionic plasma frequency. The excitation spectrum for the non-interacting one dimensional electron-phonon system at zero temperature is shown in Fig.50. Only one low frequency phonon branch is shown for clarity, although in general for any direction of propagation there are two other branches. The sound velocity determines the slope of the phonon branch and the Fermi velocity determines the slope of the electron branch near q = 0. These slopes are drawn approximately to scale in Fig.50. The electron Fermi velocity shown is for the TCNQ band calculated by Berlinsky et al (1974). The shaded area is the locus of e(k+q) - c(k) with k as a parameter and the additional requirement that the initial state k be full and the final state k+q be empty. e(k) is the energy of an electron in the state k. The electron-phonon interaction H;nt (see Part A, Chapter I, Section 1.3) will couple a phonon with wavevector q to all the electronic excitations with the same wavevector. In terms of Fig. 50, the electronic excitations lying along a vertical line directly above the phonon of interest are coupled to the phonon. To second order in perturbation theory this coupling reduces the phonon frequency to m * * o 1 V l<k l"intlk+q>l2 [7] T.o.q - T»a>q - ^ £ F(k+q) - e(k) - ' The unperturbed phonon frequency is co° and the occupancy n(q) of the phonon mode q is assumed to satisfy n(q) » 1. We substitute the squared matrix elements of Hint " 9 I 1-fi (aq " a-q) Ck-qa °k a a given in Part A, Section 1.3 where the electron-phonon coupling constant g = C//B^~ . BS is the adiabatic bulk modulus and C is a deformation potential. With these substitutions [7] reduces to rcl . * o V f(k) - f(k+g) [8] tlcoq = flag " 92 -f- I e(k+q) - e(k) assuming that the sound velocity is much smaller than the electron Fermi velocity. Although second order perturbation theory is probably the simplest way of approximating the effect of the electron-phonon interaction on the phonon frequencies, a more general technique is also available. In the Green's function formalism, the phonon frequency OJ^ is a pole of the perturbed phonon Green's function D(q). An expression for D(q) can be obtained with the help of Dyson's equation shown schematically in Fig.51-Fig. 51 _ Dyson's Equation This equation may be rearranged to give [9] D(q)-1 = D^q)"1 - £ n (q) . The last term includes the phonon self-energy II (q). We now substitute (Abrikosov et al 1963) for the finite temperature phonon Green's function and an identical expression for D(q) except the superscript zero is removed. If we set ico = Wq in [9] then [10] o,q2 = o)q°2 [l + g2 H(q)] . This expression for the perturbed phonon frequencies is valid to arbitrary order in perturbation theory. The first term in the dia grammatic expansion for II (q) is k k + q n°(q) Fig. 52 - Electron Gas Polarization Diagram This lowest order polarization insertion or "bubble diagram" is very well known. In the low frequency limit it can be shown that (Fetter and Walecka 1971, Doniach and Sondheimer 197*0 11 (q) " I e(k+q) - E(k) * The summation on the right is the Hartree polarizabi1ity x°(q) of the electron gas. If this expression is substituted into [10] we Fig. 53 - Hartree polarizabi1ity for the one dimensional TCNQ. band discussed in the text regain the second order perturbation theory result [8] for the per turbed phonon frequencies. It is not difficult to evaluate the polarizabi1ity x°(q) f°r a one dimensional tight binding band of electrons. The static polari-zability for a metallic band at temperatures low compared with the Fermi temperature is where N(e) is the electronic density of states, Cf is the Fermi energy and kj. the Fermi wavevector. A graph of x°(q) as a function of q is shown in Fig.53 using the same tight binding band parameters used in the excitation spectrum shown in Fig.50. The logarithmic singularity in x°(cl) at q = 2k^ is responsible for the large Kohn anomaly in the phonon spectrum and the resulting charge density wave or Peierls transition in one dimensional conductors. The divergence in x°(cl) is present at zero temperature in any material with parallel sections of Fermi surface. Note that x0^) for a tight-binding band does not approach zero for large q unlike the nearly free electron case (Andre et al 1976). Although the singularity in x°(cl) 's important in determining the phonon spectrum near 2k^, we are more interested in the long wave length phonons with q -»• 0. In the long wavelength limit, the change in the sound velocity implied by [8] is [11] 2 sin f fu In The integral in [12] approaches N(e^.) in the metallic phase at low temperatures and goes exponentially to zero in the semiconducting phase at low temperatures. From [12] one would expect a fractional increase in the velocity of sound of N(e^.) C2/(2BS) in cooling TTF-TCNQ. through its meta1 -insula tor transition. This is qualitatively the effect which is experimentally observed in the extensional modes. To see how the size of this effect compares with experiment we sub stitute an average density of states for the TTF-TCNQ bands (Berlinsky et al 1974) of 5.8 ev-1 and use Bs = 2X1011 dynes/cm2 as discussed earlier. For a fit to the observed 1.5% velocity anomaly C = O.38 ev. This number is certainly reasonable since the average bandwidth from the molecular orbital calculations (Berlinsky et al 1974) is 0.32 ev, and tight-binding bands are expected to have deformation potentials of the order of the bandwidth (M itra 1969). The much smaller anomalies observed in the shear volocities are also explained by this model because the deformation potential for shear waves is expected to be small for metals in which all of the Fermi surface is in one Brillouin zone (Kittel 1963)• There is no reason to expect any of the TTF-TCNQ Fermi surface to lie outside the first zone. It is interesting to extrapolate the experimentally observed softening in the sound velocity up to q - 2kf with the help of the 187 Fig. 5k - Zero temperature longitudinal calculated from [8] using the coupling constant acoustic phonon dispersion experimental electron-phonon 188 Fig. 55 Zero temperature longitudinal acoustic phonon dispersion calculated from [8] for an electron-phonon coupling constant which yields a Tc of 5^K q dependent polarizabi1ity expression [11]. If one assumes a q independent coupling constant g, then the perturbed longitudinal acoustic phonon spectrum is given by the solid line in Fig.54, for (hypothetical) metallic TTF-TCNQ at zero temperature. The dashed line is the unperturbed phonon frequency. The small Kohn anomaly in Fig.5** is much too small to account, in the mean field theory (Rice and Strassler 1973) for the observed metal-insulator transition temperature. This discrepancy may not be unreasonable since the mean field model is not expected to be very accurate. Nevertheless, it is interesting to pursue some of the implications of the small Kohn anomaly in Fig.54, in the framework of the mean field theory. The electron-phonon coupling constant g would have to be about a factor of five bigger, in order that the Kohn anomaly in the acoustic phonon be large enough to account for the observed metal-insulator transition temperature. If the larger value of g is sub stituted into [11] then the large zero temperature softening of the acoustic phonon mode shown in Fig.55 results. Such a drastic softening is not observed in the inelastic neutron scattering measurements of Shapiro et al (1977). In fact the Kohn anomaly observed by neutron scattering resembles the much smaller anomaly in Fig.54. These ob servations suggest that the metal-insulator transition in TTF-TCNQ is caused by a static distortion in a combination of intramolecular modes as proposed by Rice and Lipari (1977). If a metal-insulator transition can be produced by coupling of the conduction electrons to intramolecul modes in addition to the intermolecular acoustic modes, then it will be more difficult to stabilize the low temperature metallic state of an organic solid, especially if the material is composed of extended molecules. Even though the model described above seems to be in excellent agreement with the low frequency sound velocity measurements below the metal-insulator transition, the calculations were made in the quantum limit for well defined phonon and electron states. That is, the lifetime broadening of the electron and phonon states has been implicitly assumed to be negligible. In reality for the frequencies used in the experiments ql « 1 (recall I is the electron mean free path), and therefore the lifetime broadening of the electronic ex citations is enormous compared to Tito. Th i s b roaden i ng has a drastic effect on the ultrasonic attenuation where transition rates for energy conserving transitions must be calculated. However, in the present case we are only interested in virtual transitions to a continuum of electron states. The fact that the electron states are much broader than the energy change involved in an electron-phonon scattering pro cess is not important since energy does not need to be conserved in virtual transitions anyway. On the average we expect the virtual transitions to coincide with the well defined quantum transition, and the quantum result should still be valid at low frequencies. Another equally non-rigorous argument can be made for the validity of the quantum result at low frequencies. In nearly free electron models the unperturbed phonon frequency is the lattice plasma frequen cy ttn = (4frne2/M)^ where M is the ion mass. The effect of adding electrons, which will interact with the ions in such a way so as to screen the ions, is to reduce the plasma frequency to ^p/*/^q~ • The new phonon branch now has a linear dispersion down to zero frequency (Pines and Nozieres 1965)• The dielectric constant tq can be calcu lated from the polarizabi1ity using the same quantum limit approach and the same approximation that we have used in calculating the change in phonon frequency. The dielectric constant approach leads to the correct low frequency sound velocity. By analogy, our calculation of the phonon frequencies should also lead to the correct low frequency veloci ty. (ii) Thermodynamic Limit These difficulties with the applicability of the quantum formalism in the low frequency limit all vanish if instead we approach the problem from the point of view of calculating a static elastic constant. The elastic constants for a material in thermodynamic equilibrium are determined by taking second order strain derivatives of an appropriate thermodynamic potential, as discussed in Section 3-1 above. For electrons the thermodynamic limit is valid provided that qt « 1 (see Part A, Chapter I) which is always well satisfied in the experiment described here. As pointed out in the previous section it is primarily the highest occupied electron energy bands which determine the elastic constants. In order to calculate the total contribution of the electrons to the elastic constants we need to know the dependence of the electron energy bands on strain, to second order in the strain. However, if we 192 Fig. 56 - Band structure of TTF-TCNQ discussed in the text. The dashed tine shows the effect of an exaggerated b axis compress ion 193 Fig. 57 - Effect of a b axis compression on the density of occupied states for the band structure shown in Fig. 56 can be satisfied with calculating the small change in elastic con stants brought about by a metal-insulator transition, then the problem is easier. In this case it is not necessary to know the strain depen dence of all of the band parameters to second order, provided the features which lead to a change in the elastic constant at the metal-insulator transition are correctly described. We consider the following simple model for the strain dependence of the band structure. The energy bands for the TTF and TCNQ chains are taken to be one dimensional tight-binding bands with band para meters as calculated by Berlinsky et al (1974). In this band structure the TTF(TCNQ) band has a maximum (minimum) at k = 0 (see Fig. 56). The Fermi level for the two bands is determined by assuming a charge transfer p of .59. In the spirit of the right-binding approximation we assume identical strain dependences for the two bandwidths and strain independent centres of gravity. That is, a one dimensional strain in the conducting direction is assumed to scale the width of both the TTF and the TCNQ bands and their energy gaps in the insulating state, by the same factor exp(-Bc), where £ is the b axis strain and B is a dimensionless parameter. In this model a strain will have no effect on the Fermi energy but wi 11 change the Fermi wavevector k^. and hence the charge transfer. The effect of a b axis compression on the density of occupied states is illustrated in Fig.57 for the tight-binding band structure shown in Fig.56. Note the additional charge transfer from the TTF to the TCNQ band. We now investigate the effect of this small strain dependent charge transfer on the elastic modulus c22. As discussed in the previous chapter the experiment measures the adiabatic moduli, and the temperature of the sample will be strain dependent. However, the lattice specific heat is very much larger than the electronic specific heat. For this reason the temperature of the sample will be determined by the requirement that the lattice entropy be a constant and not the electronic entropy. Since the electrons are thermally coupled to the lattice, their temperature will be de termined by the phonon heat bath. Accordingly the contribution of the conduction electrons will be somewhere between adiabatic and isothermal. Since the difference between the two moduli is not expected tc be very significant we will take the easiest apprcac and calculate the isothermal modulus. The isothermal contribution to the modulus c22 's determined from the b axis strain dependence of the free energy density [13] F = i c22 r,2 - kT ln(\ +exp(- -^jj NF(e,c) + NQ (e,?)]dE where all contributions to the elastic modulus not connected with electron transfer have been incorporated into the first term. We have taken the Fermi energy to be zero for convenience. The TTF and TCNQ bands have strain dependent density of states functions given by Np(e,r.) and N^(e,c) respectively. The Brillouin zone boundary follows the strain selfconsistently. The elastic constant c22 is obtained by differentiating the free energy [13] twice with respect to £. The result is [14] c22 = c°22 - B2 j [(c + np)2 NF(e) + (e + nQ)2 NQto](- If.) de + B2 | [Nf(E) + NQ(e)J e f (e) de where rip and HQ are the energies of the centre of the TTF and TCNQ bands respectively. Since the relative position of the two bands is determined by the charge transfer p, the positions of the band centres are given by np^ = 2tp^j cos ("^jr) as shown in Fig. 57 where 4|tp| and J»|tJ are the appropriate bandwidths. We now examine the be haviour of the last two terms in [14] when energy gap opens up in the density of states and the material becomes an insulator. The last term decreases by an amount proportional to B2 i^A2 Np(o) + A2 NQ(O)J where Ap and A^ are the energy gaps on the two chains. Similarly the second last term decreases in magnitude from 32 n2- N (o) NQ(O)J to zero at zero temperature. Clearly this change is much bigger than the change in the last term in [14] s i nee (^Ap/rip^2, ^AQ/T1Q^2 « 1. If the last term in [14] is neglected the elastic modulus will increase by AC_22. = 2 ^22. & v°>+ -a vo)] c22 v22 in cooling from the metallic phase, through the meta1 -insulator tran sition towards zero temperature. We now approximate the longitudinal bulk wave velocity in the b direction, v22 by the b axis elongational velocity at T =0, and use tp = .05 ev and t^ = -.11 ev for the overlap parameters (Berlinsky et al 1974). In this case 3 = 4.1 gives agreement with the observed low temperature velocity anomaly. This value for 8 may be compared . with 3 ~ 6 inferred from the molecular orbital calculation of the overlap parameters. The agreement between the observed and calculated 8 is considered to be excellent. The much smaller anomaly in the modes involving only shear moduli can now be explained as follows. Since there is no volume change associated with shear modes, there is no first order change in the lattice constants due to strain and hence a correspondingly smaller strain dependence to the bandwidths. The charge transfer mechanism will then play only a minor role in determining the shear moduli. There are at least two possible explanations for the low temperature anomaly in the temperature dependence of the a axis elongational mode. The first possibility is that the anomaly is due to coupling to the b axis molecular overlap. This coupling could result from a libration of the molecules about, for example, the a axis caused by a strain along the a direction. A second possibility is that a strain in the a direction also tends to favour a change in the charge transfer be tween chains, perhaps by altering the electrostatic Madelung energy (Torrance and Silverman 1977). This change in charge transfer could be modelled by postulating that an a axis strain produces a rigid shift of the TTF and TCNQ. bands relative to one another. As in the b axis case the strain dependent charge transfer would disappear in the insulating phase, leading to a stiffer lattice, and a low temperature velocity anomaly. It is instructive to consider some of the implications of the above model of the low temperature elastic anomaly. The dominant term in the expression for the electronic softening of the elastic modulus in [14] is practically identical to the expression for the Paul i spin susceptibility except that u is replaced by gn and Bn-.. D r Q This similarity is more general than the particular model used here o suggests, and is due to the dependence of both phenomena on the den sity of thermally accessible states near the Fermi level. The apparent lack of an anomaly in the sound velocity near the phase transition at 38K is consistent with the corresponding spin susceptibility data (Scott et al 1974). Also the small positive curvature in the tempe rature dependence of the b axis elongational velocity in the range of 200K-300K (see Fig.46) is consistent with the temperature depen dence of the spin susceptibility over the same temperature range. Another feature of the charge transfer model proposed here is that it predicts a pressure dependence of the charge transfer and of the associated Fermi wavevector. If we neglect the pressure depen dence of the bulk modulus, use the experimentally determined value of 3 and consider only the b axis strain contribution to the charge transfer, then the pressure dependence of kp is 3*10~3 /Kbar corresponding to an additional charge transfer of 1%/Kbar. In this model one would also expect a change in charge transfer to result from thermal expansion. The expected change in kp between OK and 300K is consistent with the temperature dependence of the k kp anomaly observed in X-ray measurements by Kagoshima et al (1976) and Pouget et al (1976) . In conclusion, the elastic measurements show a small stiffening at temperatures below 52K for modes of vibration which are associated with a volume change. The stiffening is interpreted as arising from a strain induced charge transfer between the TTF-TCNQ. conduction bands. This charge transfer softens the lattice in the metallic phase above 52 K but is inhibited in the low temperature insulating phase by the appearance of an energy gap in the electron energy bands. (iii) Comment on Acoustic Absorption The charge transfer model suggests an additional acoustic loss mechanism for the longitudinal modes. The idea is that it takes time for the charge transfer between chains to occur and during this time the elastic modulus relaxes from c22 before the charge transfer can take place, to c22 (see [14]) after equilibrium is reached. The contribution of this relaxation process to the damping is (Zener 19^8) [15] •1 _ c22 ~ C22 COT C22 1 + CO^T 2T2 This expression is analogous to [13] in Chapter II, the expression for thermoelastic damping. The relaxation time x is the time re-quired for the charge transfer between chains to take place. Intuitively one would expect the charge transfer relaxation time to be comparable with the a axis dielectric relaxation time (4irea/aa) where ea = h (Barry and Hardy 1977) is the a axis di electric constant and aa is the a axis conductivity. From d.c. conductivity measurements (Tiedje 1975) oa - 3-8 (fi-cm)"1 at 60K. The corresponding dielectric relaxation time is 1.5x10_lls. At 60K we estimate from Fig. kS that Ac22/c22 's 3%- With these numbers, and the 261 khz longitudinal resonance frequency of Sample #23 we can use [15] to calculate the absorption. We find that 0."1 = 0.8x10~5 at 60K whereas the observed absorption is about IO-4. We conclude from this result that the contribution of the charge transfer mechanism to the damping is probably not important in the temperature regime in which the sample is metallic. At first glance one might expect this contribution to the ab sorption to be independent of temperature below the metal-insulator transition because Ac22/c22 decreases exponentially as the tem perature is lowered and the dielectric relaxation time increases exponentially, while the damping depends on the product. However, as discussed in the preceeding section Ac22/c22 is expected to be proportional to the spin susceptibility whereas the dielectric re laxation time will depend on the conductivity. Since the activation energy for the susceptibility (Torrance et al 1977) is lower (90K) than the activation energy (180K) for the conductivity (Eldridge 1977) one would expect the temperature dependence of the relaxation time to 7 6 h 5 O 4 r 3 0 0 10 20 30 50 60 70 TOO Fig. 58 - Low temperature loss peak for a longitudinal mode. dominate and the damping to increase below the metal-insulator transition. Eventually at very low temperatures when WT > 1 or the temperature dependence of the conductivity becomes dominated by impurities the damping should either decrease or become tem perature independent. The temperature dependence of the damping in the insulating phase is shown in Fig. 58 for the fundamental longitudinal mode of Sample #10 at low temperatures. If we postulate that the excess low tem perature absorption above the absorption minimum near 60K is due to the charge transfer contribution then we can estimate the relaxation time T. For convenience consider a temperature of 35K where Sample #10 resonates at 384 khz. At this temperature we estimate from Fig. 49 that Ac22/C22 's 0.6% and from Fig. 58 that the additional absorption is 5x10_lt. With these numbers we can use [15] to cal culate T. We find T = 3*10"8s. From d.c. conductivity measurements (Tiedje 1975) oa - 0.03 (^-cm)"1 at 35K, and the corresponding dielectric relaxation time is 2x10~9s. This relaxation time is an order of magnitude shorter than the time inferred from the absorption data. However, one could argue that this dielectric relaxation time is an underestimate of the true relaxation time because the correct con ductivity to use is smaller than the a axis conductivity. The reason is that the a axis conductivity arises from a series connection of an extended molecule which has little or no resistance and a high re sistance (low conductivity) element which determines the charge trans fer relaxation time. Although the above argument is certainly not conclusive it cannot be ruled out as a possible explanation of the anomalous low temperature sound absorption in TTF-TCNQ_. SUMMARY 1. The Main Results of this Work The electronic contribution to the attenuation of ultrasonic waves in one and two dimensional metals has been calculated for arbitrary qZ using a transport equation approach. The attenuation is found to be anomalously low and strongly temperature dependent in one dimensional metals. In the quantum limit, when the electron mean free path is long compared to an acoustic wavelength, the attenuation in one and two dimensional metals depends strongly on the direction of propagation of the acoustic wave. On the other hand the attenuation of sound in non-degenerate electron gases (semiconductors) is independent of the dimensionality of the electron gas. A much simpler method of solving the Boltzmann transport equation, to obtain the amplification of sound in the presence of a d.c. electric field, has been discovered. A capacitive technique has been developed for making sound velocity and attenuation measurements on small samples. In this technique an rf carrier signal is used as a probe to detect small displacements of the sample. The rf carrier method is shown to be superior to the conventional vibration pickup which is based on applying a d.c. bias voltage to the pickup capacitor. Three different elastic constants for TTF-TCNQ have been measured as a function of temperature. TTF-TCNQ is found to be slightly stiffer perpendicular to the conducting direction than parallel to the con ducting direction. Although the material is very anisotropic electrically, elastically it is not far from being isotropic. The strong temperature dependence of the elastic constants is attributed to the importance of molecular librations and intramolecular modes in the lattice entropy of TTF-TCNQ above 20K. The bulk modulus at zero temperature, the pressure dependence of the bulk modulus, the Gruneisen constant and the room temperature specific heat are esti mated. A small anomaly in the temperature dependence of the Young's modul at low temperatures is interpreted as being due to the freezing out of the conduction electrons below the meta1 -insulator transition. The low temperature anomaly gives a direct measure of the q -> 0 electron-phonon coupling constant. The interpretation of the elastic anomaly suggests an explanation for the temperature dependence of the 4kp spots observed by X-ray scattering. Within the resolution of the measurements, no small discontinuities in the temperature dependence of the velocities, of the type envisaged by Phillips (1977). were observed. A method is proposed for determining the c axis thermal conductivi of TTF-TCNQ by measuring the frequency dependence of the damping of low frequency flexural vibrations. For a summary of the experimental results on TTF-TCNQ see Table X below. " TABLE X Summary of the Experimental Results Modulus 1 Room Temperature Elasti c Moduli 2 (I0ndynes/cm2) Percent Increase From R.T. to OK 3 Ea C66 3.1 ± 0.7 1.27 ± 0.09 0.5 ± 0.1 37 58.0 37 Gruneisen constant 4 2.6 Room temperature heat capacity (C ) 35R Pressure derivative of bulk modulus (3B/3P)5 16±2 Average deformation potential G 0.38ev Pressure dependence of kp 7 0.5%/kbar 1 For an explanation of these labels see Part B Section 2.2. 2 These are adiabatic moduli. For the difference between the adiabatic and isothermal Young's moduli see p.148. 3 See Figs. 46 and 48. 4 See p.174. 5 See p.175 6 See p.186 7 See p.198 2. Suggestions for Further Work In the calculation of the attenuation of sound in a one dimensional metal we have assumed a perfectly one dimensional metal, with non-zero conductivity in only one direction. However, in real materials there is always some conductivity in all directions. It would be worthwhile to check that the very small attenuation in the at « 1 limit is not an artifact of the perfectly one dimensional limit that we have considered here. This calculation could be done by allowing the electronic band structure to have some three dimensional character, and then calcu lating the attenuation using the method of Rice and Sham (1970) for example. Methods for measuring small displacements have received a consider able amount of attention recently in connection with the detection of gravitational radiation. From the discussion in Section 1.2 of Part B, it appears that a displacement sensitivity of 10"16 - 10-17 /Kv cm could be obtained, using the capacitive technique described in this thesis, simply by using a larger pickup capacitance and superconducting, LC resonant circuits. This sensitivity is comparable with the highest sensitivity obtained to date in current gravitational radiation de tectors. More experiments are required to determine the real limits to the sensitivity of the capacitive vibration pickup. In Section 3.2 of Part B, we have proposed two different interpre tations of the low temperature anomaly in the Young's moduli. One is based on a quantum mechanical calculation of the velocity of sound in the presence of an arbitrary deformation potential electron-phonon interaction. The other interpretation is based on an electron-phonon interaction which is peculiar to the band structure of TTF-TCNQ. If it is legitimate to extrapolate the quantum calculation to q -*.0, then one would expect a similar velocity anomaly more or less independent of the exact nature of the band structure, in other materials which undergo Peierls transitions. On the other hand if the second mechanism (change transfer) really is necessary to ex plain the low temperature anomaly then one would only expect to see similar anomalies in materials in which there is a possibility of electron transfer from one part of the energy band to another. The ambiguity might be resolved by looking for.velocity anomalies in other materials which undergo Peierls transitions, but have a different band structure, such as TaS3 (Sambongi et al 1977). TSeF-TCNQ would also be an interesting material to look at. More measurements should be made of the damping of longitudinal modes to determine the nature of the absorption anomalies near 52K and 150K. Also careful measurements of the damping of the low fre quency flexural modes could be used to obtain the transverse thermal diffusivity. At temperatures well below the metal insulator transition the microwave conductivity of TTF-TCNQ is known to be several orders of magnitude larger than the d.c. conductivity. A strong frequency dependence might be expected if the low temperature conductivity were dominated by localized carriers hopping between pinning sites. A measurement of the frequency dependence of the conductivity in the range 0-1 Ghz could give some information about the hopping time. The measurement might be done with substantially the same apparatus as that used in the acoustic resonance experiments. APPENDIX § 1. Diode Temperature Sensor Calibration Diode #D2755 Current 10yA T(K) Voltage T(K) Voltage 4.2 2.1956 60 1.0273 5 2.1581 65 1.0143 6 2.1123 70 1.0011 7 2.0688 75 • 9877 8 2.0286 77 .9814 9 1.9917 80 • 9742 10 1.9585 90 .9468 11 1.9290 100 .9192 12 1.9023 110 .8916 13 1.8772 120 .8639 14 1.8514 130 .8361 15 1.8226 140 .8083 16 1.7503 150 .7803 17 1.7543 160 .7522 18 1.7099 170 .7241 19 1.6516 180 .6959 20 1.5791 190 .6677 21 1.4951 200 • 6393 22 1.4063 210 .6108 23 1.3228 220 .5824 24 1.2549 230 • 5539 25 1.2076 240 .5259 26 1.1735 250 .4985 27 1.1496 260 .4710 28 1.1335 270 .4434 29 1.1223 280 .4155 30 1.1140 290 .3869 32 1.1022 300 • 3578 34 1.0939 36 1.0873 38 1.0819 40 1.0765 45 1.0644 50 1.0523 55 1.0400 211 § 2. Circuit Diagrams (ii) Diode Detector Circuit TWO FERROXCUBES IN ^ lOpf 4h 500pf 3-9K FERROXCUBE TOROIDAL INDUCTOR * OUT 33K DIODE: HP 5082-2800 Schottky Barrier Diode (i i) MOSFET Preampli fier -12V OUT 0.082 15K 100K 68 0.082 0.082 15K 100K 100 ( all capacitances in microfarads ) 212 Preamplifier Specifications MOSFET Tl 3N211 3db Bandwidth .05-12 Mhz Gain 20db Output Impedance 50ft Output Power ~ -8dbm (iii) Phase Shifter (designed by S. Knotek) Speci f i cat i ons MOSFET Tetrodes 3N128 Bandwidth .05-15 Mhz Input l8-30dbm Output Impedance 500. Output Power 7dbm (Circuit diagram, next page) 213 214 (iv) Lorentzian Generator 10K(ten turn) 68K Signal Averager A/VV-- » Sweep Output 10K i—Wr DIODE: HP 5082-2800 r VCG .0015 2.1uH CRT ^.005 ? 10K i ( f0 = 2.8Mhz Q = 66 ) (v) Signal Averager Input Amplifier 12V low noise resistors BJT's 2N4403 Gain 180 Bandwidth 0.03-50 OOOhz 215 § 3« Thermal Expansion Correction (Jericho et al 1977) b axis a ar,d c axis expansion: T tJUliX) 300 0 290 .10 280 .21 270 • 31 260 .42 250 .52 240 .63 230 .73 220 .83 210 • 93 200 1.03 190 1.13 180 1.22 170 1.32 160 1.41 150 1.51 140 1.60 130 1.70 120 1 --78 110 1.87 100 1.95 90 2.02 80 2.09 70 2.16 60 2.21 50 2.25 40 2.29 30 2.31 20 2.33 10 2.34 0 2.34 «a ~ ac ~ -3 aD BIBLIOGRAPHY Abragam, A. 1961. Principles of Nuclear Magnetism, Oxford University Press, London. Abrikosov, A.A., Gorkov, L.P. and Dzyaloshinski, I.E. 1963- Methods  of Quantum Field Theory in Statistical Physics, Prentice Hall, Toronto. 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