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The optical properties of (TMTSF)₂ReO₄ and (TMTSF)₂BF₄ above and below their metal-insulator transitions Homes, Christopher C. (Christopher Craver) 1990

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THE OPTICAL PROPERTIES OF (TMTSF)2Re04 AND (TMTSF) 2BF 4 ABOVE AND BELOW THEIR METAL-INSULATOR TRANSITIONS By Christopher C. Homes B. Sc. (Hon.) McMaster University, 1983 M . Sc. University of British Columbia, 1986 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A May 1990 © Christopher C. Homes, 1990 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The reflectivity of large single crystals of protonated and deuterated (TMTSF)2Re04 and (TMTSF) 2 BF 4 has been measured from « 30 cm - 1 to m 8000 cm - 1 using a Bruker IFS 113V Fourier Transform Interferometer for E||a and E||b' above and below the metal--insulator transitions at 177 K and 39 K respectively. The infrared powder absorption spectra of protonated and deuterated (TMTSF)2Re04 has been measured from 200 cm"1 to 2000 cm"1. The Kramers-Kronig optical conductivity has been calculated from the reflectivity using Drude extrapolations to high frequency. The results for the conductiv-ity for E||a show a one-dimensional density of states, characteristic of a one-dimensional semiconductor with strong electron-phonon coupling, with the vibrations appearing as resonances below the gap and as antiresonances above. The E||b' conductivity is smaller by almost two orders of magnitude than that for E||a, but displays the same semiconduct-ing behavior. The phonons active in the E||b' polarization appear only as resonances. A normal coordinate analysis has been performed for protonated and deuterated TMTSF 0 and T M T S F + . The results have been used to infer the frequencies of vibration and the deuterium shifts of T M T S F + 0 ' 5 . The molecular frequencies of vibration have been assigned on the basis of their observed frequencies and optical polarization, as well as their deuterium shifts. Some external phonons have also been assigned. The observation that many of the internal and external vibrations are split is due to the eightfold increase in the size of the unit cell (and subsequent reduction of the Brillouin zone) below the metal-insulator transition. ii The optical properties of the semiconducting state have been modelled for a one--dimensional molecular conductor with a twofold-commensurate charge-density wave, which accurately reproduces the effects of the lattice dimerization and the potential due to the anion chains. The calculations yield the electron-molecular-vibrational coupling constants for the totally symmetric ag vibrations of the T M T S F molecule. The model also yields a transfer integral of 1400 c m - 1 for both materials and semiconducting energy gaps of 2 A = 1700 cm" 1 and 2A = 1120 cm" 1 for (TMTSF^ReCU and ( T M T S F ) 2 B F 4 respectively. The optical conductivity in the E | | b ' polarization has been discussed in -terms of a two-dimensional band structure with anisotropic transfer integrals. The band structure calculations show the same general features as the measured spectra. i i i Table of Contents Abstract i i List of Tables v i i i List of Figures , x Acknowledgements x v i 1 Introduction 1 1.1 Quasi-One-Dimensional Organic Conductors 1 1.2 ( T M T S F ) 2 X Charge-Transfer Salts 4 1.2.1 Structure of ( T M T S F ) 2 X Salts 6 1.2.2 General Properties 9 1.2.3 Band Structure and Electronic Properties 15 1.3 Optical Properties of ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 19 1.3.1 Previous Studies 19 1.3.2 New Results . 21 1.4 Thesis Outline 23 2 Experimental Techniques 25 2.1 Introduction 25 2.2 Bruker IFS 113V Interferometer 26 2.2.1 Optical Parameters 28 iv 2.2.2 Signal Analysis 29 2.3 Sample Preparation 31 2.4 Reflectivity Measurements 32 2.4.1 Reflectivity Module 34 2.4.2 Bolometer Transfer Optics 36 2.4.3 Heli-tran Refrigerator 40 2.4.4 Reflectivity Sample Mounting 44 2.4.5 Correction for Diffraction 45 2.5 Powder Measurements 49 2.5.1 Fixed Temperature Cryostat 49 2.5.2 Powder Sample Mounting . 51 3 Experimental Results for ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 53 3.1 ( T M T S F ) 2 R e 0 4 53 3.1.1 Single-Crystal Reflectivity 53 3.1.2 Optical Conductivity 67 3.1.3 Powder Absorption 74 3.2 ( T M T S F ) 2 B F 4 78 3.2.1 Single-Crystal Reflectivity 78 3.2.2 Optical Conductivity . . . 86 4 Normal Coordinate Analysis of T M T S F ( h 1 2 / d 1 2 ) ° • + 92 4.1 Introduction 92 4.2 Group Theory 93 4.3 Normal Coordinate Analysis 96 4.3.1 Normal Coordinates 96 4.4 Normal Modes of Vibration of TMTSF(h 1 2 /d 1 2 ) 0 -+ 97 v 4.4.1 Atomic Displacement Vectors of the (ag) Vibrations 103 4.5 Proportionality Factors 105 4.6 Vibrational Assignments 107 4.6.1 Internal Molecular Vibrational Assignments in ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 107 4.6.2 External Phonons in ( T M T S F ) 2 R e 0 4 120 5 Optical Properties of One-Dimensional Semiconductors 128 5.1 Introduction 128 5.2 Models of Infrared Optical Conductivity 130 5.2.1 Dimer Model 131 5.2.2 Tetramer Model 135 5.2.3 Phase-Phonon Model 140 5.2.4 Model Failures 144 5.3 One-Dimensional Systems with Twofold-Commensurate Charge-Density Waves 145 5.3.1 Application to ( T M T S F ) 2 R e 0 4 149 5.3.2 Application to ( T M T S F ) 2 B F 4 156 5.3.3 Conclusions 162 5.4 Optical Conductivity for E | | b ' 163 6 Conclusions 169 6.1 Optical Properties and Vibrational Assignments 169 6.2 Applications 170 Appendices 172 vi A Normal Coordinate Analysis 172 A . l Introduction 172 A.2 Secular Equation 172 A.2.1 Internal Valence Coordinates 174 A.2.2 G Matrix 177 A.2.3 F Matrix 180 A.3 Symmetry Valence Coordinates 182 A.4 Normal Coordinates 187 A . 5 Potential Energy Distribution 188 B Models of Infrared Optical Conductivity 190 B. l Introduction 190 B.2 Dimer Model 190 B.3 Tetramer Model 193 B . 4 Phase-Phonon Model 197 C I D Systems with Commensurate CDW's 203 C. l Introduction 203 C.2 Model Hamiltonian 203 C.3 Model Simulations 213 C.3.1 Systems without Electron-Phonon Coupling 214 C.3.2 Systems with Electron-Phonon Coupling 216 Bibliography 223 vii List of Tables 1.1 Properties of some ( T M T S F ) 2 X salts 10 2.1 Optical elements used in each wavenumber range ". . . . 28 3.1 The Drude parameters used in the high-frequency extrapolations of pro-tonated and deuterated ( T M T S F ) 2 R e 0 4 for E | | a at various temperatures 54 3.2 Parameters obtained from a fit of Lorentzian oscillators to the data in Fig. 3.6 64 3.3 The Drude parameters used in the high-frequency extrapolations of pro-tonated and deuterated (TMTSF) 2 BF 4 for E | | a at various temperatures . 80 4.1 The calculated in-plane ag and &3 f l normal modes of protonated and deu-terated T M T S F 0 98 4.2 The calculated in-plane ag and b3g normal modes of protonated and deu-terated TMTSF" 1" 99 4.3 The calculated in-plane bXu and 6 2 u normal modes of protonated and deu-terated T M T S F 0 100 4.4 The calculated in-plane 6 i u and 6 2 u normal modes of protonated and deu-terated TMTSF+ 101 4.5 The calculated out-of-plane au, big and hu normal modes of the pro-tonated and deuterated TMTSF 0-" 1" methyl groups 102 4.6 Proportionality factors for band intensities along the a, b' and the c' axes 106 viii 4.7 The interpolated frequencies and deuterium shifts of the in-plane ag, &3S, blu and 6 2 u normal modes of TMTSF" 1" 0- 5 108 4.8 Correlation diagram for the high-temperature phase of (TMTSF)2Re04 110 4.9 Internal vibrational assignments for protonted and deuterated (TMTSF) 2 -ReC-4 and ( T M T S F ) 2 B F 4 for E||a 113 4.10 Internal vibrational assignments for protonted and deuterated (TMTSF) 2 -ReC-4 and ( T M T S F ) 2 B F 4 for E||a (cont.) 114 4.11 Internal vibrational assignments for protonted and deuterated ( T M T S F ) 2 -ReO-4 and ( T M T S F ) 2 B F 4 for E||b' 117 4.12 Internal vibrational assignments for protonted and deuterated (TMTSF) 2 -R e 0 4 and ( T M T S F ) 2 B F 4 for E||b' (cont.) 118 4.13 Calculated isotope shifts of the ( T M T S F ) 2 R e 0 4 lattice modes 121 4.14 Absorption peaks of ( T M T S F ) 2 R e 0 4 for E||b' at 8 K 126 5.1 Dimer model parameters for the fit to the optical conductivity of (TMTSF) 2 -R e 0 4 and ( T M T S F ) 2 B F 4 at « 25 K and « 20 K respectively for E||a . . 132 5.2 Tetramer model parameters for the fit to optical conductivity for ( T M T S F ) 2 -R e 0 4 at T w 25 K and ( T M T S F ) 2 B F 4 at « 25 K for E||a 136 5.3 Phase-phonon model parameters for the fit to the optical conductivity of ( T M T S F ) 2 R e 0 4 at « 25 K and ( T M T S F ) 2 B F 4 at T « 20 K for E||a . . . 141 5.4 Model parameters for a fe-CDW fit to the optical conductivity of (TMTSF) 2 -R e 0 4 for E||a at T = 20 K 151 5.5 Model parameters for a 6-CDW fit to the optical conductivity of ( T M T S F ) 2 -B F 4 for E | | a at T = 25 K 158 A . l The bond lengths and bond angles for the T M T S F molecule 180 A.2 Valence force constants for T M T S F and its cation 181 ix List of Figures 1.1 The dimerization of a linear chain-material (Peierls distortion) resulting in the formation of a charge-density wave 5 1.2 The structure of the ( T M T S F ) 2 X 7 1.3 Electron density distribution the plane of a T M T S F molecule 8 1.4 The (a) d.c. resistivity and (b) resistivity at 35 GHz versus temperature for some typical samples of the ( T M T S F ) 2 X salts 11 1.5 Side view of the stacks in the ordered phase of (TMTSF^ReO^ 12 1.6 The pressure-temperature phase diagrams of (TMTSF)2Re04 14 1.7 The high-temperature band structure and Fermi contour in the first Br i l -louin zone of the reciprocal lattice of (TMTSF)2Re04 16 1.8 Model band structure for the q = ( | , | ) two-dimensional broken symmetry 17 1.9 The reflectivity and optical conductivity obtained by Jacobsen et al. and Bozio et al. for ( T M T S F ) 2 R e 0 4 at w 40 K and » 120 K 20 2.1 Optical path of the basic Bruker IFS 113V interferometric spectrometer. 27 2.2 The horizontal optical arrangement in the front sample chamber of the Bruker IFS 113v interferometer, leaving the back sample chamber free . . 35 2.3 The vertical optical transfer system for the commercial bolometer 38 2.4 Typical LT-3-110 Heli-tran system flow diagram 41 x 2.5 A vertical section through the cryogenic arrangement for the Heli-tran which allows translational interchange of the sample and the reference mirror 42 2.6 Mounting arrangement used for single-crystal reflectivity experiments . . 45 2.7 Diffraction from a single crystal of ( T M T S F ) 2 B F 4 for E||a in its metal-lic state at room temperature in the far infrared and the correction for diffraction 48 2.8 A vertical cross-section of the fixed tempertaure brass crytostat 50 3.1 The reflectivity of ( T M T S F ) 2 R e 0 4 at room temperature from 30 c m - 1 to 16000 cm" 1 for E||a and E | | b ' 56 3.2 The reflectivity of ( T M T S F ) 2 R e 0 4 at « 85 K from 30 c m - 1 to 16000 cm" 1 for E | | aand E | | b ' 57 3.3 The reflectivity of ( T M T S F ) 2 R e 0 4 at « 25 K from 30 cm" 1 to 16000 cm" 1 for E||a and E | | b ' 58 3.4 The reflectivity of ( T M T S F ) 2 R e 0 4 at T « 25 K from 0 cm" 1 to 2500 c m - 1 for E||a and E||b' 59 3.5 The high-resolution reflectivity of ( T M T S F ) 2 R e 0 4 at T « 25 K from 200 cm" 1 to 350 c m - 1 for E||a 61 3.6 The high-resolution reflectivity of ( T M T S F ) 2 R e 0 4 at T « 25 K from 30 c m - 1 to 80 c m - 1 for E||a and E||b' 62 3.7 The temperature dependence of the phonons at « 50 c m - 1 in (TMTSF) 2 -R e 0 4 for E||a . 63 3.8 The reflectivity of ( T M T S F - d 1 2 ) 2 R e 0 4 at » 25 K from 30 cm" 1 to 16000 cm" 1 for E||a and E||b' 65 xi 3.9 The reflectivity of (TMTSF-d" 1 2 ) 2 Re0 4 at T * 25 K from 0 cm" 1 to 2500 cm" 1 for E||a and E||b' 66 3.10 The optical conductivity of (TMTSF) 2 Re04 at room temperature and T « 8 5 K for E||a from 0 cm" 1 to 2500 cm" 1 69 3.11 The optical conductivity of ( T M T S F ) 2 R e 0 4 at T » 25 K for E||a from 0 c m - 1 to 2500 c m - 1 70 3.12 The optical conductivity of ( T M T S F ) 2 R e 0 4 at T w 25 K for E||b' from 0 c m - 1 to 2500 c m - 1 71 3.13 The optical conductivity of (TMTSF-d" i 2 ) 2 Re0 4 at T « 25 K for E||a from 0 c m - 1 to 2500 c m - 1 72 3.14 The optical conductivity of ( T M T S F - d 1 2 ) 2 R e 0 4 at T w 25 K for E||b' from 0 c m - 1 to 2500 cm" 1 73 3.15 Powder absorption coefficient (in arbitrary units) of protonated ( T M T S F ) 2 -R e 0 4 at room temperature and w 80 K from 200 c m - 1 to 600 c m - 1 . . . 75 3.16 Powder absorption coefficient (in arbitrary units) of protonated and deu-terated ( T M T S F ) 2 R e 0 4 at « 80 K from 200 cm" 1 to 600 cm" 1 76 3.17 Powder absorption coefficient (in arbitrary units) of protonated and deu-terated ( T M T S F ) 2 R e 0 4 at « 6 K from 500 cm" 1 to 2000 cm" 1 77 3.18 The reflectivity of ( T M T S F ) 2 B F 4 at room temperature from 30 c m - 1 to 16000 cm" 1 for E||a and E||b' 81 3.19 The reflectivity of ( T M T S F ) 2 B F 4 at « 20 K from 30 cm" 1 to 16000 cm" 1 for E||a and E||b 82 3.20 The reflectivity of ( T M T S F ) 2 B F 4 at T « 20 K from 0 c m - 1 to 2500 cm" 1 for E||a and E||b' 83 3.21 The reflectivity of ( T M T S F - d 1 2 ) 2 B F 4 at T « 20 K from 30 cm" 1 to 16000 cm" 1 for E||a and E||b' 84 xii 3.22 The reflectivity of ( T M T S F - d 1 2 ) 2 B F 4 at T « 20 K from 0 cm" 1 to 2500 cm" 1 for E||a and E||b' 85 3.23 The optical conductivity of ( T M T S F ) 2 B F 4 at room temperature and T tn 20 K for E||a from 0 cm" 1 to 2500 cm" 1 87 3.24 The optical conductivity of ( T M T S F ) 2 B F 4 at room temperature and T tn 20 K for E||b' from 0 cm" 1 to 2500 cm" 1 88 3.25 The optical conductivity of ( T M T S F - d 1 2 ) 2 B F 4 at T tn 20 K for E||a from 0 cm" 1 to 2500 cm" 1 89 3.26 The optical conductivity of ( T M T S F - d i 2 ) 2 B F 4 at T tn 25 K for E||b' from 0 cm" 1 to 2500 cm" 1 90 4.1 The eight vibrational symmetry species for a T M T S F molecule with a D^h molecular point group . 95 4.2 Atomic displacement vectors for the totally symmetric (ag) modes of pro-tonated TMTSF+ 104 4.3 Sketches of the displacements of the T M T S F molecule and the R e 0 4 molecule in possible lattice modes 123 4.4 A plane of the Brillouin zone containing the origin for a cubic lattice and the ( T M T S F ) 2 X lattice 125 5.1 Diagram illustrating the optical activity arising from out-of-phase ag modes in an isolated dimer 129 5.2 The dimer model optical conductivity and the experimental optical con-ductivity of (TMTSF) 2 Re0 4 for E||a at T « 25 K . . . 133 5.3 The dimer model optical conductivity and the experimental optical con-ductivity of ( T M T S F ) 2 B F 4 for E||a at T « 20 K 134 xiii 5.4 The tetramer model optical conductivity and the experimental optical con-ductivity of ( T M T S F ) 2 R e 0 4 for E||a at T « 25 K 138 5.5 The tetramer model optical conductivity and the experimental optical con-ductivity of ( T M T S F ) 2 B F 4 for E||a at T « 25 K 139 5.6 The phase-phonon model optical conductivity and the experimental opti-cal conductivity for ( T M T S F ) 2 R e 0 4 for E||a at ta 25 K 142 5.7 The phase-phonon model optical conductivity and the experimental opti-cal conductivity for ( T M T S F ) 2 B F 4 for E||a at « 25 K 143 5.8 Schematic drawing of the structure of a single molecular chain 146 5.9 The model reflectivity for ( T M T S F ) 2 R e 0 4 at T « 25 K calculated for a 6-CDW 153 5.10 The model optical conductivity for ( T M T S F ) 2 R e 0 4 at T « 25 K calcu-lated for a fe-CDW 154 5.11 The experimental and model reflectivity for the Vu(ag) mode in (TMTSF) 2 -R e 0 4 for E||a at T « 25 K . 155 5.12 The model reflectivity for ( T M T S F ) 2 B F 4 at T « 20 K calculated for a 6-CDW 159 5.13 The model optical conductivity for ( T M T S F ) 2 B F 4 at T « 20 K calculated forafc-CDW 160 5.14 The experimental and model conductivity for the v&(ag) mode in (TMTSF) 2 -B F 4 for E||a at T w 20 K 161 5.15 Electronic energy, e(fc), bands for a distortion wave vector Q = ( | , | ) for ( T M T S F ) 2 B F 4 166 5.16 Conductivity calculated for E||b' for direct transitions for the energy bands in Fig. 5.15 167 XIV A . l Types of internal valence coordinates 175 A.2 The symbolic designation of the internal valence coordinates used in the normal coordinate calculation of T M T S F 178 A . 3 Schematic representation of the G matrix elements 179 B. l Steps of evolution of the distorted dimeric state 192 B.2 Excited state energies which are accessible by optical transitions from the ground state 196 B. 3 The phase-phonon reflectivity and optical conductivity for rj = 0 c m - 1 and n ^ 0 cm" 1 at T = 0 K 201 L C. l The tight-binding energy bands for a one-dimensional semiconductor . . 207 C.2 the calculated optical conductivity for 6-CDW and s-CDW energy gaps . 215 C.3 Calculated reflectivity in the gap region for the case in which only the 6-CDW has a nonzero amplitude 217 C.4 Calculated optical conductivity the gap region for the case in which only the 6-CDW has a nonzero amplitude 218 C.5 The calculated reflectivity for the case where both 6-CDW and s -CDW contriubute to the total gap with equal weights 220 C.6 The calculated conductivity for the case where both 6-CDW and s -CDW contriubute to the total gap with equal weights 221 xv Acknowledgements I would like to thank my supervisor, Dr. J .E. Edridge for his encouragement and assis-tance during the many different stages of this thesis, and for his many useful suggestions. I would also like to thank Dr. G. S. Bates of the Department of Chemistry at the University of British Columbia for preparing the deuterated T M T S F , and for the use of his laboratory. I would like to acknowledge the many useful disscussions I have had with Dr. Frances E . Bates with respect to normal coordinate analysis and in particular, her assistance with the symmetry coordinates for T M T S F . I would also like to thank Kevin Kornelsen for his assistance in the lab from time to time. I am grateful to Dr. J . Gronholz, of Bruker Analytische Messtechnik GmbH, for assistance with software development and Ms. Janice L . Howe of Bruker Spectrospin Canada for help with file-transfer programs for the Aspect 3000. I would like to thank Peter Haas and the technicians in the machine shop for their assistance and advice in fabricating the mirrors used in the bolometer focusing optics in these experiments, as well as their many other contributions to this thesis. I would like to acknowledge useful discussions with Dr. Moreno Meneghetti from the Department of Chemical Physics at the Univerity of Padova, Italy and Dr. V . M . Yartsev from the Department of Physics at Chelyabinsk State University, USSR. I am also deeply grateful to Tom Nicol of the University of British Columbia Computing Services for access to the I B M 3090/150S, and to Mary-Ann Potts for access to the departmental Sun 3/60. I would like to thank the electronics shop for their expertise in designing the interlocks for the pumping station, and for the many numerous jobs they were called upon to xvi perform in getting the Bruker commissioned. Finally, many thanks to Mona, who put up with far more than anyone should ever have to. During this research I have been supported by the University of British Columbia and the Province of British Columbia. This work was also supported by Grant No. A5653 from the Natural Sciences and Engineering Research Council of Canada. xvii Chapter 1 Introduction 1.1 Quasi-One-Dimensional Organic Conductors Organic molecular solids display an amazing diversity of forms and have many different properties. One of the physical properties that organic solids seldom display, however, is electrical conductivity and even more rarely, superconductivity. The first stable organic conductors were synthesized in the early 1960's. The organic molecule tetracyanoquinodimethane, or TCNQ, was prepared in 1960 by workers at E.I. du Pont de Nemours & Company [1]. Little energy is needed to introduce an extra electron into TCNQ and the negatively charged structure that results is chemically stable. TCNQ by itself, however, is not able to conduct electrons, and so in an electronically neutral system there is no tendency for electrons to move from one TCNQ molecule to another. If the TCNQ molecules can obtain a fraction of electronic charge (to become anions) from other atoms or molecules (which become cations), the material can become conductive. In the cesium salt of TCNQ, (Cs+)2(TCNQ~^)3 for example, each cesium atom gives up an average of two-thirds of an electron to each TCNQ molecule. Thus, two out of every three TCNQ molecules become negatively charged and if an electric field is imposed, the extra electrons can move from the charged TCNQ molecules to the neutral ones. Another TCNQ charge-transfer salt, tetrathiafulvalene-tetracyanotetra-quinodimethane (TTF-TCNQ), was synthesized in 1973 by Cowan and Ferraris [2], and 1 Chapter 1. Introduction 2 by Heeger and Garito [3]. In the solid form the T T F and TCNQ molecules stack in separate columns, and electrons are donated from the T T F stack to the TCNQ stack. A fractional charge of 0.59 electron per molecule is transferred from one stack to another. Because of the overlap of the molecular orbitals in the stack direction there can be a net motion of electrons along the stacks; hence the material is conductive. At room temperature T T F - T C N Q has a conductivity of « 500 - 1000 (ficm) - 1, about three orders of magnitude smaller than that of copper. The architecture of the TTF-TCNQ crystal gives rise to a striking electrical property: the material is highly conductive in one direction only. The most favourable direction is 500 times as conductive as the least favourable direction. This is because the T T F and TCNQ molecules are large planar molecules, with the valence 7r-electrons located above and below the planar framework. They stack together like pancakes so that the orbitals of the valence ir electrons overlap, allowing the electrons to move freely up and down the stacks. The reason for the anisotropy is that the T T F and the TCNQ molecules are stacked in parallel planes, with little overlap between adjacent stacks. Compounds that display this anisotropic behavior are called quasi^ one-dimensional solids or linear-chain materials. Some of the highly conducting inorganic linear-chain solids have been known for quite some time. Tetracyanoplatinate, or TCP, was first synthesized in 1842 and the material poly(sufumitride), (SN)X, in 1910. However, the properties of TCP and (SN)* were not examined until the 1970's. There are several good historical reviews available [4]. The search for new linear-chain compounds was prompted by W. A. Little, who in 1964 proposed an alternative method for the formation of Cooper pairs [5]. Within the formulation of superconductivity by Bardeen, Cooper and Schriffer [6], superconduc-tivity results when electrons in a metal form into bound pairs, or Cooper pairs, that are mediated by lattice distortions. Little proposed a way that Cooper pairs might be Chapter 1. Introduction 3 produced without lattice distortions. He considered a large organic molecule consisting of two parts: a long chain of carbon atoms called the "spine" and a series of arms or side chains attached to the spine. Each molecule would be a hydrocarbon with loosely bound valence electrons, such as an organic dye. An electron travelling along the spine of the carbon atoms would repel the outer electrons of the hydrocarbon molecules, thus creating a region of net positive charge on the parts of the side chains nearest to the spine. A second electron moving along the spine in the opposite direction would be attracted by the relatively dense positive charge and indirectly to the first electron thus forming a Cooper pair. The region of enhanced positive charge is created by the displacement of electrons rather than atoms in a lattice. Since the electron is about 100,000 times less massive than a typical ion, and since a general relation exists for the critical temperature Pi) [6] T < * v m ( u ) then Tc for this material should be increased by a factor of « 300. Such a material would be superconducting at room temperature, and in principle the superconducting state could be stable up to 2000 K. It is known from theoretical calculations that it is impossible to sustain the formation of Cooper pairs in a strictly one-dimensional conductor at any temperature above absolute zero. Thus the strictly one-dimensional superconductor cannot exist. In any real molecular system, however, the conductors are only approximately one-dimensional, and there will be some interaction between the chains that will allow a Cooper pair to jump from one stack to another. Hence the theoretical obstacle to superconductivity in one dimension is not a practical one. While no one has succeeded in constructing the sort of organic molecule that Little envisioned, the idea has generated a strong interest in organic superconductors and linear-chain organic systems (some of the problems of superconductivity in systems with one-dimensional Chapter 1. Introduction 4 anisotropy have been discussed [7]). 1.2 (TMTSF) 2 X Charge-TVansfer Salts The organic conductor TTF-TCNQ is a good electrical conductor at room temperature. As the temperature is lowered the conductivity increases in a metallic fashion until at about 58 K, the uniform molecular spacing in the stacks gives way to a more energetically favourable structure in which the separation of adjacent molecules in the stack changes. This is referred to as a Peierls distortion [8]. The result of this structural change is to alter the electronic density along the chain. The charge density undergoes a periodic concentration and rarefaction, that is called a charge-density wave (CDW). Structural studies of TTF-TCNQ indicate that there are three sucessive phase transitions [9]; the CDW transition in the TCNQ stacks occurs at 54 K, the CDW transition in the T T F stacks at 49 K and at 39 K three-dimensional order is established. This is attributed to the CDW's becoming pinned due to significant interchain coupling [10]. In other words, the oppositely-charged CDW's on adjacent stacks are pinned to each other by coulomb attraction. At low temperatures the periodic lattice distortion associated with the Peierls transition is incommensurate along the stacking direction with a periodicity of « 3.4 times the molecular spacing along the stacking direction. When a commensurate CDW involves the pairing of the molecules in the chain this transition is referred to as dimerization of the molecules, as illustrated for a linear-chain material in Fig. 1.1 In the case of a simple half-filled band for a one-dimensional metal, the onset of a CDW resulting in a dimerization of the molecular stack causes the electron conduction band to split into two bands separated by an energy gap 2A; the lower band is completely filled, the upper band is completely empty, as shown in Fig. 1.1. Conduction may occur Chapter 1. Introduction 5 MOMENTUM Figure 1.1: The dimerization of a linear chain-material (Peierls distortion) resulting in the formation of a charge-density wave. Note that the CDW has a periodicity twice the interatomic spacing. The diagrams at the right depict the creation of a band gap due to band splitting (after Reference [4]). only if electrons are promoted across the energy gap by thermal activation or photo-absorption. Thus, the material changes suddenly from a conductor to a semiconductor or an insulator. While the valence electrons are no longer available for free conduction, an alternate mechanism for conduction is the sliding CDW [11], which slides along the lattice stacks. It is this effect which is responsible for the enhanced conductivity in T T F -TCNQ at « 60 K [12, 13]. As the temperature is lowered below 58 K for TTF-TCNQ, there is a decrease in the conductivity due to the pinning of the CDW. The Peierls distortion in TTF-TCNQ motivated the search for a new organic conduc-tor which would have small molecules in the anion stacks that would not be susceptible to CDW formation, thus allowing sliding CDW's along the cation stacks and suppressing pinning due to interchain coupling. In 1979, a fascinating new series of charge-transfer Chapter 1. Introduction 6 salts were synthesized by Bechgaard [14]. The fcts-tetramethyltetraselenafulvalene-X compounds have the chemical formula (TMTSF)2AT, where X is any one of a number of relatively small inorganic radical anions, e.g. hexafluorophosphate (PFe), hexafluoroarse-nate (AsFe), hexafluoroantimonate (SbFe), tetrafluoroborate (BF4), perchlorate (C104), perrhenate (ReCu), nitrate (NO3), etc., and are known as the Bechgaard salts. 1.2.1 Structure of ( T M T S F ) 2 X Salts The crystal structure of the (TMTSF) 2 X salts is triclinic with a Pl(Cj) space group [15]. The anion positions are centers of inversion symmetry, but no other symmetry operations apply to the crystal lattice. The crystal structure of (TMTSF)2Br04 [16] is shown in Fig. 1.2. The planar TMTSF molecules are slightly dimerized as they stack together in a zig-zag fashion to form a linear chain. The chain direction is the a axis. The stacks are tightly arranged along the interchain or b axis, forming a sheet in the ab plane. The anions form a well defined lattice and sit in the cavities left by the TMTSF molecules. The c axis lies along the anion lattice direction that is almost normal to the ab plane. Other sets of axes that are routinely used are the orthogonal set (a, b' and c') and the set of principal magnetic axes (a*, b* and c*). In the orthogonal set b' is orthogonal to a and in the ab plane, and c' completes the set. The principal magnetic axes are determined by the principal axes of the magnetic susceptibility tensor and coincide with the principal axes of the conductivity (or resistance) tensor; a and a* are the same; and b* and c* are close to, but not the same as b and c respectively. The TMTSF molecules stack almost perpendicular to the a axis, deviating from perpendicularity by w 1.1° [17]. The stack units repeat by inversion with overlap dis-placements alternating along the a axis. The value of the lattice parameters for the unit cell will vary depending on the anion Chapter 1. Introduction 7 Figure 1.2: The structure of (TMTSF) 2A\ (a) The TMTSF molecule, (b) Illustra-tion of the crystal structure of the conductor (TMTSF^BrCU viewed down the stacking direction and, (c) showing the zigzag stacking of the TMTSF molecules and the crys-tallographic coordinates. The positions of the oxygen atoms of the anion are shown for both orientations of the anion which is disordered at room temperature. Note the Se-Se contacts (d « 3.9 — 4.0 A) and that the dimerization of the TMTSF stack is very weak. The dimensions of the unit cell for (TMTSF^BrC^ at room temperature are a = 7.282(2) A, b = 7.714(2) A, c = 13.425(4) A, a = 83.74(2)°, 0 = 86.18(2)°, 7 = 70.17(2)° and Vc = 707.2(3) A 3 (the number in parenthesis indictes the uncertainty in the last significant figure) (after References [20, 24]). Chapter 1. Introduction 8 Figure 1.3: Electron densities in the plane of a TMTSF molecule for (TMTSF)2AsF6 . Contours are 0.05 e/A 3 per contour line starting at 0.05 e/A3; negative contours are shown by broken lines and the error is 0.05 e/A 3 for all density maps. The high electron density about the central Cl—C6 carbon double bond is of special interest, (after Reference [25]). Chapter 1. Introduction 9 substitution. The room-temperature crystal structures (and in most cases the crystal structure at w 120 K) of (TMTSF) 2 PF 6 [17], (TMTSF) 2AsF 6 [18], (TMTSF) 2Re0 4 [19], (TMTSF) 2 FS0 3 [16], (TMTSF) 2 BF 4 [20], (TMTSF) 2C10 4 [21] and (TMTSF) 2 N0 3 [22] have been reported and there have also been several temperature-dependent crystal structure studies [16, 23]. There are two TMTSF molecules per unit cell, and there is a transfer of one electron from two TMTSF molecules to one anion giving a half-filled band if we take a to be two molecular spacings. Conduction along the TMTSF stacks is by normal band conduction of holes. Unlike most other charge-transfer salts the anion stacks do not have overlapping atomic orbitals and thus conduction electrons do not move up and down the anion stacks. The anions are important in controlling lattice parameters. The electron density distribution in the plane of the TMTSF molecule has been measured using X-ray diffraction data for crystals of (TMTSF)2AsF6 [25], shown in Fig. 1.3. The density map obtained for TMTSF in (TMTSF) 2AsF 6 may be extended to TMTSF molecules in different salts. The electron density was oberved to be highest for the C2—C3 and CI—C6 double bonds and lower for the C7—C8 double bond, in line with the observed bond lengths [18]. 1.2.2 General Properties The Bechgaard salts display a variety of interesting low-temperature states [26]. Despite having the same stoichiometry, the compounds having centrosymmetric (e.g. octahe-dral symmetry) anions usually have different physical properties than those with non-centrosymmetric (e.g. tetrahedral, planar, etc.) anions. At room temperature, all the (TMTSF) 2 X salts are good conductors, but the conductivity is highly anisotropic (the anisotropy is as large as a:b:c« 10s : 400 : 1 in some salts at low temperature). At low temperatures, the ground states can be either insulating, metallic or superconducting. Chapter 1. Introduction 10 Table 1.1: Properties of some (TMTSF) 2JV salts. Anion Anion rpa 1MI Transition rpb symmetry C K) (° K) (kbar) P F 6 Octahedral 12 SDW 1.4 8.5 A s F 6 Octahedral 12 SDW 1.4 9.5 SbF 6 Octahedral 17 SDW 0.38 10.5 TaF 6 Octahedral 11 SDW 1.3 11 B F 4 Tetrahedral 39 AO _ _ B r 0 4 Tetrahedral 250 AO - -Re0 4 Tetrahedral 177 AO 1.3 8 C10 4 Tetrahedral 24 AO 1.2 -3.5 SDW - -FSO3 Asymmetric 86 AO 2d 5d N 0 3 Planar 40 AO _ 12 ? - -° TMI is the temperature of the spin-density-wave (SDW) transition or the anion-ordering (AO) transition. 6 TC is the temperature at which the material becomes a superconductor. c Pc is the minimum pressure required for superconductivity to occur. d The superconducting transition is thought to be incomplete in (TMTSF) 2 FS03 • Some of the physical properties of the (TMTSF) 2 X salts are shown in Table 1.1, and the d.c. resistivity versus temperature shown in Fig. 1.4(a). The resistivity at 35 GHz is also shown in Fig. 1.4(b) for comparison. The non-centrosymmetric anions include the tetrahedral anions B F 4 , Re0 4 , C10 4 , the asymmetric FSO3 anion and the planar anion NO3. The Bechgaard salts with non-centrosymmetric anions are all good conductors at room temperature, but they undergo metal-insulator (MI) transitions as they are cooled. The tetrahedral anions sit on in-version centers in the crystal lattice and may have two different orientations. At room Chapter 1. Introduction 11 B o m3 "53 N o CO 2.10-5 10 - i — i — i i i i — r (b) ( T M T S F I j X o o o x = N 0 3 ~ o o o X t P F 6 " AAA X = B F 4 ~ • i I I 30 100 300 Temperature (K) Temperature (K) Figure 1.4: The (a) d.c. resistivity and (b) resistivity at 35 GHz versus temperature for some typical samples of the (TMTSF) 2 X salts. Notice the logarithmic temperature and resistivity scales (after Reference [14]). temperature, the anions are dynamically disordered, flipping back and forth from one orientation to another and thus there is no long-range order. Neutron diffraction [28] and x-ray diffuse scattering [27] studies have shown that the MI transitions are due to the ordering of the anions, thus forming a superlattice structure. As (TMTSF)2Re04 and (TMTSF)2BF4 are cooled they have MI transitions at 177 K and 39 K respectively when anions order along all three lattice directions. The unit cell changes from a x b x c to 2a x 26 x 2c, corresponding to a modulation wave vector of cji = (|, | , |). The new Chapter 1. Introduction 12 Figure 1.5: Side view of the stacks in the ordered phase of (TMTSF)2Re04. The directions of the displacements are marked with arrows from the Re atoms and from the TMTSF molecules. Centers of symmetry are marked with a cross (after Reference [30]). unit cell results in the formation of an energy gap at the fermi level due to band split-ting. The anion-ordering (AO) transition also results in a periodic lattice distortion in which the molecules in the unit cell are shifted and the lattice dimerization grows larger resulting in a commensurate CDW [29, 30, 31, 32]. Fig. 1.5 illustrates the positions of the anions in the ordered state of (TMTSF)2Re04 and the displacements of the molecules that occur below the MI transition. If (TMTSF)2Re04 is quenched (cooled at a rate of « 25 K/min or greater) then anion disorder is frozen in [33]. The anions are ordered in (TMTSF) 2 BF 4 at 20 K even after quenching [31]. This is not too suprising, since the BF4 anion is smaller than ReO^. The salts with octahedral anions (PFe, AsFe, SbFe and TaFe) display large conductiv-ities at low temperature. In particular, the conductivity of (TMTSF)2PF6 is 10s (flcm) -1 along the a-axis at w 20 K. All of the octahedral anion salts undergo MI transitions at Chapter 1. Introduction 13 T\fj « 12 K at ambient pressure due to the formation of a spin-density-wave (SDW) state [34, 35, 36], making these materials itinerant antiferromagnets. The SDW state is interesting because a Peierls instability is more common among low-dimensional materi-als. The octahedral anions do not display anion-ordering because they are symmetric; all the allowed anion orientations are the same. Superconductivity occurs in the octahedral anion compounds under pressure ( « 10 kbar). The first organic material observed to exhibit superconductivity was (TMTSF^PFe [15], which becomes a bulk superconduc-tor at Tc « 1 K with the application of > 8.5 kbar of pressure. Evidence for type II superconductivity includes both zero resistance [15] and a partial Meissner effect [37, 38]. In (TMTSF)2Re04 the metal-insulator transition due to the qi AO that exists at ambient pressure is supressed above 8 kbar in favour of a metallic phase in which the anions are ordered with a unit cell a x 2b x 2c and a wave vector q2 = (0, 5,5) [39]. The respective ground states are a non-magnetic anion-driven insulator and a super-conductor [40]. However, if the q2 AO is maintained at pressures lower than 8 kbar, a new semiconducting phase is stabilized as the ground state [41]. The pressure depen-dence of the transition strongly resembles that of the SDW ground state found in the octahedral-anion compounds, and also that of (TMTSF) 2 N03 [42, 43], indicating that the low temperature q2 semiconducting ground state in (TMTSF)2Re04 may be anti-ferromagnetic (although no direct proof has yet been given with magnetic experiments). The pressure-temperature diagrams for (TMTSF) 2Re04 are shown in Fig. 1.6(a) (above 50 K only) and 1.6(b) (full range of temperatures and pressures). Efforts to eliminate the need for pressure to achieve superconductivity led to the sub-stitution of very small anions in order to decrease the interstack distance. When this was done with the perchlorate anion in 1980, the resulting salt became superconducting Chapter 1. Introduction 14 Figure 1.6: The pressure-temperature diagrams of (TMTSF) 2Re0 4. (a) above 50 K (after Reference [39]) and (b) over the full range of temperatures below 17 kbar (after Reference [41]). Open and full squares represent cooling and warming, respectively. After pressure-temperature cycling along path L: open triangles for cooling, full triangles, circles and squares for warming. Open and full hexagons for decreasing and increasing pressure, respectively. The path L is fixed by the melting curve of helium 4, (shown in line H). at Tc « 1.2 K at ambient pressure [44, 45], making (TMTSF)2C104 the first ambient--pressure organic superconductor! The superconductivity in (TMTSF)2C104 was con-firmed by the observation of Meissner signals [46] and specific heat measurements using an a.c. calorimetric technique [47], indicating a type II character. The molar specific heat displays a very large anomaly around 1.22 K. The superconducting state in (TM-TSF)2C104 is very sensitive to the magnetic field. The critical fields are H£2 = 150 G and H* 2 = 10 kG [48]. In (TMTSF) 2C10 4 the anion-ordering transition is at 24 K, but unlike (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 the anions only order along the b axis, thus the unit cell changes to a x 26 x c [28]. If (TMTSF) 2C10 4 is quenched, the anion disorder present at room temperature is "frozen in" and a SDW state is observed below « 3.5 K [49] with no superconducting transition. This effect is reversible, and when the sample Chapter 1. Introduction 15 is cooled slowly below 50 K then the usual AO transition is observed at 24 K with a superconducting transition at T c w 1.4 K. The anion sites do not interact directly, but rather through the cation-anion Se-X interaction for (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 [28] and through the CE3-X inter-action for (TMTSF)2N03 [50]. While the strength of these interactions is thought to determine how the non-centrosymmetric anions will order [51], the specific role of the anions with respect to the ground state is far from being understood. 1.2.3 Band Structure and Electronic Properties The electronic energy bands of the (TMTSF) 2X organic conductors are nominally quarter-—filled due to a charge delocalisation on the organic chain. However, the three-dimensional environment of a stack makes the band half-filled. The half-filled and quarter-filled sys-tems play a central role in the study of polymeric and molecular organic conductors [52]. Their electronic structure is commensurate with the underlying lattice in the ratio G/2kF (G is the reciprocal-lattice vector) which is a simple integer: 2 in the former case; 4 in the latter. Grant has performed a tight-binding calculation for a unified single-particle model [53]. Rather than solving the secular equation, however, for 76 valence bands the calcu-ations have been simplified by only considering the highest occupied molecular orbital (this reduces the number of bands to just two). The model is solved in two dimensions because the anisotropy is basically two dimensional; the difference between the b and c axis being minor with respect to that between the a and b axis. The room-temperature band structure of (TMTSF) 2Re0 4 (calculated from Grant's model [54]) is shown in Fig. 1.7. If a one-dimensional conductor is considered, then this would correspond to the electronic band structure along the T —> X path. The location of the Fermi level Ep is below the top of the conduction band, indicating that conduction Chapter 1. Introduction 16 8000.0 4000.0 -6 ID 0.0 -4000.0 -8000.0 r x v Y r ( 0 . 0 ) (n/a. 0 ) (n/a. n/b) ( 0 . n/b) ( 0 , 0 ) Figure 1.7: The high-temperature band structure and Fermi contour in the first Bril-louin zone of the reciprocal lattice for (TMTSF^rveCV using six transfer integrals to characterize the system [54]. Notice that a small gap is. present below the Fermi level ( E F = 4142 cm - 1 , after Reference [53]). by electrons in the band is possible. At room temperature there is a small gap due to the slight dimerization of the stacks, but it is below the Fermi level E F and does not affect the electrical transport properties. This sort of band structure is representative of the room-temperature behavior of all of the (TMTSF^X salts. The formation of a lattice distortion (such as a CDW or an AO) at low temperatures changes the lattice parameters and alters the reciprocal lattice vectors and the shape of the first Brillouin zone thus changing the band structure of the system. Such a change in the lattice parameter is referred to as a broken symmetry. The band structure shown in Fig. 1.8 is for the highly commensurate two-dimensional broken symmetry that involves Chapter 1. Introduction 17 r x v Y r Figure 1.8: Model band structures for the q = (1/2,1/2 two-dimensional broken sym-metry, with t\\/tx = 10 and 2A = 4tx. t\\ is the transfer integral parallel to the stacks, t\. is the transfer integral transverse to the stacks and A is half of the semiconducting energy gap (after Reference [53]). a doubling of the lattice parameters in both the chain and the transverse directions, resulting in a modulation wave vector of q = (|, |). The result in Fig. 1.8 is calculated for t\\/t± = 10 and 2A = 4t±. t\\ is the transfer integral parallel to the stacks, t± is the transfer integral transverse to the stacks and 2A is the semiconducting energy gap. (The energy gap, expressed as 2A for largely historical reasons, is alternatively written as Eg). The energy axis indicates the dependence of the major features on the parameters 2A, t\\ and tj.. The wave vector axes are not to scale; however the Brillouin zone insets are. A doubling of the lattice parameter along both the stack and transverse directions occurs in (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 due to AO transitions resulting in a 2a x Chapter 1. Introduction 18 26 x 2c unit cell. This gives a modulation wave vector of qi = (|, | , |) and corresponds to the q = (j, |) broken symmetry. From Fig. 1.8 a direct gap of 2A is created in the band structure. The electrical resistivity of (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 has been measured along the a-axis as a function of temperature from T « 300 K to T « 20 K; the semiconducting energy gap in (TMTSF) 2Re0 4 is calculated to be 2A = 1330 cm - 1 (but thermopower measurements give a somewhat larger value of 2A = 1850 cm - 1 [56]), while the gap for (TMTSF) 2 BF 4 is 2A « 1200 cm"1 [14]. The high-temperature Fermi contour, indicated by the dashed lines in the inset of Fig. 1.8, shows a strong 2kp nesting in the [1,1] direction and provides a natural mecha-nism for SDW formation in centrosymmetric anion compounds. The Fermi contour is a line that joins all the points in the first Brillouin zone where the electron energy is equal to the Fermi energy. This is the only symmetry that yields an insulator for every finite value of 2A, since E F is in the middle of the gap. This gap should not be removable by pressure without some further change in the symmetry. In fact, a different phase exists in (TMTSF) 2Re0 4 above 8 kbar with a wave vector of (0, \ , \) [29] and this state is metal-lic, in agreement with the Grant's prediction. The direct gap is created by the removal of a virtual crossing near X by a lattice distortion. The inset in Fig. 1.8 shows the Fermi contours for the electron-hole band extrema. These bands form a closed "trench" about the X - V direction. This model for the band structure provides some useful insights into the electronic transport properties of the (TMTSF) 2 X compounds and allows an understanding of how different anions and the anion-ordering transitions lead to different physical properties. Chapter 1. Introduction 19 1.3 Optical Properties of (TMTSF ) 2 He0 4 and (TMTSF) 2 BF 4 1.3.1 Previous Studies A number of reflectivity studies have been performed on the Bechgaard salts, but they have focused mostly on the ambient pressure organic superconductor (TMTSF)2C104 [57, 58, 59], or on those salts with octahedral anions, such as (TMTSF) 2 PF 6 [57,60], (TM-TSF) 2SbF 6 or (TMTSF) 2AsF 6 [61, 62]. There have been two studies of the reflectivity of (TMTSF)2Re04 below the metal-insulator transition, the first by Jacobsen et al. [60] and subsequently by Bozio et al. [63]. No data for (TMTSF) 2 BF 4 had been reported until the preliminary results in this thesis were published [64], and no reflectivity studies of deuterated (TMTSF)2Re04 or (TMTSF) 2 BF 4 have been previously performed. Jacobsen et al. [60] measured the reflectivity from single crystals and mosaics of crystals for E||a at T = 300 K and « 40 K. They fitted their results to a "dimer model" due to Rice [65]. Bozio et al. also measured the reflectivity of a mosaic of single crystals for E||a at T « 120 K. They fitted their results to a modified "tetramer model", originally developed by Yartsev [66] (these models are discussed in Appendix B, and some results are shown in Chapter 5). Neither set of measurements include the far-infrared reflectivity below 400 cm - 1 and both are for the E||a polarization only. Furthermore, neither model was entirely satisfactory in explaining the experimental results, and the assignments of several features remained questionable. The experimental reflectivity and conductivity obtained by Jacobsen et al. for (TMTSF) 2Re0 4 for E||a at T » 40 K are shown in Fig. 1.9(a) and 1.9(b) respectivily. The conductivity obtained by Bozio et al. for (TMTSF)2ReC*4 at « 120K for E|ja is shown in Fig. 1.9(c). The resolution of the optical conductivity at « 40 K is lower than that reported at « 120 K, as several new features in Fig. 1.9(c) indicate. Both of the spectra display an asymmetry (the peak at « 1200 cm - 1 is higher Chapter 1. Introduction 20 Figure 1.9: The (a) experimental reflectivity and (b) optical conducivity obtained by Jacobsen et al. (after Reference [60]) and (c) the optical conductivity obtained by Bozio et al. indicated by the scale on the right of the figure; the scale on the left is for the fit to the tetramer model (after Reference [63]) for (TMTSF) 2Re0 4 at « 40 K and « 120 K for E||a. The resolution in (a) and (b) is lower than in (c), as several missing features indicate. The optical conductivity in (c) is a factor of two too large. The dashed line in (c) denotes a fit to the tetramer model. (The tetramer model described in Appendix B). Chapter 1. Introduction 21 than that at « 1850 cm - 1), which becomes more pronounced at low temperature. Powder absorption measurements have been performed on (TMTSF)2Re04 from « 250 cm - 1 to 2000 cm - 1 by Bozio et al. [67] and in the far infrared (< 100 cm - 1) on both (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 by Eldridge et al. [68]; some assignments of the external and low-frequency internal molecular vibrations have been made. 1.3.2 New Results The salts (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 are interesting to study because both have tetrahedral anions and they both have MI transitions leaving 2a x 26 x 2c superlattices resulting in semiconducting states with fairly large energy gaps (in both materials, 2A is expected to be above 1000 cm - 1) and commensurate CDW's. The creation of a semicon-ducting energy gap below the metal-insulator transition eliminates the free carriers which normally mask the optical properties because of their strong absorption. This allows the normally weak external phonons, as well as the internal ag modes, which are known to be strongly active in one-dimensional organic conductors, to be examined. The anion-ordering transition is also accompanied by an eightfold increase in the size of the unit cell, thus there is a corresponding reduction of the first Brillouin zone. Zone-boundary acoustic phonons which are normally infrared inactive are mapped back to the zone ori-gin where they may become infrared active. Furthermore, optically active internal and external modes may be split due to dispersion at the zone face, resulting in the splitting of modes which are already optically active. The power reflectivity of single crystals of protonated and deuterated (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 has been measured from « 30 cm - 1 to » 8000 cm - 1 for the E||a and E||b' polarizations above and below the metal-insulator transitions with a resolution of 2 cm - 1 . Measurements performed on single crystals have a number of advantages over those performed on oriented crystal mosaics (discussed in detail in Chapter 2). The Chapter 1. Introduction 22 ability to perform far-infrared reflectivity measurements on small single crystals is, at the present time, restricted to only a few labs. The results from (TMTSF+Re0 4 for E||a may be compared to previous results above 400 cm - 1 . The results at low temperature have a much higher resolution than other previous work and the reflectivity has been extended below 400 cm - 1 into the far infrared. The optical properties of protonated and deuterated (TMTSF)2Re04 and (TMTSF)2-B F 4 for E||b' appear here for the first time. The reflectivity of both materials reveals strong electron-phonon coupling due to the internal ag modes, and a very strong lattice vibration for the E||a polarization in the insulating state at « 50 cm - 1 . The optical properties have been calculated using a Kramers-Kronig anaylsis for both the E||a and E||b' polarizations above and below TMI- In the insulating state, the optical conductivity for E||a behaves as a one-dimensional semiconductor; however, the internal ag modes are strongly activated. The optical properties along the stack direction for both (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 in the insulating state have been calculated from a model developed by Bozio et al. [69] for one-dimensional semiconductors with twofold--commensurate CDW's and electron-phonon coupling. By allowing only the size of the energy gap to vary and by holding the vibrational and the other electronic parameters as fixed, the model produces results which are in excellent agreement with the experi-mental results [2A = 1700 cm"1 and 1120 cm"1 in (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 respectively]. The conductivity for the E||b' direction also displays an energy gap, but its shape is unusual. Calculations using Grant's two-dimensional band structure [53] model show that the effective mass of the carriers in the energy band for E||b' leads to a distorted conductivity, but the semiconducting energy gap is isotropic. The optical conductivity has also been calculated for the deuterated salts for both polarizations, allowing vibrational assignments to be made. The powder data allow the E||c vibrations to be studied. The assignments and the group theory for the external Chapter 1. Introduction 23 vibrations in ( T M T S F ) 2 R e 0 4 presented in this thesis have been previously reported [70]. 1.4 Thesis Outline In Chapter 2, the Bruker IFS 113V interferometer and the optical parameters used in the experiments are described, along with the reflectivity module and the sample mounting and cryogenic arrangements. The optical constants discussed in the thesis are introduced. The effects of diffraction due to a single crystal are treated and compared to the effects produced by a mosaic of crystals. The experimental results for the reflectivity and optical conductivity of protonated and deuterated ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 for E | | a and E||b' are presented in Chapter 3. The group theory and normal coordinate analysis calculations for the internal vibra-tions of the T M T S F + molecule are developed and presented in Chapter 4, and vibrational assignments are made for the internal and external vibrations. In Chapter 5 a model developed by Bozio et al. [69] for the optical properties of one--dimensional organic semiconductors with twofold-commensurate CDW's is presented, and it is shown that the E||a optical properties below TMI for both ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 may be explained using this model by effectively changing only the size of the semiconducting gap. The E||b' optical properties of ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 -B F 4 in the insulating states will be discussed in terms of Grant's two-dimensional band structure model [53] and it is shown that the large effective mass of the carriers for the band for E]|b' leads to a distorted optical conductivity; however, the semiconducting gap is isotropic. The conclusions of the thesis are reviewed in Chapter 6, and some suggestions for future experiments. Chapter 1. Introduction 24 Appendix A provides an introduction to some of the details of normal coordinate analysis calculations, and provides the details for the calculations performed in Chapter 4. Appendix B provides details of the dimer, tetramer and phase-phonon models which have been used to calculate the infrared optical properties of (TMTSF^ReC^ and (TMTSF) 2-B F 4 in Chapter 5. Appendix C deals with the development of Bozio's model for one-dimensional organic conductors with twofold-commensurate charge-density waves. Some model simulations are also performed to illustrate systems with and without phonons, as well as the effect of changing the size of and the nature of the semiconducting energy gap in systems with (internal and external) phonons. Chapter 2 Experimental Techniques 2.1 Introduction Reflectivity measurements were performed on large, single crystals of protonated and deuterated (TMTSF) 2Re0 4 and (TMTSF) 2 BF 4 for E||a and E||b' above and below their metal-insulator transitions from « 20 cm - 1to « 8000 cm - 1 . A Kramers-Kronig analysis of the reflectivity yields the optical properties, which contain information on the electronic transport properties as well as information on the normal modes of vibration. Reflectivity measurements form the body of this work. Powder absorption measurements were performed on samples of protonated and deu-terated (TMTSF) 2Re0 4 in the far and mid infrared above and below the metal-insulator transition. Powder absorption measurements are useful for examining vibrations in the E|jc polarization, which are not accessible through reflectivity measurements. Powder absorption measurements are not selective, however, in that they do not discriminate between the vibrations polarized for E||b and E||c, as both of these crystal directions have low reflectivities in both the metallic and the insulating states. The E||a polariza-tion is not accessible at all using powder measurements because of the high reflectivity of (TMTSF)2Re04 for this crystal direction above and below the metal-insulator tran-sition. Absorption measurements were also performed on oriented crystal mosaics to provide information on the vibrations polarized for E||b', but the information that they 25 Chapter 2. Experimental Techniques 26 provide is consistent when considered together with the reflectivity results and will not be reported. The fact that reflectivity measurements can determine the optical properties for E||a and E||b' is the reason that they are used rather than powder or crystal mosaic absorption measurements, which are useful for vibrational measurements but difficult to interpret for anything else. 2.2 Bruker IFS 113V Interferometer The single-crystal reflectivity, powder absorption and criented-crystal mosaic absorption measurements were performed on a Bruker IFS 113V Fourier transform interferometric spectrometer. The Bruker IFS 113V is a Genzel-type interferometer [71] designed to operate under vacuum and covers the full range from the far infrared (w 10 c m - 1 ) to the near infrared ( « 15000 cm - 1 ) with a maximum resolution of w 0.03 c m - 1 . The optical system of a basic Bruker IFS 113V is shown in Fig. 2.1. In the Genzel interferometer, the radiation is focused on the beamsplitter which allows the beamsplitters to be very small ( « 2 cm in diameter). In a conventional interferometer the radiation is not focused at the beam splitter and as a result the beamsplitters are quite large (12-20 cm in diameter). These large beamsplitters vibrate slightly, resulting in a diffusion of the infrared beam and increased spectral noise. This is called the "drum-head'' effect. The small size of the beamsplitters in the Genzel interferometer greatly reduces the "drum-head'' effect. Another advantage of this design is that the angle of incidence at the beamsplitters is only 14°. This low angle of incidence compared with other interferometers results in increased light throughput and decreased polarization effects. The small size of the beamsplitters also allows them to be mounted on a rotatable carrier so that they may be changed while the instrument is under vacuum. The IFS Chapter 2. Experimental Techniques 27 Figure 2.1: Optical path of the basic Bruker IFS 113V interferometric spectrometer. I Source Chamber; a - Tungsten/Halogen/Quartz lamp, glowbar, mercury arc lamp; b -automated aperture (4). II Interferometer Chamber; c - optical filters (4); d - automatic beamsplitter changer (6); e - two-sided moving mirror; f - control interferometer; g -reference laser; h - remote control alignment mirror. Ill Sample Chamber; i - sample focus; j - reference focus. IV Detector Chamber; k - far-infrared (deuterated triglycerine sulfate (DTGS) detector, mid infrared (MCT and InSb) detectors and near infrared (Si diode) detector. 113V also allows the automatic selection from three sources, four apertures, four optical niters, two sample chambers and four detectors. This ability to change spectral regions without opening the optical bench to atmospheric pressure preserves the thermal stability of the spectrometer. The two beams which combine at the beamsplitter are incident on opposite sides of a movable double-sided plane mirror. The mirror is supported on an dual gas bearing that uses dry nitrogen and it is driven by a linear induction motor. Translation of this mirror Chapter 2. Experimental Techniques 28 Table 2.1: Optical elements used in each wavenumber range. Range c m - 1 Source Beam splitter Detector Polarizer Optical filter 20-400 200-650 400-1200 850-5000 2000-10000 Hg arc Hg arc Globar Globax Tungsten Mylar 3.5 nm Mylar Ge/KBr Ge/KBr S i /CaF 2 4 K Bolometer 4 K Bolometer 77 K M C T 77 K M C T 77 K InSb IGP 223(2) IGP 225 IGP 225 IGP 228(2) IGP 228(2) Polyethylene Polyethylene None None None therefore gives twice the change in path difference that is obtained with a conventional Michelson interferometer. This has the advantage that to obtain a given resolution, only half of the usual mirror movement is required. Successive increments of path difference are determined by fringe counting using a Helium-Neon laser reference interferometer; a white light source is used in addition to determine the mirror position corresponding to the centerburst, or zero-path difference. 2.2.1 Optical Parameters Some of the optical components used in the experiments are listed in Table 2.1. The near-infrared measurements in the 2000-10000 c m - 1 region were made using a Tung-sten/Halogen/Quartz source without any optical filters and a S i /CaF 2 beamsplitter. The beam was polarized with two IGP 228 polarizers1 which have an effective range of 850-10000 c m - 1 . An Infrared Associates2 InSb photovoltaic detector cooled to 77 K in a DMSL-8-1M model dewar with a sapphire window was used in this region. The active area of the detector is w 1 mm 2 . At its operating temperature the responsivity is 2.79 A / W and the specific detectivity D* at 1 kHz is 2.24 x 10 u cmHz*/W. The mid-infrared measurements in the 400-5000 c m - 1 region were made using a ] IGP polarizers are manufactured by Cambridge Consultants Ltd, Cambridge, England. 'Infrared Associates Inc., 12A Jules Lane, New Brunswick, N.J., U.S.A. 08901. Chapter 2. Experimental Techniques 29 globar source without any optical filters and a Ge/K3r beamsplitter. The beam was polarized with either an IGP 225 or two IGP 228 polarizers (the IGP 225 polarizer has an effective range of 200-5000 cm - 1 , but it is only used in the 200-1200 cm"1 region where it has an efficiency of over 98%). An Infrared Associates HgCdTe (MCT) pho-toconductive detector in a DMSL-6-1M model dewar with a KRS-5 window was used in this region. The active area of the detector was « 1 mm2. At its operating tem-perature and a bias current of 50 mA the responsivity is 189 V / W and D" at 1 kHz is 4.43 x 109 cmHz^/W. Limited use has also been made of a room temperature deut-erated-triglycerine-sulfate (DTGS) pyroelectric detector with a KBr window, but with its characteristic low responsivity it has proven suitable only for powder measurements. The far infrared measurements in the 20-60 cm - 1 , 30-120 cm - 1 , 100-400 cm - 1 and 150-650 cm"1 regions were made using an IR-7L d.c. mecury arc lamp with a blue polyethylene film optical filter and 50 fim, 23 / im, 6 fim and 3.5 fim mylar beamsplitters respectively. The optical filter is intended primarily to block ultraviolet radiation which might otherwise damage the thin mylar beamsplitters. The beam was polarized with either two IGP 223 polarizers or an IGP 225 polarizer (the IGP 228 polarizers have an effective range of 10-400 cm - 1). A commercial Infrared Laboratories3 4 K composite bolometer of Gallium doped Germanium on a sapphire sheet with a thin bismuth film was used to detect the radiation. At its operating temperature of 4.2 K the bolometer has a responsivity of 8.75 x 103 V / W and a noise equivalent power of 4.8 x 10"13 W/Hz J at 80 Hz. 2.2.2 Signal Analysis For most of the experiments, a mirror velocity of 0.333 cm/sec is used. [The configuration of the Genzel interferometer results in an actual mirror velocity (or optical velocity) of 3 I n f a r e d Laboratories, I n c . , 1 8 0 8 E. 1 7 t h S t r e e t , T u c s o n , A r i z o n a , U . S . A . 8 5 7 1 0 . Chapter 2. Experimental Techniques 30 1.330 cm/sec]. A moving mirror results in the optical radiation measured at the detector being modulated at audio frequencies [72]: where / p is the modulation frequency in Hz, v is the optical velocity of the mirror (cm/sec) and v is the wave number in cm - 1 . This relation is also useful for determining where a feature due to mechanical noise will appear in the spectra. It is important to ensure that the velocity is such that the audio frequencies are high, so that the effects of 1// noise on the spectrum are minimized. The modulated signal from the detector passes through electronic high and low-pass filters, which are adjusted using the Bruker ATS88B4 data collection software to lie within the wavenumber range being sampled so as to avoid aliasing. The filtered signal is then amplified. The automatic gain settings have been used. A 16-bit ADC then converts the analog interferogram into a digital interferogram. Using the Bruker ATS88B data collection software and an ASPECT 3000 computer the interferograms from each scan are coadded and stored. The resulting interferogram, often highly asymmetric, is corrected for phase using the Mertz method [73] and apodized using a ramp function from the start of the data collection to a distance that is twice the distance from the start as the centerburst; for the remainder of the interferogram a Blackman-Harris three-term apodization function is used where I n e u, is the new interferogram intensity, l0u is the recorded interferogram intensity and 6 is the angle from 0 to 180 degrees from the end of the ramp to the end of data col-lection. This apodization function is a considerable improvement over the Happ-Genzel 4for more information on the adjustable parameters contained within the FT-IR application program ATS88B for the ASPECT 3000, see the Bruker instruction manual. fp = vv (2.1) Ine» = W0.42323 + 0.49755 cos(0) + 0.07922 cos (25)], (2.2) Chapter 2. Experimental Techniques 31 function that was previously used. The apodized interferogram is Fourier transformed and the result stored on hard disk. 2.3 Sample Preparation Samples of the protonated (TMTSF-/i 1 2) and deuterated (TMTSF-d 1 2) salts (TMTSF) 2-Re04 and (TMTSF) 2 BF 4 were prepared by electrocrystallization using a modified H-cell [74]. The reaction was typically carried out under a dry nitrogen atmosphere in anhy-drous dichloromethane (5 x 10 - 2 M in electrolyte, 4 x 10 - 3 M in TMTSF at the anode). To facilitate the growth of large single crystals, initial currents of < IfiA were used. Once a seed had been started the current was increased to w 2fiA [75]. The electrol-ysis was discontinued at w 50% conversion. The R 4 N X electrolytes were purified by recrystallization prior to use. Initially, the TMTSF was prepared in our laboratory, but commercially available 5 material is now routinely used. The TMTSF-di 2 was prepared from deuterated 3-methanesulphonyl-2-butanone (from 2,3-butadione-d6) using a mod-ification of the literature procedure [76] by Gordon Bates of the Chemistry Department at the University of Brisith Columbia. Deuterium-incorporation levels in the intermedi-ate products were estimated by proton magnetic resonance and/or high-resolution mass spectrometry. The deuterium content of the 4,5-dimethyl-l, 3-diselenole-2-selone, the immediate precursor to TMTSF-di 2 , was estimated from an analysis of the C4 m/z mass spectral cluster. The level of deuterium incorporation of the TMTSF-di 2 (w 85%) is assumed to be that of the 2-selone precursor. This estimate has also been verified by a comparison of the relative integrated strengths of the C D 3 / C H 3 vibrational clusters observed in the TMTSF powder spectra. The crystals were washed in anhydrous dichloromethane and stored in a dessicator in 8 S t r e m C h e m i c a l s , I n c . , 7 M u l l i k e n W a y , D e x t e r I n d u s t r i a l P a r k , P . O . B o x 108 N e w b u r y p o r t , M A , 0 1 9 5 0 U . S . A . Chapter 2. Experimental Techniques 32 a atmosphere of nitrogen. 2.4 Reflectivity Measurements Optical reflectivity measurements are important because they can be used to determine any of the response functions commonly used in electromagnetic theory; such as the complex index of refraction (n = n + ik) or the dielectric constant (e = e' + it"). The complex reflectivity of a material is defined as where R is the power reflectivity and 6 is the phase difference between the incident and reflected waves. For large samples with high-quality surfaces a number of techniques exist where R and 6 may be measured directly, such as ellipsometry, dispersive reflection spectroscopy and asymmetric interferometry. If the sample is small, then these techniques cannot be used and only the power reflectivity may be measured directly. In this case a Kramers-Kronig relation [77] can be used to calculate the phase in terms of the reflectivity. The Kramers-Kronig relation is an expression of the causality condition and can be applied to the real and imaginary parts of any causal response function (such as n and A; or c' or c"); however, a knowledge of the measured quantity over the entire spectral range is required. Another, equivalent expression employing Fourier transforms [78, 79] yields It has been assumed here that the reflectivity changes very little above a cutoff frequency f = y/Rei{ (2-3) (2.4) ( 2 . 5 ) Chapter 2. Experimental Techniques 33 [RQ = R(U>H)], allowing the integral to be truncated. This expression for the Kramers-Kronig relation can use the fast-Fourier Transform algorithm; it can be faster by several orders of magnitude than the standard method. For light normally incident on a reflecting surface surrounded by vacuum the relation h — 1 „. r = — (2.6) n -+• 1 leads to R - (i + ny + k* ( 2 7 ) and 2k 6 — arctan (2.8) n 2 + fc2-l The complex index of refraction may be obtained by inverting the above two equations: n = ]—A= (2-9) 1 + . R - 2 ^ 0 0 8 0 and 2 ^ s i ° * . (2.10) 1 + R-2y/Rcosd The complex dielectric function is another quantity which is commonly used to describe the response of a material to an electromagnetic field. The dielectric function is related to the optical constants by e = (n + ik)2. (2.11) Equating the real and imaginary parts gives c' and c" in terms of n and k (which are already known in terms of R and 0 from equations 2.9 and 2.10 respectively) gives c,_n* h2_ ( l - f l ) 2 - 4 W ( f l ) [l + R-2y/Rcos(0)]2 K * } j ^ W L . ( 2 . 1 3 ) Chapter 2. Experimental Techniques 34 The optical conductivity of the material can be related to the imaginary part of the dielectric function by (2-14) where v is in cm - 1 and the optical conductivity will have the units of (Pxm) - 1. 2.4.1 Reflectivity Module The reflectivity measurements were performed using a reflectivity module designed by J. E. Eldridge specifically for the Bruker IFS 113V that has recently been described elsewhere [80]. The prime consideration in the design of the system is the ability to change components and align the optics externally, while the spectrometer is under vacuum. This avoids venting the spectrometer to atmospheric pressure which can be followed by a waiting period of up to two hours after reevacuation before the instrument returns to thermal equilibrium. The optical arrangement of the reflectivity module is shown in Fig. 2.2. An extension of one of the sample chambers allows enough room for the optics while keeping the second sample chamber free for other experiments. The front plate of the extension is plexiglass so that the illuminated sample can be observed. Two plane mirrors and two toroidal mirrors are used. When a new sample is placed in the reflectivity module, the first toroidal mirror must be moved so that the focused image of the aperture is in the plane of the sample or reference (a piece of paper tucked in beside the mirror is useful for this). The second toroidal mirror must then be moved so as to keep the focus of the beam leaving the front sample chamber the same as that of the beam leaving the back sample chamber. This procedure ensures that the beam is focused at the detectors. The first toroidal mirror has an external adjustment to allow for a change in the sample height once the instrument is evacuated. This adjustment is only made once at the beginning of an experiment. The second toroidal mirror has external Chapter 2. Experimental Techniques 35 ! i N i J LJ L  1—x - Figure 2.2: The horizontal optical arrangement in the front sample chamber of the Bruker IFS 113v interferometer, leaving the back sample chamber free. A,B - reference mirror and sample; C - radiation shield; D - vacuum shroud; £ - rotatable polarizers; F - adjustable rectangular slit; G - plane mirrors; H - toroidal mirrors; I - chopper with motor; J - external rod to move chopper into beam and activate it with a microswitch; K - extension to sample chamber; L - plexiglass lid; M - illustration of translational degree of freedom; N - back of sample chamber (after Reference [80]). Chapter 2. Experimental Techniques 36 adjusters in order to optimize the signal from the sample or the reference mirror. This is a critical alignment since all of the detectors have fairly small detection elements. For those detectors requiring an a.c. signal, a chopper is moved into the beam where it is activated by a microswitch. A n adjustable rectangular aperture is placed at the first intermediate focus. This is important not only because it matches the sample shape better than the Bruker circular apertures, but also because it is located after the interferometer chamber. With the Bruker apertures, both the source radiation passing through the aperture as well as the radiation from the warm aperture itself are modulated. This makes it impossible to restrict the size of the focused beam. Evidence of this effect is the presence of an interferogram when the source is off, and the detector is cold. This is not the "inverse sample interferogram" that occurs in some rapid-scan machines with a cold detector and no beam chopper [81], since the Genzel interferometer eliminates that effect. It is merely the large room-temperature source before the beamsplitter. With the small rectangular source positioned as in Fig. 2.2, no interferogram is obtained with the source off. Furthermore, it is better to have the movable chopper positioned before the aperture rather than after it, when aligning the toroid, for much the same reason. The polarizers are placed at the second intermediate focus, and may be rotated through 90° using external adjustors. The absolute value of the reflectivity was obtained by comparison with the aluminum mirror and the published values of the aluminum reflectivity [82]. 2.4.2 B o l o m e t e r T r a n s f e r O p t i c s The bolometer is located on the top of the spectrometer. The transfer optics for this, con-sisting of two custom made aluminum toroidal mirrors, are shown in Fig. 2.3. Previously, Chapter 2. Experimental Techniques 37 a light pipe had been used to collect the far-infrared radiation, but this arrangement re-sulted in over 95% of the signal being rejected at the entrance to the light cone in the bolometer. The light pipe also required that the aperture in front of the light cone be large ( « 8 mm) in order to allow as much of the collected signal into the light cone as possible. This also allowed more room-temperature radiation to strike the bolometer, thus heating it and reducing its sensitivity. The use of a single toroidal mirror focusing system was an improvement, but this system had magnification of 2.5, and very poor off-axis focusing. In order to capture any reasonable amount of the focused radiation, the aperture to the light cone had to be the same size as it was in the case of the light pipe, resulting in the same problem of a lowered bolometer sensitivity. The two mirror focusing system was designed using a raytracing program to give a magnification of close to unity (1.14). The system was to take the beam from the in-termediate focus at the entrance of the detector chamber and focus the radiation at the entrance of the light cone in the bolometer using an intermediate focus. Initially, the focusing problem was solved exactly for aspheric surfaces, and then a toroidal surface was fit to the aspheric case. The parameters for a toroidal mirror consist of two radii of curvature, one for the vertical focusing (Ri), the other for the horizontal focusing (R2). The resulting approximation was then tested with the raytracing program and the parameters optimized to give the best compromise between on- and off-axis focusing. The parameters for the mirror that sits in the sample chamber of the Bruker (Mi) are Ri=13.5 cm and R2=6.7 cm. The magnification from the first stage is 0.75. The pa-rameters for M2 (whose position is shown in Fig. 2.3) are Ri=14.0 cm and R2=6.7 cm. The magnification from the second stage is 1.52, giving a total magnification of 1.14. The mirrors were turned out on a lathe in the Machine Shop at the University of British Columbia using a custom built mirror jig and tool holder. The mirrors were made from Chapter 2. Experimental Techniques 38 I F i g u r e 2.3: T h e v e r t i c a l o p t i c a l transfer system for the commercial bolometer. A -B r u k e r chamber following t h e sample chamber; B - B r u k e r detector chamber; C - Infrared Labs. Dewai; D - fixed t o r o i d a l mirror; E - adjustable toroidal mirror; F - bolometer filters at 290 K, 77 K and 4 K; G - 4 K light cone; H - bolometer element; I - dewar s u p p o r t / v i b r a t i o n isolation; J - B r u k e r lids (after Reference [80]). Chapter 2. Experimental Techniques 39 extruded Aluminum 6061-T6 alloy. The cuts were made perpendicular to the extruded face. Mirrors cut parallel to the extruded face resulted in a poorer surface. The mirrors were polished by hand using a fine jewelers rouge, and then finished with Brasso and Silvo pastes to yield a bright polish. The finished surfaces were very smooth and provided a good optical image with which to focus the system. The bolometer is equipped with three filters that allow only the far-infrared radiation to strike the detector. One filter is at room temperature, the other two are cooled. The room-temperature filter is wedged white polyethylene with a black polyethylene film melted on. TJiis blocks out the visible radiation. The cold filters at 77 K and « 4 K are wedged white polyethylene (n w 1.5) with 15 /xm diamond dust (n « 2.4) thermally embedded into both sides of the filter. The diamond-dust scattering filters were constructed using techniques referred to in the literature [83]. The scattering filters are low-pass filters that transmit only radiation below w 650 cm - 1 , thus blocking room-temperature and mid-infrared radiation that would otherwise heat the bolometer and lower its sensitivity. The cutoff of the filter is not sharp due to a distribution of the size of the diamond dust particles. The superior focusing properties of the two mirror system ensure that little signal will be rejected at the aperture, which may be made smaller (diameter « 4 mm) due to the lower magnification. When this is considered with the diamond-dust scattering filters, little room-temperature radiation is allowed to strike the bolometer and thus sensitivity of the bolometer is greatly increased. The bolometer is electrically isolated from the Bruker spectrometer. The signal from the bolometer goes through a Infrared Laboratories low-noise preamplifier before it reaches the Bruker amplifier board. The bolometer is microphonic, therefore it is also vibrationally isolated. With the bolometer in place, the other two cold detectors, the MCT and the InSb, can still be selected. Chapter 2. Experimental Techniques 40 2.4.3 Heli-tran Refrigerator The single crystal reflectivity was measured between 10 and 300 K using an Air Products, Advanced Products Department6 (APD) LT-3-110 Heli-tran liquid transfer refrigerator together with an Model APD-K cryogenic microprocessor temperature controller with dual sensors. The cooling power in the refrigerator is supplied by either liquid nitrogen or liquid helium, giving ultimate temperatures of w 77 K and « 5 K respectively. With a flow rate of 0.7 liquid litres per hour of helium the cooling power is 500 milliwatts at 4.2 K, 3 watts at 20 K and 7 watts at 50 K. The cryogenic liquid is transferred to the refrigerator continuously by an evacuated room temperature transfer siphon. Heat leak to the cryogenic flow stream within the transfer line is minimized through the use of an internal suspension system and the interception of incoming heat by the shield flow circuit which surrounds the central flow. The transfer line bayonet tube is placed into a pressurized dewar ( « 5 psi nominal), the other end of the bayonet tube is placed into the Heli-tran refrigerator. The flow rate is regulated by a needle valve at the tip of the refrigerator cold end bayonet. The flow is directed through a heat exchanger in the refrigerator which serves as the mount or "cold finger" for the sample holder. A typical Heli-tran system flow diagram is shown in Fig. 2.4. The temperature controller has one temperature sensor just below the cold finger, the other is at the bottom of the sample holder. The stability (automatic) is ±0.01 K from 2 K to 300 K. The temperature is controlled between 2 K and 300 K by adding heat from a small resistive heater wrapped around the neck of the cold finger. The temperature controller may be programmed to cool the crystals slowly (< 1 K/min) through the metal-insulator phase transitions. Since the machine takes at least an hour to stabilize after evacuation, and since there 6 A i r P r o d u c t s , A d v a n c e d P r o d u c t s D e p a r t m e n t , P . O . B o x 2 8 0 2 , A l l e n t o w n , P A 1 8 1 0 5 Chapter 2. Experimental Techniques 41 Figure 2.4: Typical LT-3-110 Heli-tran system flow diagram. The Heli-tran is shown in the configuration in which it is sold. The vacuum shroud has been modified for reflectivity experiments (shown in the next figure). Chapter 2. Experimental Techniques 42 SIDE VIEW CLAMPING RING PUMPING LINE V SLIDIN£\ HELI TRAN. k n I i i • n 1 ! I ! SUPPORTS I6T.RI 24TPI / SAMPLE CHAMBER LID VACUUM y\Ax SHROUD RADIATION SHIELD I t j l ; r L ! J i ' l 1 i i i I — V — 1 — ~1 j l l i l I \ I XPINS TEFLON RING WINDOW SAMPLE REFERENCE MIRROR Figure 2.5: A vertical section through the cryogenic arrangement for the Heli-tran which allows translational interchange of the sample and the reference mirror. Not shown in this diagram is the transfer siphon, one end of is inserted into the Heli-tran, the other into a pressurized vessel of liquid nitrogen or liquid helium (after Reference [80]). Chapter 2. Experimental Techniques 43 can be long-term drift, it is essential to run sample and reference scans alternately. In switching between the sample and the reference, it was decided that a small translation could be made more accurately than a 180° rotation. Accordingly, the cold finger is moved sideways under vacuum when switching. The arrangement used to accomplish this is shown in Fig. 2.5. The Heli-tran is inserted into a tube which is attached to a sliding plate. Double O-rings on the Helitran provide a vacuum seal. The top end is also clamped to supports which are bolted to the sliding plate. The plate has a regular O-ring and a teflon ring between it and the sample chamber lid, which is screwed to the sample chamber. (The Bruker sample chamber walls need to be drilled and tapped for these screws, otherwise the height of the sample chamber lid decreases when the instrument is evacuated). By means of a differential thread and a sensitive dial indicator, the sliding plate can be slowly and reproducibly translated sideways. Four pins in slots guide the movement. The dial indicator (not shown) measuring to 0.0001", reads the lateral position. The vacuum shroud surrounding the cold finger is made as small as possible in order to minimize outgassing and cryopumping by the cold sample and reference. (The presence of icing and surface contamination is easily detected when one tries to evacuate a large volume, such as a sample chamber, especially if it contains plastics and paint.) The single window on the front is either 2 mil polypropylene for wavenumbers below 600 cm - 1 , or Csl for wavenumbers above 200 cm - 1 . The polypropylene window is mounted on an aluminum holder using Versamid 140 resin epoxy. The vacuum shroud is evacuated by a Varian HS2 diffusion pump with a large liquid-nitrogen cold trap. A radiation shield made from thick, OFHC copper surrounds the sample inside the vacuum shroud. The optical port in the radiation shield is made as small as possible to avoid any excess heating from room-temperature radiation. An activated-charcoal getter is attached to the inside surface of the radiation shield. The radiation shield is cooled by conduction from the helium (or Chapter 2. Experimental Techniques 44 nitrogen) exhausting at the heat exchanger. 2.4.4 Reflectivity Sample Mounting Large single crystals approximately 4.0 x 1.0 x 0.05 mm (with the largest direction indi-cating the a axis) were attached using silver paint to an oxygen-free high conductivity (OFHC) copper mounting block, as is shown in Fig. 2.6. The top of the crystal was pressed firmly against the mounting block by a thin peice of paper to prevent any bend-ing or twisting of the crystal upon cooling, but the crystal remains free to contract. Next to the crystal a freshly prepared aluminum mirror is attached using a spring clip which is bolted to the mounting block using 00-90 stainless screws in order to avoid any movement. Initially, for the (TMTSF)2Re04 experiments, the mounting block was bolted to a polished OFHC copper plate which has an aperture behind the crystal in order to avoid radiation that does not strike the crystal being reflected back into the optical path. The disadvantage to this design was that it allowed room-temperature radiation from the vacuum shroud to strike the crystal from behind and heat it. The lowest temperature achieved using this configuration was ss 35 K, adequate for (TMTSF)2Re04 with a metal-insulator transition at 177 K, but inadequate for (TMTSF)jBF4 with a metal-insulator transition at 39 K. (The temperature of the crystal was determined by measuring the far-infrared reflectivity as it passed through the metal-insulator transition, an event which is marked by a sharp decrease in the far-infrared reflectivity. The transition temperature was then compared to the calibrated thermometers in the sample holder and the temperature of the crystal deduced). In later experiments, for both (TMTSF) 2-Re04 and (TMTSF) 2 BF 4 , the OFHC plate with an aperture was replaced by an OFHC copper block covered with microwave absorber, and the back aperture to the radiation shield was covered. The assembled holder was screwed into the heat exchanger of the Chapter 2. Experimental Techniques 4 5 OFHC Copper Mounting Plate OFHC Copper Backing Plate Figure 2.6: Mounting arrangement used for single-crystal reflectivity experiments. A -Organic crystal sample; B - aluminum reference mirror; C - spring clip; D - paper retainer; E - silver paint; F - OFHC copper mounting block; G - holes for 00-90 stainless bolts; H - microwave absorber. Heli-tran refrigerator using an indium washer and Air Products cry-con grease to ensure good thermal contact. The lowest temperature achieved using this configuration was « 20 - 25 K. 2.4.5 Correction for Diffraction In the past, studies have been performed on mosaics of crystals rather than on single crystals because of inadequate sensitivity. It is preferable to measure the reflectivity from single crystals rather than mosaics of crystals, because of misalignments between crystals that occur during mounting and also from diffraction effects due to the mosaics. Chapter 2. Experimental Techniques 46 The usual method for correcting these effects is to evaporate a layer of gold onto the mosaic after the measurements are complete, and use the reflectivity from the gold-coated sample to obtain a background. This technique works well at room temperature. As the mosaics are cooled, however, each crystal acts as a bimetallic strip, seriously reducing the reflectivity [61]. Some attempts have been made at tn situ evaporation using lead, but this still involves heating the crystal to « 80 K and then cooling it back down, again allowing the crystals to act as bimetallic strips. The combination of the Bruker interferometer with the optical design of the reflectivity module and bolometer transfer optics has allowed us to work with small single crystals in the far infrared. No other group is so far capable of such measurements. The reflectivity of a single crystal also displays diffraction effects below 50 cm - 1 or so in the far infrared, but they are much simpler than those of a mosaic. The effect is very distinct, and may be modelled very simply. The source is focused at the aperture in the reflectivity module; thus the aperture looks like a source with diffraction. The beam is focused on a small part of the toroidal mirror for small wavelengths (large v). When the wavelength is of the order of the size of the aperture, then diffraction effects become important. Instead of using- a small part of the mirror, the beam is spread out so that more intensity goes onto the edges, where the focusing properties of the mirror are poorer. Because the image is now smeared out, some of it will miss the crystal while the much larger reference mirror will reflect the image without clipping any of it. The effects of diffraction in the system may be considered without explicitly calculating the properties of the mirror. Instead, two surfaces, a sample and a reference, close to an aperture axe considered. Note that this introduces an "effective separation" between the aperture and surfaces that represents the actual separation in the reflectivity module. For convenience, the aperture is assumed to be circular. The diffraction pattern for a Chapter 2. Experimental Techniques 47 circular aperture for a fixed wavenumber is: 1(9) = Io 2 2Ji(2waPsin8) (2.15) 2nau sin 9 where 70 is the intensity of the radiation incident on the aperture, a is the radius of the aperture, 0 is the angle between the normal line from the center of the aperture to the surface and the position on the surface where the intensity is being measured, Ji(x) is the Bessel function (of the first kind) of order one and 1(9) is the intensity at an angle 9. In the system we are working with, both the crystal and the reference mirror have a finite width. The total intensity reflected is calculated by integrating the intensity over the angle subtended by the surface some "effective distance" Ho away, as shown in the inset of Fig. 2.7. The ratio of the reflected intensities from the crystal and the mirror will then be the ratio of the two intensity integrals, D(P) = [6l I(0)d6/ f h1(9)d0 (2.16) Jo Jo where 6\ and 02 are the half-angles from the sample and the reference respectively. The single crystal diffraction effects are most noticeable at room temperature, when the sample is metallic. In Fig. 2.7, the reflectivity (Roht) of (TMTSF) 2 BF 4 for E | | a at room temperature, shown by the open circles, is attenuated in the far-infrared. The diffraction effects may be modelled using the diffraction correction function [D(p)], shown by the dotted line. The correction function has also been scaled by a factor of 0.78, shown by the dashed line, to show more clearly how the function follows the reflectivity of the sample. The true reflectivity (iltnie) is /2true = i2ob.I> ( P ) - 1 . (2.17) and is shown by the solid line in Fig. 2.7. The diffraction correction function is reduced to one adjustable parameter, the effective separation 7£o, the other parameters are constant Chapter 2. Experimental Techniques 48 1.0 0.9 -0.8 0.7 -• r 0.6 . j-H % 0.5 PCj 0.4 0.3 0.2 0.1 0.0 o a o 9' O,' 0 ' ******* o I- d p Aperture Sample or Reference ( T M T S F ) 2 B F 4 T = 295 K E l l a * 0 20.0 40.0 60.0 80.0 100.0 Wave Number (cm"*1) Figure 2.7: Diffraction from a single crystal of (TMTSF) 2 BF 4 for E||a in its metallic state at room temperature in the far infrared and the correction for diffraction. The open circles represent the observed reflectivity . The dotted line is the curve generated by the model for the diffraction correction function when the aperture width is 1 mm, the crystal width is 1 mm, the mirror width is 5 mm and the effective distance from the aperture to the sample/reference is Ho = 4.0 mm. The dashed line is the correction function after a scaling factor of 0.78 has been applied. The solid line is the corrected reflectivity. The inset shows how the angles in the integrals are defined in terms of the aperture and sample widths and the effective distance. Chapter 2. Experimental Techniques 49 (the aperture width is 1 mm, the crystal width is 1 mm and the mirror width is 5 mm). In this example, Ho — 4.0 mm. The smaller the crystal, the larger the value of the effective separation Ho. Being able to correct for the diffraction of a single crystal in the far infrared allows vibrational features to be fit more accurately and is also very important when extrapo-lations are being performed for Kramers-Kronig analysis. 2.5 Powder Measurements 2.5.1 F i x e d Temperature Cryostat The powder and oriented crystal mosaic absorption measurements were performed in a fixed-temperature, custom-built brass cryostat. This cryostat operates at two tempera-tures, w 80 K (using liquid nitrogen) and « 8 K (for liquid helium). The cross-section of the cryostat is shown in Fig. 2.8. A liquid-nitrogen reservoir surrounds the inner liquid-helium. The bottom of the outer can is threaded to allow a radiation shield to be attached. The tail of the inner can ends in a threaded section to allow sample holders to be attached. A calibrated silicon diode located below the sample mount is used to moni-tor the temperature to using a Keithly 602 Electrometer. This cryostat does not contain a heater at the tip of the inner can (heat exchanger), which is why only two temperatures are accessible. The lowest temperature accessible using this cryostat is « 6 K. The tail of the cryostat sits at the focus of the spectrometer, and is surrounded by a vacuum shroud with optical ports. The tail of the cryostat may be rotated on a bearing support while cold to bring either the sample or the reference into the focus of the interferometer. The samples of (TMTSF^ReO^ were cooled slowly through the anion-ordering transition by transferring liquid-nitrogen through a needle valve to the inner can from T « 200 K to T w 160 K at a rate of « 2 K/min. The vacuum shroud Chapter 2. Experimental Techniques 50 J Figure 2.8: A vertical cross-section of the fixed temperature brass cryostat used for the powder and oriented crystal mosaic absorption measurements. A - Vacuum port to helium chamber; B - Helium transfer line; C - Rotating mount for inner can; D -Adjusting/support bolts; E - Felixble bellows; F - Liquid nitrogen fill tubes; G - Liquid helium; H - Liquid nitrogen; I - High vacuum port to sample chamber; J - Bruker lid; K -Vacuum shroud; L - Optical ports/polarizer mounts; M - OFHC copper liquid nitrogen cold shield; N - OFHC copper sample/reference holder. Chapter 2. Experimental Techniques 51 surrounding the sample mount is again made as small as possible in order to minimize outgassing and cryopumping by the cold sample and reference. A radiation shield made from thick, OFHC copper with optical ports surrounds the sample mount and is cooled from the liquid-nitrogen reservoir. An activated charcoal getter is attached to the inside surface of the radiation shield. The windows on the optical ports are either 2 mil polypropylene (measurements below 600 cm - 1) or Csl (for measurments above 200 cm - 1). The vacuum shroud is evacuated by a Varian HS2 diffusion pump with a large liquid-nitrogen cold trap. 2.5.2 Powder Sample Mounting For powder absorption measurements in the far infrared below 450 cm - 1 the crystals were ground up in Nujol mineral oil with a mortar and pestle and the resulting mull transferred to a wedged ( « 2°) piece of TPX (the Mitsui and Co., Ltd. trademark for methyl pentene polymer). Nujol has a number of absorptions in the mid infrared occuring at 1375 cm - 1 , 1465 cm - 1 , 2850 cm""1and 2930 cm - 1 , with the last two due to C-H vibrations being very strong. This makes it difficult to study some vibrational features, especially those of the methyl vibrations. Instead, for measurements above 500 cm - 1 , the crystals were ground together with Csl powder with a mortal and pestle and pressed into a thin sheet seated in a thick OFHC copper jacket using a custom built press. The lower limit for measurements on the Csl sheets were determined by the interference fringes, which usually became noticeable below 600 cm - 1 . The small particles in the powder had a typical diameter < 1 fim and the largest particles had a typical diameter of < 3 fim. The Nujol mulls and the Csl sheets were bolted onto an OFHC copper sample holder located at the base of liquid helium reservior. The sample holder in turn was screwed onto the shaft of the liquid helium reservior and good thermal contact was achieved using Chapter 2. Experimental Techniques an indium plug and a small amount of Air Products cry-con grease. Chapter 3 Experimental Results for (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 3.1 (TMTSF ) 2 R e 0 4 The crystal mosaic reflectivity of (TMTSF^ReC^ has been measured below the metal-insulator transition in the mid infrared by Jacobsen et al. [60] and by Bozio et al. [63], but only for the E||a polarization and only above 400 cm - 1 . No information exists for the E||b' polarization in this material. A powder absorption study has been performed by Bozio et al. [67] from « 250 cm - 1 to 2000 cm - 1 and some preliminary assignments made. No powder absorption studies have been made on the deuterated systems, which when used in conjunction with normal coordinate analysis calculations, serve as a powerful tool for making vibrational assignments. 3.1.1 Single-Crystal Reflectivity The single-crystal reflectivity of (TMTSF) 2Re0 4 for the E||a and E||b' polarizations at room temperature from 30 cm - 1 to 16000 cm - 1 is shown in Fig. 3.1. The reflectivity was smoothly continued to unity below « 40 c m - 1 for both polarizations, after applying the diffraction correction function to the spectra below « 100 cm - 1 , with %Q = 3.2 mm. The E||a reflectivity has been extended above « 8000 cm - 1 out to « 16000 c m - 1 by fitting the reflectivity above w 5000 cm - 1 to a Drude model, where complex dielectric function 53 Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 54 Table 3.1: The Drude parameters used in the high-frequency extrapolations of proton-ated and deuterated (TMTSF) 2Re04 for E||a at various temperatures0. (TMTSF) 2Re0 4 Polarization r (K) up (cm - 1) T (cm"1) E||a 295 9736 1274 85 10248 1028 25 10255 1111 E||b' 295 2611 4145 d\2 E a 25 10247 1100 ° For the E||a and E||b' Drude model fits to the reflectivity, Coo = 2.60 and 3.45 respec-tively. of the Drude model is £ = e o o - ( (3.1) u: is the frequency, up is the plasma frequency, T is a damping term and is the high-frequency core dielectric term. Table 3.1 lists the Drude parameters used in the high-frequency extrapolations for protonated and deuterated samples of (TMTSF) 2Re04 for E||a at various temperatures. At room temperature, the parameters used for the Drude extrapolations are up = 9736 cm - 1 , T = 1274 cm - 1 and £«, = 2.6. The E||b' data has been extended above « 4000 cm - 1 to w 8000 cm"1 by assuming a constant value for the reflectivity (Ro = 0.085). The reflectivity of (TMTSF) 2Re0 4 at « 85 K from 30 to 16000 cm"1 for E | | a and E||b' is shown in Fig. 3.2. The reflectivity below w 40 c m - 1 corrected for diffraction effects and continued smoothly to 0.54 for Ej|a and 0.16 for E||b' at zero frequency. The E||a data has been extended above « 8000 cm - 1 out to « 16000 c m - 1 using a Drude model with the parameters up = 10248 c m - 1 , T = 1028 cm"1 and = 2.6. The E||b' data has been extended above w 4000 c m - 1 to w 8000 cm - 1 by assuming a constant value for R (Ro = 0.08). Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 55 The reflectivity of (TMTSF) 2Re0 4 at « 25 K from 30 cm - 1 to 16000 cm - 1 for E||a and E||b' is shown in Fig. 3.3. The reflectivity below « 40 cm - 1 was corrected for diffraction effects and continued smoothly to 0.56 for E||a and 0.22 for E||b' at zero frequency. The E||a data has been extended above w 8000 cm - 1 out to w 16000 cm - 1 using a Drude model with the parameters u>p = 10255 cm - 1 , T — 1111 cm - 1 and £«> = 2.6. The E||b data has been extended above « 4000 cm -Ho sa 8000 cm - 1 by assuming a constant value for R (Ro = 0.08). The reflectivity of (TMTSF) 2Re0 4 at T « 25 K from 0 cm - 1 to 2500 cm - 1 for E||a and E||b' is shown in Fig. 3.4. This 3xpanded plot of the low-temperature reflectivity contains most of the vibrations and much of the fine structure above 1000 cm - 1 which was not visible in logarithmic plot of the reflectivity which was shown in Fig. 3.3. At room temperature and T « 25 K the spectra for E||a are similar to those in Ref. [60] which go down to 400 cm - 1 , but differ in magnitude by up to 10%. At room temperature, except for a large feature at « 1250 cm - 1 , the E||a reflectivity contains little in the way of vibrational features, and displays behavior characteristic of a Drude metal. The E||b' room temperature reflectivity has a number of vibrations in the mid in-frared, but little below 1000 cm - 1 and it also behaves like a Drude metal with a smaller plasma frequency and a much larger damping factor. The fits are satisfactory, but the large damping is not physical as it leads to a mean-free path for the carriers less than the interstack distance. Below the phase transition, the change in the E||a and E||b' reflectivity demonstrates the dramatic nature of the phase transition; many of the broad, weak vibrations have become very sharp and very strong. Furthermore, many of the vibrations are observed to split into two or more components; this splitting of some of the vibrations becomes more pronounced as the temperature is lowered. The most striking feature in the reflectivity Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 56 y i o 1 lrf io 3 io4 Wave Number (cm - 1 ) Figure 3.1: The reflectivity of (TMTSF)2Re04 at room temperature from 30 cm - 1 to 16000 cm - 1 for E||a and E||b' with a resolution of 2 cm - 1 . The lines below 40 cm"1 are continued smoothly to unity at zero frequency. The E||a data has been extended above w 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters ujp = 9736 cm - 1 , T = 1274 cm - 1 and = 2.6. The E||b' data has been extended above « 4000 cm - 1 to « 8000 cm - 1 by assuming a constant value for R (RQ = 0.085). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)3BF4 57 1.0 0.9 0.8 0.7 PC? ^0.6 -»-? > *43 0.5 o cp *g 0.4 0.3 0.2 0.1 0.0 1 1 1 1 1 1 1 1 1 — ~ 1 1 1 1 1 1 ( T M T S F ) 2 R e 0 4 T « 8 5 K -3*ltf lrf 103 Wave Number (cm"1) ltf Figure 3.2: The reflectivity of (TMTSF) 2Re0 4 at « 85 K from 30 cm - 1 to 16000 cm - 1 for E||a and E||b' with a resolution of 2 cm - 1 . The lines below 40 cm - 1 are continued smoothly to 0.54 for E||a and 0.16 for E||b' at zero frequency. The E||a data has been extended above « 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters u>p = 10248 cm - 1 , T = 1028 cm - 1 and = 2.6. The E||b' data has been extended above w 4000 cm - 1 to « 8000 cm - 1 by assuming a constant value for R (Ro = 0.08). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 58 1.0 0.9 0.8 0.7 M i l l I I I I I I I I I I T ( T M T S F ) 2 R e O , T « 25 K l ( f 10° Wave Number (cm-1) Figure 3.3: The reflectivity of (TMTSF) 2Re0 4 at « 20 K from 30 cm - 1 to 16000 cm - 1 for E||a and E||b' with a resolution of 2 cm - 1 . The lines below 40 cm"1 are continued smoothly to 0.56 for E||a and 0.22 for E||b' at zero frequency. The E||a data has been extended above w 8000 cm - 1 out to ~ 16000 cm - 1 using a Drude model with the parameters wp = 10255 cm - 1 , T — 1111 cm - 1 and = 2.6. The E||b' data has been extended above ~ 4000 cm - 1 to « 8000 cm - 1 by assuming a constant value for R (Ro = 0.095). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 59 1.0 0.9 0.8 0.7 0.0 ( T M T S F ) 2 R e 0 4 T « 25 K \ 0.0 500.0 1000.0 1500.0 2000.0 2500.0 Wave Number (cm"1) Figure 3.4: The reflectivity of (TMTSF) 2Re0 4 at T « 25 K from 0 cm"1 to 2500 cm - 1 for E)|a and E||b' with a resolution of 2 cm - 1 . This portion of the reflectivity contains all of the vibrational features (except for the symmetric and antisymmetric methyl vibrations) and much of the fine structure above 1000 cm - 1 which was not visible in the logarithmic plot. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 60 is the structure between 1000 cm - 1 and 2000 cm - 1 . The reflectivity is observed to drop sharply from « 80% down to « 10% at « 1500 cm - 1 , and then rise back up to « 90% with numerous features superimposed in this region. In the T « 25 K E||b' reflectivity the vibrations become sharper, and they are also observed to split. At low frequencies the vibrations appear as Fano-like antiresonances in the reflectivity which is very unusual. The low-frequency external phonons are numerous for E||b', and create very complicated structure below as 200 cm - 1 . Fig. 3.4 shows the expanded region of the reflectivity from 0 cm - 1 to 2500 cm - 1 , which allows some of the fine structure (difficult to distinguish in the logarithmic plot of the reflectivity) to be observed. Figs. 3.5 and 3.6 are high-resolution (0.2 cm - 1) far-infrared studies of the 200 — 350 cm - 1 and 30 — 80 cm - 1 regions, showing the splitting of an internal-molecular vibration of the TMTSF molecule and an external phonon respectively. The temperature dependence from T « 25 K to T w 180 K of the external phonon for E||a polarization are shown in Fig. 3.7. Above the transition temperature the external mode cannot be distinguished, but below TMI a single mode becomes visible quickly and proceeds to split into seven components, four of which are strong and three are weaker. In order to quantify the oscillator strengths, the cluster was fitted to a set of seven Lorentzian oscillators. The dielectric function for a set of N Lorentzian oscillators is written as N n2-e » = eoo + £ r i ' , (3.2) % (^2 - u]) + »W7J where cjj, jj, fij and £«> are the frequency, damping, oscillator strength and core dielectric terms respectively. The results of the fit are given in Table 3.2. The phonons in the E||b' polarization are not observed to split, suggesting that there is no relation to the features seen in the E||a polarization. The oscillator strength and the side bands clearly grow quickly below the phase transition, and therefore the vibrations must be coupling to the Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 61 0.9 -0.2 - Ella o.i -o.o I ^ 1 ' 1— ' 200.0 250.0 300.0 350.0 Wave Number (cm - 1) Figure 3.5: The far-infrared high-resolution reflectivity of (TMTSF) 2Re0 4 at T w 25 K from 200 cm"1 to 350 cm - 1 for E||a (resolution 0.2 cm - 1). The fundamental appears to be at « 275 c m - 1 and is split into at least three and possibly as many as seven components. The feature at 219 cm - 1 is is not associated with this structure. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 6 2 30.0 40.0 50.0 60.0 70.0 80.0 Wave Number (cm"1) Figure 3.6: The fax-infrared high-resolution reflectivity of (TMTSF) 2Re0 4 at T « 25 K from 30 cm"1 to 80 cm"1 for E| |a and E||b' (resolution 0.2 cm - 1). In this region, the main feature is a cluster of seven lines (each indicated by an arrow) due to an external phonon centered at « 50 cm - 1 . The phonons in the E||b' spectra axe not related to the single phonon observed in the E||a polarization. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 1.0 63 40 50 60 WAVE NUMBER (cm -1) Figure 3.7: The temperature dependence of the phonons at w 50 cm - 1 in ( T M T S F ) 2 -Re04 for E||a from 30 — 80 cm - 1 . The bottom left-hand scale is the scale for the 25 K spectrum, while the top is for the 180 K spectrum. The sidebands of the phonon grow quickly below T\fj. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 64 Table 3.2: Parameters obtained from a fit of Lorentzian oscillators to the E||a reflectivity data in Fig. 3.6, £«> = 33.15. Polarization u> (cm - 1) 7 (cm 1) Q (cm"1) E||a 60.07 0.50 23.84 56.41 0.52 19.66 53.63 0.34 45.09 49.51 0.47 40.27 48.17 0.36 90.43 43.38 0.47 72.16 39.82 0.22 20.74 lattice distortion, and hence to the CDW, caused by the anion ordering. The reflectivity of (TMTSF-di 2) 2Re0 4 at T « 25 K from 30 to 16000 cm"1 for E||a and E||b' is shown in Fig. 3.8. The deuterated crystals tended to be smaller than the protonated samples, often resulting in some of the beam missing the crystal and thus giving low values for the reflectivity. In these cases a uniform scaling factor ( « 1.17) has been applied to the reflectivity, using the reflectivity of protonated (TMTSF) 2Re0 4 in the mid to near infrared as a reference point. The smaller crystals lowered the signal--to-noise ratio and increased the diffraction effects somewhat in the far infrared. The incomplete level of deuterium incorporation has the effect of broadening the vibrations and lowering their intensity. The reflectivity in Fig. 3.8 below « 30 cm - 1 was corrected for diffraction effects using TZo — 7.5 mm (the larger value indicates that the diffraction effects were more pronounced in this sample) and then continued smoothly to 0.55 for E||a and 0.2 for E||b' at zero frequency. The E||a data has been extended above « 8000 c m - 1 out to w 16000 cm - 1 using a Drude model with the parameters u»p = 10247 cm - 1 , T = 1100 cm - 1 and €<» = 2.6. The E||b' data has been extended above ~ 4000 cm - 1 to « 8000 cm - 1 by assuming a constant value for R (Ro — 0.1). The reflectivity for (TMTSF-d 1 2 ) 2 Re0 4 at T « 25 K for E||a and E||b' from 0 cm"1 Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 65 1.0 0.9 0.8 i ~ i i i 11 i i i I I 11 r ( T M T S F - d 1 2 ) 2 R e 0 4 I i I i i 1 1 1 1 T « 2 5 K o.o i ' M I I l l I l l l l l l J 1 3 * l ( j l ( f 1 0 3 W a v e N u m b e r ( c m - 1 ) Figure 3.8: The reflectivity of (TMTSF-di 2) 2Re0 4 at « 20 K from 30 cm - 1 to 16000 cm - 1 for E||a and E||b' with a resolution of 2 cm - 1 . The lines below w 40 cm - 1 are continued smoothly to 0.56 for E||a and 0.2 for E||b' at zero frequency. The E||a data has been extended above « 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters u>p = 10255 cm - 1 , T = 1111 cm - 1 and €<» = 2.6. The E||b' data has been extended above « 4000 cm"1 to « 8000 cm - 1 by assuming a constant value for R (Ro = 0.095). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 66 1.0 0.9 0.8 1 I 1 I 1 0.3 0.2 0.1 0.0 ( T M T S F - d 1 2 ) 2 R e 0 4 T « 25 K 0.0 500.0 1000.0 1500.0 2000.0 2500.0 Wave Number (cm - 1) Figure 3.9: The reflectivity of (TMTSF-^12)2Re04 at T « 25 K from 0 c m - 1 to 2500 c m - 1 for E||a and E||b' with a resolution of 2 cm - 1 . This portion of the reflectivity contains many of the vibrational features and much of the fine structure above 1000 c m - 1 which was not visible in the logarithmic plot. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 67 to 2500 cm - 1 is shown in Fig. 3.9. This expanded portion of the reflectivity allows some of the fine structure (difficult to distinguish in the logarithmic plot of the reflectivity) to be observed. Some of the vibrational features in the E||a reflectivity have been shifted. In par-ticular, much of the fine structure in the 1300 — 1500 cm - 1 range has shifted down to the 1000 — 1200 cm - 1 region. The large antiresonances between 1000 cm - 1 and 2000 cm - 1 have not been altered except for some fine structure which has been shifted down. This region of the reflectivity resembles the protonated spectra, suggesting that the phe-nomenon responsible for this structure is not affected by the incorporation of deuterium into the TMTSF molecule. The vibrations for E||b' are broader and weaker than in the protonated reflectivity, but the behavior of the internal and external phonons below « 500 cm - 1 is similar. 3.1.2 Optical Conductivity The Kramers-Kronig optical conductivities have been calculated (using equations 2.5, 2.13 and 2.14) for (TMTSF) 2Re0 4 for E||a at room temperature, 85 K and 25 K from the extrapolated reflectivities in Figs. 3.1, 3.2 and 3.3 respectively. Fig. 3.10 shows the conductivity of (TMTSF) 2Re0 4 for E||a at room temperature and T w 85 K from 0 cm - 1 to 2500 cm - 1 and Fig. 3.11 shows the conductivity of (TMTSF) 2Re0 4 for E||a at T « 25 K from 0 cm - 1 to 2500 cm"1. The room-temperature conductivity is similar to other (TMTSF) 2 X salts and dis-plays a broad peak at finite frequency ( « 1000 cm - 1) which must be due to intraband transitions. The weak dimerization of the stack creates a small gap, but it is below the Fermi surface and should not contribute to the electronic transport properties. A large antiresonance at w 1380 cm - 1 stands out clearly from the electronic background, but the remaining features are very weak. The extrapolated d.c. conductivity of w 300 (Jlcm) -1 Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 68 is in good agreement with measured values [56]. The optical conductivity at T « 85 K has changed dramatically from its room-temperature counterpart. The conductivity has been suppressed below « 1200 c m - 1 , indicating the presence of a semiconducting energy gap, 2 A 0 - Many of the features that were weakly activated or not seen at all in the room temperature conductivity have become strongly activated and have split into two or more components. The conductivity at T « 25 K is similar to that at T « 85 K , but the conductivity above the gap is much larger and the splitting has become much more pronounced. The splitting occurs due to the zone folding that takes place below the metal-insulator transition, depending on the dispersion of the phonon branch through the Brillouin zone. The conductivity is suppressed below « 1200 c m - 1 , but the strong features in the 1200 — 1500 c m - 1 range, which appear to have the characteristics of both a resonance and an antiresonance, makes the determination of the size of the semiconducting energy gap difficult. [Within the context of the phase-phonon model (discussed in Chapter 5 and Appendix B) vibrations below the gap appear as resonances while vibrations above the gap appear as antiresonances. Thus, the observed features may be one or the other, depending on the size of the gap]. The conductivity of ( T M T S F ) 2 R e 0 4 for E | |b ' at room temperature and « 25 K is shown in Fig. 3.12. At room temperature the conductivity is level at « 25 (ficm)" 1 , except for the vibrations superimposed upon it. At w 25 K the vibrations that were active at room temperature have become sharper and are split into two or more components; a number of new modes which are overtones and combination bands are also observed. The conductivity is suppressed below w 1700 c m - 1 , above which it rises slowly to reach a maximum at w 2130 c m - 1 , suggesting an energy gap 2Ab> in the 1700 — 2100 c m - 1 range. The gap for E||b', however, should be the same as the gap for E||a. The size of the gap for E||b' indicates that the gap for E||a is 2A « 1700 c m - 1 , and that the structure Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 69 5000.0 4000.0 3000.0 I a b 2000.0 fl O ^ 1000.0 h 0.0 (TMTSF ) 2 Re0 4 E| |a T = 295 K .1 A. 1 0.0 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) Figure 3.10: The Kramers-Kronig optical conductivity of (TMTSF)2Re04 at room temperature (dashed line) and T « 85 K (solid line) for E||a from 0 cm - 1 to 2500 cm - 1 . At T « 85 K the asymmetry in the conductivity is already clearly developed and several of the resonances appear to have been split into several components. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 70 5000.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm-1) Figure 3.11: The Kramers-Kronig optical conductivity of (TMTSF)2Pve04 at T « 25 K for E||a from 0 cm - 1 to 2500 cm - 1 . The splittings are much sharper and the asymmetry and the magnitude of the conductivity are much stronger than at T « 85 K. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 71 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm-1) Figure 3.12: The Kramers-Kronig optical conductivity of (TMTSF) 2Re0 4 at T w 25 K for E||b' from 0 cm"1 to 2500 cm"1. At low temperature, the vibrations have split into doublets (mostly). Below « 500 cm - 1 the vibrations do not have the characteristic resonance behavior. The overall conductivity is low from 0 cm"1 to « 1800 cm"1, where it is seen to slowly increase until it reaches a maximum at « 2130 cm"1. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 72 5000.0 ,—,4000.0 h i—i I s o £L 3000.0 b •5 2000.0 h o S3 O ^ 1000.0 0.0 (TMTSF-d i 2 ) 2 R e 0 4 E||a I T « 25 K A . 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm-1) Figure 3.13: The Kraxners-Kronig optical conductivity of (TMTSF-di2)2Re04 at T ss 25 K for E||a from 0 cm - 1 to 2500 cm - 1 . The conductivity is very similar to that of (TMTSF)2rte04 at T w 25 K, but the splittings are weaker and broader than in the protonated case due to the incomplete level of deuterium incorportation. The strong features in the 1200 — 1500 cm - 1 are unshifted, while the fine structure present in the protonated spectra has been shifted clearly showing that the observed structure is due to just two strong features. The strong asymmetry in the conducivity has not been altered. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 73 100.0 80.0 o ^60 .0 • i—( > *43 40.0 cj O ^ 20.0 0.0 ( T M T S F - d 1 2 ) 2 R e 0 4 E l l b ' T « 2 5 K 0.0 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) Figure 3.14: The Kramers-Kronig optical conductivity of (TMTSF-di2)2Re04 at T « 25 K for E||b' from 0 c m - 1 to 2500 cm - 1 . The vibrations are weaker and broader than in the protonated case due to the incomplete level of deuterium incorporation. The spectra is much noisier below « 500 c m - 1 than the protonated case, but the same an-tiresonance behavior of the vibrations is observed. The overall conductivity is low from 0 cm - 1 to « 1800 cm"1, where it is seen to rise faster than in the protonated case until it reaches a maximum at « 2160 cm - 1 . Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 74 between 1200 and 1400 cm" 1 is acting as a resonance. The shape of the conductivity peak in the E | | a polarization at « 1800 c m - 1 is basically that of the combined density of states for a ID band. The Kramers-Kronig optical conductivity of ( T M T S F - d i 2 ) 2 R e 0 4 at T » 25 K has been calculated from the extrapolated reflectivities in Fig. 3.8 and is shown in Figs. 3.13 and 3.14 for E | | a and E | | b ' respectively. The large-scale features of the conductivity are similar to those of the protonated system, but many of the vibrations have been shifted and the incomplete level of deuterium incorporation has broadened the vibrations. The structure between 1200 c m - 1 and 1600 c m - 1 , is unchanged except for a decreased magnitude, possibly due to some new structure superimposed on top of it. The large-scale features in the E | |b ' conductivity are also very similar to that of the protonated system. The vibrational features have been shifted down and broadened somewhat. The conductivity remains low until « 1800 c m - 1 , above which it rises to reach a maximum at ss 2130 c m - 1 . This was also observed in the protonated system, hence the estimate for the gap in the deuterated system is the same as for the protonated system. 3.1.3 Powder Absorption The powder absorption spectra of protonated and deuterated (TMTSF) 2 Re04 have been measured from 200 c m - 1 to 2000 c m - 1 at room temperature and « 80 K and ~ 6 K . The powder absorption spectra for protonated (TMTSF) 2 Re04 in Nujol on Csl at room temperature and T « 80 K from 200 c m - 1 to 600 c m - 1 is shown in Fig. 3.15. The features at room temperature are broad and weak when compared to those at low temperature. The low temperature absorption peaks are very sharp and in several cases have split into two or more components; the low temperature peaks have also shifted up several wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 75 j , i . i 200.0 300.0 400.0 500.0 600.0 W a v e N u m b e r ( c m - 1 ) Figure 3.15: Powder absorption coefficient [log10(io/^)> in arbitrary units] of protonated (TMTSF) 2Re0 4 powder in Nujol on Csl at room temperature and « 80 K from 200 c m - 1 to 600 cm - 1 . The positions of the peaks in the absorption spectra are indicated by the arrows. The curves have the same scale and have been displaced for clarity. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 76 i , i , i 2 0 0 . 0 300 .0 4 0 0 . 0 500 .0 600 .0 W a v e N u m b e r ( c m - 1 ) Figure 3.16: Powder absorption coefficient [log10(/o/^)> in arbitrary units] of protonated and deuterated (TMTSF) 2Re0 4 powder in Nujol on Csl at w 80 K from 200 cm - 1 to 600 cm - 1 . The positions of the peaks in the absorption spectra are indicated by the arrows. The curves have the same scale and have been displaced for clarity. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 77 2 ( T M T S F ) 2 R e 0 4 T « 6 K i i i i i 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) Figure 3.17: Powder absorption coefficient [log10(70/7), in arbitrary units] of protonated and deuterated (TMTSF) 2Re0 4 powder in a Csl pellet at « 6 K from 500 cm - 1 to 2000 cm - 1 . The positions of the peaks in the powder absorption spectra are indicated by the arrows. The curves have the same scale and have been displaced for clarity. Chapter 3. Experimental Results for (TMTSF)2Re04 and ( T M T S F ) 2 B F 4 78 The powder absorption spectra for (TMTSF) 2Re0 4 and (TMTSF-di 2) 2Re0 4 in Nujol on Csl at T m 80 K from 200 cm - 1 to 600 cm - 1 is shown in Fig. 3.16. With the exception of the quartet at « 320 cm - 1 associated with the ReO^ anion, and the two weak modes at 454 cm - 1 and 466 cm - 1 , the modes have been shifted down in the deuterated sample. In one case, a new strong mode has appeared in the deuterated sample at « 289 cm - 1 . The powder absorption spectra for (TMTSF) 2Re0 4 and (TMTSF-d 1 2 ) 2 Re0 4 in Csl pellets at T PS 6 K from 500 c m - 1 to 2000 cm - 1 is shown in Fig. 3.17. The results for the protonated powder spectra agree quite well with previous results [67]. The features in the protonated spectra are quite sharp, but those in the deuterated spectra have been broadened due to the incomplete level of deuterium incorporation. The most striking feature in this spectra is the structure from 1200 cm - 1 to 1600 cm - 1 ; this is due to the v4{ag) mode is strongly activated for E||a and causes the reflectivity to be lowered to approximately the same levels as those for E||b and E||c in this region. The deuterium shifts from the powder spectra in the far infrared are especially useful in making vibrational assignments because the E||b' conductivity is noisy,and does not show clearly where the vibrations occur. 3.2 (TMTSF)2BF4 There have been no previous optical measurements of either protonated or deuterated (TMTSF) 2 BF 4 . Some powder absorption results have been published [68], but that study only investigated u> < 100 cm - 1 . 3.2.1 Single-Crystal Reflectivity The single-crystal reflectivity of protonated and deuterated (TMTSF) 2 BF 4 has been measured above and below the metal-insulator transition (TMI = 39 K) for the E||a and Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 79 E||b' polarizations from « 30 cm - 1 to 8000 cm - 1 with a resolution of 2 cm - 1 . The reflectivity of (TMTSF) 2 BF 4 at room temperature from 30 to 16000 cm - 1 for E||a and E||b' is shown in Fig. 3.18. The reflectivity below « 40 cm - 1 was corrected for diffraction effects using TZo = 3.2 mm and continued smoothly to unity at zero frequency for both polarizations. The E||a data has been extended above « 8000 cm - 1 out to w 16000 cm - 1 using a Drude model with the parameters u>p = 9973 cm - 1 , T = 1288 cm - 1 and Coo = 2.6. The Drude parameters used in the high-frequency extrapolations for protonated and deuterated (TMTSF) 2 BF 4 at various temperatures is shown in Table 3.4. The E||b' data has been extended above «s 4000 ';m_1to « 8000 cm - 1 by assuming a constant value for R (Ro = 0.08). The reflectivity at of (TMTSF) 2 BF 4 at « 20 K from 30 to 16000 cm - 1 for E||a and E||b' is shown in Fig. 3.19. The reflectivity below « 40 cm - 1 corrected for diffraction effects and continued smoothly to 0.65 for E||a and 0.14 for E||b' at zero frequency. The E||a data has been extended above « 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters up = 10941 cm - 1 , T = 857 cm - 1 and = 2.6. The E||b data has been extended above « 4000 cm _ 1to « 8000 cm - 1 by assuming a constant value for R (Ro = 0.09). Fig. 3.20 shows the expanded region of the reflectivity from 0 cm - 1 to 2500 cm - 1 , which allows some of the fine structure (difficult to distinguish in the logarithmic plot of the reflectivity) to be observed. The E||a room temperature reflectivity for (TMTSF) 2 BF 4 is somewhat higher below w 1000 cm - 1 than it is for (TMTSF) 2Re0 4, but otherwise the two spectra are almost identical. The E||b' reflectivity is also similar to the that of (TMTSF) 2Re0 4, but the sharp peak observed at « 920 cm - 1 in (TMTSF) 2Re0 4 has shifted to « 1070 cm - 1 , suggesting that this feature is due to the anion. The reflectivity of (TMTSF) 2 BF 4 at T « 20 K for E||a is different than that of (TMTSF) 2Re0 4. The strong antiresonance in the reflectivity between 1000 cm - 1 and Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 80 Table 3.3: The Drude parameters used in the high-frequency extrapolations of proton-ated and deuterated (TMTSF)2BF4 for E | | a at various temperatures0. ( T M T S F ) 2 B F 4 Polarization T ( K ) Up ( cm - 1 ) T (cm" 1) hi2 E||a 295 9973 1288 50 10783 1103 20 * 10941 857 E||b' 295 2901 4005 50 1846 275 d\2 E||a 20 10855 767 ° For the E | | a and E | | b ' Drude model fits to the reflectivity, = 2.60 and 3.75 respec-tively. 2000 c m - 1 is as not strong in ( T M T S F ) 2 B F 4 , with a reflectivity minimum of « 50%. The reflectivity of ( T M T S F ) 2 B F 4 is also higher below w 1000 c m - 1 . The strong modes that were active in ( T M T S F ) 2 R e 0 4 are also active in ( T M T S F ) 2 B F 4 , but in some cases the splittings are different or not as clear. The reflectivity of ( T M T S F - d 1 2 ) 2 B F 4 at « 20 K from 30 to 16000 cm" 1 for E||a and E||b' is shown in Fig. 3.21. The reflectivity below ss 40 c m - 1 was corrected for diffraction and continued smoothly to 0.65 for E||a and 0.14 for E | | b ' at zero frequency. The E||a data has been extended above « 8000 c m - 1 out to » 16000 c m - 1 using a Drude model with the parameters u>p = 10941 c m - 1 , T = 857 c m - 1 and e M = 2.6. The E||b data has been extended above « 4000 cm _ 1 to w 8000 c m - 1 by assuming a constant value for R (Ro = 0.08). The reflectivity is shown for ( T M T S F - d 1 2 ) 2 B F 4 at T w 20 K from 0 cm" 1 to 2500 c m - 1 for E||a in Fig. 3.22. This expanded plot of the reflectivity shows much of the fine structure in the mid infrared which was lost in the logarithmic plot. The far-infrared reflectivity is noisy. The E||a reflectivity is shows that many of the vibrations in the mid infrared have been shifted down, leaving the two strong antiresonances between 1200 c m - 1 and 1500 cm" 1 unobscured. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 81 0.0 I I I 1 T i l ( T M T S F ) 2 B F T = 295 K 3*ICJ lrf io 3 W a v e N u m b e r ( c m - 1 ) ltf Figure 3.18: The reflectivity of (TMTSF) 2 BF 4 at room temperature from 30 cm - 1 to 16000 cm - 1 for Ej|a and E||b' with a resolution of 2 cm - 1 . The lines below w 40 cm"1 are continued smoothly to unity at zero frequency. The E||a data has been extended above « 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters u>„ = 10272 cm - 1 , T - 1263 cm - 1 and £«» = 2.6. The E||b' data has been extended above w 4000 cm - 1 to w 8000 cm - 1 by assuming a constant value for R (Ro — 0.085). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 82 1.0 0.9 0.8 0.7 >;o .e > t £ 0.5 o> 0.4 0.3 0.2 0.1 0.0 i r i i i i i i i I i i i ( T M T S F ) 2 B F i i i 1 1 1 T w 20 K E l l b ' J J . I I I M i l l J I S ' l O 1 l t f i o 3 W a v e N u m b e r ( c m - 1 ) lCf Figure 3.19: The reflectivity of (TMTSF) 2 BF 4 at « 20 K from 30 c m - 1 to 16000 cm - 1 for E||a and E||b with a resolution of 2 cm - 1 . The lines below 40 cm - 1 are continued smoothly to 0.65 for Ej |a and 0.14 for E||b' at zero frequency. The E | | a data has been extended above ~ 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters up = 10941 cm - 1 , T = 857 cm - 1 and £«, = 2.6. The E||b' data has been extended above « 4000 cm - 1 to « 8000 cm - 1 by assuming a constant value for R (Ro = 0.08). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 83 0.0 500.0 1000.0 1500.0 2000.0 2500.0 W a v e N u m b e r ( c m " 1 ) Figure 3.20: The reflectivity of (TMTSF) 2 BF 4 at T « 20 K from 0 c m - 1 to 2500 c m - 1 for E||a and E||b' with a resolution of 2 cm - 1 . This portion of the reflectivity contains many of the vibrational features and much of the fine structure above 1000 c m - 1 which was not visible in the logarithmic plot. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 84 0.1 -I I I I I I I I I I II I ( T M T S F - d 1 2 ) 2 B F 4 I I I I I I I I | T « 2 0 K 3*ltf l t f 103 W a v e N u m b e r ( c m - 1 ) 10* Figure 3.21: The reflectivity of (TMTSF-tf 1 2) 2BF 4 at T « 20 K from 30 cm"1 to 16000 cm - 1 for E||a and E|jb' with a resolution of 2 cm - 1 . The lines below w 40 cm - 1 are continued smoothly to 0.65 for E||a and 0.14 for E||b' at zero frequency. The E||a data has been extended above « 8000 cm - 1 out to « 16000 cm - 1 using a Drude model with the parameters up = 10855 cm - 1 , T = 767 cm - 1 and £«, = 2.6. The E||b' data has been extended above « 4000 cm - 1 to w 8000 cm - 1 by assuming a constant value for R (Ro = 0.085). Note the logarithmic scale for the wave numbers. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 85 0.2 -0.1 -. ( T M T S F - d 1 2 ) 2 B F 4 T « 2 0 K 0.0 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) 2500.0 Figure 3.22: The reflectivity of (TMTSF-d 1 2) 2BF4 at T w 20 K from 0 cm - 1 to 2500 cm - 1 for E||a and E||b' with a resolution of 2 cm - 1 . This portion of the reflectivity contains all of the vibrational features and much of the fine structure above 1000 cm - 1 which was not visible in the logarithmic plot. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 86 3.2.2 Optical Conductivity The Kramers-Kronig Optical conductivity of (TMTSF)2BF 4 at room temperature and T « 20 K is shown from 0 cm_ 1to 2500 cm - 1 for E||a in Fig. 3.23. The room temperature conductivity is very similar to that of (TMTSF)2Re04 but the shape of the optical conductivity at T « 20 K is far different in (TMTSF) 2 BF 4 than it is in (TMTSF) 2-Re0 4. The conductivity is suppressed until « 900 cm - 1 , at which point a sharp series of resonances are encountered and the conductivity rises rapidly until w 1100 cm - 1 , after which it decreases in a series of complicated resonances and antiresonances. The m:;t striking features in the conductivity, aside from its remarkable asymmetry (e.g., suppressed conductivity below w 1000 cm - 1 , large conductivity above) are the two large antiresonances present between 1200 cm - 1 and 1500 cm - 1 . The strength and position of these features suggests they are the same features which were seen as resonances in protonated and deuterated (TMTSF) 2Re0 4. The size of the semiconducting energy gap in (TMTSF)2BF 4 appears to be in the 900 - 1200 cm - 1 range. Once again, the complicated structure in the region of the energy gap makes its exact determination difficult. The optical conductivity of (TMTSF) 2 BF 4 at room temperature and f « 20 K is shown from 0 c m - 1 to 2500 cm - 1 for E||b' in Fig. 3.24. At room temperature the conductivity is level at « 35 (flcm) - 1, somewhat higher than it is for (TMTSF) 2Re0 4, and the strong feature seen at 920 cm - 1 in (TMTSF) 2Re0 4 has moved up to 1050 cm"1 indicating that this feature is due to the anion. At T « 20 K, the vibrations that were active at room temperature have become sharper and some have been split; a number of new modes are also observed. The conductivity is suppressed below « 1100 cm - 1 , after which it increases slowly until it reaches a maximum at « 1360 cm - 1 and then decreases slowly. This suggests a semiconducting energy gap in the 1100 — 1300 cm - 1 range, but Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 87 W a v e N u m b e r ( c m - 1 ) Figure 3.23: The Kramers-Kronig optical conductivity of (TMTSF) 2 BF 4 at room tem-perature (dashed line) and T « 20 K (solid line) for E||a from 0 cm - 1 to 2500 cm - 1 . At T « 20 K the asymmetry in the conductivity is already clearly developed and several of the resonances appear to have been split into several components. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 88 100.0 80.0 a 2 : 6o.o b >> -*^ > • 1—1 > *43 40.0 o rt rt O ^ 20.0 0.0 T = 295 K y , — ^ ^ T « 20 K ( T M T S F ) 2 B F 4 E l l b ' 0.0 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) Figure 3.24: The Kramers-Kronig optical conductivity of (TMTSF) 2 BF 4 at T » 20 K for EJjb' from 0 cm"1 to 2500 cm - 1 . At low temperature, the vibrations have split into doublets (mostly). Below « 500 cm - 1 the vibrations do not have the characteristic resonance behavior. The overall conductivity is low from 0 cm - 1 to w 1100 cm - 1 , where it is seen to increase until it reaches a maximum at « 1360 cm - 1 . Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 89 5000.0 4000.0 o 2^3000.0 b >> 2000.0 o rt rt o ^ 1000.0 0.0 i ( T M T S F - d 1 2 ) 2 B F 4 T » 2 0 K 0.0 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) Figure 3.25: The Kramers-Kronig optical conductivity of (TMTSF-<fi 2) 2BF 4 at T « 20 K for E | | a from 0 c m - 1 to 2500 cm"1. The conductivity is similar to that of (TMTSF)2BF4 at T « 20 K, but the sharp asymmetry is obscured by a number of vibrations in the region of the gap. The splittings are weaker and broader than in the protonated case due to the incomplete level of deuterium incorporation. The antireso-nances between 1200 cm - 1 and 1500 cm - 1 are unshifted, while the fine structure present in the protonated spectra has been shifted clearly showing that the observed structure is due to just two strong vibrations. Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 90 100.0 n 1 — i 1 i | ,—, 80.0 S 60.0 • r-t ;§ 40.0 o O ^ 20.0 0.0 ( T M T S F - d 1 2 ) 2 B F 4 E l l b ' T « 2 0 K iLArtu ill 0.0 500.0 1000.0 1500.0 2000.0 W a v e N u m b e r ( c m - 1 ) Figure 3.26: The Kraxners-Kronig optical conductivity of (TMTSF-d 1 2 ) 2 BF 4 at T « 25 K for E||b' from 0 cm - 1 to 2500 cm - 1 . The vibrations are weaker and broader than in the protonated case due to the incomplete level of deuterium incorporation. The spectra is much noiser below « 500 c m - 1 than the protonated case, but the same an-tiresonance behavior of the vibrations is observed. The overall conductivity is low from 0 cm - 1 to « 1100 cm - 1 , where it is seen to slowly faster than in the protonated case until it reaches a maximum at « 1350 cm - 1 . Chapter 3. Experimental Results for (TMTSF)2Re04 and (TMTSF)2BF4 91 the width of the conductivity maximum makes the exact determination of the energy gap difficult. The Kramers-Kronig optical conductivity of (TMTSF-<f 1 2 ) 2 BF 4 at T » 20 K has been calculated from the extrapolated reflectivities in Fig. 3.21 and is shown from 0 c m - 1 to 2500 c m - 1 in Figs. 3.25 and 3.26 for E||a and E||b' respectively. The large-scale features of the E | | a conductivity are similar to those of the protonated system, but the asymmetry of the conductivity is obscured by several vibrations which occur near the gap, but the conductivity is clearly suppressed below w 900 c m - 1 . Many of the vibrations have been shifted down in frequency and the incomplete level of deuterium incorporation has broadened many of the vibrations. The two strong antiresonances between 1200 c m - 1 and 1500 c m - 1 are virtually unchanged, indicating that these features are not affected by deuterium incorporation. This is the same behavior that was displayed by the strong resonances in the same region in protonated and deuterated ( T M T S F ) 2 R e 0 4 . The optical conductivity of ( T M T S F - ( f 1 2 ) 2 B F 4 at T w 20 K for E||b' displays the same shape of the electronic continuum as was observed in the protonated case, but many of the vibrations have been shifted and broadened. While some features show signs of splitting, it is difficult to distinguish because of the broadening due to incomplete deuterium incorporation. The conductivity remains low until « 1100 c m - 1 , above which it rises to reach a maximum at « 1350 c m - 1 . This was also observed in the protonated system, hence the estimate for the gap in the deuterated system is the same as for the protonated system. Unlike the E||a polarization, the vibrations that occur above the gap are resonances, indicating that the normally infrared active vibrations in the E||b' polarization do not couple with the conduction electrons. Chapter 4 Normal Coordinate Analysis of TMTSF(h 1 2 /d i 2 ) ° ' + 4.1 Introduction A normal coordinate analysis (NCA) yields the eigenvalues and the eigenvectors, that is the frequencies of the normal modes of vibration and the vibrational motions of all of the atoms respectively, of a molecule. The solutions are given in terms of a set of forces that define the resistance of each bond to stretch, torsion and all of the interbond angles and deformations; second order effects of these forces on one another are also recognized. The geometry of the molecule is obtained from structural studies of the molecule, while the force constants may be fixed initially, but then refined to perform a fit to the known frequencies. A normal mode of vibration is one in which each atom reaches its position of maximum displacement at the same time, and each atom passes through its equilibrium position at the same time. A normal coordinate analysis was performed on protonated and deuterated TMTSF 0 and T M T S F + to calculate the isotopic frequency shifts. Only the in-plane vibrations have been considered, except for the methyl groups, where the out-of-plane vibrations have been calculated. The normal coordinate calculations were based on a modified valence force field (MVFF) using Wilson's GF matrix method [84] with the programs developed by Furher et al [85] at the National Research Council of Canada. The programs were adapted by the author for an Amdahl 5860 computer. A complete NCA 92 Chapter 4. Normal Coordinate Analysis of TMTSFfaw/du) 93 of protonated T M T S F and its radical cation has been performed by Meneghetti et al. [86]. The information for the deuterated compound, however, does not exist. The notation introduced by Meneghetti et al. has been adopted for this work. 4.2 Group Theory-Group theory is a powerful tool when analysing the normal modes of vibration of a molecule. The symmetry that the molecule possesses is referred to as the molecular point group. By examining the effect of the symmetry operations in the molecular point group, a reducible representation for the molecule in Cartesian coordinates may be determined. The reducible representation may be decomposed into its irreducible representations of the symmetry types using standard group theory techniques discussed in some detail by Wilson et al. [84]. From the irreducible representation, the translational (3) and rota-tional (3) degrees of freedom of the molecule may be removed, leaving just the symmetry species of the molecular vibrations. Group theory may also be used to construct symmetry coordinates which simplify the problem of calculating the solution to the secular equation (as shown in Appendix The anions in ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 , R e O ; and B F J both have tetra-hedral point group symmetry (Td). The irreducible representation of the anions may be A) . written as r tot = «i + e + A + 3 / 2 (4.1) of these rrot — / l r, trans = h Chapter 4. Normal Coordinate Analysis of TMTSF (hl2/dx2f<+ 94 rvib = aa + e + 2/2 (4.2) where a is nondegenerate, e is doubly degenerate and / is triply degenerate. Only the vibrations of the anion with the symmetry f2 are expected to be infrared active. The TMTSF molecule has D2h symmetry. The D2h group consists of the identity element E, an inversion center t (in the center of the molecule), three twofold-rotation axes about each of the principal axes of the molecule (Chapter 1, Fig. 1.2), C 2 x , C 2 y and C2z), and three reflection planes (crxy, oxz and <ryz). The irreducible representation of TMTSF is r t o t = 12aB + 7blg + 8b2g + 12b3g + 7au + 126lu + 1262u + 8b3u (4.3) where all the symmetry species are nondegenerate. The subscript g denotes a gerade or symmetric vibration (Raman active) and u denotes a ungerade or antisymmetric vibration (infrared active). Of these r r ot = + b2g + b3g Ttrans = &lu + &2u + &3u Tvib = 12afl + 6&l5 + 7&2fl + ll& 3 a + 7au + ll&i u + l l6 2 u + 7&3u (4.4) There are therefore 29 infrared active modes and 36 Raman active modes for the TMTSF molecule (the au modes are not optically active). The eight symmetry species for the internal vibrations of a TMTSF molecule are shown in Fig. 4.1. The vibrations are not necessarily the normal modes of the molecule, merely ones which transform according to the symmetry species of the irreducible rep-resentations. The in-plane ag vibrations are symmetric under all symmetry operations in the group. The out-of-plane b\g modes are antisymmetric under C 2 y , C^*, 0XZ and <TyZ symmetry operations. The out-of-plane b2g modes are antisymmetric under C2z, Figure 4.1: The eight vibrational symmetry species for a TMTSF molecule with a D2/1 molecular point group. The arrows indicate the direction of the atomic displacement, + indicates a displacement out of the page and — a displacement into the page. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°'+ 96 C 2 l , crxy and <Jyz. The in-plane b3g mode is antisymmetric under C2z, C2y, oxy. The out-of-plane a u mode is infrared inactive and is antisymmetric under i, o~xy, oxz and ayz. The in-plane b\u vibration is antisymmetric under the i , C2V, C2x and axy operations. The in-plane b2u mode is antisymmetric under i, C2x, C2z and o~yz. The out-of-plane b2u mode is antisymmetric under i , 0>2y, Q2z and ayz. 4.3 Normal Coordinate Analysis An introduction to the theoretical aspects and the procedure used to carry out a normal coordinate analysis (NCA) is described in detail in Appendix A . The secular equation, whose solutions represent the normal modes of vibrations of a molecule, is introduced and described. The internal valence coordinates used to describe the configuration of the molecule and the symmetry valence coordinates (linear combinations of internal valence coordinates which transform like one or another of the irreducible representations of the molecular point group) described and listed for the T M T S F molecule. The two matrices introduced in the secular equation, the F matrix (the matrix of force constants which describes the interaction of an internal coordinate with itself or a neighbour during a bond stretch or an angle bend) and the G matrix (describing the geometric configuration of the molecule with respect to the internal coordinates involved in a bond stretch or angle bend) are described. The values used for the F matrix for the neutral and ionized T M T S F molecule are listed. A l l of these elements must be calculated in order perform an N C A . 4.3.1 Normal Coordinates A normal mode of vibration will produce a displacement in an internal coordinate. The internal coordinates are defined in such a way that many different normal modes will involve the same internal coordinate, thus the motion of an internal coordinate depends Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°* 97 on many different frequencies. A normal coordinate (J, is one which describes only one normal mode and therefore depends only on one frequency. Normal coordinates are a complicated mixture of internal coordinates, which are generally not known in advance. Normal coordinates are discussed in more detail in Appendix A . 4.4 Normal Modes of Vibration of TMTSF(hi 2 /d i2 ) 0 ' + The secular equation has been solved numerically using symmetry coordinates. From the previous discussion of the secular equation it is seen that there are 73 modes of vibration, however, the frequencies are not necessarily distinct; some of the roots of the secular equation may occur more than once and are said to be degenerate. These degeneracies have been ignored. The initial calculations were performed for protonated T M T S F 0 and TMTSF" 1" were compared with previous calculations [86] and an overall agreement of better than 0.1% was achieved. The vibrational frequencies for deuterated T M T S F were then calculated using the same set of force constants from the protonated molecule and using a G matrix that was constructed using the same equilibrium geometry as the protonated molecule, however, the hydrogen atoms have been replaced by deuterium. The results of the normal coordinate analysis for the in-plane ag and b3g vibrations for protonated and deuterated T M T S F 0 and TMTSF+ are shown in Tables 4.1 and 4.2 respectively. The results for the in-plane 6 l u and b2u vibrations for T M T S F 0 and TMTSF+ are shown in Tables 4.3 and 4.4, respectively. The out-of-plane au, big, b^g and 63,, vibrations for methyl groups are shown in Table 4.5. The methyl vibrations do not show any dependence on the formal charge on the molecule. The rest of the out-of-plane vibrations have not been calculated because the out-of-plane force constants are not well known. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°'+ 98 Table 4.1: The calculated in-plane ag and b3g normal modes of protonated and deuter-ated TMTSF 0 . TMTSF 0 N (cm"1) PED(%)° Vibration hn di2 8 hi2 di2 ag 2946 2200 -746 # 6 ( 1 0 0 ) # 6 ( 9 8 ) v2 2849 2053 -796 # 6 ( 9 9 ) # e ( 9 7 ) "3 1623 1612 -11 # 2 ( 8 2 ) # 2 ( 8 4 ) "4 1538 1537 -1 # i ( 8 8 ) # i ( 8 6 ) 1438 1037 -401 # 6 ( 9 1 ) # e ( 9 2 ) 1370 1066 -304 # s ( 6 ) , # 6 ( 4 7 ) , # 7 ( 5 9 ) # 5 ( 4 1 ) , # 6 ( 3 9 ) , # 7 ( 5 0 ) Fl6(-U) F 1 6 ( - 3 4 ) "7 1056 962 -95 # 4 ( H ) , # 5 ( 4 1 ) , # 7 ( 5 0 ) # 4 ( 2 4 ) , # 5 ( 3 5 ) , # 7 ( 1 9 ) "8 917 737 -179 # 5 ( 3 9 ) , ff7(39) tf5(6),#7(79) "9 448 425 -23 # 4 ( 7 4 ) # 4(70) "10 283 279 -6 # 3 ( 5 1 ) , # 5 ( 3 5 ) # 3 ( 6 7 ) , # 5 ( 1 2 ) " l l 256 226 -30 # 3 ( 1 9 ) , # 2 ( 1 2 ) , t f 5 ( 6 0 ) / < T 3(3),# 2(7),# 5(79) "12 133 131 -2 # 2 ( 3 8 ) # 2 ( 3 7 ) hg "55 2945 2200 -746 # 6 ( 1 0 0 ) # 6 ( 9 8 ) "56 2849 2052 -797 # 6 ( 9 9 ) # e ( 9 7 ) "57 1441 1036 -405 # e(91) # 6 ( 9 5 ) "58 1382 1053 -328 # 6 ( 4 5 ) , # 7 ( 5 7 ) , F 1 6 ( - 4 ) # 6 ( 4 5 ) , # 7 ( 5 7 ) , F 1 6 ( - 1 6 ) "59 1146 1135 -11 # 5 ( 7 5 ) , # 3 ( 2 9 ) # 5 ( 7 8 ) , # 3 ( 3 3 ) "60 1045 868 -178 # 5 ( 7 ) , # 7 ( 7 4 ) # 5 ( 2 4 ) , # 7 ( 6 3 ) " e i 958 958 0 # 3 ( 6 8 ) , # 4 ( 6 6 ) , F 1 3 ( - 2 7 ) # 3 ( 6 8 ) , # 4 ( 6 6 ) , F13(-27) "62 684 612 -71 # 4 ( 5 3 ) , # 5 ( 1 8 ) , # 7 ( 4 1 ) # 4 ( 4 4 ) , # 5 ( 2 3 ) , # 7 ( 2 8 ) "63 405 380 -25 # 4 ( 2 5 ) , # 3 ( 2 3 ) , # 5 ( 3 8 ) # 4 ( 3 0 ) , # 3 ( 2 2 ) , # 5 ( 2 8 ) "64 297 278 -19 # 4 ( 2 8 ) , # s ( 4 0 ) # 4 ( 1 9 ) , # 5(45) "65 176 171 - 6 # 3 ( 4 4 ) , # 4 ( 3 3 ) # 3 ( 4 1 ) , # 4 ( 3 1 ) 0 Potential energy distribution. Force constant and interaction force constant contribu-tions of greater than 20% and their corresponding values have been listed to show how the nature of the vibration is altered upon deuterium incorporation. Interaction force constants of less than 20% have also been selectively introduced to help characterize the nature of the vibration. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°>+ 99 Table 4.2: The calculated in-plane ag and b3g normal modes of protonated and deuter-ated TMTSF+. TMTSF+ (cm"1) PED(%)° Vibration hn d i 2 6 h 1 2 d i 2 ag vi 2946 2202 -744 tfe(lOO) K6(99) v2 2849 2053 -796 7f6(99) 7^(97) "3 1568 1553 -15 K2 (80) 7<2(86) V\ 1399 1399 0 #i(87) ^1(88) 1439 1038 -401 776(90) 776(93) v$ 1SS9 1066 -303 #s(7),#6(46),#7(58) 7iT5(43),776(38),777(49) F 1 6(-15) 7\ 6(-34) 1063 970 -93 7^(43), 777(47) 7^(24), 7^(32), 7/7(20) v8 914 737 -177 7f5(35),777(40) 7^5(6), 777(77) fg 452 429 -23 7^(71) 7^(67) Vio 312 296 -16 K3(22), 7/5(66) /r3(57),775(21) V\\ 278 253 -25 7^(45), 772(20),775(26) 7f3(10),772(10),775(67) V\2 137 134 -3 772(38) 772(38) hg "55 2946 2201 -745 7f6(100) 7^(98) "56 2849 2053 -796 #e(99) 7T6(97) V57 1441 1036 -405 776(92) 776(95) 1383 1055 -328 776(44),777(57),F16(-4) 776(44),7/7(58),F16(-21) V59 1170 1156 -14 7^(61), 773(24),775(28) 7^(67), 773(29),775(28) "60 1057 904 -153 7^(25), 7/7(63) 775(29),777(50) "61 998 998 0 7^3(69),774(61),F13(-25) 7i:3(69),774(61),F13(-25) 728 632 -96 7^(43), 775(43),777(14) 7^(35), 7/5(21),777(40) 428 397 -31 773(26),775(40) 773(24),775(32) "64 314 296 -18 7^(45), 775(26) 7^4(35), 775(31) "65 182 177 -5 7i:3(41),774(38) A:3(39),774(36) 0 Potential energy distribution. Force constant and interaction force constant contribu-tions of greater than 20% and their corresponding values have been listed to show how the nature of the vibration is altered upon deuterium incorporation. Interaction force constants of less than 20% have also been selectively introduced to help characterize the nature of the vibration. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/du)0'+ 100 Table 4.3: The calculated in-plane biu and b2u normal modes of protonated and deu-terated TMTSF 0 . TMTSF 0 (cm' 1) PED(%)° Vibration h n di2 6 h n di2 &lti "26 2945 2200 -745 tfe(lOO) # 6 (98) "27 2849 2053 -796 # 6 ( 9 9 ) #e(97) "28 1619 1606 -13 # 2 (87) # 2 ( 9 2 ) "29 1438 1037 -401 # 6 (91) #e(92) "30 1370 1066 -304 # 5 ( 6 ) , # 6 ( 4 7 ) , # 7 ( 5 9 ) # 5 ( 4 2 ) , # 6 ( 3 9 ) , # 7 ( 4 9 ) F 1 6 ( - 1 4 ) F 1 6 ( - 3 4 ) "31 1058 965 -93 # 4 ( 1 1 ) , # 5 ( 4 1 ) , # 7 ( 4 9 ) # 4 ( 2 4 ) , # 5 ( 3 3 ) , # 7 ( 1 9 ) "32 918 737 -181 # 5 ( 3 7 ) , # 7 ( 4 0 ) # 5 ( 6 ) , # 7 ( 7 9 ) "33 661 659 -2 # 3 ( 6 6 ) # 3 (67) "34 448 425 -23 # 4 (73) # 4 (70) "35 263 225 -39 # 5 ( 8 6 ) , F „ ( - 2 0 ) #5 ( 8 6 ) , . F n ( - l l ) "36 251 254 3 # 3 ( 2 6 ) , # 2 ( 4 2 ) # 3 ( 2 6 ) , # 2 ( 4 6 ) b2u I/44 2945 2200 -745 # 6 (100) # 6 ( 9 8 ) "45 2849 2052 -797 #e(99) #e(97) "46 1441 1036 -405 #e(91) # 6 (95) "47 1382 1053 -329 # 6 ( 4 5 ) , # 7 ( 5 7 ) , F 1 6 ( - 4 ) # 6 ( 4 5 ) , i / 7 ( 5 7 ) , F 1 6 ( - 1 6 ) "48 1146 1135 -12 # 5 ( 7 5 ) , # 3 ( 2 9 ) # 5 ( 7 9 ) , # 3 ( 3 3 ) "49 1045 868 -177 # 5 ( 7 ) , # 7 ( 7 4 ) # 5 ( 2 4 ) , # 7 ( 6 2 ) "50 750 746 -4 # 3 (106) # 3 (114) "51 679 609 -70 # 4 ( 5 1 ) , # 5 (41) , # 7 ( 8 ) # 4 ( 4 4 ) , # 5 ( 2 4 ) , # 7 ( 2 7 ) "52 404 379 -25 # 4 ( 2 6 ) , # 3 ( 2 4 ) , # 5 ( 3 6 ) . # 4 ( 3 0 ) , # 3 ( 2 3 ) , # 5 ( 2 6 ) ^ n ( - 2 7 ) F n ( - 2 5 ) "53 289 266 -23 # 4 ( 3 0 ) , # 5 (47) , F „ ( 2 5 ) # 4 ( 2 1 ) , #5(55), J F „ ( 2 3 ) "54 55 52 -3 # 4 ( 9 9 ) # 4 (98) 0 Potential energy distribution. Force constant and interaction force constant contribu-tions of greater than 20% and their corresponding values have been listed to show how the nature of the vibration is altered upon deuterium incorporation. Interaction force constants of less than 20% have also been selectively introduced to help characterize the nature of the vibration. Chapter 4. Normal Coordinate Analysis of TMTSF(hi2/d12)°>+ 101 Table 4.4: The calculated in-plane &i„ and b2u normal modes of protonated and deu-terated TMTSF+. TMTSF+ (cm"1) PED(%)° Vibration hi2 dn 6 hn di2 blu "26 2946 2202 -744 #6(100) 7<T6(98) "27 2849 2053 -796 7^(99) #6(97) "28 1565 1550 -15 7<T2(82) #2(88) "29 1438 1038 -400 #e(91) 776(93) "30 1369 1067 -302 7^(';),776(47),777(59) #5(43),776(38),777(48) F\&(-lh) F 1 6(-34) "31 1065 973 92 7t'4(12),7f5(42),7/7(46) #4(24), 7iT5(30), 777(21) "32 915 737 -178 7iT5(34),777(41) 7^(6), 777(77) "33 690 688 -2 #3(67) #3(67) "34 452 429 -23 #4(71) #4(67) "35 300 270 -30 772(l),775(89),Fn(-12) 77 2(34),77 5(33),F„(-11) "36 264 252 -12 7f3(28),772(49),775(5) 7^3(11), 772(16),775(55) b2u U44 2946 2201 -745 # 6 (100) 7iT6(98) "45 2849 2052 -797 tf6(99) 7^(97) "46 1441 1038 -403 776(92) 776(95) "47 1383 1055 -328 776(44),777(57),F16(-4) 776(44),777(58),F16(-21) "48 1170 1156 -14 7<T5(61),773(24),775(28) #5(67),773(30),775(29) "49 1056 905 -151 7^(26), 775(8),7/7(63) 7^(13), 775(29),777(50) "50 795 793 -2 7vT3(109) #3(112) "51 725 630 -95 7^(42), 775(44),777(14) #4(35),775(21),777(40) "52 425 395 -30 773(27),775(38) 773(25),775(29) "53 308 287 -21 7^(47), 775(31),F„(24) #4(37),775(38),FU(22) "54 55 53 -2 774(99) 774 (98) 0 Potential energy distribution. Force constant and interaction force constant contribu-tions of greater than 20% and their corresponding values have been listed to show how the nature of the vibration is altered upon deuterium incorporation. Interaction force constants of less than 20% have also been selectively introduced to help characterize the nature of the vibration. Chapter 4. Normal Coordinate Analysis of TMTSF(hu/d12)°'+ 102 Table 4.5: The calculated out-of-plane a u , big, b2g and b 3 u normal modes of the proton-ated and deuterated T M T S F 0 , + methyl groups. Vibration TMTSF+ (cm"1) h n dn 6 PED(%)° h i 2 dn "l7 2944 2195 -749 #6(100) #e(99) "18 1440 1034 -406 #e(92) #e(95) "19 1017 786 -231 #7(92) #7(95) b2g "20 2944 2195 -749 #6(100) #e(99) "21 1440 1034 -406 #e(92) #e(95) "22 1017 786 -23.1 #/(92) #7(95) au "23 2944 2195 -749 #6(100) #e(99) "24 1440 1034 -406 #e(92) #e(95) "25 1017 786 -231 #7(92) #7(95) hu "71 2944 2195 -749 #6(100) #6(99) "72 1440 1034 -406 #e(92) #6(95) "73 1017 786 -231 #7(92) #7(95) ° Potential energy distribution. Only contributions greater than 20% are included. The inspection of the potential energy distribution (PED) of the force constants gives an indication of the modes which will have large isotope shifts upon deuteration, and which vibrations will remain unchanged. Any vibration with a large potential energy contribution from a methyl group force constant will be expected to have a large isotope shift. A comparison of the potential energy distributions for the protonated and deuterated materials allows a determination of the new frequency of vibration and the size of the isotope shift. Force constant and interaction force constant contributions of greater than 20% and their corresponding values for the protonated or deuterated case have been listed in the tables to show how Chapter 4. Normal Coordinate Analysis of TMTSF(hi2/d12)°>+ 103 the nature of the vibration is altered upon deuterium incorporation.Interaction force constant contributions of less than 20% have also been selectively introduced to help characterize the nature of the vibration. Some vibrations, such as v4(ag), v61(o3g)t i ^ & i u ) and vso(b2u) have very small ( < 0.3%) or nonexistant deuterium shifts, while other modes involving the methyl groups have shifts of up to several hundred wave numbers. The PED also indicates that the nature of the low-energy ring-breathing vibrations, such as the "35(&iu) and f36(fciu) vibrations, are altered substantially as much of the energy goes from the ring distortion into the wagging of the methyl groups. Because of the importance of the ag modes in determining the optical properties of the solid (as discussed in Chapter 5), the nature of these vibrations will be illustrated in the next section. One item of interest is that while the deuterium shift is usually negative, this is not always the case. In TMTSF 0 , the v 3 6 ( b \ u ) mode has a positive deuterium shift. 4.4.1 Atomic Displacement Vectors of the (ag) Vibrations The totally symmetric ag modes are of particular interest because they are observed to become strongly activated optically in (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . It is useful to know the atomic displacement vectors for these vibrations in order to be able to discuss the nature of the vibration and which atoms are involved in the vibration. The atomic displacement vectors for the totally symmetric (ag) modes of T M T S F + are shown in Fig. 4.2. The displacement vectors have not been drawn to scale, they are simply intended to indicate which atoms are involved in the vibration (assigned through the potential energy distribution) and the direction of the motion (determined by the eigenvectors of the internal coordinates). The high-frequency V\{ag) and v2{ag) modes involve the an-tisymmetric and symmetric vibrations of the methyl groups respectively. These modes Chapter 4. Normal Coordinate Analysis of TMTSF'(hu/dw)0'* 104 Figure 4.2: Atomic displacement vectors for the totally symmetric (ag) modes of pro-tonated T M T S F + . Only projections onto the molecular plane are shown for the methyl group atoms. Below each vibration the frequencies for the protonated and deuterated molecules are listed and the isotope shift is indicated in parentheses. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/dn)°'+ 105 have very large isotope shifts (> 700 cm"1) as do the i/s(ag), ue(ag), V7{ag) and the j/8(afl) (> 100 cm - 1 ) vibrations which all involve the methyl groups to one extent or another. The four low-frequency vibrations, v9(ag), vio(ag), U\\(ag) and V\2(ag) are all very complicated vibrations that involve many force constants and may be character-ized as ring-breathing and methyl-wagging vibrations. Upon deuteration, all of these vibrations have shifts, but because the vibrations involve the displacement of the entire methyl group rather than just a C-H (C-D) stretch or angle bend, the total isotope shift is relatively small (< 30 cm - 1 ) . There is one ag vibration that has no isotope shift, the v4(ag) mode. This vibration involves only the central carbon-carbon stretch and as such is affected very little by deuteration. It will be shown that the Vi(ag) vibration plays a central role in the electronic properties of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . 4.5 Proportionality Factors The proportionality factors describe the degree of optical activity for a normally infrared active molecular vibration along the basis vectors of the unit cell (as shown in Chapter 1, Fig. 1.2). The TMTSF molecules in the (TMTSF^-X" crystal lattice are more or less uniformly oriented, so that radiation polarized along a lattice direction may couple to the dipole moment of one vibration more strongly than another. The proportionality factors may be determined for the infrared active u modes. Each of the u modes is associated with a particular direction in the molecular basis (as shown in Appendix A, Fig. A.2) due to the orientation of the dipole moment created by the vibration; biu(z), &2u(y) and 03U(x). The molecular basis is projected onto the set of axes chosen to define the crystal directions, and the proportionality factors for vibrational coupling for radiation polarized along the crystal directions are cos2(0<), where 0, is the Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°'+ 106 Table 4.6: Proportionality factors for band intensities along the a, b' and c' axes for the normally infrared active modes0. Symmetry species Axes a b' c' hu{y) 0.000 0.217 0.783 0.000 0.869 0.131 0.999 0.000 0.000 0 The b' and the c' axes are perpendicular to the a axis (see Figure 1.2 in Chapter 1). The x,y and z labels refer to the molecular basis, which are shown in Appendix A, Fig. A.2. angle formed between the crystal axis and the molecular basis (i = 1,2,3 for each axis). The proportionality factors for the band intensities along the a, b' and c' axes are shown in Table 4.6. Note that because the plane of the TMTSF molecule is nearly perpendicular to the molecular stacks, the b3U modes are active only for E||a, while the 6i„ and the b2u modes are active for both the E||b and the E||c polarizations. The fact that we can associate certain infrared active vibrations with crystallographic directions, when considered with the highly anisotropic nature of the Bechgaard salts, has important consequences for absorption measurements with powder spectra. Little light will be transmitted for Ej|a, because of the high reflectivity in the a-axis direction, making the 6 3 u modes difficult to observe in the powder spectra. The reflectivity along the b and c axes is much lower, thus a powder will transmit light preferentially along these directions and the &i„ and the b2u modes will be observable.. Chapter 4. Normal Coordinate Analysis of TMTSF (hi2/dl2f'+ 107 4.6 Vibrational Assignments The vibrational assignments of the internal molecular vibrations and the external lattice vibrations have been made on the basis of isotope shifts and optical activity. The assign-ments of the internal vibrations are made for both (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . The analysis of the external vibrations has been restricted to the (TMTSF) 2 Re0 4 salt. 4.6.1 Internal Molecular Vibrational Assignments in ( T M T S F ) 2 R e 0 4 and ( T M T S F ) 2 B F 4 The separation of the reflectivity, and hence the optical conductivity, into the E | |a and E||b' components lends itself naturally to the assignment of the vibrational modes that will be active for the polarization parallel to the stacks such as the out-of-plane b3u modes. The in-plane ag modes are also active in the stack direction (shown later in Chapter 5 and Appendix B to be activated through coupling of the molecular vibrations to the local oscillations of the charge—density wave). The in-plane &i„ and b2u modes will be active for the polarization transverse to the stacks. Because the symmetry axes of the TMTSF molecule do not correspond to the b' or c' axes directions, the E||b' polarization will consist of both the biu and 6 2 u modes, although the 6 2 u modes will be more strongly activated. Powder absorption spectra, on the other hand, will consist predominantly of the vibrational modes polarized for E||c', the biu modes. The normal coordinate calculations have been performed for the neutral TMTSF molecule and its radical cation, T M T S F + . The formal charge of the TMTSF molecule in the unit cell, however, is q = +0.5. There is evidence that the frequency of the ag modes depends linearly on the charge density on the molecule [87, 88]. The final frequencies and deuterium shifts used in the vibratinal assignments have been interpolated from those of Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°'+ 108 Table 4.7: The interpolated frequencies" and deuterium shifts of the in-plane ag, 63, blu and b2u normal modes of TMTSF+ 0 5 . Vibration hi2 di2 TMTSF+ 0 5 6 Vibration h i 2 di2 6 ag vx 2946 2201 -745 biu (E||c) "26 2946 2201 -745 v2 2849 2053 -796 "27 2849 2053 -796 "3 1596 1583 -13 "28 1592 1578 -14 "4 1469 1469 0 "29 1438 1038 -400 vs 1439 1038 -401 "30 1370 1067 -303 ve 1370 1066 -304 "31 1062 969 -93 V7 1060 966 -94 "32 917 737 -180 VB 916 737 -179 "33 676 674 -2 v9 450 427 -23 "34 450 427 -23 VlO 298 288 -10 "35 282 248 -34 "11 267 240 -27 "36 258 253 -5 V12 135 133 -2 b3S "55 2945 2201 -745 b2u (E||b) "44 2946 2201 -745 "56 2849 2053 -796 "45 2849 2052 -797 "57 1441 1036 -405 "46 1441 1037 -404 "58 1383 1054 -329 "47 1383 1054 -329 "59 1158 1146 -12 "48 1158 1146 -12 "60 1051 886 -165 "49 1051 887 -164 "61 978 978 0 t"50 773 770 -3 "62 706 622 -84 "51 702 620 -82 "63 417 389 -28 "52 415 387 -28 "64 306 287 -19 "53 299 277 -22 "65 179 174 -5 "54 55 53 -2 0 Al l wave numbers are in c m - 1 . Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°<+ 109 TMTSF 0 and TMTSF+ to yield the estimated frequecies and shifts for TMTSF" 1" 0 5 and are shown in Table 4.7. The group theory of the TMTSF molecule has been discussed, but the effects of plac-ing the molecule into a molecular crystal have not been examined. At room temperature the (TMTSF)2-X" salt crystallizes in the triclinic system, space group P1(C 1) with the one formula unit per unit cell. Table 4.8 shows the correlation between the irreducible representations of the TMTSF molecular point group and those under the site and unit-cell groups. This is referred to as a factor-group analysis and is treated in detail by Turrell [89]. Also given are the number of modes of each symmetry species for each type of molecule including the external modes. Because the two TMTSF molecules sit at general sites, each internal mode can be both infrared active and Raman active in the solid, depending on whether the two molecules are in phase or 180° out of phase. [In the Au (ir) modes the ungerade vibrations are in phase while the gerade vibrations are out of phase. The opposite holds for the Ag (Raman) modes]. Any frequency splitting between the Au and Ag modes resulting from the intermolecular interaction is known as Davydov, correlation-field or dynamic splitting. Since the ReO^ anion has no inversion center but occupies a site with inversion symmetry, then it must be dynamically disordered in the metallic phase. However, since C, is not a subgroup of the point group Td, then we have assumed a general site (1) in the correlation table. Because the normally infrared inactive ag modes in the molecule may expressed as Au and Ag vibrations in the solid (as the correlation table demonstrates), it is possible for the ag vibrations (or any gerade vibration) to become infrared active in the transverse polarization. (This mechanism for ag mode activity is not to be confused with the coupling to the CDW, which results in infrared activity parallel to the stacks). In the low-temperature ordered phase there are eight times as many molecules per Chapter 4. Normal Coordinate Analysis of TMTSF(bt2/di2)0'+ 110 Table 4.8: Correlation diagram for the high-temperature phase of (TMTSF) 2 Re0 4 . R = Raman active; ir = infrared active; Q = internal; L = libration; T and T' = optical and acoustic translations, respectively. Molecular point group Site group Unit cell group D2h 1 Q (12)a,(JQ ( ~ ) M * ) \ (S)b29(R)\\ ( 1 2 ) ^ ( H ) ^ ^Ag{R)(12Q + ZT + 31) )A(R + ir){78Q)< ><4tt(»r){Z2G + 3(T + I") + 3L] (12)blu(ir)7/ (12)M*r)/ {&)hu(iry 1 Ci ( l ) e ( i ? ) X ( l ) / i ( » > ) ^ M(/2 + ir)(15Q)-—A(R + tr)[9Q + 3(T + J') + 3L) unit cell as in the room-temperature phase, and therefore eight times as many k = 0 normal modes for the internal vibrations. E| |a The assignments for E| |a will be dealt with first and are shown in Tables 4.9 and 4.10. Not shown in the Figures in Chapter 3 is vi(ag) mode at 2946 c m - 1 , which is weak and probably a doublet. The v2(ag) mode, calculated at 2849 cm" 1 (hi2) and 2053 c m - 1 (di2) is very weak and is seen only in (TMTSF) 2ReQj. The strong antiresonance at 1845 cm" 1 in Fig. 3-11 must be a combination band. It has a slightly positive deuterium shift. The calculated negative shift of vs(ag) + V\i(ag) Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°'+ 111 is greater than that observed, but since the vn(ag) mode is observed to have a negligible shift this is a possible combination. The v${ag) mode is assigned to the three, or possibly four lines between 1604 c m - 1 and 1541 c m - 1 . In the deuterated spectra, a new line appears at « 1681 c m - 1 which may be a residual line. The fs(ag) mode is assigned to the weak doublet around 1450 c m - 1 , but this may also be the normally active ^72(63*) out-of-plane methyl vibration. The u5(ag) and the i/72(63„) mode both have isotope shifts of « 400 c m - 1 . The feature at w 1050 c m - 1 is observed to become a weak doublet in (TMTSF-<fi2) 2Re04, but becomes buried in the BF^ v3(}2) anion resonance at 1070 c m - 1 [90]. The 1^  (afl) vibration involves only the central C=C bond which couples very strongly to the conduction band electrons (as discussed in Chapter 4). It has no isotope shift and it is thus responsible for two very broad antiresonances at fa 1430 c m - 1 and « 1345 c m - 1 . In the deuterated spectra of the optical conductivity in Fig. 3.13 the sharp feature at 1385 c m - 1 is merely the result of this antiresonance doublet and not a vibration. While the Ki(a f l) antiresonance behavior is obvious in (TMTSF) 2 BF 4 , in (TMTSF) 2 Re0 4 the optical conductivity can be interpreted as having a large resonant peak, near « 1290 c m - 1 due to the 1430 c m - 1 component, and an antiresonant feature at « 1370 c m - 1 due to the 1345 component of the V4(ag) vibration. It is difficult to guess the coupling constants, and as the calculation in Chapter 4 showed, the e-mv coupling constant for 1430 c m - 1 is more than twice that for the 1345 c m - 1 feature. In the protonated spectra this sharp feature due to the " 4 ( 0 ^ ) doublet has the v&{ag) mode superimposed upon it as an antiresonance. The v$(ag) is shifted by almost the calculated amount in the deuterated spectra, however, the v^{ag) mode appears in the protonated spectra as a doublet, but in the deuterated spectra it is a quartet. Because this vibration involves the displacement of the methyl groups, the incomplete deuterium incorporation will result in the observed splitting of this mode. Chapter 4. Normal Coordinate Analysis of TMTSF(hl2/d12)°>+ 112 The two medium sized antiresonances in (TMTSF) 2 Re0 4 shown in Fig. 3.11 at 1376 c m - 1 and 1368 c m - 1 are probably combination bands, possibly t^r(ag) + 'Ui0(ag) [although vio(ag) is not observed in (TMTSF) 2Re0 4] and v8(ag) + v9(ag). In (TMTSF) 2 BF 4 the features are somewhat larger (especially in the deuterated spectra) due to the fact that they are in regions of high conductivity. Two close doublets, one weaker than the other, well below the "4(a f l) mode have been assigned to the f7{ag) vibration. In (TMTSF) 2 BF 4 , this same mode is much stronger due to the fact that it is now on the leading edge of the single-particle conductivity. A similar situation holds for the v&(ag) vibration, which consists of a group of five lines in (TMTSF) 2 Re0 4 , three of which are due to the vs(ag) mode, and two of which are due to the ReO^ vz(f2) anion vibration (the lines are separated by observing the splitting of the anion mode in the E| |b' direction). In the deuterated spectra in Fig. 3.25 the correspond-ing lines have been shifted to fa 750 c m - 1 and have become weak. In (TMTSF) 2 BF 4 , the three v%{ag) lines are three strong antiresonances just below the gap. The deuterium shift takes them below the gap and they become much weaker. The origin of the dip at 915 c m - 1 is not understood, but it may be a residual protonated line in an incompletely deuterated sample. This would be consistent with other results. The infrared-active internal anion modes are the triply-degenerate ^ ( / i ) (bond stretching) and ^ 4(/ 2) (angle bending), previously observed [90] at 920 c m - 1 and 331 c m - 1 in ReOj and 1070 c m - 1 and 533 c m - 1 in B F J . In the conductivity spectra strong "3(/2) doublets are observed at 907 c m - 1 and 925 c m - 1 in (TMTSF) 2 Re0 4 and 1054 c m - 1 and 1045 c m - 1 in (TMTSF) 2 BF 4 , but the vz(ag) modes (angle bending) modes were only observed in the powder spectra, where theywere split into four components. The doublets in the conductivity are presumably due to the two anion-orientations in the low-temperature unit cell. The assignments in Tables. 4.9 and 4.10 list only one feature in the B F 4 compound since the h\2 spectrum is very complicated in that region Chapter 4. Normal Coordinate Analysis of TMTSF'(hu/du)0'* 113 Table 4.9: Internal molecular vibrational assignments for protonated and deuterated (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 for the E||a polarization (ag and b3u modes). (TMTSF) 2 Re0 4 (TMTSF) 2 BF 4 Calculated 6 di2 6 mode h\2 d\2 8 2940* vw 2194*vw 2188*vw -746 -752 2940* vw 2190* vw -750 u^ag) 2946 2201 -745 2815vw 2019* vw 2005*vw -796 -810 i/2(ag) 2849 2053 -796 1845*s 1851*m 6 1849*w 1853* vw 4 v3(ag) + vn(ag 1862 1822 -40 1617w 1606* vw v3(ag) 1595 1580 -15 1604m 1594w -10 1605*m 1598" -7 1595w 1583w -12 1586*m 1576*m -10 1541s 1537s -4 1550*s 1549*s -1 1462vw 1046w -416 1437w 1052*br -385 z/5(afl) or u72(b3u) ? 1434w 1028vw -406 1430w 1439 1038 -401 1430*vs 1430*vs 0 1411*vs 1417*vs 5 u4(ag) 1469 1469 0 1362*vs 1375*vs 13 1350*vs 1370*vs 20 1389*w 1114w -275 1388*w 1118*s -270 v&{ag) 1370 1066 -304 1384sh 1106m 1096w 1082w -278 -290 -304 1382*m 1110's 1095*w 1085*w -272 -292 -302 1376m 1270*m -106 1376*w 1261*m -115 v7(ag) + u10(ag 1358 1254 -104 1368w 1178vw -190 1367*w 1163*w -204 Mas) + Mag) ? 1162w -206 1366 1164 -202 1088vw 974vw -114 1110*w 986*m -124 v7(ag) 1063 970 -93 1085w 967vw -118 1088*w 978*m -110 1067w 943w -124 1070*s 962* s -108 1063w 943vw -115 1061*m 946*m -115 1037vw 778vw -259 1039*w 785w -254 "73(&3u) 1017 786 -231 1021vw 771w -250 1030*w 770w -260 0 Al l wave numbers are in c m - 1 . In the case of doublets or multiplets, which are grouped together, the assignment is given on the first line only. The symbols signify; * - antires-onance dip, vs - very strong, s - strong, m - medium, w - weak, vw - very weak and sh - shoulder. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°>+ 114 Table 4.10: Internal molecular vibrational assignments for protonated and deuterated (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 for the E||a polarization (ag and b3u modes) (cont.). (TMTSF) 2 Re0 4 (TMTSF) 2 BF 4 Calculated hi2 d12 6 hi2 d\2 8 mode h12 du 8 1054"m 1052*m B F 4 «/3(/2) 1070 1046*m 923w 923w Re0 4 v3{f2) 920 908m 907w 918m 755w -163 937*s 770w -167 914 737 -177 920s 749w -161 928* s 762m -166 904m 915*vs 749 -166 676w 676w 0 679w hu 668w 667w -1 673w 672w -1 467w 678w 1 464w hu 454w 454w 0 444m 420m -24 440s 418m -22 Mat) 542 429 -23 437m 292w 310w 291w -19 298 288 -10 286w 286w 279m -7 vn(ag) 267 240 -27 283w 280w 277w 270w -8 276w 269sh -7 268m 260w -8 261m 261w 0 260m 254m -5 257m 250w -5 255m 245m -10 253m 244m -9 219w 218w -1 223w 223w 0 hu 158w 151w -7 158w 153w -5 ul2(ag) or b3u ? 153w -2 135 133 -2 0 All wave numbers are in c m - 1 . In the case of doublets or multiplets, which are grouped together, the assignment is given on the first line only. The symbols signify; * - antires-onance dip, vs - very strong, s - strong, m - medium, w - weak, vw - very weak and sh - shoulder. Chapter 4. Normal Coordinate Analysis of TMTSF (hl2/dl2f<+ 115 and the d\2 feature at 1052 c m - 1 is rather broad. There are two weak doublets, one at « 670 c m - 1 , the other at w 460 c m - 1 , which exhibit no deuterium shift and do not correspond to any of the calculated ag modes or their combinations. They have been assigned as normally active out-of-plane &3„ modes. The remaining two prominent features in the spectra have been assigned to the v9(ag) and fn(a s). ^(ap) is a doublet in Fig. 3.11 but a singlet elsewhere. The v\i(ag) mode has a least six, and possibly seven components. The Uio{ag) mode is very weak and is not observed in protonated (TMTSF^ReO-j. The sharp line at 219 c m - 1 displays a small deuterium shift, and is assigned to a 63a mode. The remaining line at RS 155 c m - 1 is very weak, and has a deuterium shift of 2 c m - 1 . It could be the v\2{ag) mode, which is calculated to occur at 135 c m - 1 and have a shift of 2 c m - 1 , but the large (14%) difference between the calculated and observed frequencies indicates that it could also be a normally active 63^ mode, which is also expected to occur at low frequencies and have a small deuterium shift. E||b' The assignments for the E||b' polarization are much more difficult than those for E||a. Instead of having just the very strong ag modes and their combinations, or the much weaker 6 3 u modes in E||a, for E||b' both the 6 l t l and the b2u modes may be activated. Also, there will be a number of possible combination bands that will be active. The cross products from the character table for D2h indicates that any even representation in combination with an odd representation will produce an odd representation (which will be infrared active in general, e.g. b\u ® 63, = &2„, etc.). Furthermore, the ag modes are allowed by symmetry to become Au infrared active, as shown in the correlation table (Table 4.8). This means that there are now many different combination bands that may occur, as well as the large number of fundamentals. The assignments for the E||b' Chapter 4. Normal Coordinate Analysis of TMTSF(hn/du)0,+ 116 polarization are shown in Tables 4.11 and 4.12. In many cases, there is more than one assignment for a given mode. The lines at w 2910 c m - 1 and w 2847 c m - 1 , which are not in the Figures in Chapter 3, are assigned to the V44(b2v) and the "45(621*) m ° d e respectively. These modes are degenerate with two 6i„ modes, the v2&{biu) and the v27(b\u) modes respectively, but the &2„ modes should have more intensity than the &iu modes, thus the vibrations have been assigned as b2u. The two weak vibrations at 2741 c m - 1 and 2725 c m - 1 are seen to occur only in (TMTSF)2Re04. They have been assigned as a combination band of "47(&iu) + ve(ag), which has the symmetry b\u. The quartet of lines between 1540 c m - 1 and 1610 c m - 1 has been assigned to both the v2s(biu) and the totally symmetric vs(ag) vibration, which is allowed to be infrared active in the unit cell. While the deuterium shifts may be followed in (TMTSF^ReO^, in (TMTSF)2BF 4 there is only one line in the protonated spectra at 1561 c m - 1 which then becomes a triplet in the deuterated material. The weak mode at 1447 c m - 1 and the weak doublet at 1383 c m - 1 and 1375 c m - 1 display either no isotope shift, or a small shift, respectively. The only calculated in-plane vibration at this frequency that have no deuterium shift is the " 4 ( 0 ^ ) vibration, which was also observed to split into two components in the E||a polarization with a separation of « 85 c m - 1 , which is the same as that observed for these two vibrations. These modes have therefore been assigned as the V4(ag) vibration. The next two doublets are the strongest features (excluding the anion vibrations) in the spectra. The first doublet is at 1420 c m - 1 and 1438 c m - 1 has been assigned to the "46(&2u) mode and the second doublet at 1357 c m - 1 and 1361 c m - 1 has been assigned to the "47(&2u) mode. Both of these modes are degenerate with the v2$(biu) and the i/3o(blu) modes respectively. These two b2u modes are very unusual in that Chapter 4. Normal Coordinate Analysis of TMTSF(h12/du)°'+ 117 Table 4.1.1: Internal molecular vibrational assignments for protonated and deuterated (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 for the E||b' polarization. (TMTSF) 2ReO 4 (TMTSF) 2 BF 4 Calculated du 6 du 6 mode hi2 d\2 6 2914vw "44(&2u) 2946 2201 -745 2906w 2177w -729 2908 2901w 2164w -737 2847vw 2053vw -749 2053vw "45(&2n) 2849 2052 -797 2741vw "47(&2u) + "e(a5) 2752w 2124w -602 2753 2120 -633 1606w 1594w -12 "28(t2 U ) ! , i/3(ag) mixed 1597vw 1583w -14 1575w 1592 1578 -14 1556w 1550w -6 1561w 1554w -7 1596 1583 -13 1544w 1540w -4 1545vw 1447w 1447w 0 1447w u4{ag) 1469 1469 0 1438s 1043m -395 1455m 1040w -415 "46(&2U)'- 1441 1037 -404 1419w 1435s 1032w -403 1383w 1373m -10 1384w 1372m -12 v4(ag) 1345 1345 0 1375w 1362w -12 1378w 1359w -19 1361s 1012w -349 1364s lOllw -353 VAi{b2u)\ 1383 1054 -329 1357s lOlOw -348 1359s 1157w 1138w -19 1156w l l l l w -45 V4&{b2u) 1157 1146 -12 1149m 1106w -43 1149w HOOw -39 1079w 985vw -94 1097w "3l(&lu) 1062 969 -12 1074w 965w -109 1088w 974w -102 1066w 950w -116 1076w 959w -117 1054vs 1058br B F 4 i/ 3(/ 2) 1070 1046vs 1048sh 0 Al l wave numbers are in c m - 1 . In the case of doublets or multiplets, which are grouped together, the assignment is given on the first line only. The symbols signify; * - antires-onance dip, vs - very strong, s - strong, m - medium, w - weak, vw - very weak and sh - shoulder. The ! indicates intensity transfer. Chapter 4. Normal Coordinate Analysis of TMTSFfiu/du)0, 118 Table 4.12: Internal molecular vibrational assignments for protonated and deuterated (TMTSF)2Re04 and (TMTSF)2BF4 for the E||b' polarization (cont.). (TMTSF)2ReO 4 (TMTSF)2BF4 Calculated hi2 di2 8 hi2 6 mode hi2 di2 8 1037w 874w -163 1038vw 877w - 161 M M 1051 887 -164 1023vw 864w -159 1032m 1014vw 844vw -170 1019w 844w 175 1007w 840w -167 lOllw 827w - 184 925vs 923m -2 Re0 4 i / 3 ( / 2 ) 920 907vs 907vs 0 920m 754w -166 931vw 750vw - 181 " 3 2 ( & i u ) 917 737 -180 914vw 739vw - 175 792vw 789w -3 777vw 776w -1 "50(&2u) 773 770 -3 773vw 769w -4 767vw 766w -1 683w 680w -3 683vw 681vw -2 "33(ilu) 676 674 -2 675w 672w -3 664vw 466*w 4666w 0 b3u from E||a 454fcw 454bw 0 443fcm 419bm -24 444w 426w -18 v34(biu) 450 427 -23 436w 407w -29 441w 419w -22 398bw 3786w -20 396*w 374* w -22 "52(&2u) 415 389 -28 338fcm 3376m -1 ReOr u4(f2) 320 3306m 3306m 0 3246m 324fcm 0 318fcn 3196m -1 283bw 2486sh -35 2836w "35(fc2u) 282 248 -34 268fcm 2616w -7 264bm " 3 5 ( & i u ) 258 253 -5 261bm 252bw -9 2586m 49m 47m -2 "54(621*) 55 53 -2 0 All wave numbers are in c m - 1 . In the case of doublets or multiplets, which are grouped together, the assignment is given on the first line only. The symbols signify; * - antires-onance dip, vs - very strong, s - strong, m - medium, w - weak, vw - very weak and sh - shoulder. The ! indicates intensity transfer. 6 Taken from powder absorption data. Chapter 4. Normal Coordinate Analysis of TMTSF(hi2/di2)0,+ 119 they have much larger amplitudes than any other modes of the TMTSF molecule, the J/46(&2II) and the y47(2>2u) modes. These modes also occur at almost the same energy as the very strong components of the ^(a^) mode, seen in the E||a polarization. This is no coincidence. There is a mechanism, introduced by Horowitz et al. [91], by which an antisymmetric vibration may gain strength from a totally symmetric vibration if the two are degenerate. This is referred to as the "intensity transfer syndrome". The two 62u modes are nearly degenerate with the two components of the f 4 (a 5 ) mode and will therefore borrow some intensity from these vibrations, giving them larger amplitudes. The fact that the f 4(a a) mode has a much larger e — mv coupling constant than any of the other ag modes (shown in Chapter 5) means that even though intensity transfer may occur for other antisymmetric modes that are degenerate with ag modes, it will be to a much lesser degree. This raises the possibility that the degenerate 6 i u modes could also be contributing to these features. Those cases where intensity transfer is suspected are indicated by an next to the assignment. The weak doublet at 1157 c m - 1 and 1138 c m - 1 is assigned to the i/4&(b2u) vibration. The larger than expected value for the deuterium shift in both materials may be the result of an interaction with the anion. The i/3i(blu) vibration is observed as a weak triplet between « 1050 c m - 1 and « 1100 c m - 1 . The two strong peaks at in (TMTSF)2BF 4 at 1046 c m - 1 and 1054 c m - 1 are due to the ^ 3(72) vibration of the B F J anion. In the deuterated material, this vibration has been broadened, with the fundamental at 1058 c m - 1 and a weak shoulder associated with the doublet at 1048 c m - 1 . The quartet of lines between 1000 c m - 1 and 1040 c m - 1 in the protonated compounds have been assigned to the ^9(6211), which is calculated to occur at 1051 c m - 1 (hi2) and 887 c m - 1 (di2). The measured deuterium shifts match almost exactly the calculated shifts of « 165 c m - 1 . Chapter 4. Normal Coordinate Analysis of TMTSF(h12/di2)°'+ 120 The two strong peaks in (TMTSF) 2 Re0 4 at 925 c m - 1 and 907 c m - 1 are due to the "3(/*2) vibration of the ReOj anion. The next two weak doublets at « 780 c m - 1 and « 680 c m - 1 have been assigned to the vso(l>2u) and the 1/33(0)lu) vibrations respectively. The doublets are very weak, and there is some question if the positive deuterium shifts in (TMTSF)2BF4 are correct. The remaining assignments have been made mostly on the basis of the (TMTSF)2ReC>4 powder absorption data. The E||b' conductivity is very difficult to interpret below « 500 c m - 1 . The four sharp modes observed only in (TMTSF) 2 Re0 4 at 338 c m - 1 , 330 c m - 1 , 324 c m - 1 and 318 c m - 1 are the v4(f2) modes of the ReO^ anion, which are expected to occur at 320 c m - 1 [90]. The remaining two vibrations are both 6 l u modes, assigned from the powder data. The feature at 283 c m - 1 is assigned as the "35(61,,) vibration and the doublet at w 265 c m - 1 is assigned to the f 3 6(&iu) vibration. The two modes at 466 c m - 1 and 454 c m - 1 are the b3u mode seen in the E| |a polariza-tion. The dominant feature at 443 c m - 1 is the ^ ( f t i u ) vibration. The "52(6211) vibration is also expected to be active in this region, with calculated frequencies of 415 c m - 1 ( h i 2 ) and 387 c m - 1 ( d i 2 ) . This mode may or not be the antiresonance structures seen in the reflectivity at w 400 c m - 1 and « 385 c m - 1 respectively. The presence of a weak mode in the powder absorption spectra of (TMTSF) 2Re04 at 398 c m - 1 and at 378 c m - 1 in the deuterated spectrum leads to the conclusion that the "52(62,1) mode is being observed, but it is very weak. 4.6.2 External Phonons in ( T M T S F ) 2 R e 0 4 The external phonons at « 50 cm" 1 observed in the E| |a reflectivity of (TMTSF) 2 Re0 4 (Fig. 3.6) and (TMTSF) 2 BF 4 (Fig. 3.20) at low temperature are strongly activated, much more so than ordinary lattice modes. The activity in these features is comparable to that Chapter 4. Normal Coordinate Analysis of TMTSF(hl2/dx2)0'+ 121 Table 4.13: Calculated shifts of the (TMTSF) 2 Re0 4 lattice modes". Type Molecules involved vvjvd Translational Translational Libration Rx Libration Ry Libration Rz 2 TMTSF, 1 Re0 4 TMTSF only TMTSF only TMTSF only TMTSF only 1.003 1.103 1.044 1.052 1.024 a The coordinate system is shown in Fig. 1.2. up and are the frequencies of the proton-ated and deuterated features, respectively. of an ag mode. The temperature dependence of this feature, shown for (TMTSF) 2 Re0 4 in Fig. 3.7, indicates that the oscillator strength and the side bands grow quickly below the phase transition, and therefore these vibrations must be coupling to the CDW caused by the dimerization of the stack and the anion potential. The splitting is presumably due to Brillouin-zone folding resulting from the increased size of the unit cell. It is clearly of interest to determine the nature of this vibrational mode, which couples so strongly to the CDW. The temperature dependence of the frequency of the central feature seen in Fig. 3.7 is characteristic of a lattice mode, rather than of a pure internal mode. The E||a oriented crystal mosaic absorption spectra of protonated and 90%-deuterated compounds [64] gives frequency ratios of vvlv& of 1.010, 1.011 and 1.016 for the three major peaks. Table 4.13 shows the calculated ratios for some of the possible forms of lattice modes. Any mode in which the light anion has a large displacement will show a very small deuterium shift. Large shifts indicate modes involving mainly TMTSF molecules, and the largest shifts are due to librations of the TMTSF molecules. The two lowest frequency internal modes also have high calculated isotope shifts, 3.9% [^ S4(^ 2w)] and 7% ["72(&3U)]- The approximately 1% shifts are observed for the 50 cm_1cluster would therefore seem to indicate a pure translational motion of the TMTSF Chapter 4. Normal Coordinate Analysis of TMTSF (hx2/dl2)°* 122 molecule alone. In a preliminary report [64], it was suggested that this feature might be a zone-boundary longitudinal acoustic phonon, activated by zone folding. However, in an earlier study of (TMTSF+A" powders [68], four singlet resonances were observed below 90 c m - 1 in the octahedral-anion compounds, which develope no CDW. One of these was an internal mode, and three were lattice modes. Two of these latter were strong and involved mainly TMTSF molecules, one near 50 c m - 1 with a 1% deuterium shift, and one near 70 c m - 1 with a 2.6% shift. Since the 50 c m - 1 mode has also been seen in the E||a polarized bolometric spectrum [61] of (TMTSF) 2 AsF 6 and (TMTSF) 2SbF 6 , then it is very likely, since it has the same frequency shifts and polarization, that the 50 c m - 1 mode is the same vibration that we are observing in Fig. 3.6 and is a normally active (optic) k — 0 lattice mode. Any pure translation of the TMTSF molecule alone produces no optical activity, since the Au (introduced in the correlation table) mode, in which the molecules move together, would be an acoustic mode, while the Ag mode, in which the molecules move against one and other, would be only Raman active. If the anions participate in the Au(k = 0) mode, as shown in Fig. 4.3(a), then in order to keep the center of mass stationary, the anion displacement would be large and the isotope shift would be very small (as shown in Table 4.13) and the frequency would shift a great deal from one compound to another, which it does not [68]. The mode must, therefore, include some librational and/or internal vibrational motion. Since the librations and translations have the same symmetry, they will undoubtedly be mixed. Furthermore, it was shown in the case of biphenyl [92] that the low-frequency lattice modes were strongly mixed with the internal torsional mode (au), which twists the rings about the central bond and has the same shift as an Rx libration (2.4% in this case). The mode is presumably a mixture of such motions, which coincidentally leads to an overall 1% isotope shift. Some of the possible combinations are shown in Fig. 4.3. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/du)0'+ 123 (a) Y *-± ^ (b) # + °+» Y Y ^ + - + t T o 4 +• + • o 4_ + ±-# >v Y o t + ± - i Y ( O Y i A Y f — » *4v - f - _ J | —» T 2a t « , * - J ~ Figure 4.3: Sketches of the displacements of the TMTSF molecule (straight line) and the Re04 molecule (Y) in possible lattice modes, (a) Au mode involving translations only (o = inversion center, longer arrows indicate larger displacements); (b) Av mode involving translation and Ry libration; (c) the [100] zone-boundary mode of the Au vibration in (b), which is folded back to the zone origin in the new unit cell (after Reference [70]). Chapter 4. Normal Coordinate Analysis of TMTSF (hl2/dl2f 124 In Fig. 4.3(b) is shown an Au mixture of translation of both ions, with an fly libration of the TMTSF molecule. This combination would produce optical activity not only from the net ionic displacements, butalso from the positively charged selenium atoms and the negatively charged central carbon atoms in the laterally displaced molecule. The two dipoles are of opposite sign in Fig. 4.3(b), but would probably be unequal. Alternatively, if the a u torsional mode is involved, which is more likely, then optical activity may result simply from the general site of the TMTSF molecule, or possibly because of the modulation of the short chemical bond between the selenium atom and the 0 /F atom in the anion, which has been emphasized as a factor in the low-temperature anion ordering [51]. In the room-temperature phase of the correlation table indicates that there should be six infrared-active lattice modes. In the low-temperature ordered phase there are eight times as many molecules per unit cell as in the room-temperature phase, and therefore eight times as many k = 0 normal modes. Each optic mode will split into eight new optic modes and each acoustic mode will become one acoustic plus seven optic modes. In reciprocal space these new optic modes are seen to be the zone-boundary (ZB) modes reflected back to the zone origin when the Brillouin zone is halved in each direction. Fig. 4.4 shows planes containing the origin of a cubic Brillouin zone and that of the (TMTSF) 2 X family. The ZB modes reflected back to the origin come from the [100], [010], [110], [001], [101], [011] and [111] boundaries, some of which are labeled in Fig. 4.4. The [100] ZB mode of the mixed model of Fig. 4.3(b) will resemble the displacements drawn in Fig. 4.3(c), in which pairs of molecules in the new unit cell vibrate against each other. An early report by Moret et al. [29], and a later high-resolution X-ray analysis of the low-temperature structure of (TMTSF^ReC^ by Rindorf et al. [30], shows that below the metal-insulator transition there are translational displacements of both the TMTSF Chapter 4. Normal Coordinate Analysis of TMTSF(hl2/d12)°'+ 125 Figure 4.4: A plane of the Brillouin zone containing the origin for (a) a cubic lattice and (b) the (TMTSF) 2 X lattice. The zone boundary for the ordered phase is drawn inside. Symmetry points reflected back to the origin upon ordering are labeled (after Reference [70]). and Re0 4 molecules, as well as the Re0 4 orientational ordering. Their results have been reproduced in Fig. 1.5, which shows that pairs of TMTSF molecules have moved in the opposite direction to adjacent pairs. The translational component of the [100] ZB mode shown in Fig. 4.3(c) is also a component of the distortion which has produced the CDW and will therefore strongly modulate it and produce optical activity for E||a. The other ZB modes involve out-of-phase oscillations of the dimers in other directions in the new unit cell, and the frequency splitting between them is essentially the same as normal Davydov splitting in a large unit cell. The magnitude of the components of the ZB wave vectors along a will presumably determine the strength of each sideband. This would produce large oscillator strengths for [100], [110] [V and X in Fig. 4.3(b)], [101] and [111], but little strength from [010] [Y in Fig. 4.3(b)], [001], [011] and the original in-phase optical mode. This may explain the four strong and three weaker components in Fig. 3.6. Chapter 4. Normal Coordinate Analysis of TMTSF(h12/dn)0,+ 126 Table 4.14: Absorption peaks of (TMTSF ) 2 R e 0 4 for E||b' at 8 K. (TMTSF-/i 1 2 ) 2 R e 0 4 (TMTSF -< i 1 2 ) 2 Re0 4 "prot/"deut Assignment (calc vvjv&) 28.44 (vw) 28.20 (m) 1.008 Lattice 33.98 (vw) 33.74 (m) 1.007 Acoustic 39.53 (s) 39.29 (m) 1.006 Acoustic 44.35 (s) 43.63 (m) 1.016 T M T S F "54(62u)(1.039) 48.93 (s) 47.25 (s) 1.035 Vhi{b2v) 56.17 (m) 55.44 (m) 1.013 J"54(&2u) 61.45 (m) 60.26 (m) 1.019 T M T S F #,(1.024) 68.70 (m) 64.12 (sh) 1.071 #,(1.052) 73.04 (s) 68.21 (s) 1.071 #*(1.043) 76.89 (m) 74.49 (m) 1.032 79.79 (m) 83.16 (m) 87.26 (w) 87.50 (m) 0.997 Lattice 91.12 (m) 90.88 (m) 1.003 Both ions 96.66 (m) 95.70 (m) 1.010 104.37 (m) 102.69 (sh) 1.016 Lattice 110.88 (m) 106.55 (m) 1.041 T M T S F #x(1.043) 116.67 (w) 114.50 (m) 1.019 138.36 (w) 139.81 (w) 0.989 T M T S F 144.87 (m) 152.58 (m) 0.949 (hu) 178.14 (w) 182.71 (vw) 0 The b' and the c' axes are perpendicular to the a axis (see Figure 1.3 in Chapter 1). The activation of ZB modes by zone folding explains the lack of any strong 50 c m - 1 cluster absorption in the powder spectrum of the (TMTSF) 2C10 4 compound, since the anion ordering at 24 K produces a new unit cell with doubling only in the b direction and no CDW formation. Because (TMTSF) 2C10 4 remains metallic normally active optic phonon is not observed due to the high reflectivity for E||a. The E||b' polarization conductivity for (TMTSF-/ i 1 2 ) 2 Re0 4 (shown in Fig. 3.12) and Chapter 4. Normal Coordinate Analysis of TMTSF(h12/d12)°'+ 127 (TMTSF-iii2)2Re04 (Fig 3.14) display many phonons at low temperatures. The con-ductivity for protonated and deuterated (TMTSF)2BF4 is similar. At room temperature the high conductivity has broadened all the vibrations so that none are visible. They appear, however, in the energy gap at temperatures below the phase transition. It is clear that there are more features in the tetrahedral compounds than in the octahedral--anion case and that zone-folding is responsible. Only the strongest components are easily visible. Table 4.14 lists the frequencies and strengths of all the features'seen in the protonated and deuterated spectra together with some assignments. These were aided by the previously measured powder-absorption temperature dependence [68]. Neverthe-less, it is difficult to be unambiguous since the split modes overlap each other, and if the character of the vibration changes across the Brillouin zone, then the isotope shifts will also change. Groups of features have therefore been assigned in Table 4.14. Of the three strong features that appear below 50 c m - 1 , the one at 48.93 c m - 1 is assigned to the internal mode Vs4(b2u), because of the isotope shift and the temperature dependence. The temperature dependence indicates that the strong 44.35 c m - 1 and the weaker 56.17 c m - 1 may have the same origin. The very narrow 39.53 c m - 1 peak is a lattice mode, most probably a ZB acoustic phonon. The strong 30 c m - 1 feature, seen in the powders but not here, attributed to a zone-boundary transverse-acoustic mode must therefore be active for E||c. The strong lattice mode near 70 c m - 1 , as in the octahedral compounds, involves mainly TMTSF libration as does the medium group near 110 c m - 1 . The group near 90 c m - 1 involves both ions. The features at « 145 c m - 1 and « 180 c m - 1 are seen to be stronger in the E| |a and are tentatively assigned to the low-frequency TMTSF bzu out-of-plane modes. As expected, the E| |a cluster does not appear in the E| |b ' spectra since b' is perpendicular to a. Chapter 5 Optical Properties of One—Dimensional Semiconductors 5.1 Introduction The infrared optical properties of systems containing molecular stacks with a lattice dimerization (LD) have been analyzed using either molecular cluster models, such as the "dimer charge oscillation" model developed by Rice (1980) [65] and the "tetramer" model developed by Yartsev (1984) [66] (which deals with pairs of dimers), or a "phase-phonon" model, also due to Rice (1976) [93] which takes into account the one-dimensional electronic energy bands of the solid. More recently, Bozio et al. [69] have developed a model to calculated the infrared properties of linear organic conductors with twofold-commensurate charge-density waves, but this model has yet to be applied to a physical system. The dimer and tetramer models are characterized by Lorentzian shaped resonances [also referred to as a charge-transfer (CT) band] in the optical conductivity, centered in the mid infrared. The shape of the optical conductivity phase-phonon model is different; the conductivity is small below the gap, and above the gap displays a 1/y/t behavior characteristic of a joint one-dimensional density of states. A common feature of each of these models is that they provide a mechanism by which the totally symmetric ag modes, normally infrared inactive, may be activated. 128 Chapter 5. Optical Properties of One-Dimensional Semiconductors 129 < • Figure 5.1: Diagram illustrating the optical activity arising from out-of-phase ag modes in an isolated dimer. As the lower molecule contracts and the upper molecule expands, charge is forced from one molecule to another, creating a dipole moment (indicated by the arrow). Notice that the dipole moment is perpendicular to the molecular vibration, which is in the plane of the molecule. The distribution of charge is very sensitive upon the configuration of the molecule. In a simplified picture, when a bond contracts then the charge is forced away from the bond. In a large planar molecule, such as TMTSF, then during a vibration the charge may move from one part of the molecule to another. If one part of the molecule contracts while the other expands (an antisymmetric vibration), then an excess of charge will be formed on one side of the molecule resulting in a dipole moment; this mode will be infrared active. In the case of a totally symmetric vibration, the entire molecule is contracting or expanding at the same time thus the charge is either pushed off or gathered back onto the molecule. Alone, this is not enough to create an infrared active mode. If a pair of molecules is considered, as is the case in the dimer model (and, by extension, the tetramer model), and if the ag modes are 180 degrees out of phase then as one molecule is contracting and the other is expanding the charge is being pushed from one molecule Chapter 5. Optical Properties of One-Dimensional Semiconductors 130 to another, as indicated in Fig. 5.1. A dipole moment is formed, and thus the ag modes become infrared active, in the stack direction. In the phase-phonon model a charge-density wave (CDW) is present (as shown in Fig. 1.1). If there are local oscillations of the phase of the CDW, then the charge density is shifted from one molecule to the other and dipole moments are formed, parallel to the stack. The preferential frequencies at which this charge transfer will occur will be the frequencies of the ag modes. Thus, the phase oscillations of the CDW occur at the frequencies of the ag modes and are referred to as "phase-phonons". These models allow the electron-molecular-vibrational e — mv coupling constants to be determined. In the dimer and tetramer models, near the CT band, the ag modes occur as antiresonances, while far away from the CT band they occur as resonances. The phase-phonon model differs in that the ag modes occur as antiresonances above the gap and as resonances below the gap. This antiresonance behavior is often referred to as the Fano effect [94, 95]. 5.2 Models of Infrared Optical Conductivity (TMTSF) 2 Re0 4 and (TMTSF)2BF4, physically and electronically, are very similar ma-terials. Any model which describes one material should also describe the other. In the following sections, the three models previously discussed (The dimer, tetramer and the phase-phonon model) will be used to analyse (TMTSF) 2 Re0 4 at T sa 25 K and (TMTSF) 2 BF 4 at T « 20 K for E||a. It will be shown that while none of these mod-els can explain the optical properties of both (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 , the character of the conductivity reveals that the phase-phonon mechanism is correct. The Hamiltonians of each of these models and the parameters used to describe the optical properties are discussed in detail in Appendix B and the reader is referred there for further information. Chapter 5. Optical Properties of One-Dimensional Semiconductors 131 5.2.1 D imer Model The dimer model has previously been applied to the reflectivity of the insulating state of (TMTSF) 2Re04 for E||a by Jacobsen [60], although he questions the interpretation of the results. The dimer model has been used to fit the optical conductivity reported in this thesis for (TMTSF) 2 Re0 4 at w 25 K and ( T M T S F ) 2 B F 4 at tn 20 K for the E||a polarization using MINUIT, a non-linear least squares fitting routine developed at C E R N . Only the bulk features are fit, and no attempt is made to account for the fine structure [e.g. the v&{ag) mode and the observed splittings]. The results of the fitting are shown in Table 5.1, and the optical conductivity calculated by the dimer model are shown, along with the experimental data, in Figs. 5.2 and 5.3 for (TMTSF) 2 Re0 4 and ( T M T S F ) 2 B F 4 respectively. For (TMTSF) 2 Re0 4 , Table 5.1 gives the fitted values of the charge-transfer frequency as UQT = 1738 c m - 1 and the transfer integral as t = 1472 c m - 1 , but it must be true that UCT > 2t (from Appendix B), thus U>CT > 2800 c m - 1 ! Clearly, this is not the case. This problem is even worse in ( T M T S F ) 2 B F 4 , where U>CT = 1200 c m - 1 and t = 1421 c m - 1 . This value of the transfer integral again requires that UCT > 2800 c m - 1 . Decreasing u>* results in a lower value of the transfer integral, but it also lowers the strength of the charge-transfer excitation giving a lower overall conductivity. A Drude analysis at room temperature gives t fa 0.2 — 0.25 eV [60], which is close to the fitted value of t fa 0.18 eV but much smaller than that calculated by Grant [53] for this material at room temperature, t fa 0.36 eV. The optical conductivity calculated by the dimer model for (TMTSF) 2 Re0 4 appears to generate a reasonable fit, but it fails to display the asymmetry of the optical conductiv-ity observed experimentally, and above fa 2000 c m - 1 the calculated optical conductivity Chapter 5. Optical Properties of One-Dimensional Semiconductors 132 Table 5 . 1 : Dimer model parameters for the fit to the optical conductivity of (TMTSF) 2-Re0 4 and (TMTSF) 2 BF 4 at m 25 K and « 20 K respectively for the E||a polarization0. (TMTSF) 2 Re0 4 (TMTSF) 2 BF 4 V 8265 c m- 1 9600 c m - 1 UlcT 1738 c m - 1 1200 c m - 1 t 1472 c m - 1 1421 c m - 1 T 274 c m - 1 325 c m - 1 uQ 7a 9o 6« 7a 9c - 6* Mode (cm"1) (cm"1) (cm"1) (cm' 1) (cm"1) (cm"1) (cm-1) (cm"1) "3 + " l l 1845 5 18 4.1 1845 5 12 0.5 "3 1607 4 29 -9.0 1607 4 21 2.8 1546 2 46 -16.2 1551 3 20 2.8 "4 1457 2 182 -135.9 1419 10 103 84.5 1362 8 60 -14.9 1362 8 60 44.3 "7 1066 1 18 -0.9 1066 1 10 -2.2 "8 925 2 24 -2.6 925 3 35 -13.6 910 2 40 -3.2 910 2 40 -16.9 "9 440 2 30 -1.6 440 2 14 -1.1 "11 270 2 45 -3.4 270 2 18 -1.6 ° In the calculation e2 = 1.1615 x 10"3, Snne2a2 = 5.4770 x 104 cm" 1. For (TMTSF) 2-Re0 4 a = 3.55 A, Vm = 341 A3 [19] and the dimer density is n = 1.466 x 1021 c m - 3 . For (TMTSF) 2 BF 4 and a = 3.51 A and Vm = 334 A3 [20] and n = 1.497 x IO" 2 1 cm" 3. In both calculations too = 2.6. decreases too quickly. The vibrational feature due to the "4(a f l) mode at w 1290 c m - 1 is too small, while the oscillator strength due to the charge-transfer excitation above 1700 c m - 1 is too large. The same problems occur when the dimer model is used to fit the data for (TMTSF) 2 BF 4 , but the best fit is much worse. The high-frequency phonons in (TMTSF) 2 BF 4 , (e.g. « 1850 cm - 1 ) are also calculated to be resonances, when they are in fact observed to be antiresonances. There is no mechanism by which the dimer model may generate the numerous split-tings that are observed in the reflectivity and the conductivity. The dimer model does Chapter 5. Optical Properties of One-Dimensional Semiconductors 133 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) — 5000.0 I 1 1 1 1 • i 1 1 i Figure 5.2: The (a) experimental optical conductivity of (TMTSF) 2 Re0 4 for E| |a at T « 25 K (introduced in Chapter 3) and (b) the dimer model optical conductivity cal-culated with the parameters listed in Table 5.1 with (solid line) and without (dashed line) phonons. Comparing the calculated conductivity with the experimental conductiv-ity, several differences may be observed: the conductivity does not display the observed asymmetry and the vibrational feature due to the v4(ag) mode at « 1290 c m - 1 is too small, while the oscillator strength due to the charge-transfer excitation above 1700 c m - 1 is too large. Chapter 5. Optical Properties of One-Dimensional Semiconductors 134 1 5000.0 cm) 4000.0 b 3000.0 >^  > • »—H 2000.0 u d 1000.0 o O 0.0 1 5000.0 If 4000.0 £ b 3000.0 >^  -1-5 > • 1—< 2000.0 o C o 1000.0 o 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) Figure 5.3: The (a) experimental optical conductivity of (TMTSF)2BF 4 for E||a at T « 20 K (introduced in Chapter 3) and (b) the dimer model optical conductivity cal-culated with the parameters listed in Table 5.1 with (solid line) and without (dashed line) phonons. Comparing the calculated conductivity with the experimental conductiv-ity, several differences may be observed: the conductivity does not display the observed asymmetry and the vibrational feature due to the 1/4(0,) mode at w 1290 cm - 1 is too small, while the oscillator strength due to the charge-transfer excitation above 1700 cm - 1 is too large. Chapter 5. Optical Properties of One-Dimensional Semiconductors 135 not yield consistent results for the transfer integral and the charge—transfer frequency and fails to explain the optical properties of the insulating states of (TMTSF) 2Re04 and (TMTSF) 2 BF 4 . 5.2.2 Tetramer Model A variant of Yartsev's tetramer model [66] has been used by Bozio et al. [63] in an attempt to explain the optical properties of (TMTSF) 2 Re0 4 for E| |a at « 120 K, the result of which was shown in Chapter 1. Table 5.2 gives a list of the parameters used to calculate the optical conductivity using Yartsev's tetramer model. The parameters represent an initial refinement only. The lowest charge-transfer frequency, U>CTI was fitted by parameterizing the variables t\U,V and 2A in terms of the transfer integral: t/t' = 0.8, U/4t = 1.0, V/U = 0.2 and A = t/4 and then fitting the transfer integral. These ratios aire common for linear organic conductors [66] and are approximately the same as determined by Bozio et al. for (TMTSF) 2 Re0 4 at T ta 120 K [63]. The values for UCTI were chosen to correspond to the best-fit values as determined by the dimer model, UCTI = 1750 c m - 1 for (TMTSF) 2 Re0 4 and UCTI = 1200 for (TMTSF) 2 BF 4 . The results of the fit to the optical conductivity of (TMTSF) 2 Re0 4 at T ta 25 K and (TMTSF) 2 BF 4 at T ta 20 K for E| |a are shown in Table 5.2. The optical conductivity calculated by the tetramer model are shown, along with the experimental data, for (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 in Figs. 5.4 and 5.5 respectively. In this calculation, it has been assumed that D& = DB, that is, the charge densities on the molecules A and B are assumed to be equal. There are four charge-transfer frqeuencies for (TMTSF) 2 Re0 4 , UCTI = 1750 c m - 1 , wen — 3563 cm" 1 , UCTS — 7612 c m - 1 and UCTA = 9274; and four charge-transfer frequencies for (TMTSF) 2 BF 4 , u>Cri = 1200 vbar ,u>cr2 = 2443 c m - 1 , UCTZ = 5220 Chapter 5. Optical Properties of One-Dimensional Semiconductors 136 Table 5.2: Tetramer model parameters for the fit to the optical conductivity" for (TMTSF) 2 Re0 4 at T « 25 K and (TMTSF) 2 BF 4 at « 20 K for the E||a polarization. (TMTSF) 2 Re0 4 (TMTSF) 2 BF 4 t,t' u,v 2A r e 1737 cm" 1 1259, 1007 cm" 1 5033,1007 cm" 1 629 cm" 1 250 c m - 1 1200 cm" 1 863, 690 cm" 1 3451,690 cm" 1 431 cm" 1 250 c m - 1 Mode ua (cm"1) 7 a (cm - 1) ga (cm - 1) iva (cm - 1) 7Q (cm"1) ga (cm - 1) "3 + " l l "3 "4 "7 "8 "9 "11 1845 2 30 1605 2 30 1545 2 70 1425 6 450 1067 2 40 920 2 100 440 2 110 270 3 140 1845 2 30 1605 2 30 1545 2 70 1425 6 450 1067 2 40 920 2 100 440 2 110 270 3 140 ° The coefficients of the ground-state wavefunction are a\ = 0.047, a\ = 0.158, a\ = 0.158, a\ = 0.336, a\ = 0.248 and a\ = 0.369, the overlap parameters are A = 5.77 A , B = 1.89 A , and the number of tetramers per unit volume is the tetramer density Nt = 7.331 x lO 2 0 cm" 3 (Vm = 341 A 3 [19]) for (TMTSF) 2Re0 4 and Nt = 7.485 x 1020 cm" 1 {Vm = 341 A 3 [20]) for (TMTSF) 2 BF 4 . In both calculations, = 2.6. c m - 1 and UJCTA — 6360 c m - 1 respectively. Only the lower two will have any noticeable intensity, and the charge-transfer band at 1752 (1200) c m - 1 is characteristically much stronger than the one at 3874 (2443) c m - 1 . In (TMTSF) 2 Re0 4 , the conductivity from U > C T 2 produces a noticable bump in that is not observed experimentally. A similar feature is not observed in (TMTSF) 2 BF 4 . In the tetramer model, the interpretation of the structure from w 1200 — 1600 cm" 1 is that the resonances at « 1290 cm" 1 and « 1370 cm" 1 in (TMTSF) 2 Re0 4 and the antiresonances at « 1370 cm" 1 and » 1420 c m - 1 in (TMTSF) 2 BF 4 are due to the vA(ag) mode, which is split as a result of coupling with UCT2 as well as UCTI due to the large Chapter 5. Optical Properties of One-Dimensional Semiconductors 137 value of g4 (450 cm - 1); other ag vibrations with smaller coupling constants do not have large splittings. This does not agree with the results observed in (TMTSF ) 2 Re0 4 and (TMTSF) 2 BF 4 , where the v\\(ag) mode, in particular, is observed to have split into at least three, and possibly as many as seven, components. The nature of the splittings observed experimentally raises an important question about the mechanism by which they occur. In the calculation, it was assumed that D\ = Z?B» but in a more realistic scenario the charge densities on molecules A and B are expected to be different (in fact they are calculated to be = 0.40 and = 0.60). Yartsev has suggested [96] that as a result U > 0 A 7^  W o B i (there is evidence that the frequency of the ag modes depends linearly on the charge density on the molecule [87, 88]). If the frequency of the ag mode is very sensitive to the ionicity of the TMTSF molecule, as is the case for vz(ag), i/ 4(a f l), t/io(ag) and vu{ag) (which have shifts from TMTSF 0 to TMTSF+ of -52 cm"1, -140 cm"1, 41 cm - 1 and 22 cm - 1 respectively [86]), then these vibrations should be split into doublets. There are modes [e.g (^a )^] which are insensitive to the ionicity of the TMTSF and that do not couple strongly to UCT2 which, regardless, show splitting anyway. The v4{ag) mode does split into two components with approximately the right separation, but the splitting of the other ag modes is not reproduced. For both (TMTSF ) 2 Re0 4 and (TMTSF) 2 BF 4 the Lorentzian charge-transfer band at UJCTI cannot reproduce the intensity in the v4{ag) mode and the oscillator strength above « 1700 cm - 1 at the same time. In this way, the tetramer model suffers from the same problems as the dimer model. The dimer and the tetramer model produce reflectivities that are too large above the charge-transfer frequencies and too small at low wave numbers; both the dimer and the tetramer model fail to reproduce the asymmetry in the optical conductivity and both produce conductivities that are too low. This may be due to the fact that neither model takes into consideration the electronic band structure of these materials, the effects of Chapter 5. Optical Properties of One-Dimensional Semiconductors 138 (a) (TMTSF) 2 Re0 4 El la T « 25 K i J L 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) ^ 5000.0 w 4000.0 b 3000.0 -«^> •5 2000.0 § 1000.0 0.0 — 5000.0 § 4000.0 3000.0 > ^ ..—< •5 2000.0 o 1 1000.0 o 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) Figure 5.4: The (a) experimental optical conductivity of (TMTSF) 2 Re04 for E| |a at T « 25 K (introduced in Chapter 3) and (b) the tetramer model optical conductivity calculated using the parameters listed in Table 5.2 with (solid line) and without (dashed line) phonons. Comparing the calculated conductivity with the experimental conductiv-ity, several differences may be observed: the conductivity does not display the observed asymmetry and the Lorentzian charge-transfer bands at UJCT\ and WCTI cannot repro-duce the intensity in the "4(0,) mode and the oscillator strength above w 1700 c m - 1 at the same time. Chapter 5. Optical Properties of One-Dimensional Semiconductors 139 5000.0 S 4000.0 ^3000.0 >^  £ 2000.0 -o 1 1000.0 h o O 0.0 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) _ 5000.0 1 § 4000.0 T 3000.0 > 2000.0 h 13 1000.0 h o o 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) Figure 5.5: The (a) experimental optical conductivity of (TMTSF) 2 BF 4 for E| |a at T « 25 K (introduced in Chapter 3) and (b) the tetramer model optical conductivity calculated using the parameters listed in Table 5.2 with (solid line) and without (dashed line) phonons. Comparing the calculated conductivity with the experimental conductiv-ity, several differences may be observed: the conductivity does not display the observed asymmetry and the Lorentzian charge-transfer bands at UCTI and u>cx2 cannot repro-duce the intensity in the v4{ag) mode and the oscillator strength above « 1700 c m - 1 at the same time. Chapter 5. Optical Properties of One-Dimensional Semiconductors 140 which will be considered in the discussion of the phase-phonon model in the next section. 5.2.3 Phase-Phonon Model The phase-phonon model has been fitted to the optical conductivity of (TMTSF) 2 Re0 4 at T « 25 K and (TMTSF) 2 BF 4 at T « 20 K for the E||a polarization. The results of the fit are shown in Table 5.3. The optical conductivities calculated using the phase-phonon model are shown, along with the experimental conductivity in Figs. 5.6 and 5.7 respectively. Some of the parameters have been adjusted by hand when MINUIT failed to converge within 2000 iterations, and as such the parameters represent an initial refinement only. The phase-phonon model reproduces the asymmetry observed in the optical conduc-tivity of the insulating states of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . Setting T? ^  0 in order to remove the singular behavior near 2A results in values for the conductivity below the gap which are too high and a nonzero value for the d.c. conductivity. If the reflectivity is calculated, this results in the conductivity going to unity at zero frequency rather than to a constant value as would be the case for a semiconductor or an insulator (see Appendix B). One of the novel features of the phase-phonon model is that since it requires a lattice distortion to create a CDW, the Brillouin zone must be reduced. This leads to zone-boundary modes being mapped back to the zone origin, where vibrational modes which are normally inactive may become active. If the mode is subject to a large amount of dispersion within the zone, this may result in a dramatic splitting of the vibrations. In (TMTSF)2Re04 the feature at w 1290 c m - 1 is assumed to be due to the Vi{ag) mode which has been shifted from its bare frequency of 1435 c m - 1 due to the large amount of coupling (<74 = 470 cm - 1), but the feature has been anomalously broadened (74 = 28 cm - 1 ) . The resonance at « 1370 c m - 1 is due to a mode at w 1350 c m - 1 , assumed to be Chapter 5. Optical Properties of One-Dimensional Semiconductors 141 Table 5.3: Phase-phonon model parameters for the fit to the optical conductivity0 for (TMTSF) 2 Re0 4 at « 25 K and (TMTSF) 2 BF 4 at T « 20 K for the E||a polarization. (TMTSF) 2 Re0 4 (TMTSF) 2 BF 4 9800 c m - 1 11624 cm" 1 2A 1775 cm" 1 1087 cm" 1 V 75 c m- 1 45 c m - 1 V/A A(A0) 0.47 0.33(0.18) 0.91 0.085(0.0) Mode u?a (cm - 1) 7c, (cm - 1) ga (cm"1) u>a (cm" *) 7 a (cm - 1) ga (cm"1) "3 + " l l 1845 2 60 1845 2 61 "3 1607 2 77 1607 2 73 1592 2 77 1592 2 72 1548 2 121 1548 2 96 " 4 1425 28 430 1425 12 590 1345 2 187 1340 6 355 "7 1068 2 77 1068 4 84 "8 925 2 104 925 4 70 910 2 139 910 6 134 "9 440 3 128 440 3 114 "11 270 3 211 270 4 163 ° In this calculation A = 105.7 A 2 and d = 3.55 Afor (TMTSF) 2 Re0 4 and 3.51 Afor (TMTSF) 2 BF 4 . The term Ao has been included to account for the effects of low-energy acoustic phonons that are coupling to the CDW. For (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 c s is 46 and 87 respectiviely, and €«, = 2.6 for both materials. an activated branch of the "4(a f l) mode, interfering with itself. This combination yields the right intensity, but even with the anomalously broadened "4(a$) feature it still does not reproduce the shape of the conductivity and the structure between « 1200 — 1500 c m - 1 observed in the insulating state of (TMTSF) 2 Re0 4 . The phase-phonon model describes the conductivity of (TMTSF) 2 BF 4 much better than either the dimer or the tetramer models. The asymmetry in the conductivity is reproduced,'and the structure between 1100 — 1600 c m - 1 is reproduced by assuming a Chapter 5. Optical Properties of One-Dimensional Semiconductors — 5000.0 142 « 4000.0 a 3000.0 -•§ 2000.0 g 1000.0 h 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) 0.0 ^ 5000.0 « 4000.0 b 3000.0 2000.0 o g 1000.0 o 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) Figure 5.6: The (a) experimental optical conductivity of (TMTSF) 2 Re0 4 for E| |a at T « 25 K introduced in Chapter 3) and (b) the phase-phonon model optical conductivity calculated with (solid line) and without (dashed line) phonons using the parameters listed in Table 5.3. The phase-phonon model reproduces the asymmetry observed in the experimental conductivity, but setting r) ^ 0 to remove the singular behavior at 2A results in a{y) ^ 0 below the gap. The split ^(a^) has been shifted down from 1435 c m - 1 due its large coupling constant (470 cm - 1 ) , but the feature has been anomalously broadened (74 = 28 cm - 1 ) ; however, it still does not reproduce the shape of the observed conductivity and the structure between « 1200 — 1500 c m - 1 . Chapter 5. Optical Properties of One-Dimensional Semiconductors 143 7 5000.0 0 o 4000.0 t> 3000.0 >^  *> 2000.0 T3 8 o 1000.0 O 0.0 5000.0 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) o 4000.0 -^ 3 0 0 0 . 0 •J3 2000.0 o 1 1000.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) Figure 5.7: The (a) experimental reflectivity of (TMTSF) 2 BF 4 for E||a at T w 25 K in-troduced in Chapter 3) and (b) the phase-phonon model optical conductivity calculated with (solid line) and without (dashed line) phonons using the parameters listed in Ta-ble 5.3. The phase-phonon model reproduces the asymetry observed in the experimental conductivity. Setting rj ^  0 to remove the singular behavior at 2A results in a{p) ^ 0 below the gap. The split v4{ag) mode is responsible for the two large antiresonances in the 1200 — 1600 c m - 1 range. For u < 2A, the conductivity is not reproduced very well. Chapter 5. Optical Properties of One-Dimensional Semiconductors 144 split "4(a f l) mode with frequencies and intensities similar to those used to fit (TMTSF) 2-Re04- In this case, however, the v4{ag) mode is not anomalously broadened. Near the gap, the model breaks down. The conductivity is far too high ( « 15,000 (ficm) - 1) at the gap and the modest amount (45 cm - 1 ) of damping has resulted in large values for the conductivity below the gap. In both cases the coupling constants are larger than in the dimer model, but comparable to the tetramer model and similar to the known values for TMTTF and TTF [103]. The phase-phonon model calculates the asymmetry in the optical conductivity, but because for rj ^ 0 the conductivity below the gap is non-zero indicates that the interband damping has not been introduced in the correct way. 5.2.4 Model Failures The dimer and the tetramer models fail to reproduce the asymmetry and the magnitude of the E| |a optical conductivity in (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . The phase-phonon model reproduces the asymmetry in the conductivity, but because it does not consider any interband damping it cannot reproduce the electron-phonon coupling or the shape of the conductivity near the gap (attempts to introduce interband damping give rise to large values for the d.c. conductivity below the gap). The asymmetry of the conductivity, however, indicates that the materials are one-dimensional (ID) semiconductors and thus the phase-phonon mechanism is the correct one to pursue. A model which incorporates the phase-phonon ideas, but which is much more rig-oursly developed [69], has proven to be the most useful when examining one—dimensional semiconducting systems where the ag modes are activated by phase oscillations of the CDW, such as the insulating states of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . Chapter 5. Optical Properties of One-Dimensional Semiconductors 145 5.3 One-Dimensional Systems with Twofold-Commensurate Charge—Density Waves A comprehensive model for the optical properties in the near to far infrared of ID molecular conductors (systems of noninteracting molecular-ion chains) with twofold-commensurate CDW's has been developed by Bozio et al. [69]. This model has been successfully applied to the E| |a conductivity of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 be-low the metal-insulator transitions by the author. The important features of this model are as follows: (i) It consists of an assembly of noninteracting molecular chains composed of N molecules (one of these chains is depicted in Fig. 5.8). (ii) Each chain may be subject to various kinds of periodic lattice distortions (PLD's) whose period is bound to be twice the regular chain spacing d (the wave vector of the PLD is qo = ir/d). These distortions may be either intermolecular or intramolecular in nature, i.e., they may correspond to either a lattice dimerization (LD) or an alternating molecular distortion (AMD), whose amplitudes are specified by uo (which is the longitudinal displacement of the molecules from their positions in the undistorted chain) and 8qi (the set of intramolecular nor-mal coordinate displacements onto which the actual molecular distortion is projected) respectively. An important distinction exists between the two types of distortion based on their different symmetry-breaking effects. The LD breaks the inversion symmetry between the molecular sites (site interchange symmetry), whereas the AMD breaks the symmetry on the molecules, (iii) The presence of nearby counterion chains is taken into account by introducing an effective potential Vx = Sxcos(qond) + Bxsm(q0nd) (where n is the site index from 0 to N — 1) which consists of two components; the first with its extremes on the sites, the second with its extremes on the bonds, (iv) The electrons are coupled.to an arbitrary number of internal modes and to one external longitudinal acoustic phonon branch. The reflectivity and the optical conductivity of (TMTSF) 2 Re0 4 Chapter 5. Optical Properties of One-Dimensional Semiconductors 146 Figure 5.8: Schematic drawing of the structure of a single molecular chain. Thick horizontal bars represent the planar molecular units. Their width alternates around a mean value indicated by the dashed lines. This represents a molecular distortion whose projections on the intramolecular normal modes are given by the set of the 6qi amplitudes. it0 is the amplitude of the longitudnal displacement of the molecular centers from their location in the undistorted lattice (open circles) whose period is d. Open and solid triangles indicate nearby counterion chains which generate a periodic potential Vx of the period 2d (after Reference [69]). and (TMTSF) 2 BF 4 which have been measured can be reproduced very accurately using this model, (with uo # 0, Bx ^ 0), and {6qi} = Sx = 0). The parameters used in the fit are in agreement with results from other experiments performed on these materials [60, 15]. The development of Bozio's model follows that of Schulz [97] for the phonon dynamics and infrared properties of commensurate ID conductors with coupling to one acoustic branch only. The model presented by Bozio is more sophisticated than that due to Schulz, but it has some important limitations: (i) Direct electron-electron interactions are ignored, by using a tight-binding model for independent electrons. Modifications of the same calculation allow the model to be extended to the opposite case where a large on-site effective Coulomb term is added and the electrons behave as spinless fermions. (ii) Only interband optical transitions are calculated so that the model gives a complete Chapter 5. Optical Properties of One-Dimensional Semiconductors 147 picture of the infrared properties only for systems where the Fermi level lies within a gap determined by the presence of an LD or an AMD and by the counterion potential. These are two criteria that both (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 satisfy in their insulating states. One of the important features of this model is the formulation of "selection rules" for the infrared activity of the external and the internal phonon modes. Such an activity occurs through the phase oscillations of the CDW components with extremes on the sites (s-CDW) for the former modes and on the bonds (6-CDW) for the latter. The relationship between the vibronic intensities and the amplitude of the CDW will be developed, and its application to the analysis of experimental spectral data allows the nature and the origin of the optical gap in the materials being studied to be determined. The model is developed in detail in Appendix C, and the reader is referred there for a discussion of the parameters used to calculate the complex dielectric function. The final form of the dielectric function is C"(U,) = £~ + - N T T where x ( w ) i s the dielectric susceptibility for the single-particle excitations across the gap and XM + £'aM(3a(")>/£(") (5.1) <Qa(u,)) = D0(u) I £ TaJ,{u)(QfiW) + W c M (5-2) is the average value of the Fourier-transformed coordinate oscillating at the same fre-quency as the applied field is given by the system of coupled inhomogeneous linear equa-tions. Dite(u) are the phonon propagators for the internal and external phonons, Taip(u>) is the matrix of coupling constants and J0(w) is an amplitude for a mode with a, /? = i, e indexing the internal and external vibrations. Finding an analytic solution to the linear system hidden in Eq. (5.1) is a very tedious job for systems of n > 2, but once the equivalence with a linear system of equations Chapter 5. Optical Properties of One-Dimensional Semiconductors 148 A(u>)X(u;) = B(w) is recognized [98], the resort to numerical algorithms allows the solution of the system for large n, where A*p(u) = Sep - D0(u) £ Tap(u) (5.3) 0=1 Ba(u) = E(u)Da(u)Ia(u) (5.4) and Xa{u) = (Q(u)) (5.5) where Q , fi = i , e, as mentioned previously. A(u>) is often nearly singular, therefore numerical attempts to solve the system by inverting the matrix A are prone to failure. The technique used here is LU decomposition [99] (modified for complex arithmetic). The initial solutions to the system of linear equations are then improved iteratively. The iterative improvement may be called several times, but in model simulations comparisons with analytical solutions indicate that only one iteration is required, even for large n. Equation (5.1) describes the optical properties of a distorted chain in which the PLD consists of an LD or AMD or both. The chain is also subject to an external potential which mimics that due to the presence of counterion chains. The electrons are coupled to many intramolecular and one intermolecular (acoustic) phonons. The spectroscopic effects of this coupling are accounted for by the second term in the square brackets in Eq. (5.1). Note that despite the doubling of the phonon branches due to the Brillouin-zone folding, the summation includes the q = 0 phonon of one single branch per coupled mode. This result is different than that of Schulz [Eq. (5.2) of Ref. [97]] in that the latter author lets the summation run over all the branches generated by a single acoustic mode in the reduced zone of the distorted chains of twofold or higher commensurability (e.g., trimerized, tetramerized). In both models, however, the phonons are optically active via phase oscillations of the CDW components if the factor Ia{u) is nonvanishing. Chapter 5. Optical Properties of One-Dimensional Semiconductors 149 Although the model has been developed in the absence of direct electron-electron correlations, numerical calculations will be performed for the case of a strongly correlated (U —* oo) quarter-filled system on account of the fact that a number of intermediate- to high-conductivity charge transfer crystals (such (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 at room temperature) are thought to approach this limit [52]. The required modifications to Eq. (5.1) are simply the omission of the summation over the spin quantum numbers and the use of distribution functions accounting for the modified statistics of spinless fermions [100]. What is calculated in this way is the contribution to the infrared spectra due to interband transitions and phase phonon modes in a half-filled system of spinless electrons or holes. 5.3.1 Application to ( T M T S F ) 2 R e 0 4 The experimental power reflectivity and optical conductivity of (TMTSF) 2 Re0 4 for E| |a at T tn 25 K has been fitted using Bozio's model. The parameters used in the fit are shown in Table 5.4. The structural parameters have been fixed to the literature values [19, 30] and the electronic and vibrational parameters have been fitted by hand, and as such represent a preliminary fit only. The ag modes and their splittings have been fitted. The calculated reflectivity and optical conductivity are shown in Figs. 5.9 and 5.10 respectively. The values for the e — mv coupling constants reported here are larger than those determined using the dimer model [60], but similar to those reported using a preliminary model based on the one employed here but for a quarter-filled, slightly dimerized organic molecular conductor characterized by a narrow optical gaps which has been used to fit four ag modes in (TMTSF)2C104 at room temperature [102]. The values for the #'s are somewhat smaller than the available estimates for T M T T F and TTF [103]. An external phonon at 50 c m - 1 has been included to account for an acoustic phonon branch, but because the energy gap is due to a 6-CDW this phonon is not infrared Chapter 5. Optical Properties of One-Dimensional Semiconductors 150 active. (This agrees with our previous observation that the activated phonon seen in both (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 is a normally active optic mode). The calculated dimensionless coupling constant for the acoustic branch of (d/t)(dt/du)o = 3.77 is well within the range of values physically acceptable for the materials under consideration [101, 54]. The relatively weak LD results in a value for Bx of «' 785 c m - 1 , thus the contribution of the anion potential to the energy gap, which has been fitted to 2A = 1700 c m - 1 , is 92%. The fitted value of 2A = 1700 c m - 1 is larger than that determined by d.c. resistivity measurements, but is consistent with thermopower measurements [60]. The vibrational features in the region of the semiconducting energy gap are sensitive on the size of the gap, thus the value of the gap is accurate to ±60 cm - 1 . The value of the transfer integral of t = 1400 c m - 1 is larger than that predicted for the (TMTSF) 2 X salts by cluster models, which give t « 1200 [63], but smaller than that derived from a simple Drude analyses of the plasma edge in the reflectivity t « 1600 — 2000 c m - 1 [60]. In Figs. 5.9 and 5.10, the calculated spectra with electron-phonon coupling is indi-cated by a solid line, while the dashed line shows the reflectivity or conductivity of a one-dimensional band in the absence of electron-phonon coupling. The agreement be-tween the experimental and model reflectivities and conductivities is excellent below the semiconducting energy gap, but the model yield values for the conductivity that are high above the gap. The large resonances at w 1290 c m - 1 and « 1370 c m - 1 below the gap in the conduc-tivity are due to the splitting of the v^Cg) mode into two lines at 1345 c m - 1 and 1430 c m - 1 with coupling constants of <j4t0 = 160 c m - 1 and g^b = 420 c m - 1 respectively. In order to reproduce the observed data, these two features have larger damping, 3 c m - 1 and 8 c m - 1 respectively, than the other modes in the spectrum (which all have damping of 1 c m - 1 or 2 cm - 1 ) . This effect is a consequence of the large degree of coupling that this Chapter 5. Optical Properties of One-Dimensional Semiconductors 151 Table 5 . 4 : One-dimensional twofold-commensurate CDW model parameters for a 6-CDW for the fit to the optical conductivity of (TMTSF) 2Re0 4 for E||a at T = 20 K° d 3.550 A t 1400 cm - 1 UQ 0.022 A 2A 1700 cm"1 vm 341 A 3 r 100 cm"1 M 7.437 x 10 - 2 3 kg Coo 2.6 9a 6« Mode (cm - 1) (cm - 1) (cm"1) (cm"1) u3(ag) + vu(ag)l 1845 1 40 0.3 vz{ag) 1605 1 50 0.5 1595 1 30 0.2 1550 1 90 1.5 1430 8 420 33.0 1345 3 160 4.8 Mag) 1365 1 30 0.2 v 7 ( a g ) 1065 1 50 0.5 v8(ag) 921* 1 140 3.7 914 1 55 0.6 905 1 40 0.3 v9{ag) 448 1 110 2.3 438 1 70 0.9 vn(ag) 288 1 50 0.5 284 1 55 0.6 279 1 60 0.7 274 1 150 4.2 264 1 65 0.6 257 1 50 0.5 ° In this calculation the number of divisions in the BZ was 256 and the temperature was set at 25 K. A lattice mode at 50 cm - 1 has been included to account for acoustic phonons. The vibrational assignments for the ag modes are discussed in detail in Chapter 5. Chapter 5. Optical Properties of One-Dimensional Semiconductors 152 mode displays. The conduction electrons couple through the modulation of the charge distribution that results from the change in the C = C bond (a region of high electron density [25]) due to the v4{ag) vibration. By coupling to the vibration, the conduction electrons may also act as scatterers thereby reducing the lifetime of the vibration and increasing the damping. This observation is consistent with other vibrations that have large coupling constants, but none of them have a coupling constant as large as the Vi{ag) doublet. Most of the ag modes are split into doublets, but some modes are not split and other modes appear to have many (> 3) components. While the splitting of the v4{ag) mode is clearly visible in Figs. 5.9 and 5.10, much of the fine structure associated with the splitting of the other ag modes is difficult to observe because of the wide wave number range. The model fit to the v\i(ag) mode is shown from 200 c m - 1 to 350 c m - 1 in Fig. 5.11. The group of vibrations is characterized by one strong feature at « 274 c m - 1 with a coupling constant of 150 c m - 1 , while the other five vibrations have much smaller coupling constants (« 60 cm - 1 ) . The dashed line in Fig. 5.11 shows just the strong central feature. The splittings are much weaker than the fundamental at 274 c m - 1 , but because it is shifted down by w 4 c m - 1 , the two lower vibrations interfere with it very strongly to produce the multiplet structure observed. The coupling and interference effects produced by the splitting tends to mask the true intensity of the central feature in the reflectivity, as the dashed line demonstrates. The behavior of the optical conductivity produced by the " 9 ( 0 , ) doublet is a good example of how direct measurements of peak position and intensity (see Chapter 4, Tables 4.5 and 4.6) can be misleading. While the optical conductivity indicates two peaks of similar intensity, this structure is actually the product of a weak splitting coupling to the stronger fundamental (as indicated in Table 5.4). Thus, the model is invaluable for providing a true insight as to the positions and relative strengths of the ag vibrations. Chapter 5. Optical Properties of One-Dimensional Semiconductors 153 0.0 500 .0 1000 .0 1500.0 2 0 0 0 . 0 W a v e N u m b e r ( c m - 1 ) Figure 5.9: The model reflectivity for (TMTSF) 2 Re0 4 at T w 25 K calculated for a 6-CDW using the parameters listed in Table 5.4. The calculation has been performed with (solid line) and without (dashed line) electron-phonon coupling to an fc-CDW with 2A = 1700 cm" 1 (indicated by the arrow). Chapter 5. Optical Properties of One-Dimensional Semiconductors 154 5000.0 4000.0 6 b • i—I > • r-( O o O 3000.0 -2000.0 1000.0 0.0 Commensurate CDW Model 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm-1) Figure 5.10: The model optical conductivity for (TMTSF) 2 Re0 4 at T « 25 K calcu-lated for a 6-CDW using the parameters listed in Table 5.4. The calculation has been performed with (solid line) and without (dashed line) electron-phonon coupling. The large resonance below the gap (2A = 1700 cm - 1 ) is due to the i/4(ag) mode at « 1430 c m - 1 with </4 = 420 c m - 1 , interacting with the edge of the gap. Chapter 5. Optical Properties of One-Dimensional Semiconductors 155 1.0 0.8 h X 0.6 * 0.4 0.2 0.0 (a) (TMTSF)2Re04 T « 25 K El la 200.0 250.0 300.0 Wave Number (cm-1) 350.0 200.0 250.0 300.0 Wave Number (cm-1) 350.0 Figure 5.11: The (a) high-resolution experimental reflectivity for the v\\(ag) mode in (TMTSF) 2Re0 4for E| |a at T = 25 K and (b) the model reflectivity calculated from the model by fitting six vibrations to the "n(a9) mode for a 6-CDW. The resolution of the experimental data is 0.2 c m - 1 . The dashed line indicates just the fundamental. The splittings are much weaker than the fundamental at 274 c m - 1 , but because it is shifted down by « 4 c m - 1 , the two lower vibrations interfere very strongly to produce the multiplets observed. Chapter 5. Optical Properties of One-Dimension al Semiconductors 156 This model reproduces the asymmetry in the conductivity while preserving the shape of the conductivity near the gap, something which other models that have been applied to linear organic conductors (described in Appendix B) fail to do. The experimental conductivity above the gap is, however, lower than that which is calculated, and it does not obey the l/\/e behavior expected of a one-dimensional density-of-states above the energy gap. This may be due to the fact that the band structure has been assumed to be simple cosine bands. The true band structure near the zone origin may have a larger energy difference than 2yjAt2 + A 2 , which would result in lower conductivities at high wave numbers due the energy difference occuring in the denominator of the dielectric susceptibility. The model reproduces the vibronic due to the ag modes observed in the reflectivity (and optical conductivity) very well, as well as the bulk features. 5.3.2 Application to ( T M T S F ) 2 B F 4 The experimental power reflectivity and optical conductivity of (TMTSF) 2 BF 4 for E||a at T fa 20 K has been fitted using Bozio's model. The parameters used in the fit are shown in Table 5.5. The structural parameters have been fixed to the literature values [20, 31]. The electronic and vibrational parameters, except for the semiconducting energy gap, have remained essentially unchanged except for some small changes in the frequencies (less than 2%) and the coupling constants (less than 10%). The transfer integral and the interband damping have not been changed. The calculated reflectivity and optical conductivity are shown in Figs. 5.12 and 5.13 respectively. As in the previous case, n external phonon at 50 c m - 1 has been included to account for an acoustic phonon branch, but because the energy gap is due to a 6-CDW this phonon is not infrared active. The calculated dimensionless coupling constant (d/t)(dt/du)0 = 4.29. The relatively weak LD results in a value for Bx of fa 485 c m - 1 , thus the contribution of the anion potential Chapter 5. Optical Properties of One-Dimensional Semiconductors 157 to the energy gap, which has been fitted to 2A = 1120 c m - 1 , is 87%. The fitted value of 2A = 1120 c m - 1 is consistent with d.c. resistivity measurements [14]. The vibrational features are very senstive on the size of the gap (more so than in (TMTSF) 2 Re0 4 bacause of the larger number of vibrations in the region of the semi-conducting energy gap); changing the size of the gap by as little as 20 c m - 1 produces a much different result, thus the value of the gap is accurate to ±40 c m - 1 . The solid lines in Figs. 5.12 and 5.13 show the spectra for electron-phonon coupling, while the dashed lines show the spectra for a one-dimensional band in the absence of any electron-phonon coupling. The calculated reflectivity is almost identical to the experimental reflectivity. In par-ticular the split " 4 ( 0 , ) structure is reproduced quite well. The experimental reflectivity goes from w 80% above the v4{ag) features to w 90% just below, before slowly decreasing. The agreement between the experimental and model conductivities is excellent; better, in fact, than it was for (TMTSF)2Re04. Above the semiconducting energy gap, the calculated and experimental conductivity both obey the same 1 /y/e behavior indicative of one-dimensional energy bands. The split v4{ag) modes, at 1340 c m - 1 and 1430 c m - 1 , which appeared as resonances below the semiconducting gap in (TMTSF) 2Re04 are seen to be antiresonances in (TMTSF)2BF4 where they occur above the semiconducting gap. The coupling constants have not changed. This sort of behavior is a trademark of the "phase-phonon" model. As in the case of (TMTSF) 2Re04, these features are also broader than the other activated modes, with damping of 3 c m - 1 and 8 c m - 1 respectively. This model successfully reproduces highly asymmetric conductivity while preserving the shape of the conductivity near the gap, something which other models that have been applied to linear organic conductors (described in Appendix B) fail to do. The v&(ag) mode was split into three vibrations in (TMTSF) 2 Re0 4 , but the <v3(/2) anion vibration masked this effect. In (TMTSF) 2 BF 4 , the anion mode has shifted up to Chapter 5. Optical Properties of One-Dimensional Semiconductors 158 Table 5.5: One-dimensional twofold-commensurate CDW model parameters for a 6-CDW for the fit to the optical conductivity of (TMTSF) 2 BF 4 for E||a at T = 25 d 3.510 A t 1400 c m - 1 "0 0.022 A 2A 1120 cm" 1 vm 334 A 3 r 100 c m - 1 M 7.437 x 10~23 kg Coo 2.6 W a 7a 9a Mode (cm - 1) (cm - 1) (cm - 1 ) (cm"1) u3(ag) + vn(ag) 1840 1 40 0.4 1606 1 50 0.5 1586 1 50 0.5 1551 1 80 1.2 1430 8 420 32.2 1345 3 180 5.9 u6(ag) 1365 1 30 0.2 v7(ag) 1071 1 35 0.3 uB{ag) 939 1 130 3.1 927 1 55 0.6 913 1.5 75 1.0 v9{ag) 440 1 110 2.2 vn{ag) 288 1 50 0.5 284 1 55 0.6 279 1 60 0.7 272 1 120 2.2 260 1 75 1.0 253 1 70 0.9 0 In this calculation the number of divisions in the BZ was 256 and the temperature was set at 25 K. A lattice mode at 50 c m - 1 has been included to account for acoustic phonons. The vibrational assignments for the ag modes are discussed in detail in Chapter 5. Chapter 5. Optical Properties of One-Dimensional Semiconductors 159 1.0 0.9 0.8 0.7 ft?. ^ 0.6 > • 1 0.5 o <D OH 0.4 0.3 0.2 0.1 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1) Figure 5.12: The model reflectivity for (TMTSF) 2 BF 4 at T nt 20 K calculated for a 6-CDW using the parameters listed in Table 5.5. The calculation has been performed with (solid line) and without (dashed line) electron-phonon coupling to an 6-CDW with 2A = 1120 c m - 1 (indicated by the arrow). Chapter 5. Optical Properties of One-Dimensional Semiconductors 160 5000.0 Figure 5.13: The model optical conductivity for (TMTSF) 2 BF 4 at T m 20 K calculated for a 6-CDW using the parameters listed in Table 5.5. The calculation has been per-formed with (solid line) and without (dashed line) electron-phonon coupling. The large resonance below the gap (2A = 1700 cm"1) is due to the vA{ag) mode at « 1430 c m - 1 with 04 = 420 c m - 1 , interacting with the edge of the gap. Chapter 5. Optical Properties of One-Dimensional Semiconductors 161 5000.0 I § 4000.0 "b 3000.0 !> 2000.0 o p c 1000.0 o o 0.0 _ 5000.0 1—* I 6 o O O 0.0 ( T M T S F ) 2 B F 4 El la T « 2 0 K 800.0 850.0 900.0 950.0 Wave Number (cm-1) 4000.0 -b 3000.0 > 2000.0 -1000.0 -800.0 850.0 900.0 950.0 Wave Number (cm-1) Figure 5.14: The (a) experimental optical conductivity for the v%{ag) mode in (TMTSF) 2 BF 4 for E| |a at T = 20 K and (b) the optical conductivity calculated from the model by fitting three vibrations to the fs(ag) mode for a 6-CDW. The resolution of the experimental data is 2 cm" 1. The dashed line indicates just the fundamental. The splittings are much weaker than the fundamental at 939 cm" 1 , but couple very strongly to the conduction electrons because of their proximity to the gap. Chapter 5. Optical Properties of One-Dimensional Semiconductors 162 w 1070 c m - 1 and the v&(ag) mode is now clearly visible as three lines [see Fig 5.14(a)]. The structure that these three lines produce is complicated due to the fact that they lie so close to the edge of the gap, but the model reproduces this complex optical conductivity extremely well. The results of the fit to the v&(ag) mode is shown in Fig. 5.14. Once again, the group consists of a strong fundamental at 939 c m - 1 , and two weaker splittings at 927 c m - 1 and 913 c m - 1 . The frequencies of some of the ag modes are slightly different in the two materials. The structure of the TMTSF molecule in the two materials is virtually identical. The anion potential, reflected by the size of the semiconducting energy gap, has changed a great deal in the two materials. The anion potential may therefore affect the molecule and produce the slight differences observed. This model successfully reproduces highly asymmetric conductivity while preserving the shape of the conductivity near the gap, something which other models that have been applied to linear organic conductors fail to do. 5.3.3 Conclusions Unlike other models, in which many of the parameters would have to be refit in order to reproduce the reflectivity or the conductivity, the only parameter that has been altered to any significant degree is the size of the semiconducting energy gap. The vibrational parameters are not expected to change [since the structure of the TMTSF molecules does not change from (TMTSF) 2 Re0 4 to (TMTSF) 2 BF 4 ] , nor are the other electronic para-meters such as the transfer integral or the interband damping since the overall structure in the stack direction for the two materials is very similar. The major differences arise from the type of anion, and therefore in the nature of the anion potential. It is reasonable to expect that the system may be parameterized in terms of the semiconducting energy gap, as this model has shown. Chapter 5. Optical Properties of One-Dimensional Semiconductors 163 5.4 Optical Conductivity for E||b' The model discussed in the previous section provides a detailed understanding of the optical properties and the nature of the intermolecular and intramolecular electron-phonon coupling in the stack direction. It does not, however, offer any insight into the optical properties transverse to the stack for the E||b' polarization. The optical conductivity for E||b' is particularly interesting because of the fact that in both materials the conductivity is suppressed below 2A a , it then rises slowly above 2A„ but does not reach a maximum for another « 350 c m - 1 . An attempt to understand the optical conductivity for E||b' may be made by calculat-ing the optical properties from the two-dimensional band structure of the (TMTSF)2^ salts [53] using the method of Eldridge et al. [104] (this method does not allow for any electron-phonon coupling). An expression for the frequency-dependent conductivity, a,j(u>), in the i direction when E || j , is the Kubo-Greenwood formula [105], given by *[<w)l = v E"(**)[l - n(cy)][-ftk | V,- | *l>k.){xl>k, 1 V,- | V*)] x [6(ek> -ek- nu) - 6{tk. - ek + hu)] (5.6) where V is the volume of the unit cell, m* is the effective mass, ek is the electron energy at wave vector k and n(ek) has been previously defined as the Fermi-Dirac distribution. For simplicity, T = 0 has been assumed so that the valence bands with energy ck are occupied and the conduction bands with energy tk> are unoccupied. Assuming only vertical transitions (k = k'), and considering only conductivity parallel to the electric field, one obtains = J I (V>k ! V i | T M I2 6{ek, -ek- hu>)dk. (5.7) The CDW couples electrons with a wave vector k to those with wave vector k + Q, where Q is the single wave vector of the low-temperature superlattice due to the CDW. Chapter 5. Optical Properties of One-Dimensional Semiconductors 164 Writing (r | *k) = a k e ' k r + fe^^' (5.8) and (r | ipk>) = 6ke*'r - a k e , ( k + Q ) r (5.9) one obtains *[*«(«)] = / 4 ^ ( 4 - e* - M ^ - (5-10) The integral is the combined density of states for vertical transitions and it should be noted that it is independent of the polarization i . The polarization enters through Qi, the component of Q along the E field, which will affect the magnitude of <ru{u>), but mainly through the effective mass m* which will have a considerable effect on the magnitude of <T,,(W) and will also influence the shape. The electronic energy bands for the for the low-temperature structure of the (TMTSF^A" salts have been calculated by Grant [53] for the case in which the there is a doubling of the direct lattice in both the a and b directions, corresponding to a Q = ( | , | ) . Solving the 4 x 4 determinant gives for the energy e(k) [e(k) 2 - A 2 ] 2 - A[t(k) 2 - A 2 ] — B = 0 (5.11) where A = 4*2cos(k.a) + 4^cos(k-b) + 4(*2 + <2J, (5.12) B = 8(^2-Ocos(k-a) + 8(*^-^)co8(k.b) +4t2atl cos[k • (a + b)] + U\t\ cos[k • (a - b)] -2t\ cos(2k • a) - 2t\ cos(2k. b) + St\tl - 6t4a - 6<{, (5.13) where 4ta and U are the bandwidths. The value of the transfer integral calculated by Grant using band structure methods is ta « 2930 c m - 1 [53], but the fitted values from Chapter 5. Optical Properties of One-Dimensional Semiconductors 165 the previous section yield a lower value ta « 1400. c m - 1 . The transfer integral in the transverse direction was taken assuming that ta/h « 10, so that tt ~ 140 c m - 1 . The energy bands obtained for ta = 1400 c m - 1 , h = 140 c m - 1 and 2A = 1120 (values appropriate for (TMTSF) 2 BF 4 ) are shown in Fig. 5.15. Also shown in Fig 5.15 is one--half of the reduced Brillouin zone, showing the symmetry points, and relevant angles and vectors. The position of the energy gap is drawn as a "trench" from V to either side of X. When computing a according to Eq. (5.10), the optical conductivity for E| |b' was obtained by integrating over that part of the BZ containing the trench (i.e., where the energy was 5000 c m - 1 or less, as shown in Fig. 5.16), in the b' direction, so that the effective mass m\, could be calculated from with k parallel to b'. The effective mass for E||b' is much larger than that for E||a. Values of a = 7.297 A, b = 7.711 Aand 7 = 70.01° were used to calculate the zone boundaries. (These give 0 = 33.3°). In order to give a sufficiently smooth result, a wave-vector grid of approximately 50 x 108/zone was required when performing the numerical integration of the Brillouin zone. The resulting conductivity is shown in Fig. 5.16. The results are not very good. The conductivity rises much more quickly above 2A than is observed experimentally, where the conductivity is seen to rise linearly above the gap. The maximum occurs at w 1500 c m - 1 , but it is seen at w 1350 c m - 1 in the E||b' conductivity of (TMTSF) 2 BF 4 at « 20 K. The difficulty in using this technique to calculate the conductivity lies in the effective mass, which is accurate only when the energy bands are parabolic. This is appropriate near the gap where the bands are reasonably approximated by paraboloids, but optical transitions that occur at higher wave numbers do so in a region where the energy bands are flat. Thus, the approximation for m* breaks down. The result in this case is that the shape of the conductivity above (5.14) Chapter 5. Optical Properties of One-Dimensional Semiconductors 166 4000.0 0.0 • ' 1 ' 1- ' — 1 1 1 • r x v Y r Figure 5.15: Electronic energy, e(fc), bands [53] for a distortion wave vector Q = (|, | ) . Input values of t a = 1400 c m - 1 , ij, = 140 c m - 1 and 2A = 1120 (appropriate for (TMTSF)2BF4) were used. The valence bands are obtained by reflecting the energy bands about the Fermi energy Ep. Half of the reduced Brillouin zone, drawn to scale, is also shown, with the relevant vectors and angles. Chapter 5. Optical Properties of One-Dimensional Semiconductors 167 CO -4-3 o o o 0.0 1000.0 2000.0 3000.0 Wave Number (cm-1) 4000.0 Figure 5.16: Conductivity calculated for E||b' for direct, transitions using the energy bands in Fig. 5.15 and effective masses from (d2t/dk?)~l. Constant arbitrary values have been assumed for a* and 6*. The initial histogram has been smoothed using a 17 point smoothing function. Chapter 5. Optical Properties of One-Dimensional Semiconductors 168 the gap does not have the characteristic l/y/e shape; it does not decrease quickly enough. While the result is unsatisfactory, the maximum in the conductivity is shifted signifi-cantly above the gap due to the large effective mass for E||b', (an effect that was observed to a lesser extent in the (TMTSF) 2 X narrow-gap semiconductors [104]) indicating that the energy gap can be isotropic and still produce the observed behavior. Chapter 6 Conclusions 6.1 Optical Properties and Vibrational Assignments The organic conductors (TMTSF^ReO^ and (TMTSF)2BF4 are very similar materials; both undergo metal-insulator transitions (at 177 K and 39 K respectively) and become semiconductors. The metal-insulator phase transition results in a new low-temperature unit cell of 2a x 26 x 2c in both materials. A normal coordinate analysis was performed on protonated and deuterated T M T S F 0 , + , and a linear interpolation was performed to extract the frequencies and the deuterium shifts for T M T S F + 0 5 . Vibrational assignments were made on the basis of optical polar-ization, frequency and the deuterium shift for the ag, biu and b2u vibrations. Al l of the ag vibrations appear to have been observed. A phenomenon referred to as "intensity trans-fer" between the symmetric and the antisymmetric modes was observed for the i/4(afl).. vibration, and the ungerade modes that were degenerate in energy with it: the "46(&2u) and i/47(&2u) modes. The optical properties of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 are dominated in the E| |a polarization by the totally symmetric ag modes, particularly by the " 4 ( 0 , ) vibration, which involves the modulation of the central carbon-carbon bond. A survey of the various types of models used to interpret the spectra of linear-chain organic conductors and 169 Chapter 6. Conclusions 170 semiconductors was performed and the one-dimensional twofold-commensurate charge-density wave model by Bozio et al. [69] was determined to yield the best explanation of the optical properties of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 . A transfer integral of t = 1400 c m - 1 was fit to both materials, and semiconducting energy gaps of 2A = 1700 c m - 1 and 2A = 1120 were calculated for (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 respectively. The "4(a f l) vibration was split into two components at 1345 c m - 1 and 1430 c m - 1 at low temperature due to the reorganization of the unit cell below the phase transition, and displayed the largest electron-molecular-vibrational e-mv coupling constants, « 170 c m - 1 and 420 c m - 1 , respectively. The optical properties for the E||b' polarizations are explained using a two-dimensional anisotropic band structure due to Grant [53]. A summation over the Brillouin zone yields the shape of the conductivity for E||b', but the approximation of the band mass leads to the result that the shape of the conductivity is poorly reproduced. 6.2 Applications The successful calculation of the optical properties of the semiconducting states of (TM-TSF) 2 Re0 4 and (TMTSF) 2 BF 4 indicates that Bozio's model may be used to calculate the optical properties of other one-dimensional linear-chain organic semiconductors. From these calculations the transfer integral, semiconducting energy gap and the e-mv coupling constants (all of which are important electronic parameters) may be determined. One of the strengths of Bozio's model is that it accurately reproduces the potential produced by the anion chains. Different anion substitutions are known to have a profound effect on the electronic properties of the (TMTSF)2.Y salts. By studying different systems it may be possible to correlate the size of the gap with the anion potential. One promising candidate for study is (TMTSF) 2N03. Chapter 6. Conclusions 171 Understanding the role that the anions play in lattice instabilities and the formation of semiconducting energy gaps is also important in the search for a high-temperature organic superconductor. The suppression of a semiconducting energy gap is essential if a superconducting ground state is to be achieved. Appendix A Normal Coordinate Analysis A.l Introduction The principles used in normal coordinate analysis (NCA) to determine the normal modes of vibration of the T M T S F 0 , + molecules in Chapter 4 are described in detail in this Appendix. Among the topics discussed are the formulation of the secular equation, the internal valence, symmetry and normal coordinates, and the G and the F matrices. The potential energy distribution and its importance will also be discussed. The force constants and the symmetry coordinates used in the NCA calculation are listed. A.2 Secular Equation The secular equation is determined by the Lagrangian, thus expressions for the kinetic and potential energies T and V are required. Cartesian coordinates are useful because of the simple way in which the kinetic energy can be written. In general, it is convenient at this stage to introduce mass-weighted coordinates 9,-, where 91,92,93, • • • » 9 3 J V refers to y/rh~[Axi, y/fh~i&yi, v /mTAzi, • • •, ^ /m^Azjv, where mj is the mass of the t t h atom, and the i i , y i , z i , - • • are the cartesian coordinates of the atom. The kinetic energy may be 172 Appendix A. Normal Coordinate Analysis 173 written as 3N 2T = y £ q > (A.l) »=i where «jt- is the velocity of the t t h atom along one of the cartesian coordinates. The analytical form of the potential energy V of the system is unfortunately not known. The only information available on it is that it must be some function of the displacement coordinates. Under these conditions, for small displacements, the potential energy may be expanded in a power series of the displacement coordinates 2V = 2V0+2 £ — qt+ ]T ^ - T - q i q j + - J2 *«,•«*+• • • (A.2) Since the absolute value of the potential energy is not known, we can shift the zero of the energy scale so that the energy of the equilibrium configuration Vo is zero. Further, since the equilibrium configuration is by definition at a minimum of the potential energy, and the qi are all independent, it must be true that For small excursions of the atoms around their equilibrium positions, the higher order terms of equation (A.2) can be neglected and the potential becomes 3N 2V=J2 fijqiqj. (A.4) In this equation, the terms fij are constants which are normally called "force constants" since they represent the proportionality factors between the displacements of the nuclei and the restoring forces acting on them. There are in principle 3N(3N + l ) /2 force constants since, V being a continuous function, the order of differentiation is immaterial, thus fii = fa (A.5) Appendix A. Normal Coordinate Analysis 174 Because T is a function of the velocities only and V is a function of the displacements only, the equation of motion in Lagrangian form for a vibrating molecule may be written *©* * - • The expression for the kinetic energy is particularly simple when expressed in Carte-sian coordinates. It is customary to express the interatomic potential V in terms of interatomic bond lengths and bond angles; these are the internal valence coordinates. The vibrational analysis may thus be carried out by converting T into internal valence coordinates [106]; or by converting V into cartesian coordinates [107]. The former ap-proach is used here, thus it is necessary to define a system of internal valence coordinates and to express T in terms of them. A.2.1 Internal Valence Coordinates Changes in the bond lengths and in the angles between chemical bonds provide the most significant and physically meaningful set of coordinates for the description of the potential energy. They are called "internal coordinates" because they describe (being unaffected by translations and rotations of the molecule) just the internal motions of the molecule, or the molecular vibrations. The types of internal coordinates which are generally used in vibrational problems are shown in Fig. A . l . The calculations that will follow, however, use only the bond stretching coordinate (variation in length of a chemical bond) and the in-plane bending coordinate (variation of the angle between two chemical bonds having on atom in common). A molecule has ZN — 6 internal degrees of freedom (ZN — 5 for a linear molecule). The number of internal coordinates chosen is seldom equal to the number of internal degrees of freedom. In general, a certain number of redundant coordinates are included, and the number of redundancies increases rapidly with the molecular dimensions [108]. Appendix A. Normal Coordinate Analysis 175 Figure A . l : Types of internal valence coordinates, (a) Bond stretching; (b) valence an-gle bending; (c) out-of-plane wag; (d) torsion; (e) bending (linear connection in-plane); (f) bending (linear connection out-of-plane) (after Reference [85]). A set of M internal valence coordinates Ri,R2,R3,- • • ,RM is introduced such that a transformation B exists Rt = ^Buxi, (A.l) i=l where Rt is the tth internal coordinate and x, is the ith coordinate, or R = B x in matrix form where R and x are column matrices, also R = Dq (A.8) where q is the column matrix of the mass-weighted coordinates, it follows that (A.9) Thus, the kinetic energy from (A.l) may now be written as 3JV 2r = £ 9 2 = qtq- (A.10) «=i Appendix A. Normal Coordinate Analysis 176 It is not possible to make the substitution q = D - 1 R into (A.10), because D is usually not square and therefore has no inverse. Converting to momentum coordinates gives B - g - i . <*•»> thus 2T = pfp. (A.12) The internal valence coordinates may be introduced through the relation dT _ dTdkt Oqj t dRt Oq, Pi = 32PtDti (A.14) t or ptrrPD and p = D+P (A.15) 3 N 3 N 3 N = 712 BtjmJ1 By; = G«. *,*' = 1,2,3,- .- ,M (A.16) thus 2T = P fGP. (A.17) The elements Gtt< are referred to as the elements of the G matrix. It may further be shown that [84] 2T = R+G^R. (A.18) Appendix A. Normal Coordinate Analysis 177 The G matrix may therefore be be used to change the kinetic energy from cartesian coordinates into internal coordinates. The potential energy may be written 2V = £ FwRtRf = R f F R (A.19) w The form of the secular equation is det |F — G _ 1 A | = 0. Because the inverse of the G matrix is a tedious calculation, it is easier to mulitply by the determinant of G , thus giving the final form of the secular equation det | G F - AE| = 0. (A.20) where E is the identity matrix and A is an eigenvalue of the system. Solving the secular equation requires the determination of the G and F matrices using internal valence coordinates. A.2.2 G Matr ix The elements of the G matrix are defined in Eq. (A. 16) as 37V 'Gtv^BtjmfBvj (A.21) where B relates the cartesian to internal coordinates. The computation of the elements of G is facilitated by a number of simplifications and tabulations [84] and can be accom-plished with great convenience employing computers whose requisite input is just the equilibrium molecular geometry, the masses, and the coordinate types. The symbolic designation of the 73 internal coordinates used in the normal coordinate calculation of TMTSF is shown in Fig. A.2. In this case the number of internal coordinates is slightly larger than the number of internal degrees of freedom. The internal coordinates all lie in the plane of the molecule, with the exception of the U, a, and the /?,- (i = 1,2,3) internal coordinates associated with the methyl groups. As a result, the normal modes of Appendix A. Normal Coordinate Analysis 178 Figure A.2 : The symbolic designation of the 73 internal valence coordinates used in the normal coordinate calculation of TMTSF for; (a) the TMTSF molecule (i==l,2,3); (b) the methyl group in the upper right quadrant; (c) the methyl group in the lower right quadrant; (d) the methyl group in the lower left quadrant and (e) the methyl group in the upper left quadrant (after Reference [86]). Appendix A. Normal Coordinate Analysis (*) 179 0>) G\r Figure A.3: Schematic representation of the G matrix elements. The double circles represent the atoms common to each internal coordinate. Each G matrix element relates two internal coordinates; (a) Gff.r relates a bond stretch with itself; (b) G\R relates two bond stretches; (c) G2.^ relates a bond stretch with an angle bend and (d) G^ relates an angle bend with itself (after Reference [84]). vibration will be in plane, except for some out-of-plane vibrations involving the methyl groups. A more general way of writing an element of G for a bond stretch or an angle bend is GRR, where R\ and R2 denote the internal coordinates involved and n denotes the number of atoms common to both coordinates, as shown in Fig. A.3. The elements are mi 77l 2 G j r = —cosfa) mi m2 Gr<t, = — 1 1 m 2 rf 2 m\r\2 + m 3 r | 3 + f4. +4. + 2£-JflY \r\2 r23 r12r23 / . (A.22) (A.23) (A.24) (A.25) Appendix A. Normal Coordinate Analysis 180 Table A . l : The bond lengths and bond angles for the TMTSF molecule used to construct the cartesian coordinates0. Bond Lengths Bond Angles Bond Length(A) Bond Angle (deg) C(l)-C(6) 1.89 C(6)-C(l)-Se(2) 123.2 C(l)-Se(2) 1.91 C(l)-Se(2)-C(3) 93.9 Se(2)-C(3) 1.45 Se(2)-C(3)-C(5) 114.4 C(3)-C(5) 1.08 C(3)-C(5)-H 109.5 H-C(5)-H 109.5 ° The labels used to define the bond lengths and the angles are defined in Chapter 1 in Fig. 1.3. The internal valence coordinates were assigned from the cartesian coordinates which were calculated using the values for bond lengths and the bond angles for the TMTSF molecule listed in Table A . l . The structure was determined by averaging several struc-tures of the TMTSF molecule for different anions [17, 18, 19, 20]. The methyl groups were assumed to be perfect tetrahedrons frozen in place with the edges pointing out (as shown in Fig. A.2), and the C-H bond length was taken to be 1.08 A. Only one quadrant of the molecule was calculated using the values in Table A . l , the result is reflected to form the complete molecule with D2h symmetry. There are 26 atoms in the TMTSF molecule giving 72 internal degrees of freedom. The masses used are the atomic masses for pure isotopes, rather than the average atomic weights. A.2.3 F Mat r ix The F matrix is the matrix of force constants. The forces considered resist the extension or compression of valence bonds, together with those that resist the bending or torsion of bonds; forces between nonbonded atoms are not directly considered. The rows and columns of the F matrix are indexed by the internal coordinates. The diagonal elements Appendix A. Normal Coordinate Analysis 181 Table A.2: Valence force constants for TMTSF and its cation. Coordinate0 TMTSF 6 Coordinate" TMTSF" Symbol involved force constant Symbol involved force constant Ki R2R2 7.86 (6.26) Fi R2Sl 0.330 K2 Ri Ri 8.62 (7.79) F2 Ris2 0.304 K3 SiSi 2.94 (3.25) F3 SiS4 0.181 K4 S2S2 2.99 (3.08) F4 R^i 0.237 K5 r i r i 4.47 Fs s2ri 0.208 K6 UU 4.63 Fe R26i 0.100 Hi <f>i<f>i 0.414 (0.418) F7 8i<f>i = Si<j)2 0.409 H2 <t>2<f>2 1.361 (1.541) F8 S2<f>2 0.413 Hz <f>3<j>3 1.100 (0.978) F9 &2<t>3 0.684 H4 SiSi 0.700 (0.710) Fio Rl<f>3 0.627 H5 7*7* 0.749 (1.002) Fu S2Kl 0.685 He a, a, 0.511 Fi2 Rl12 0.180 H7 m 0.651 Fi3 s\Si 0.294 Fu r\l2 0.471 Fi5 n i l 0.528 Fie rift 0.373 Fir 7273 0.129 Fis 0.022 0 t,j = 1,2,3; k = 1,2. 6 The modified values of the diagonal force constants used for the radical ion are reported in parenthesis. Stretch constants are in units of N m _ 1 x 10 - 2 ; stretch bend, N rad - 1 x 108; bend N m rad - 2 x 101 8. refer to the interaction of a force constants with itself, while the off-diagonal elements refer to the interaction between different force constants. The force constants have differ-ent dimensions depending on their type. The units for bond stretching interactions are N m - 1 ; for angle bending interactions N m r a d - 2 and for bond stretch and angle bending interactions the dimensions are Nrad - 1 . When considering a new structure, it is convenient to use general force constants that are associated with a specific type of chemical bond. It is found that force constants Appendix A. Normal Coordinate Analysis 182 that occur are at least roughly characteristic of the type of bond [e.g. carbon-hydrogen bond (C-H) or a double carbon bond (C = C)], regardless of the molecule in which it occurs. These force constants yield the approximate frequencies and the subsequent addition of interaction force constants and refinement yields force constants appropriate for that molecule. The success of transferring force constants from one molecule to another is extremely sensitive to the differences between environments of the bond in the two molecules and also to the number of interaction constants employed in the potential functions [109]. Approximate force constants for bond stretching and bond angle bending have been tabulated for some common chemical bonds [109, 110]. The valence force constants used in this calculation were those calculated for TMTSF and its radical cation by Meneghetti et al. [86]. The force constants are similar to those of TMTTF [86], but have been modifies using a least-squares procedure to obtain the best fit to the observed frequencies. For the calculation of the frequencies of the radical cation, only the value of diagonal stretching and bending force constants have changed in order to reproduce the observed frequencies. The final values of the force constants are shown in Table A.2. Because of the slightly different geometries used for the methyl groups, the force constants H^atou) and Hy(fiifii) used in this work are slightly different than those in Ref. [86]. The force constants represent the refined values. A.3 Symmetry Valence Coordinates Symmetry valence coordinates are linear combinations of internal valence coordinates such that each combination (or new coordinate) belongs to one of the symmetry species of the molecular point group. In order to deal with the molecules of any reasonable size, such as TMTSF, sim-plifications resulting from symmetry are essential. Using internal valence coordinates, Appendix A. Normal Coordinate Analysis 183 solving the secular equation requires consists of finding the eigenvalues for a 73 x 73 determinant. Transforming to symmetry coordinates, however, reduces the determinant to several smaller determinantes. The transformation-from internal coordinates R to symmetry coordinates S is S = U R (A.26) where U is orthogonal and unitary, U - 1 =11*. To transform F and G into symmetry coordinates, G , = U f G U (A.27) F , = L T t F U (A.28) thus, the symmetrized secular equation is | G . F , - AE| = 0 (A.29) The reduction of the secular equation into smaller matrices reduces the computation time and organizes the calculated frequencies in terms of the symmetry species of the molecule. The 73 symmetry coordinates used in the normal coordinate analysis of TMTSF are listed below. Qg Si(ag) = -L(fl 1 + Jfe) (A.30) S2(as) = R2 (A.31) = | ( r i + r 2 - r r 3 + r 4) (A.32) S A M = i(S 1 + S 4 + S s + S8) (A.33) S * M = | ( 5 2 + 5 3 + 5 6 + 5 7) (A.34) Appendix A. Normal Coordinate Analysis 184 S e M = + * 4 + <io + <T) (A.35) S7(dg) = ^ ( < 2 + '3 + *5 + <6 + <8 + *9 + <ll+<12) (A.36) S » { a B ) = ^ ( Q ! + a 4 + a 7 + a 1 0) (A.37) S9(ag) = £ ~ ^ ( A 2 + A 3 + Q 5 + <*6 + a 8 + a 9 + an + ai 2) (A.38) SioM = |(A + fc + fr + 0io) (A-39) S u ( a g ) = ^ { h + h + k + l k + l k + fa + fa) (A.40) S12K) = + S2 + 63 + 64) (A.41) 513 ( 0 , ) = ^ ( 7 1 + 7 4 + 7 5 + 7 8 ) (A-42) S u M = ^ ( 7 2 + 7 3 + 7 6 + 7 7 ) (A.43) S i * M = | (* i + *0 (A.44) S u M = + + ^ 7 + 4>io) (A.45) Sn(as) = ^ 3 + ^ 4 + ^ 8 + ^9) (A.46) S l & ( b l u ) = - L ^ - f l , ) (A.47) 5 i 9 ( M = \(ri+r2-r3-r4) (A.48) S»(*u) = i(5i + S4-ft-5.) (A.49) 52i(felu) = | ( 5 2 + 5 3 - 5 6 - 5 7 ) (A.50) 5 2 2(ti.) = | ( t i + i 4 - t T - * i o ) (A.51) SasC&u) = ^(<2 + <3 + t 6 +<6-t 8 -<9-i i i -< 1 2) (A.52) 5 2 4 (6i„) = i(tt! + a 4 - a 7 - a 1 0) (A.53) Appendix A. Normal Coordinate Analysis 185 = 5(012 + a 3 + a 5 + oc6 — a 8 - Qg - au — Q12) (A.54) = l(01 + 04-07-0lo) (A.55) = ^=(& + 03 + A + A - 08 - 09 ~ 011 - fa) (A.56) 5*28 (blu) = 5(61+62-63-64) (A.57) "SwC&lu) — 5(71 + 74 - 7s - 7s) (A.58) S3o(blu) — 5(12 + 73 - 76 - 77) (A .59) S3l(blu) = 5(^1 - 4>&) (A.60) S32(blu) = -^=(<f>2 + <t>5 — 4>7 ~ <f>lo) (A.61) £33(61 u) = 2 (<t>2 + <f>4 - <t>8 ~ <l>9) (A.62) & 2 u 534 (&2u) = 5^1 - r2 - r3 + r4) (A.63) S3b(b2u) = \(S1-S4-S5 + S8) (A.64) S36(b2u) = 5^2 - S3-S6 + S7) (A.65) £ 3 7 ( & 2 u ) = | ( < l - < 4 - . < 7 + <10) (A.66) S3s(hu) = ^ ^ ( ^ 2 + <3 — *5 — *6 — <8 _ 9^ + Ul + h 2 ) (A.67) S39 (&2u) ^=(a j - a 4 - a 7 + Q i 0 ) (A.68) 54o(&2u) = ^ ^ ( o 2 + a 3 - or5 - a 6 - a 8 - <*9 + ctn + ctn) (A.69) £ 4 l ( & 2 u ) ±(01-04-07 + 0lo) (A.70) £42(6211) = ^(02 + 03— 0s - fit — 0s- 09 + 0\\ + 0n) (A.71) = 5(61-62-63+64) (A.72) Appendix A. Normal Coordinate Analysis 186 S^ihu) = 2-(7i-74 - 7 5 + 78) (A.73) £45(62*) = 5 ( 7 2 - 7 3 - 7 4 + 77) (A.74) £ 4 6 ( 6 ^ ) = 5(^ 2 - ^ 5 — ^ 7 + ^ 1 0 ) (A.75) £47(62*) = 5 ( ^ 3 - ^ 4 - ^ 8 + ^9) (A.76) hg S4&{ha) = i ( n - r 2 + r 3 - r4) (A.77) £49(63,) = 5 ( 5 i - £ 4 + £ 5 - £ 8 ) (A.78) £50(63,) = 5 ( £ 2 - £ 3 + £ 6 - £ 7 ) (A.79) £51(63,) = \{h + U + h - t10) (A.80) £52(635) = 5 ^ 2 + h ~ h - U + h + U - t n - t 1 2 ) (A.81) £53(630) = | ( a x - a 4 + a 7 - a 1 0) (A.82) £54(63,) = 2~/|(Q2 + <*3 - <*5 - <*6 + a 8 + a 9 - a n - a i 2) (A.83) £55(630) = \{h-fc+h-ho) (A.84) £56(630) = ^(fo + k-fo-fo + fo + h-lhi-fa) (A.85) £57(63,) = 5(^1-£2 + £ 3 - ^ ) (A.86) £58(63,) = ^(71 - 7 4 + 7 5 - 7 8 ) (A.87) £59(63,) = 5(72-73 + 7 6 - 7 7 ) (A.88) £60(63,) = 5(^2-^5 + ^ - ^ 1 0 ) (A.89) £61(63,) = 5(^3-^4 + ^ 8 - ^ 9 ) (A.90) Appendix A. Normal Coordinate Analysis 187 S62(au) = ~ ^ ( * 2 - < 3 - * « + <6 + < 8 - ' 9 - < n + <i2) (A.91) Stticiu) = ^ ^ ( a 2 - Q 3 - or5 + ot6 + a 8 - a 9 - a n + ct12) (A.92) SM(au) = - ^ ( f o - f o - h + fo + lh-fr-fiii+M (A.93) 5 6 5 (&i f l ) = ~ ^ ( < 2 - *3 - U +16 - ts + ts + hi - tu) (A.94) Seeihg) = ^ ~ / = ( Q 2 - " 3 - « 5 + <*e - « s + " 9 + a n - c*i 2 ) (A.95) 5 6 7 ( 6 l f l ) = ^{fa-fa-fc + fa-h + fo + fa-fa) (A.96) Sesihg) = ^ ( < 2 - * 3 + < 6 - < 6 - < 8 + * 8 - < n + * « ) (A.97) Sesihg) = ^ | ( a 2 - » 3 + " 5 - a 6 - a 8 + a 9 - Q n + Q 1 2 ) (A.98) 5 T O ( ^ ) = ^={fo - fo + fc - fo - h + h~ fa + fa) (A.99) S 7 i (6 3 u ) = ^ = ( < 2 - <3 + fs - t6 + U - U + tu -t12) (A.100) £72(6311) = J ^ ( a 2 - o 3 + Q 5 - " 6 + ^ 8 - « 9 + ctu - Q12) (A.101) Snihu) = r ^ ( f o - f o + fo-fo + fo-fo + fa-fa) (A.102) A.4 Normal Coordinates In tenns of the normal coordinates, the kinetic and potential energy are expressed simply as 2T = JTQl (A.103) Appendix A. Normal Coordinate Analysis 188 2V = £ > Q 2 , (A.104) fc=i The expression for the total energy in terms of the normal coordinates is E = T + V where A,- = uff and the mass has been included in the Q,'s. Since the total energy must be invariant under any symmetry operation and since the energy contains only squared terms, then it follows that Qi —» ± # , under any symmetry operation. However, symmetry coordinates are either symmetrical or antisymmetrical with respect to a sym-metry operation, thus normal coordinates will either be symmetry coordinates or linear combinations of them. The transformation that converts normal coordinates into symmetry coordinates is Si = J2LikQk fc=i where k denotes the kth root, A*, or in matrix notation S = L Q . (A.106) The transformation between the normal coordinates and the symmetry coordinates, L , is required in order to define the potential energy distribution. A.5 Potential Energy Distribution A convenient type of information on the contribution of each force constant Fij to the normal frequencies of vibration is furnished by the so-called potential energy distribution (PED). From the previous section, in a normal coordinate basis the potential energy is Appendix A. Normal Coordinate Analysis 189 given by 2V='£\kQl (A.107) k=l which may be expressed in symmetry coordinates as 2 V = E E (F.hLikLjkQl (A.108) k t j=i from which it follows that Afc = E (F,)ijLikLjk (A.109) The fractional contribution of the diagonal and off-diagonal elements of the F matrix to the potential energy may then be written as (F, ),,£,?• A ; and (F,)ijLjkLjk A * respectively, or ( P E D y o ) . ^ ^ ' ' ^ ' ^ ' 1 4 ^ " 1 0 0 (A.110) A* The potential energy distribution is important in characterizing the nature of a molecular vibration. Appendix B Models of Infrared Optical Conductivity B.l Introduction In Chapter 5, the insulating states of (TMTSF) 2 Re0 4 and (TMTSF) 2 BF 4 were fit to the dimer, tetramer and phase-phonon models. The following sections describe the Hamiltonians and the complex dielectric functions for each model. B.2 Dimer Model The Dimer Model was originally developed by Rice (1980) [65] to study the optical properties of unpaired electrons located on dimers. The Hamiltonian in the dimer model is H = He + Hv + 1£ganiQa,i - E • p (B.l) where He = {Eo + A C K + (EQ - A c ) n 2 + £ t{a\,aal<a + a\t0a2,a) (B.2) Hv = T,\(Pli + Ql>° (B.3) p = |ea(n, - n 2) (B.4) He is the Hamiltonian of the unpaired electron in the absence of vibronic coupling and consists of the interaction of the two monomer 7r molecular orbitals. (Eo+Ac) denotes the 190 Appendix B. Models of Infrared Optical Conductivity 191 energy of the (unperturbed) rr orbital of the first monomer, while (EQ — A c ) denotes that for the second (it is assumed that A c -C EQ). EQ is the common molecular orbital energy that would result in the absence of the asymmetric cation arrangement, t is the transfer integral and a\g and a l ) f f denote the site fermion creation and annihilation operators for the unpaired electron, i labels the site (i=l,2) and a the electronic spin. The occupation number for the site i, n< = a \ ^ a i , o - Hv is the Hamiltonian of the totally symmetric (ag) internal modes of vibration. Pa,i and Qaj denote the dimensionless normal mode coordinates for the vibration a and site i denned in terms of the vibrational creation and annihilation operators. The ga are the usual e — mv coupling constants. After Fourier transforming the electronic Hamiltonian and solving the equations of motion for the system [65], the complex dielectric function along the stack direction may be expressed as • 47re 2a 2(iV /n)(2i>cr) , R °° + UCT[l - D(u)] -U>*- iuT 1 J where a is the distance between the molecules in the stack, and N/Q is the number of dimers divided by the volume of the unit cell, or the dimer density (B.6) U>CT is the frequency of the charge-transfer excitation, t is the transfer integral, T is the relaxation rate and £«, is the core dielectric constant. The function D(u>) describes the ag modes and the coupling strengths = E / 8 A ° r ° • V (B.7) where ua and 7 a are the bare frequency and the damping of the mode, respectively, and \ a is a dimensionless coupling constant. This quantity is related to the e - m v coupling constant ga by 1 (<\gl Appendix B. Models of Infrared Optical Conductivity 192 Figure B.l: Steps of evolution of the distorted dimeric state; one-electron energy levels of the dimeric unit with (a) no site inequivalence, (b) stearic-site inequivalence and (c) stearic-site inequivalence and vibronic coupling (after Reference [65]). Physically, the charge-transfer excitation represents the energy required to move an electron from one molecule in the dimer to the other. The parameter u*p = 8Tne7a2(t2/ucT) (B.9) may be introduced, where u* is not the plasma frequency (which is defined as up = 47rne2/m), but rather a measure of the strength in the charge-transfer band. The transfer integral may written in terms of u* and UCT as t = \ "?UC* (B.10) The observed shift of a vibration may be calculated by finding the zero's of the real part of the dielectric function. The frequency shift 6a [111] of ua depends on the degree of coupling and the size of the charge-transfer frequency and the position of the frequency Appendix B. Models of Infrared Optical Conductivity 193 with respect to L>CT I2 + Xa^l^CT (B.11) The shifts are always away from the central charge-transfer band. A lower bound may be placed on the size of the charge-transfer frequency from the one-electron energy levels of the dimeric unit where t has been prevously denned, and A c and A„ are gaps due to stearic site inequiva-lence and vibronic coupling respectively. The charge-transfer frequency is the transition from one energy level to another as shown in Fig. B . l . By assuming that t » A c + A„, then the lowest possible value for u>cr is 2t, or Alternatively, the charge-transfer frequency may be expressed in terms of the transfer integral and the on-site Coulomb repuslion U [65] For linear chain organic materials U « At is often chosen. In the limit U —• 0, the relation between LOOT and t remains unchanged. The dimer model has also been used to calculate d.c. activation energies due to hopping between molecular clusters for dimers with one and two radical electrons per dimer [112]. B.3 Tetramer Model It is not too suprising that a theory for two isolated molecules fails to explain the effects of collective behavior in a solid. If clusters of dimers, or tetramers, are considered, then (B.12) U>CT > 2i (B.13) (B.14) Appendix B. Models of Infrared Optical Conductivity 194 a closer approximation to the behavior of the system may be reached, since the dimers must interact with each other. The tetramer developed by Yartsev (1984) [66] considers pairs of interacting dimers A B B A A B B A A B B A . . . , where the transfer integrals between the neighboring molecules are assumed to statisfy the relations | IAA tsB 1= t\ I tAB |= t where t > t' so that the tetramers are isolated ( A B B A ) . The Hamiltonian for the tetramer model is H = He + Hv+ ^9*^0^-E-p (B.15) where HE = A(TH + n4) + A (n 2 + n3) + ^ A ,^>nt-,_(T +V(nin2 + n 2 n 3 - f n3n4) + <'E( a2,<7 a3,<7 + at,<ra2,*) 0 +* E ( f l l , * a 2 , « + a 2 , < 7 a l , " + ^ L ^ c r + a l ^ « 3 , » ) ( B -16) o ^ = E k - - + ^ K (B.17) t,a describe respectively the radical electrons and the symmetric molecular vibrations of each monomer in the absence of vibronic coupling. The e-mv coupling is described by the third term in H. The tetramer model is an extended variant of the Hubbard model [113]. The parameters used to describe the optical properties of the system are the on site coulomb repulsion U, the coulomb repulsion between electrons on neighbouring sites, V. The energy inequivalence between the monomer sites A (the same as the dimer) and the transfer integrals for the molecular pairs A B and B B are t and t'. The optical conductivity in the tetramer model is defined as X i i ^ 2 +2X17AB + X22B2 + (X2l2 - XnX22)(A2DB + B2DA)' a(u) = —iu}Nte2 where DA,B(<*>) axe 1 - XII^A - X22DB + ( 11X22 - X 1 2 2 ) D A £ B (B.18) B A J B M = E 2 9lA'BT,B • (B.19) Appendix B. Models of Infrared Optical Conductivity 195 The ^ Q A , B ? 7 Q A , B and the <70A,B a r e the frequency, damping and the coupling strength. The subscripts A and B allow the vibrations to have different frequencies, damping and coupling constants due to different charge densities on the A and B molecules. A and B are defined as A = (2a + a')/2 and B = a'/2, where a and a' is the molecular overlap for AB and BB respectively. The reduced charge-transfer polarizabilities Xij are given by | (0 | n x - n 4 | 1) | 2 2u>pi x n H = E w| a - up- -2 "*>7e X 2 2 ( c ^ — 2 - ^ w / 8 l — W 2 — l^7e (B.20) (B.21) (B.22) in which the ujpi represent the frequency of the transition from an excited state 0 (7, 8, 9, 10) to the ground state (1) and 7 e is the damping for the charge-transfer band. The components of the ground-state expectation values are (0 | m - n 4 | 1) = 4(ajaf + a\al - a\4), (B.23) (0\n2-n3\l) = A(a\4 + a\a0b + a 6a£). (B.24) The a] are the coefficients of the ground state wave function. The energy and the coefficients of the ground-state wave function are the eigenvalues and eigenvectors of the equation [114] U + 2A-E 0 2* 0 0 0 0 U - 2 A - E 2t 0 0 2f t t V - E t' 0 0 0 0 t' -E t t 0 0 0 2t 2A-E 0 0 2f 0 2t 0 V - 2 A - E ( = 0 (B.25) Appendix B. Models of Infrared Optical Conductivity 196 8 2 0 -2 ioy y / v" ~ y y y y > y s y* y s / — ^^rzrz^^II 0 05 ,10 •U/4\t\-15 Figure B.2: Excited state, energies (7 to 10) which are accessible by optical transi-tions from the ground state (1) for the two-electron system in a tetramer: t/t' = 0.4, A/t = 0.25, V = 0 (dashed lines); 0.4C7 (solid lines) (after Reference [66]). and the af are the coefficients of the excited state wavefunctions. The coefficients of the excited-state wavefunction are listed in the Appendix of Ref. [66]. The excited state energies are obtained from the equation [114] U + 2A-E 0 2t 0 0 U-2A-E 2t 0 t t V-E t' 0 0 t' -E = 0 (B.26) It is also possible using the coefficients to calculate the charge density on qA on molecule A and ge on molecule B. qA = 2[(a})2 + (a*)2 + (a 1) 2 + (a})2] (B.27) Appendix B. Models of Infrared Optical Conductivity 197 q B = 2[(o\)2 + (a\)2 + (a\)2 + (al)2}. (B.28) One of the results of the tetramer model is that instead of only one strong charge-transfer band, there are now four. The parameters typically chosen for linear organic conductors indicate that two of these peaks will be in the mid infrared and the other two will be at much higher frequencies. Some of the possible excited energy states are shown in Fig. B.2. B.4 Phase—Phonon Model The phase-phonon model considers the electronic interband transitions for a one-di-mensional (ID) semiconductor. The ag modes are activated as phase phonons [phase oscillations of the stabalizing charge-density wave (CDW) distortion] which arise from the linear coupling of the totally symmetric modes with the conduction electron molec-ular orbital. The model Hamiltonian for a linear-chain semiconductor with conduction electron-phonon coupling is given by Rice (1976) [93] as H = He + Hv + -±=Yl gnQn(q)p-q (B.29) where He = 2> 0 ( e B - | ek \)alak + V{pqo + p.go) (B.30) k Hv = £ l 6 n ( ? ) * n ( ? ) + ^ ( C J ) . (B.31) lit. describe a system of n conduction electrons per unit length with energies tk = (| k I —kp)vF lying within the range | c |<C CB (relative to their Fermi energy) moving in a periodic potential V (| V \ / C B assumed small) of wave vector <2o = 2kp. They represent a simple model of the semiconductor, in the absence of electron-phonon cou-pling, kp, VF denote the conduction electron Fermi wave vector and velocity and <4(a*) Appendix B. Models of Infrared Optical Conductivity 198 denote the creation (annihilation) operator. The operator pq = alak+q creates an electronic density fluctuation of wave vector q. Hv is the Hamiltonian for a set of non-interacting phonon bands where b^n(bn) are the boson creation (annihilation) operators for the phonon. The potential V may be considered to simulate the effect of a static periodic modulation of the conduction-electron molecular-orbital energy caused by the presence of a donor chain structure [e.g. the TEA (where TEA is triethyl ammonium) in TEA(TCNQ)2]. The third term in H describes the coupling of the conduction electrons of a set of phonon bands, specifically those totally symmetric (ag) internal vibrations of the stack molecules which induce a modulation of the local conduction-electron, molecu-lar-orbital. Qn(q) = bn(q) + bl(—q) denotes the dimensionless normal mode displacement operator associated with the nth phonon band and gn denote the symmetry allowed e-mv coupling constants. In the absence of electron-phonon coupling the semiconducting state arises from the action of the periodic potential V of inducing the conduction electrons to condense into a CDW of wave vector g0. The expectation value of />,, 6pq — (pq), is nonvanishing for q = ±go- The phase of the CDW is fixed by the (fixed) phase of V and the single-electron energy states are E° = ±eicVt2 + V 2 . The frequency dependent conductivity, O~Q, is due simply to the single-electron transitions between the two sub-bands contained in E°. The expression for the optical conductivity of a one-dimensional semiconductor was first described by Lee et al. [115]. The contribution of the single-particle term together with the diamagnetic term at zero temperature is iuM (B.32) where (B.33) Appendix B. Models of Infrared Optical Conductivity 199 and E 2 = £ 2 + A 2 and = ek — CF and 2A is the semiconducting gap; thus 2 f°° d{ / ( w ) = -2A v/^ + A j [ ( u ; + ^ ) 2 _ 4 ^ 2 + A a ) ] - ( B - 3 4 ^ There are two cases; (i) TJ = 0 and (ii) v ^ 0. The parameter has been introduced in a phenomenological way and may be thought of as a damping factor (or the inverse lifetime of single particle in the gap). If n = 0, then the function /(w) is calculated to be f(u) = { u S [ r \ 1 + 5 ; j x t ~ (B.35) u2S —1 [arctan (j^) + arctan fez^)] w < 2A . where S - J l - ( f ) . (B.36) It is important to note that when w < 2A, /(a;) is real, while for u > 2A, f(u>) is a complex function. For the case r\ ^ 0, f(ui) is complex everywhere. The real and imaginary parts of f(u>) axe W f r ^ l - T / 2 ^cos(g)-a6co S 3 (g)^ - io cos<(0)-2aicos2(0) + a 2 ( B > 3 7 ) where 4A 2 a = -7—^ (B.39) u; 2 + r/2 b = (B.40) a;2 + r/2 Rice shown that the normally infrared-inactive ag modes may be activated in quasi-one-dimensional semiconductor by charge exchange between molecules in a stack arising from oscillations of the phase of the CDW [93]. Appendix B. Models of Infrared Optical Conductivity 200 The optical conductivity for a semiconductor in the absence of phonons is ne 2 aQ{u) = - /(0)] (B.41) turn The contribution due to phonons due to phase oscllations of the CDW is where Dv(u) is a phonon-like propogator for the phase oscillations, given by D-\.) = + 1 - I + A ( £ ) * / ( £ ) (B.43) with in which u>a and 7 a denote the frequencies and the linewidths of the original non-interacting phonon states. The zeros of D~l(u) determine the frequencies of the collective oscillations. Also A = £ Xa (B.45) Or is the total phonon coupling to the CDW in the system where the A a adjustment prin-cipally fixes the positions of the narrow absorption bands below 2A. Also, (B.46) A [(c. - l)(2A/u; p) 2 - 2/3] where et is the static dielectric constant. The optical conductivity in the phase-phonon model is expressed as tr(u) = <TQ(U) + (Tc(u) " 0 ['<">" /(0)" / M A J W (B.47) where uv is the plasma frequency. The e-mv coupling constants ga may be written as Up hAua\a . Appendix B. Models of Infrared Optical Conductivity 201 ••g 0.6 -£ 0.4 -0.2 -0.0 — 5000.0 o 4000.0 *> 3000.0 -•s 2000.0 -g 1000.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) 0.0 0.0 500.0 1000.0 1500.0 2000.0 Wave Number (cm - 1 ) Figure B.3: The phase-phonon (a) reflectivity and (b) optical conductivity for n — 0 c m - 1 (dashed line) and n = 100 c m - 1 (solid line) ior T = 0 K . The other parameters used to generate the curves are uv = 10,000 c m - 1 , and 2A = 1,000 c m - 1 , tt = 50 and too = 2.6. Note that by setting n ^  0 that a ^  0 below the gap. The overall behavior is similar to that of a Drude-Lorentz system. Appendix B. Models of Infrared Optical Conductivity 202 where a is the separation between the molecules and A is area of the molecule perpen-dicular to the stack in the unit cell. The phase-phonon reflectivity and optical conductivity are calculated in Fig. B.3 for the case where n = 0 c m - 1 (dashed line) and the case where n = 100 cm - 1 ; the other parameters are u>p — 10,000 c m - 1 , 2A = 1,000 c m - 1 , ea = 50 and £<» = 2.6. The for rj = 0 the calculated reflectivity shows singular behavior near the gap. When 77 ^ 0, this singularity is removed, but the behavior near the origin is similar to that of a Drude metal with large damping; it is no longer remotely similar to the n = 0 case. For 77 = 0 c m - 1 , the conductivity is zero below the gap, and there is a singularity in the density of states at u> = 2A. Above the energy gap, the conductivity falls off with a characteristic 1/y/e behavior. For 77 = 100 c m - 1 , the conductivity below the gap is nonzero, and slowly rises near u> = 2A. The conductivity no longer displays any singular behavior at the gap. Above the gap, the conductivity quickly becomes equivalent with the 77 = 0 c m - 1 case, indicating that damping is important at and below the gap, but not above. It is impractical to consider systems with 77 = 0 c m - 1 because of the singular behav-ior near the gap. Often, when the phase-phonon model is applied phonons with large coupling constants are introduced near the gap in order to remove the singular behav-ior [93], however, in many systems where we expect this model to be applicable there are no phonons near the gap. Because the damping parameter has been introduced as an afterthought, even though it removes the singular behavior at the gap, it produces conductivities below the gap which are unacceptably high for even modest values of 77. The phase-phonon model does provide information on the behavior of phonons that is useful in deciding if it is appropriate for a system or not; if the phonons are below 2A, then they appear as resonances, however, if they occur above 2A, then they appear as antiresonances interfering with the electronic background. Appendix C One—Dimensional Systems with Twofold-Commensurate Charge-Density Waves C. l Introduction In this Appendix, the framework for Bozio's model for one-dimensional organic conduc-tors with twofold-commensurate charge—density waves is developed. Simulations have also been performed in order to demonstrate some of the features of the model. C.2 Model Hamiltonian The following pages are taken directly (with various additions) from the paper by Bozio et al. [69]. The site representation of the model Hamiltonian is H = HU + Hev + Hv (C.l) where Hie = -<E(°Ul,<,an,ff + #.C.) n,cf +BxJ2(-ir(ai+1<can,e + H.c.) + S * B - l ) n < ^ n , . ( C 2 ) is the one-electron Hamiltonian for a ID tight-binding electron subjected to the potential Vx of the nearby counterion chains. ani<T(a^CT) is the annihilation (creation) operator 203 Appendix C. ID Systems with Commensurate CDW's 204 for electron of spin a = ± j on the n site, t is the transfer integral and Bx and Sx are the amplitudes of the anion potential components acting on the bonds and sites respectively. The H.c. terms in the Hamiltonian denote the Hermitian conjugate of the previous operators. Now n,c i=l \ UH* / o - E " + H-c-) (C-3) is the interaction Hamiltonian whose first and second terms account for the linear coupling of the electrons to an arbitrary number v of intramolecular modes and to one longitudnal mode respectively. The e-mv coupling constants = (den/dqi)o are expressed as the first derivative of the unpaired electron molecular orbital energy cn = c with respect to the z'th dimensionless normal coordinate 9, [65, 93]. The derivative is calculated at the equilibrium geometry of the molecules in the absence of an AMD when they carry a formal negative (positive) charge of 7 electrons (holes) per molecule with 7 = 2fcjr<i/7r. The normal coordinate is bound by symmetry to be totally symmetric (ag) in order to give a nonzero linear coupling constant with a nondegenerate molecular orbital [65, 93, 116]. The acoustic phonon couples through the modulation of the transfer integral by the longitudinal displacement u„ (in units of length) of the molecules from their location in the undistorted chain [93, 117]. The last term in the Hamiltonian is + 5 £ { ( O M ) + lMu.fo„)/4]K+1 - «.)'} (C.4) z n is the vibrational Hamiltonian in the harmonic approximation with the addition of elastic energy and strain terms associated with the chain distortions, a?,- is the unrenormalized Appendix C. ID Systems with Commensurate CDW's 205 frequency of the tth intramolecular mode, uje(q0) is the unrenormalized frequency of the acoustic mode at the BZ boundary and M is the mass of the TMTSF molecule. After Fourier transforming the H\e and Hv to reciprocal space, LD and AMD's of arbitrary amplitude «o and 8qi are introduced as nonzero average values of the phonons at qo in each branch, (QiM) = ^ . - A i o o (C5) where ^•(9) = ^ E 9 n , e " * ' 9 n d (C-6) and <2e(g) = ^ E " n e - * , m < (C7) With the use of displaced phonon coordinates = (C8) then equation (C.l) transforms into H = H'u + Hev-rH'v, (C.9) where k,o k,o + £ iAb «n ( f cd ) (« I + »^ f l ^ " #-c')> (CIO) k,a is the Hamiltonian of split tight-binding band where o ^ - E ^ e * - (C. l l ) n with ib extending over the Brillouin zone from —7r/d to ir/d; and cfc = -2tcos(kd), (C.12) Appendix C. ID Systems with Commensurate CDWs 206 are the tight-binding cosine bands and A . = 2>*ft + y , (C.13) = (So' u 0 + Bx. (C.14) i are the gaps for the site and bond-centered CDWs [note that the gap At is not the same as A(,< (introduced in Chapter 3), which defines the energy gap in the E||b' direction]. The electronic Hamiltonian can be diagonalized by the tranformation o*+W = E/»'(*M*».a (M = 0,1) (C15) where and r ( k \ _ -F,(n,k)  M {[F0(n,k)Y+\F1(k)\^ F/(n,*) = < tk + Ekn 1 = 0 At = A , + iAb sin(fcd) I = 1 (C.17) where Ekin = (-\r^y/el+\Ak | 2 (C.18) are the energies of two tight-binding bands separated by an energy gap of 2^/A 2 + A 2 located at the boundaries (—7r/2d, 7r/2d) of the reduced zone of the distorted chain. Equation (C.10) then reduces to ^ = E ^ < 5 4 , (C.19) kn,o The transformed vibrational Hamiltonian reads 'M IT) £ [ 0 e (9 + vqo)Qe(-q - vq0) + u\{q + vq0)Qe(q + vq0)Qe(-q - vq0)] -rMwl(q0)Qe(q0)SQe + ( y ) u2e(q0)(6Qe)2 Appendix C. ID Systems with Commensurate CDW's 207 1500.0 -1500.0 2d 0 Wave Vector, k 7T 2d Figure C . l : The tight-binding energy bands for a one-dimensional semiconductor with the parameter values t = 500 c m - 1 and the energy gap 2A = 1000 c m - 1 . The solid lines indicate the energy bands for an energy gap due to a fr-CDW (A = A 0 ) , and the dotted lines indicate the energy bands for an energy gap due to an s-CDW (A = A , ) . The energy bands are separated by 2A = 2 y A 2 + A 2 at the edges of the reduced Brillouin zone, and by 2^At2 + A 2 at the zone origin. + + i(q + vq0)Qi{-q - vq0)/(Jf + Qi{q + vq0)Qi(-q - vq0)] (C.20) with v = 0,1 and we(q) = we(qo) \ 8in(qd/2) |, represents a model chain system where only the effects of the static distortions are allowed for. It is convenient to apply the transformation used to diagonalize HU to the interaction Appendix C. ID Systems with Commensurate CDWs 208 Hamiltonian Hev to get the coupling terms. This yields k,q,<r v,n,m {v^m(k,q)Qe(q + vq0) + £ VJnm(fc,q)Qi(q + vqQ)^ 4 +,,n,.>W (C.21) where the coupling terms are Kenm(k,q) = (^ =) E9e(k-rlq0,q + vq0)rnHv(k + q)fm,l(k), (C.22) VLm(k,q) = EAVv(* + *)/«.»(*) (C23) where the usual term for the coupling constant for acoustic phonons is ge(k,q) = 2i^~j {sm(kd)-sm[(k + q)d]} (C.24) has been introduced [117]. The dynamic effects are included in Hev and will be considered perturbatively. The optical properties of the system are denned through the transverse (complex) dielectric function in the self-consistent field approximation as [77] e ( w ) = eco + (C.25) where e<jo is the core dielectric constant which is usually taken as real, E(u) is the electric field acting on the system (neglecting local effects), P(u>) is the induced macroscopic polarization which can be calculated through the standard relation P M = (^r) Tr(p/)). (C.26) In Eq. (C.26) Vm is the volume per molecule, p is the dipole moment operator, and p is the density matrix operator. In the framework of linear-response theory, p may be calculated to first order using the perturbation Hamiltonian Hint = Hev-p(u>)E(uJ). (C.27) Appendix C. ID Systems with Commensurate CDW's 209 The calculation of the polarization thus requires the calculation of the matrix elements of the dipole moment operator. Following Bozio's development, however, it is more convenient to use the relation (m\p\n) = -ih(m \ j \ n)/(Em - En) (C.28) since the current operator j can be unabiguously denned for a distorted chain as (see Appendix of Ref. [69]) +ijccos(kd)(f:ifm0 - rn0fmi)]A\ntOAkm,a (C.29) with j , = - ( - ^ ) [ t d + u 0 A f c ] , (C.30) ; c = - ( | ) [ A 6 d + u0t]. (C31) The calculation of the matrix elements of Hev implies working with the operators which are products of phonon coordinates and electron-hole operators [cf. Eq. (C.21)]. In the equations of motion for the phonon coordinates the electron-hole operators are replaced by their expectation values calculated in the linear response [93, 97, 98, 91]. The average value of the Fourier-transformed coordinate oscillating at the same frequency as the applied field is given by the system of coupled inhomogeneous linear equations <QaM> = Da{u) 12 Ta,^)(Q^)) + £(u>)J 0M J (C.32) with the a,/? = i,e index the internal and external vibrations respectively, and Dt{u) = (1/M)[(u, + t 7 e ) 2 - u£(*JT\ (C33) Di = (2u*/»)[(w + *7,f - w ? ] - 1 , (C34) Appendix C. ID Systems with Commensurate CDW's 210 k,o n , m ^ k n •VS»(*,0)V1iN(*,0), (C.35) ik n(Ekn) - n (£ f c m ) ^n(*.0)*-m(M)- (C.36) £ * n - £ * m Ekn - Ekm - f l U - ihT n(e) is the Fermi-Dirac distribution function, 7 e , 7 t and T are the phenomenological damp-ing parameters for the intermolecular phonons and for the electronic interband transi-tions, respectively. The matrix elements of the current operator are The equations for (C.32) constitute a set of coupled inhomogeneous linear equations which must be solved in order to get the required values of (Qcr(u>))'s. The latter are used in place of the phonon coordinate operators in the calculation of the needed matrix elements of H'ev. The introduction of an interband damping parameter is a great im-provement over previous "phase-phonon" theories [93, 97] which only allowed for T = 0 (as discussed in Appendix A). When phonons are introduced to the system, the list of adjustable parameters for the fitting of experimental infrared spectra must include the basic interactions of the elec-tronic and electron-phonon Hamiltonians, Eqs. (C.2) and (C.3), as well as the amplitudes of the AMD components t5g, (or the corresponding q0 phonon amplitudes 6Qi),the "ef-fective" external phonon frequency u)e(qo), and the damping parameters for the phonons and electron-hole excitations. A futher reduction of the number of adjustable parameters can be achieved using the condition that the ground state expectation value of all the displaced phonon coordinates at qo must be zero by their definition in Eqs. (C.6) and (C.7). It may be shown [69] that i n m ( M ) = (*n,<r \j | km,<r). (C.37) 7rMu>2(t7o)(uo)y/4<2 + A , 4A 6*[ff(*) - £($)] (C.38) Appendix C. ID Systems with Commensurate CDWs 211 where and #($) and E($) are the complete elliptic integrals of the first and second kind re-spectively. Equation (C.38) links the value of the electron-intermolecular-phonon cou-pling constants to that of the parameters of the electronic structure and the amplitudes of the LD. Note that the amplitude of the anion potential between sites Bx appears as a fundamental interaction parameter in Eq. (C.l), however, the total fr-CDW gap A;, will be used a a fitting parameter and the contribution due to the anions will be deduced through Eq. (C.14) in conjunction with Eq. (C.38). From the condition of the vanishing ground-state expectation value of the displaced intramolecular phonon coordinates it would be possible to derive an equation similar to Eq. (C.38) for each coupled mode. Bozio solves only for the AMD components (as they are usually unknown), this yields •KhUiyJU2 + A 2 and eliminates the amplitudes Sqi from the set of adjustable parameters. Again, Eq. (C.13) can be used, together with the set of Eqs. (C.40) with i = 1 to u, to evaluate the contribution of the anion potential Sx to the gap A , due to the s-CDW. For some of the key molecular structures that constitute organic charge transfer con-ductors the values of the e-mv coupling constants gi are well known [118], and should be regarded as fixed input variables, but this is often a fitting parameter. The frequencies of the infrared active renormalized phonons (phase phonons) are given by the zeros of the demoninator in the dielectric response function. The frequency shifts with respect to the bare phonon frequencies ua{<lo) are determined by the functions Tap. When only At (A«) is nonzero and only the intramolecular (intermolecular) phonon is active, its frequency shift is determined solely by Ta (Tee). It may be shown, assuming Appendix C. ID Systems with Commensurate CDW's 212 that the bare phonon frequencies are widely separated from each other and much lower than the CDW gap, that the shift for an intramolecular vibration u>,- is [69] * = ^ t f ( l - A 2 /4* 2 ) (C.41) and the shift for an intermolecular mode is s'=7^Ml[Km-E{n (C-42) where K(V) and E(V) are the complete elliptic integrals of the first and second kind, and A 2 * = 1 — — . (C.43) In the case of many coupled intramolecular modes, there are Tn> terms which induce a mutual attraction of all the intramolecular modes. The terms Tie is proportional to the product A 4 A 0 and therefore therefore both contributions to the gap (and to the CDW) must be present for it to be nonvanishing. This term is responsible for a mixing of intramolecular and intermolecular modes which might be relevant for low-frequency intramolecular modes. A term of the same nature has been introduced by Horowitz et al. [91] to account for a possible mixing between the totally symmetric and out-of-plane anti-symmetric modes. The coupling with intermolecular phonons would give rise to additional contributions to A&. The final form of the.complex dielectric function is 4TT where i2 x M + E7«M<<2>)>/£M (C.44) x(«> = E E ( E \ y j g ^ ' l f ^ f l i r 1 ' ™ ( M ) | 2 ( a 4 5 ) is the dielectric susceptability for the single-particle excitations across the gap. Appendix C. ID Systems with Commensurate CDWs 213 The dielectric function contains information about the optical activity of the phonons. Whether the frequency dependent functions Ia, (a = i, e), which determine the optical ac-tivity of the modes, vanish or not depends on the parity properties of Vim n(fc, 0)jnm{k, 0). It is therefore possible to formulate some simple "selection rules": (i) is nonvan-ishing and the intramolecular modes are vibronically active only if there is a nonzero component of At that is a contribution to the gap due to a 6-CDW. The latter is in turn made up of two contributions, one induced by the anion potential through the Bx term in the Hamiltonian (C.2), the other due to the LD which might might be simply a response of the system to an already existing 6-CDW (e.g., induced by a Coulomb potential). This would be a secondary effect that enhances the total fe-CDW amplitude. On the other hand the LD might also be directly due to a Peierls type transition, (ii) Ie(w) is nonzero, leading to vibronically active intermolecular modes, only if there is a nonvanishing value of A , that is a contribution to the gap due to a s-CDW. The same considerations developed for the possible components of a 6-CDW apply to this case for the s-CDW with the replacement of Bx by the Sx term in Eq. (C.2) and of the LD by the AMD. C.3 Model Simulations It is useful to examine the optical properties of the model in order to compare the model simulations with those of Bozio et al.. In the absence of any internal or external phonons, the electronic continuum may be calculated and compared to that generated by the conventional phase-phonon model discussed in Appendix B. The effect of altering the nature of the CDW from a bond- to a site-centered CDW (e.g. A = Ab to A = A „ denoted by 6-CDW or s-CDW respectively), while leaving the size of the gap the same, may also be investigated. Appendix C. ID Systems with Commensurate CDWs 214 For systems with eleetron-phonon coupling, the effect of changing the size of the gap, as well as the nature of the gap may be investigated and compared to previous simulations. C.3.1 Systems without Electron-Phonon Coupling In order to illustrate the properties of this model, it is useful to examine the optical prop-erties in the absence of any intermolecular or intramolecular eleetron-phonon coupling. In this simplified picture, the parameters for the model are N and T, the structural pa-rameters d, Vm and Uo, and the electronic parameters t, 2A, T and The calculations have been performed with the BZ divided into N = 256 divisions, and at T = 0 K. The strucural parameters have been chosen to be roughly consistent with the unit cells of the TTF and TMTSF based organic conductors, with the intermolecular stack separa-tion of d = 3.5 A , a volume per molecule of Vm = 340 A 3 and a lattice dimerization of Uo = 0.02 Ais assumed to be present. The calculated optical conductivity spectra for fr-CDW and s-CDW gaps is shown in Fig. C . l . The electronic parameters are t = 1000 c m - 1 , 2A = 1000 c m - 1 and T = 100 c m - 1 . The solid line represents the calculated optical conductivity when the gap is due to a fc-CDW gap and the dotted line repre-sents the conductivity when the gap is due to a s-CDW. The broad structures in the conductivity in Fig. C.2 behave as expected for interband transitions in narrow gap ID semiconductors. The shape of the conductivity at 2A is that of the combined density of states of the ID energy bands shown in Fig. C . l . The weak maximum that occurs in the conductivity for a 6-CDW at At is due to the ID divergence of the joint density of states for transitions at k = 0 from the bottom of the lower band to the top of the upper band. The corresponding transistion matrix element joi(k = 0) depends only on Aj, and is therefore nonvanishing only in the presence of a 6-CDW, thus in the case of a s-CDW this structure is not present, as the dotted line indicates in Fig. C . l . The Appendix C. ID Systems with Commensurate CDWs 215 3000.0 Figure C.2: The calculated optical conductivity for 6-CDW and *-CDW energy gaps with t = 1000 cm - 1 ,2A = 1000 cm - 1 and T = 100 cm - 1 . The solid line represents a gap due entirely to a fc-CDW or 2A& = 1000 cm"1 and the dotted line represents a gap due entirely to a s-CDW or 2A, = 1000 cm - 1 . Appendix C. ID Systems with Commensurate CDWs 216 introduction of an interband damping parameter is a great improvement over previous "phase-phonon" theories [93, 97] which only allowed for T = 0. Successful application of these theories required the presence of strong intramolecular eleetron-phonon coupling near the energy gap in order to remove the singular behavior of the conductivity at the gap and to broaden the feature, however, in the case where no phonons satisfied this condition damping was still observed. Attempts to introduce T ^ 0 into Rice's model [119] result in the addition of a Drude-like contribution to the optical properties [120] and a sizeable conducivity below the gap. C.3.2 Systems with Electron-Phonon Coupling The reflectivity and the optical conductivity have been calculated for T = 0 K and a transfer integral of t = 1000 c m - 1 , a 6-CDW gaps 2Ab of 300 cm" 1 and 700 cm" 1, the interband damping is T = 300 c m - 1 . The electrons are coupled to three phonon modes: two internal modes at 150 cm" 1 and 500 c m - 1 , with coupling constants of 200 c m - 1 and 400 cm" 1, respectively, and one external mode at 50 cm" 1. The damping is 5 cm" 1 and 2.5 cm" 1 for the internal and the external phonons respectively. The structural parameters have not been altered from the previous section. The calculated reflectivity and optical conductivity for these parameters in the region of the gap are shown in Figs. C.3 and C.4 respectively. The solid line is for 2Aj, = 300 cm" 1 and the dashed line is for 2Af> = 700 c m - 1 . The arrows in Figs. C.3 and C.4 indicate the location of the bare internal phonon frequencies. The internal mode at 500 c m - 1 in the optical conductivity changes its line shape from an asymmetric peak to an antiresonance dip when the gap amplitude 2A is decreased from 700 c m - 1 to 300 c m - 1 . The line at 150 cm" 1 increases in its intensity, but since it is still below the gap it remains an asymmetric peak. For 2Aj = 700 c m - 1 the effective coupling constant for the 50 cm" 1 intermolecular phonon is (d/t)(dt/du)0 = 5.78 (in dimensionless units) and the contribution to the gap through Appendix C. ID Systems with Commensurate CDW's 217 0.3 0.2 0.1 0.0 A = A ; 0.0 500.0 1000.0 Wave Number (cm-1) Figure C.3: The calculated reflectivity in the gap region for the case in which only the 6-CDW has a nonzero amplitude, 2Ab = 300 c m - 1 (solid line) and 2A = 700 c m - 1 (dashed line). The other parameters are t = 1000 c m - 1 , T — 300 cm" 1 and too = 2.5. There is one acoustic phonon with ut = 50 c m - 1 , 7 e = 2.50 cm - 1and the calculated dimensionless coupling constant is (d/t)(dt/du) — 5.78 (2A = 700 cm - 1 ) and 9.23 (2A = 300 cm - 1 ) . There are two internal phonons, u^ i = 150 c m - 1 , 7u = 5.0 c m - 1 , G,i = 200 c m - 1 , w,2 = 500 c m - 1 , 7,2 = 5.0 c m - 1 and </l2 = 400 c m - 1 . The arrows indicate the frequencies of the unperturbed internal modes. Appendix C. ID Systems with Commensurate CDW's 218 2 0 0 0 . 0 0.0 ^ ' 1 ' — ' 0.0 . 500 .0 1000 .0 Wave Number (cm-1) Figure C .4: The calculated optical conductivity spectra in the gap region for the case in which only the fr-CDW has a nonzero amplitude, 2A 0 = 300 c m - 1 (solid line) and 2A = 700 c m - 1 (dashed line). The other parameters are t = 1000 c m - 1 , T = 300 c m - 1 and Coo = 2.5. There is one acoustic phonon with u>e = 50 c m - 1 , fe = 2.50 cm - 1and the calculated dimensionless coupling constant is (d/t)(dt/du) = 5.78 (2A = 700 cm - 1 ) and 9.23 (2A = 300 cm - 1 ) . There are two internal phonons, = 150 c m - 1 , 7^ = 5.0 c m - 1 , G,i = 200 c m - 1 , u>i2 = 500 c m - 1 , 7,2 = 5.0 c m - 1 and gi2 = 400 c m - 1 . The arrows indicate the frequencies of the unperturbed internal modes. Appendix C. ID Systems with Commensurate CDWs 219 the anion potential is 81%. When 2A& = 300 c m - 1 the effective coupling constant is 9.23 and the contribution to the gap through the anion potential is only 30%. Note that because the gap is entirely due to a 6-CDW, the external phonon is not infrared active. The calculated shifts for the modes at 150 c m - 1 and 500 c m - 1 are ~ 10 c m - 1 and ~ 40 c m - 1 respectively, for both values of the gap (the downshifts are more sensitive to the coupling strengths of the vibrations rather than to the position of the gap). Decreasing the LD parameter u0 by a factor of 4 from 0.02 to 0.005 when 2A& = 700 c m - 1 has the effect of decreasing the coupling parameter from 5.78 to 1.47 and puts the contribution of the gap from the anion potential at over 98%. Despite this, the calculated spectra were practically indistinguishable from those in Figs. C.3 and C.4, indicating that the relevant parameter is the total fe-CDW amplitude independent of that of the LD. Another set of simulations for the most general case where both the 6-CDW and s-CDW contribute equally to the to the gap for values of 2A = 300 c m - 1 and 2A = 700 c m - 1 are shown in Figs. C.5 and C.6. Otherwise, the electronic parameters have not been changed. The LD amplitude of u0 = 0.02 yields a coupling constant of 8.20 (13.1) for 2A = 700 c m - 1 (2A = 300 cm"1). The AMD amplitudes Sqi = 0.29 and Sq2 = 0.18 for the modes at 50 cm" 1, 150 c m - 1 and 500 c m - 1 respectively. Note that the intermolecular phonon, absent in the previous calculations with A , = 0, is optically active. For 2A = 700 c m - 1 the anion potential accounts for 77% of the energy gap, which compares to 81% when the gap was due to just the &-CDW. The AMD amplitudes Sqi have been calculated, however, their precise determination experimentally may be beyone the accuracy limits of ordinary structural studies. These results of these simulations conform to the general rule of the "phase-phonon" theories that phonons below the gap appear as resonances and phonons above the gap appear as antiresonances, however, the structure of the infrared spectra depends in a complex way on the gap amplitude and therefore only fitting based on model calculations Appendix C. ID Systems with Commensurate CDW's 220 0.0 0.0 500.0 Wave Number (cm-1) 1000.0 Figure C.5: The calculated reflectivity for the case where both fr-CDW and s-CDW contriubute to the total gap with equal weights, 2A = 300 c m - 1 (solid line) 2A = 700 T = 300 c m - 1 and £«> = 2.5. There is one acoustic c m - 1 (dashed line); t = 1000 cm , phonon with ue = 50 c m - 1 , % ==• 2.50 cm" 1 and the calculated dimensionless coupling constant is (d/t)(dtjdu) = 8.20(13.1) for 2A = 300 c m - 1 (2A = 700 cm" 1). There are two internal phonons, w,i = 150 c m - 1 , 7^  = 5.0 cm" 1, Gn = 200 cm" 1 , u>;2 = 500 c m - 1 , 7i2 = 5.0 cm" 1 and #2 = 400 cm" 1. The arrows indicate the unperturbed frequencies of the internal modes. Note that the acoustic phonon, absent in the previous calculations with A , a= 0, is optically active. ,-1 Appendix C. ID Systems with Commensurate CDWs 221 Figure C.6: The calculated conductivity for the case where both fr-CDW and s-CDW contriubute to the total gap with equal weights, 2A = 300 c m - 1 (solid line) and 2A = 700 (dashed line); t = 1000 c m - 1 , T = 300 c m - 1 and = 2.6. 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