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An electronic band structure study of TTF-TCNQ and (SN)x Friesen, Waldemar Isebrand 1975

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AN ELECTRONIC BAND STRUCTURE STUDY OF TTF-TCNQ AND (SN)x by WALDEMAR ISEBRAND FRIESEN B.Sc, Brock University, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1975 In present ing th is thes is in par t ia l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my writ ten pe rm i ss i on. Department of p Af S /C-T  The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i Abstract The electronic energy band structures of two highly conducting, anisotropic s o l i d s have been calculated using the extended Hiickel method. One-, two-, and three-dimensional models of the organic charge transfer s a l t t e t r a t h i o f u l v a l i n i u m tetracyanoquinodimethan (TTF-TCNQ) and of the inorganic polymer polysulphur n i t r i d e (SN)x have been studied. The r e s u l t s indicate that the band structure of TTF-TCNQ i s well described by a tight-binding, one-dimensional model i n which interactions between stacks of molecules are neglected. The Fermi surface i s seen to consist of extremely f l a t electron and hole surfaces, the nature of which i s inconclusive i n predicting a Fermi-surface-related i n s t a b i l i t y leading to a Peierls d i s t o r t i o n . A one-dimensional model of (SN)x predicts m e t a l l i c behaviour as the Fermi energy i s found to l i e at a symmetry-induced point of degeneracy where two bands cross. The single chain i s highly unstable against a symmetry-reducing d i s t o r t i o n ; however, three-dimensional interchain i n t e r -actions appear to s t a b i l i z e the structure. Consequently, the Fermi surface i s that of a semimetal with electron and hole pockets. The essential features of the band structure can be explained by a simple tight-binding model involving SN molecular anti-bonding IT o r b i t a l s . Differences i n the reported c r y s t a l structures used i n the calculation are seen to have no q u a l i t a t i v e effect. i i i Table of Contents Page L i s t of Tables i v L i s t of Figures v Acknowledgements v i i Chapter 1 Quasi-One-Dimensional Metals 1.1 Introduction 1 1.2 TTF-TCNQ: An Organic Metal 2 1.3 (SN)x: A Polymeric Superconductor 3 1.4 One-Dimensional Theories 4 1.5 Energy Band Calculation 10 2 TTF-TCNQ 2.1 Crystal Structure 13 2.2 One-Dimensional TTF-TCNQ 13 2.3 Three-Dimensional TTF-TCNQ 19 3 (SN)x 3.1 Crystal Structure 31 3.2 One-Dimensional Band Structure 31 3.3 Three-Dimensional Band Structure 38 4 Discussion 48 Bibliography 50 Appendix (SN)x Band Structure .for the Penn Crystal Structure 52 iv L i s t of Tables Table Page 1 Molecular coordinates i n TTF-TCNQ. 23 2 Overlap integrals between molecular o r b i t a l s i n TTF-TCNQ. 23 3 Atomic coordinates i n the (SN)x unit c e l l for the structure determined by Boudeulle (1974). 34 Al Atomic coordinates i n the (SN)x unit c e l l for the structure determined by Cohen et a l . (1975). 56 L i s t of Figures Figure Page 1 Effect of a Peierls d i s t o r t i o n on the electronic energy bands of a l i n e a r atomic l a t t i c e . 5 2 Shi f t of the electronic energy gap i n an e l e c t r i c f i e l d when the l a t t i c e d i s t o r t i o n moves with the current. 8 3 Crystal structure of TTF-TCNQ. 14 4 One-dimensional energy bands for a charge transfer of 0.5 electrons per TTF molecule, (a) For the undistorted chain, (b) For the distorted chain. 17 5 Density of states for one-dimensional TTF-TCNQ. 18 6 Energy bands along ky i n the region of the band crossing i n the 2-D case. 20 7 Cross-section of the 2-D Fermi surface i n the D"-C" plane. 22 8 Effect of the F-Q coupling on the bands i n Fig 6. 24 9 Three-dimensional band structure of TTF-TCNQ. 25 10 Histogram of the density of states for the 3-D band structure. 26 11 Energy bands near the Fermi energy i n the TZ dir e c t i o n . 12a Superimposed cross-sections of the electron and hole surfaces i n two dif f e r e n t planes p a r a l l e l to the = 0 plane. 12b Cross-section of the Fermi surface i n the k = 0 plane. 13a Projection of the c r y s t a l structure of (SN)x onto a plane perpendicular to the chain axis. 27 29 30 32 VI Figure Page 13b Perspective drawing of a side view of the cry s t a l structure. 33 14 Energy bands for an isolated (SN)x chain, (a) For the observed structure, (b) For the distorted structure shown i n Fig. 16. 36 15 Density of states for the two 1-D bands nearest the Fermi energy. 37 16 Di s t o r t i o n which breaks the screw axis symmetry i n an (SN)x chain. 38 17 B r i l l o u i n zone associated with the c r y s t a l structures of (SN)x and TTF-TCNQ. 40 18 Three-dimensional (SN)x band structure. 41 19 Tight-binding energy bands. 45 20 Histogram of the density of states for the tight-binding model. 47 Al View down the b-axis of the (SN)x c r y s t a l structure determined by Cohen et a l . (1975). 53 A2 Three-dimensional band structure. 54 A3 Tight-binding analogue of Fig. A2. 57 v i i Acknowledgement s I would l i k e to thank Dr. Birger Bergersen for introducing me to the world of one dimension, and for his subsequent patient and knowledgeable supervision. The ubiquitous and stimulating presence of Dr. John Berlinsky led to many f r u i t f u l discussions and i s hereby g r a t e f u l l y acknowledged. Appreciation for valuable discussions must also be expressed to the Friends of TCNQ, Dr. Jim Carolan, Dr. Dan L i t v i n , Tom Tiedje, and Dr. Larry Weiler, and to Dr. T.M. Rice. I thank Dr. Gi l b e r t Lonzarich for the use of his Fermi surface p l o t t i n g program. The members of the West Penthouse also deserve a word of appreciation for t h e i r constant encouragement and moral support. F i n a l l y ^ am grateful to the National Research Council for t h e i r f i n a n c i a l assistance i n the form of a Postgraduate Scholarship. 1 Chapter 1: Quasi-One-Dimensional Metals  1.1 Introduction One of the most s i g n i f i c a n t developments i n s o l i d state physics i n recent years has been the dicovery that certain organic solids with a high degree of anisotropy exhibit m e t a l l i c properties. The importance of these "quasi-one-dimensional" materials i s that they provide a d i r e c t experimental check of the theory which has been worked out for various one-dimensional (1-D) models. Of p a r t i c u l a r interest are the charge transfer s a l t s of tetracyano-quinodimethan (TCNQ) which have the highest e l e c t r i c a l conductivity of any known organic s a l t s . The large planar TCNQ molecule i s a good accep-tor with the extra electron occupying, unpaired, a Tr-orbital ( i . e . a wave-function which i s odd under r e f l e c t i o n through the molecular plane). This feature, together with the arrangement of the molecules face-to-face i n stacks imparts a strong 1-D character to the electronic properties. An impetus to the research e f f o r t i n t h i s f i e l d was provided by Coleman et a l . (1973) who reported measurements on tetr a t h i o f u l v a l i n i u m tetracyanoquinodimethan (TTF-TCNQ). They observed a metallic conductivity from room temperature down to 58K where i t became anomalously high, or i n t h e i r words, superconducting. Below 58K, a sharp drop i n the conductivity indicating a phase t r a n s i t i o n to an insulating state was observed. Further work by various groups has confirmed the metallic properties and the metal-insulator t r a n s i t i o n , but has not duplicated the extremely high conduc-t i v i t y . The work on TTF-TCNQ has spawned a whole host of organic solids with similar properties; however, TTF-TCNQ remains the prototype, and 2 consequently the most studied. A good deal of the interest i n 1-D conductors stems from the concept of a high temperature superconductor f i r s t introduced by L i t t l e (1964). While attention has been focussed on the TCNQ sa l t s i n t h i s regard, other quasi-l-D systems have also been investigated. One, the inorganic polymer polysulphur n i t r i d e ((SN)x) which was known to have a high conductivity, has recently been shown by Greene et a l . (1975) to undergo a superconduc-ting t r a n s i t i o n at about 0.3K. (SN)x i s , i n some respects, quite s i m i l a r to TTF-TCNQ; i t d i f f e r s , though, i n that i t does not have a metal-insula-tor t r a n s i t i o n . Superconducting or not, quasi-l-D materials are of fundamental interest as novel, r e l a t i v e l y unstudied systems. This thesis concerns i t -s e l f with a study of the two di f f e r e n t examples already mentioned, TTF-TCNQ o and (SN)x. Some experimental results and several theoretical concepts are presented i n the remainder of t h i s chapter. Chapter 2 deals with TTF-TCNQ and Chapter 3 with (SN)x. F i n a l l y , i n Chapter 4, the re s u l t s of the calculations are discussed, and a comparison of the two systems i s made. 1.2 TTF-TCNQ: An Organic Metal In TTF-TCNQ the two constituent molecules are stacked on separate chains which interact weakly i n comparison to the intrachain coupling, r e s u l t i n g i n conduction primarily along the chain axis. Because TTF i s a good donor, a charge of ^  one electron per TTF i s transferred to the TCNQ. Each chain thus carries a net charge so that i n a simple 1-D band picture TTF-TCNQ i s expected to be a metal. Typical room temperature conductivities p a r a l l e l to the chain axis are a'' ^300 - 1000 (n-cm)-"1 (Tiedje, 1975; Schafer et a l . , 1974; R1 Etemad et a l . , 1975). As the temperature i s lowered, a 11 increases approx-imately as T ^ to a peak value of 10 - 15 aR'Jp. At 53K, a sudden drop occurs, signify i n g a phase t r a n s i t i o n to semiconducting state..Below 53K, a" decreases smoothly down to 38K, where another sharp drop occurs, marking another phase t r a n s i t i o n . The room temperature transverse conductivity, a ^ , i s only about 1.4 (ft-cm) * with a 58K peak of three times t h i s figure. The anisotropy a "/o - L i s thus greater than 300. Various other experiments have confirmed one or both t r a n s i t i o n s : magnetoresistance (Tiedje et al.,1975), thermal conductivity (Salamon et a l . , 1975), s p e c i f i c heat (Craven et a l . , 1974), magnetic s u s c e p t i b i l i t y (Tomkiewicz et a l . , 1974), and the thermoelectric power (Chaikin et a l . , 1973). The experimental conclusion i s that TTF-TCNQ i s a metal above 53K and a small gap semiconductor below. 1.3 (SN)x: A Polymeric Superconductor (SN)x has been known to be highly conducting since the experiments of Goehring (1956). I t i s only with the recent interest i n the quasi-l-D solids that (SN)x has again become subject to experiment. The structure i s s i m i l a r to that of TTF-TCNQ, consisting of p a r a l l e l chains of atoms. The room temperature conductivity p a r a l l e l to the chain d i r e c t i o n i s a^, ^ 1000 (fi-cm)" 1 (Hsu and Labes, 1974; Greene et a l . , 1975); with de-creasing temperature a" increases u n t i l the onset of superconductivity at 0.26K. The transverse conductivity, a^~, i s much lower, the anisotropy a "/a-** ranging from 50 - 1000. It should be stressed that no metal-insulator t r a n s i t i o n i s observed. An important feature of the macroscopic c r y s t a l structure relevant to any discussion of the anisotropy i s the 4 c r y s t a l l i z a t i o n of (SN)x into "bundles of f i b e r s " , with f i b e r diameters of o about 1000 A. The effect of t h i s fibrous structure on the transverse properties i s not clear. 1.4 One-Dimensional Theories The phrase "one-dimensional metal" i s i n one sense a paradox, since a 1-D non-interacting electron gas i n a periodic potential cannot be a conductor. This was shown by Peierls (1955), and can be i l l u s t r a t e d by a simple example which i s sketched i n Fig. 1. Consider a chain of atoms with one atom per unit c e l l , and l a t t i c e spacing a; i f , i n the t i g h t -binding approximation, each atom contributes one electron to the conduc-ti o n band, the band i s exactly h a l f - f i l l e d , with Fermi wavevector kp = Tt/2a. In t h i s case the band i s c l e a r l y m e t a l l i c . However, a dimer-i z a t i o n of the chain reduces the symmetry by doubling the l a t t i c e spacing to 2a, which shift;; the B r i l l o u i n Zone boundary to k = kp. A gap opens up at k„, thereby reducing the electronic energy and making the met a l l i c r chain unstable against such a periodic deformation, or P e i e r l s d i s t o r t i o n , of wavevector q = 2kp. This i n s t a b i l i t y i s not confined to a h a l f - f i l l e d band, but w i l l occur for any degree of b a n d - f i l l i n g . When electron-phonon interactions are included, the same symmetry-reducing t r a n s i t i o n may occur as the manifestation of an effect f i r s t predicted by Kohn (1959). Phys i c a l l y t h i s effect arises from the fact that electrons near the Fermi surface with - k„ can be scattered i n an r energy conserving process by phonons with wavevector q = -2k., As k kp, the f i r s t derivative of the d i e l e c t r i c function diverges, r e f l e c t i n g a sudden change i n the a b i l i t y of the electron gas to screen the l a t t i c e 5 —^ a if— o o o o o o o s\ 2ft o o o o o o o Figure 1. Effect of a Peierls d i s t o r t i o n , (a) Tight-binding energy band for the line a r chain with interatomic spacing a. (b) The bands for the chain i n which the atoms have dimerized to double the l a t t i c e parameter to 2a. vib r a t i o n . The resultant kink i n the phonon energy i s known as the Kohn anomaly. In the case when the phonon frequency goes to zero, the l a t t i c e v i b r a t i o n becomes a permanent (Peierls) d i s t o r t i o n with wavevector 2k^ ,. The strength of the effect depends on how much of the Fermi surface can be connected by the wavevector 2kL (or how well the Fermi surface nests) r In 2- or 3-D, the effect i s a weak one; i n 1-D however, the anomaly i s 6 pronounced since the Fermi surface consists of two singular points at kp, and the nesting i s perfect. A discussion of the Fermi surface i n s t a b i l i t y within mean f i e l d theory has been given by Rice and Strassler (1973). They consider a linear chain with a h a l f - f i l l e d tight-binding band. The (Fr6hlich) Hamiltonian i s written as, H - XeP2*Vt+ ^ * W W faM* Here the b's and c's are the usual operators for the unperturbed phonons of energy hu^ and Bloch electrons of energy respectively; N i s the number of ions i n the chain, and g(q) i s the electron-phonon coupling constant. The screened phonon frequencies, o, of the coupled electron-phonon system are given by, where the s u s c e p t i b i l i t y x(q>T) i s defined as, f i s the Fermi d i s t r i b u t i o n function. P 2 Defining a dimensionless parameter A = hco /2g N(0) i n which g = % g(q 0 = 2kp) and N(0) i s the density of states at the Fermi l e v e l , the phonon frequency ft i s found to be, ^ . = where k T c * Z . Z % £ f t - ' - x C5j 7 Thus ft approaches zero logarithmically as T approaches the c r i t i c a l temperature T . Below T , Rice and Str&ssler view the linear chain as a condensed c phonon state of wavevector q Q such that the expectation values <b > = + <b >= (Nu/2)6 , where u i s a dimensionless amplitude. In t h i s case an q q>qo energy gap of magnitude 2A(T) appears i n the electronic energy spectrum -2A at p = k„. From a zero temperature value A(0) = 4e ce , the gap de-creases, vanishing f i n a l l y at the T c given by (5). In the framework of th i s model, then, the high temperature metal undergoes a second order phase t r a n s i t i o n to a low temperature insulator. It has been argued that the Kohn anomaly does not always lead to an insulating state. Frohlich (1954) considered a system with the Hamiltonian given by (1) and argued that superconductivity might resu l t i n the f o l -lowing manner. When an external f i e l d i s applied the electrons are d i s -placed i n k-space with a v e l o c i t y v . If the macroscopically occupied l a t t i c e v i b r a t i o n with wavevector 2k„ moves with the electrons, a super-r l a t t i c e of p e r i o d i c i t y 2?r/2kp exists i n the frame of the electrons. Consequently, energy gaps w i l l appear at the displaced Fermi surface as in F i g . 2. At low temperatures the lower band i s completely f i l l e d , and as long as the two bands do not overlap no scattering can occur; t h i s gives r i s e to a supercurrent. In the c r y s t a l frame the electronic energy i s E(k) + hkpV g. When hkpV g becomes greater than h a l f the gap, A, the free energy i s lowered by electrons being scattered into the conduction band, thereby introducing a resistance and decreasing the supercurrent. The coupled electron-phonon mode described here i s known as a Frohlich mode. Bardeen (1973) has suggested that the Frohlich mode might be a mechanism 8 J \ 1 , / \ 1 A \% > T T i * \ /1 1 \ I i 7 i 1 \ • i i v i i 1 i -or -jr n m . "£ Figure 2. Shift of the electronic energy gap i n an e l e c t r i c f i e l d when the l a t t i c e d i s t o r t i o n moves with the current. for the high conductivity i n TTF-TCNQ. Unfortunately, mean f i e l d theory i s inaccurate i n describing 1-D systems due to fluctuations of the order parameter. Landau and L i f s h i t z (1969) have shown that for a system with short range forces, fluctuations prohibit a phase t r a n s i t i o n except at zero temperature. Lee, Rice, and Anderson (1973) have included fluctuations i n a 1-D model using the c o r r e l a t i o n length, E,, as the order parameter i n a genera-l i z e d Landau theory. They find that for T £ T c (T being the mean f i e l d t r a n s i t i o n temperature), £(T) becomes large, increasing exponentially with temperature. Although E, (T) diverges only at T = 0, i t i s large enough so that with a weak 3-D coupling a 3-D t r a n s i t i o n can occur at T ^ 1/4 T . The effect of the fluctuations on the electronic density of c states i s to change the well-defined gap which exists at T = 0 to a "pseudo-gap"; that i s , an energy i n t e r v a l i n which the density of states i s nearly zero. This result would then have some influence on the Kohn anomaly. 9 One-dimensional models are'unrealistic i n the sense that they ignore the f i n i t e , i f small, interchain interactions that are present i n real systems. Horovitz, Gutfreund, and Weger (HGW,1975) have included i n t e r -chain effects i n the model described by (1) with nearest-neighbour tight-binding of the form, = - £ p Cos fyt (6) p a r a l l e l to the chain, and £(px,p ) = ttpj ~~ q€0(t°s tap*. + tos ape) (7) transverse to the chain. Here £p, pp are the Fermi energy and wavevector for a single chain, eQ i s an energy parameter - £p, n i s the r a t i o of the interchain to the intrachain coupling (n<<l), and a i s a measure of the b a n d - f i l l i n g . For a h a l f - f i l l e d band (a = 0) the Fermi surface appears e f f e c t i v e l y f l a t to the wavevector q Q = ( 7 r/aj2pp, 7 r/a); that i s to say, the Fermi wavevector nests perfectly for q . This implies a giant Kohn amomaly at q . When the band i s other than h a l f - f i l l e d , the wavevectors connecting the Fermi surface do not have the same q^ component, res u l t i n g i n a smearing of the Kohn anomaly. There exists a c r i t i c a l value n £ of the interchain coupling strength such that for n > n c the Peierls t r a n s i t i o n i s suppressed; for example, at T = 0, n c -3.5 ^ T c(n=0)/e Q|a| , T £ again denoting the mean f i e l d t r a n s i t i o n temper-ature. It i s interesting to note that for a = 0, the interchain coupling does not. affect the i n s t a b i l i t y . A t r a n s i t i o n at q^ = (0,2pp,0) can also occur i f n < - J T £(n=0)/e o-10 In f a c t , an i n s t a b i l i t y w i l l exist at any for which the el e c t r o n -phonon coupling parameter A(q) i s a maximum. I f there i s no strong depen-dence on q , however, the q" = i n s t a b i l i t y w i l l be favoured for T those values of the interchain coupling for which r\ £ n c - T £ / i t i s roughly i n t h i s region that f l u c t u a t i o n effects are small. When fluctuations are included i n a treatment of the i n s t a b i l i t y at q Q, HGW f i n d a temperature c h a r a c t e r i s t i c of the system, = neQ/4; for a phase t r a n s i t i o n to occur i t i s required that the r e a l t r a n s i t i o n temperature be a-T^. Further, 1-D mean f i e l d theories are found to be appropriate i n the region, VIP1 o -> -> where T i s the mean f i e l d T for q = q . c c n no (8) 1.5 Energy Band Calculation In t h i s section the method used i n calc u l a t i n g the energy bands i s outlined. A linea r combination of atomic o r b i t a l s (LCAO) method i s used for (SN)x because of the covalent bonding nature of the constituents. The same method i n which the atomic o r b i t a l s are replaced by molecular o r b i t -als i s used for TTF-TCNQ. The electronic wavefunction, ^ ( r ) i s expanded as a linear combin-ation of Bloch o r b i t a l s , where 11 -> -V- -y Here d>. f r - R - r.) i s the i ' t h atomic o r b i t a l centered at position y i v m 1 r. i n the unit c e l l with postion vector , and N i s the number of unit i r m c e l l s i n the c r y s t a l . Substitution of (9) into the Schrodinger equation leads eventually to the set of coupled lin e a r equations, a i for the energies e(lc) and the c o e f f i c i e n t s a.. . The matrix elements H^ .. and S„ are given by, it*, ^ u e ha V (12) S V The integrals h™.. were calculated using the semiempirical extended Hilckel method (EHM) of Hoffman (1963). In t h i s approximation the term l u \ i s scaled with the overlap s1?. as follows: F i l ki\ - i (13) K i s the phenomenological Wolfsberg-Helmholtz parameter which i s varied to give the best f i t to experiment. Usually K is.taken to be 1.75, a value which was used i n our c a l c u l a t i o n , i s the experimentally deter-mined ioni z a t i o n potential for o r b i t a l i . If n o r b i t a l s per unit c e l l are included, the energy bands are obtained by solving the n x n eigen-value equation (11) for each wavevector. The EHM allows the f u l l symmetry of the c r y s t a l to be incorporated correctly. While the method i s somewhat crude, i t i s expected to give energy differences such as bandwidths quite w e l l . In addition, i t s sim-p l i c i t y leads to an easy interpretation of the energy eigenvectors. The EHM has been used i n si m i l a r calculations by several authors: M Cubbin and Manne (1968) and Fleming and Falk (1973) for polyethylene, Kortela and Manne (1974) for graphite, and Berlinsky et a l . (1974) for TTF-TCNQ. 13 Chapter 2: TTF-TCNQ 2.1 Crystal Structure The c r y s t a l structure as reported by Kistenmacher et a l . (1974) i s shown i n Fig. 3; molecular coordinates are given i n table 1. The dominant feature i s the segregated chains of TTF and TCNQ molecules stacked i n a monoclinic l a t t i c e . Neighbouring chains along the c-axis are related by a screw axis symmetry operation: a rotation of 180° about an axis midway between the- chains, together with a tra n s l a t i o n of b/2 along the axis takes one chain into the other. A unit c e l l thus contains two TTF and two TCNQ molecules. The space group of the structure i s P2^/c, the symmetry elements of which are a center of inversion at each molecular s i t e and the screw axis. 2.2 One-Dimensional TTF-TCNQ The stacking arrangement of the structure, together with the observed anisotropy i n the conductivity point to a model i n which there are interactions only along the chain and not between the chains. That t h i s f i r s t approximation i s indeed a good one i s borne out l a t e r when interchain coupling i s taken into account. A part of t h i s section i s based on the work of Berlinsky, Carolan and Weiler (BCW, 1974) since i t i s a necessary prelude to what follows. BCW have used the EHM to calculate the molecular o r b i t a l s (MO's) for both TTF + and TCNQ molecules, with the result that the highest occupied MO on each molecule i s a i r - o r b i t a l . The symmetry of these MO's i s such that the TTF + wavefunction i s even and the TCNQ wavefunction odd under 14 (b) Figure 3. Crystal structure of TTF-TCNQ. TTF and TCNQ are labelled F and Q respectively, while the numbers indicate those molecules whose coordinates are given i n Table 1; 1,3,4,6 are i n the same unit c e l l and 2 and 5 are in the c e l l displaced by ft. (a) View perpen-dicula r to the chain axis. The s o l i d c i r c l e s indicate those atoms positioned above a plane which passes through the center of the molecule and which i s p a r a l l e l to the U-c" plane, (b) Side view along the a-axis. 15 r e f l e c t i o n through planes bisecting the molecules along t h e i r short dimension and normal to the molecular plane. An important consequence of t h i s r e s u l t i s that the TTF-TCNQ overlap along the a-axis i s i d e n t i -c a l l y zero, thus greatly diminishing interchain effects. Following the BCW argument that i t i s v a l i d to i d e n t i f y these MO's with the valence and conduction bands i n the s o l i d , we take them as the basis functions for our energy band ca l c u l a t i o n . I t should be pointed out that the a-axis overlap i s zero only for those wavefunctions calculated for isola t e d molecules. Inclusion of the c r y s t a l f i e l d reduces the MO symmetry,resul-ting i n a f i n i t e overlap. This can be seen i n Fig. 3 where the atoms marked and S£ are situated i n d i f f e r e n t environments. The size and ef-fect of a non-zero a-axis overlap i s not known. In the 1-D approximation, the c a l c u l a t i o n reduces to that for the linear atomic chain of l a t t i c e constant b and one o r b i t a l per unit c e l l . We write the wavefunction as, where N i s the number of molecules i n the chain and <K(r - nb) i s an MO centered at the n'th s i t e . The subscript i = F or Q, which w i l l here-after denote TTF and TCNQ respectively. The energy for band i i s then given by, (14) 2X (15) 22* 16 Including only nearest-neighbour interactions gives, l~ l-Htoskk<fi(?+t)lfifr)> ( 1 6 ) Since <<J>^ (r + b) |<jK(r)> E << 1, the denominator can be expanded to get, £; = { < < f t ( r ] / t f / < £ ^ . (17) Making use of the Huckel approximation and retaining only terms to f i r s t order i n a^, we f i n a l l y have, (18) where e? i s the ion i z a t i o n potential of molecule i , and t. = (K - l)eVa.. l r I V I i The Fermi le v e l intersects both bands, with the consequence that the Fermi wavevector determines the amount of f i l l i n g of each band, or equivalently, the degree of charge transfer. Grobman et a l . (1974) i n t e r -preted t h e i r photoemissidn data as indicating a charge transfer of about one electron per TTF donor. I f t h i s i s the case the band i s h a l f - f i l l e d and e° =' e°. A l l calculations were therefore i n i t i a l l y performed for a h a l f - f i l l e d band model. Using the BCW estimate that e° = -7.5 eV, to-r gether with the calculated overlap integrals a p = -9.3x10 and = _2 2.0x10 , one obtains t c = 0.05 eV and t n = -0.11 eV, or bandwidths of 0.20 and 0.44 eV for the F and Q bands. The fact that t„ and t„ are op-posite i n sign i s s i g n i f i c a n t since i t means that the bands must cross, and hence interact i n the presence of F-Q coupling. More recently, evidence for a Pe i e r l s d i s t o r t i o n has been provided by an X-ray analysis of the c r y s t a l structure by Denoyer et a l . (1975). Below 40K they observe a 3-D superlattice with dimensions 2a x 3.7b x nc (n unknown). In the context of 1-D F and Q bands the new l a t t i c e parameter 3.7b implies a Fermi wavevector kp = fr/3.7b, which corresponds to a near-l y q u a r t e r - f i l l e d Q band. The io n i z a t i o n potentials were accordingly changed to E° = -7.58 eV and E Q = -7.35 eV. The proposed band structure i s presented i n Fig. 4a for an undistorted chain and i n Fig. 4b for a chain with a d i s t o r t i o n wavelength of 4b. Figure 4. (a) One-dimensional energy bands for a charge transfer of 0.5 electrons per TTF molecule, (b) The effect of a Peierls d i s t o r t i o n of wavelength 4b on the bands of (a). 18 The density of states for the cosine band i i s , per molecule, 1). f = ' The t o t a l density of states i s then simply the sum D(e) = Dp(e) + D Q ( E ) as shown i n Pig. 5. D ( € ) Figure 5. Density of states for the two non-interacting bands i n Fig. 4a. E„ i s the Fermi energy. 19 2.3 Three-Dimensional TTF-TCNQ Before considering the three-dimensional band structure i t might be ins t r u c t i v e to look at the 2-D case i n which we include interactions between chains of l i k e molecules. The equal-energy planes perpendicular to the b-axis of the 1-D problem disappear to be replaced by lines of equal energy normal to the b- and c-axes. Each of the doubly degenerate bands i n Fig. 4a i s s p l i t by the interchain coupling. The 4 x 4 secular determinant i n (11) assumes a block diagonal form, H 11 '21 e S l l H12 £ S 2 1 H22 eS eS 12 22 *33 " e S33 H34 " e S34 Ul " £ S43 H44 ~ £ S44 = 0 (20) where, i f only nearest neighbour overlaps, C K , on neighbouring chains are included, s l l = S22 --y -y - 1 + 2ar,cosk-b F S33 = S44 = -y -y - 1 + 2aQCOsk*b S12 * = S21 = a p ( l + e c ) ( l + e S34 * = S43 = a Q ( l + e C ) ( l + e H l l = H„„ = 22 0 O t> : e„ + 2Ke a„cosk r F F t H33 44 : E Q ° + 2K £ Q ° O - Q C O S1C " t H12 * = H21 = Ke F°S 1 2 H34 * = H43 = K GQ° S34 The res u l t i n g energy bands are approximately given by the expressions, where, S°£i± All ; Z>L-£i +Z(K-l)£i<ri coctt In the region of the band crossing the bands w i l l appear as i n Fig. 6 t CD c r UJ 2: UJ FERMI LEVEL Figure 6. Energy bands along k i n the region of the band crossing i n the 2-D case. y The points at which the Fermi level intersects the bands are determined by the magnitude of the s p l i t t i n g s E„ Since o << 1, the bands are ex-tremely f l a t . The deviation of the Fermi wavevector from k^ = ft/4b w i l l then be small, wo we may write i t as, 21 (23) Using equations (22) gives, 7» -» T (Jos (24) A cross-section of the Fermi surface i n the b-c plane i s sketched i n Fig. 7. The surfaces have been labelled F or Q depending on which band the Fermi surface crosses. It i s rea d i l y apparent as indicated by the arrows that the portion of the Fermi surface i n the upper h a l f of the zone nests perfectly with the portion i n the lower half. One would thus expect the 2-D model of TTF-TCNQ as outlined here to be unstable against a d i s -t o r t i o n of wavevector \ = (q ,qh,q ) = (iT/aa ,ir/2b, 0) , with a 1 deter- • 3. D C mining the p e r i o d i c i t y i n the a-direction. From the discussion of HGW, a would be that value maximizing the electron-phonon coupling constant X(q). A 2-D model for TTF-TCNQ i s not r e a l l y v a l i d since the interaction energy between F and Q chains i s roughly of the same magnitude as that for the F-F and Q-Q couplings. When F-Q interactions are added the degen-eracies at the crossover points are l i f t e d and some small curvature i n -y -y the a x e dir e c t i o n i s added to the equal energy l i n e s . Fig. 8 indicates what happens to the bands i n the crossover region i n a di r e c t i o n p a r a l l e l to the b-axis. Whether or not an energy gap occurs depends on the strength of the F-Q coupling. The important interchain overlaps (from BCW) are presented i n Table 2. With these overlaps, the terms i n the secular determinant not given by (21) are, ->-S31 " S42 = S * = S * -ik-b ik-b )(1 + e ik • a ) 13 24 - e 22 Figure 7. Cross-section of the Fermi surface i n the B"-c plane. The v e r t i c a l scale represents the deviation (in units of ir/b) of the surface from the planes = ir/4b and k^ = -7r/4b. Arrows show the nesting wavevectors. 23 Table 1. Molecular coordinates i n TTF-TCNQ for those molecules shown i n Fig.3. In the coordinate system used,the l a t t i c e vectors are, in A un i t s : £ = (12.298,0,0), ft = (0,3.819,0), t = (-4.61,0,17.883) X Y Z 1 6.149 0 0 2 6.149 3.819 0 3 3.843 1.910 8.942 4 . 12.298 0 0 5 12.298 3.819 0 6 . 9.992 1.910 8.942 Table 2. Overlap integrals s m. between MO's centered on the molecules l i s t e d i n Table 1 aAd shown i n Fig. 3.(taken from Berlinsky et a l . , 1974). 1 2 3 4 5 6 . 1 1 2.0xl0" 2 -3. -4 0x10 0 1 -4 4x10 1 8 x l 0 "? 2 2.0xl0" 2 1 -3. OxlO" 4 -1 -4 4x10 0 3 -4 ' 3x10 3 -3.0xl0" 4 -3.0xl0~ 4 1 1 7 x l 0 " 6 3 -4 4x10 0 4 0 -1.4xl0" 4 1 7 x l 0 " 6 1 -9 3 x l 0 " 3 4 8xl 0 " 5 5 1.4xl0" 4 0 3 -4 4x10 -9 3 x l 0 "3 1 4 8 x l 0 " 5 6 1.8xl0" 7 3.3xl0" 9 0 4 8 x l 0 " 5 4 8 x l 0 " 5 1 24 > o L±J UJ Figure 8. Effect of the F-Q coupling on the bands i n Fig. 6. '32 = S 23 541 H. 11 2 , - i t - (S+c) ilc-a, S14* ^ F Q ^ ' ^ " ^ K(,° + e°)S../2 • C25) The energy bands i n various directions i n the B r i l l o u i n Zone (Fig. 17) are shown i n Fig. 9. The most s t r i k i n g frature of the bands i s that they are very nearly f l a t , demonstrating the 1-D nature of TTF-TCNQ. A further corroboration of t h i s observation i s the appearance of the density of states which has been calculated for 5246 wavevectors i n the Zone. The histogram plotted i n Fig. 10 i s nearly i d e n t i c a l to the sketch i n Fig. 5. The inset of Fig. 10 shows the density of states calculated for a f i n e r mesh of wavevectors near the band crossing. The small peaks on either side of the dip arise from the mutual repulsion of the bands while the dip i t -s e l f i s due to the gap opened up i n most directions. 25 "Figure 9. Three-dimensional energy bands along several directions i n the B r i l l o u i n Zone which i s shown i n Fig. 17. 26 Figure 10. Histogram of the density of states for the 3-D band structure. Inset: density of states near the Fermi l e v e l . Figure 11. Energy bands near the Fermi energy i n the TZ d i r e c t i o n . 28 The crossover region along the Tz d i r e c t i o n i s shown i n expanded scale i n Fig. 11. To picture the Fermi surface i t i s convenient to view the bands throughout the B r i l l o u i n Zone as a large number of curves s i m i l a r to the ones along r z . Because of the flatness of the bands the top of the valence band, e (k_ ,k ), and the bottom of the conduction band, e (k ,k ), v a c c a c w i l l deviate only s l i g h t l y i n energy through the Zone. Since the i n d i r e c t overlap of the bands i s very small, the Fermi level l i e s very close to and e , intersecting both bands, only one band, or neither band, cre-ating hole and electron surfaces. It must be pointed out that because the overlap of the bands i s small the shape of the Fermi surface i s quite sensitive to even small changes i n the bands. In addition a precise deter-mination of the Fermi energy i s d i f f i c u l t and probably not meaningful, given the inaccuracies inherent i n the method of c a l c u l a t i o n . Nevertheless, i n order to gain a q u a l i t i t i v e idea of the Fermi surface a Fermi energy was guessed (E = -7.51 eV) using the calculated band structure, r Cross-sections of the Fermi surface i n several planes are shown i n Fig. 12; as expected the surface consists of a very f l a t electron and hole pocket i n each hal f of the Zone. While the value Ep = -7.51 eV i s l i k e l y a l i t t l e too low, a s l i g h t l y higher energy would, not make any q u a l i t a t i v e difference, decreasing the size of the electron surface some-what and increasing the hole surface. In any case, the extremely small degree of curvature i n the surfaces makes i t possible for d i f f e r e n t nest-ing vectors to connect various combinations of electron and hole surfaces. One of these p o s s i b i l i t i e s i s the wavevector with the component q^ = ir/2b l i n k i n g a hole surface with an electron surface i n the opposite h a l f of the Zone. However, i t i s not apparent why the observed d i s t o r t i o n i s (TT/a,ir/2b,iT/nc). Figure 12. (a) Superimposed cross-sections of the electron and hole surfaces in two d i f f e r e n t planes p a r a l l e l to k, = 0. Figure 12. (b) Cross-section of the Fermi surface i n the k = 0 plane. Shown are the electron and hole contours i n the'upper half of the zone. The v e r t i c a l scale i s i n units of ir/b 31 Chapter 3: (SN)x 3.1 Crystal Structure Two determinations of the (SN)x c r y s t a l structure have recently be-come available. The results of the f i r s t , an electron d i f f r a c t i o n study by Boudeulle (1974), are shown i n Fig. 13 and are used i n the calculations presented i n the remainder of the chapter. A l a t e r report by Cohen et a l . (1975) based on X-ray d i f f r a c t i o n measurements gives a somewhat dif f e r e n t structure (see Appendix) i n which the l a t t i c e constants are nearly the same but the atomic bond angles are di f f e r e n t from those of Boudeulle. Calculations incorporating the Cohen data are presented i n the Appen-dix. Despite the difference i n c r y s t a l structure the re s u l t s are essen-t i a l l y the same. Thus, the fact that the electron d i f f r a c t i o n method i s less r e l i a b l e i s of l i t t l e consequence i n our study. The c r y s t a l structure of (SN)x resembles that of TTF-TCNQ; i n f a c t , the space group P2^/c i s the same. Sulphur and nitrogen atoms stack along the b-axis forming alternating short (1.58 A) and long (1.72 X) SN bonds such that four atoms per chain are included i n the unit c e l l . Nearest-neighbouring chains i n the c-direction are related by inversion and are thus inequivalent, giving a t o t a l of eight atoms per unit c e l l . The atomic coordinates are l i s t e d i n Table 3. 3.2 One-Dimensional Band Structure As i n the preceding discussion of TTF-TCNQ, we f i r s t consider the case of isolated chains. The wavefunction \ ( r ) i s written as, 32 Figure 13. (a) Projection of the c r y s t a l structure of (SN)x onto a plane perpendicular to the chain axis. The sulphur and nitrogen atoms have been labelled S. and N. , i = 1,2,3,4 depending on the position i n the unit c e l l . 33 Figure 13. (b) Perspective drawing of a side view of the c r y s t a l struc-ture. The s o l i d rods indicate the bonds along the chain and the open rods represent the most important interchain overlaps discussed i n §3.3. Table 3. Atomic coordinates i n the (SN)x unit c e l l for the Boudeulle structure. In the coordinate system used, the l a t t i c e vectors are, i n A units: £ = (4.12,0,0), ft = (0,4.43,0), and t = (-2.55,0,7.20). The b-axis i s the chain axis. X Y Z Si -1.20 1.74 1.20 s 2 -0.08 -0.48 2.40 s 3 1.20 -1.74 -1.20 SH 0.08 0.48 -2.40 Ni -0.90 0.20 1.05 N 2 -0.38 -2.02 2.55 N 3 0.90 -0.21 -1.05 0.38 2.02 -2.55 ^-'U^^'/^?) (26) Here N i s the number of unit c e l l s , R i s the coordinate of the center of m the m'th c e l l , ?. i s the coordinate with respect to the center of the c e l l 3 of the j ' t h o r b i t a l , and . (r) i s an atomic Slater o r b i t a l . ^rK?) w a s sub-st i t u t e d into the Hiickel formula using the constants e^g = -26.0, = -13.4 eV, with Slater exponent 1.95 for nitrogen, and e^g = -20.0, e^p = -11.0 eV with Slater exponents 2.122 and 1.827 for sulphur. Inclusion of the sulphur 3d o r b i t a l s i n the basis set was found to result i n a narrowing of the bands near the Fermi le v e l with no other q u a l i t a t i v e changes i n the band structure. In some of the Huckel calcula-tions mentioned i n §1.5 the value K = 2.0 for the Wolfsberg-Helmholtz constant was found to lead to better re s u l t s that the more popular value K = 1.75. An increased K produces broader bands, an effect opposite to the inclusion of the 3d states. Thus i t was f e l t that we were safe i n tak-ing K = 1.75 and neglecting the 3d o r b i t a l s i n our c a l c u l a t i o n . Since the single chain unit c e l l consists of four atoms, (SN)^, the energies for each k-value are obtained by solving a 16 x 16 eigenvalue equation. Fig. 14a shows the energy bands which r e s u l t . Since the (SN)^ unit has 22 valence electrons, the 11 lowest bands w i l l be exactly f i l l e d at T = 0. Ordinarily one would then expect (SN)x to be an insulator. How-ever, the screw axis symmetry of the c r y s t a l structure requires pairs of bands to be degenerate at the Zone boundary; since an odd number of bands are f i l l e d there i s no gap at the Zone boundary. The density of states for the highest occupied and lowest unoccupied bands i s shown i n Fig. 15 (cf. Fig. 5). 36 Figure 14. Energy bands for an isolated (SN)x chain, (a) For the observed structure. The symmetry labels are defined i n the text while the dashed l i n e represents the Fermi l e v e l , (b) For the distorted structure shown i n Fig. 16. 37 CO UJ I— < I— CO u. o > I— CO z UJ Q -10 -9.5 E F ENERGY (eV) -9 Figure 15. Density of states for the two bands straddling the Fermi energy i n Fig. 14a. Because the chains are nearly planar, the labels s or IT may be used to indicate o r b i t a l s which are approximately symmetric or anti-symmetric upon r e f l e c t i o n through t h i s plane. The bands i n which the k = 0 o r b i t a l s have a node along the short SN bond are labelled by *, and 1 labels a k = 0 o r b i t a l with a node on the long SN bond. These results agree with the c a l c u l a t i o n of Parry and Thomas (1975) who used a s l i g h t l y d i f f e r e n t version of the EHM. It i s obvious that since the degeneracy at the Zone boundary i s due to the screw axis symmetry, any d i s t o r t i o n of the chain which breaks the symmetry w i l l open a gap at the Fermi l e v e l . One example of such a 38 d i s t o r t i o n i s i l l u s t r a t e d i n Fig. 16 (note that the size of the unit c e l l remains unchanged). undistorted distorted T o T Figure 16. Distortion which breaks the screw axis symmetry. The effect of such a d i s t o r t i o n i n which the short bondlength i s changed o by 0.1 A i s demonstrated i n Fig. 14b. We conclude, therefore, that because of t h i s i n s t a b i l i t y (SN)x should be a semiconductor. 3.3 Three-Dimensional Band Structure Boudeulle has pointed out that some of the S-S and N-N distances between atoms on di f f e r e n t chains are less than twice the Van der Waals r a d i i for sulphur and nitrogen. The interchain coupling should then be r e l a t i v e l y large, implying that the 1-D approximation i s not very good. Since there are two chains per unit c e l l , the 1-D bands are doubly degen-erate, the degeneracy being removed by the interchain coupling. The s p l i t -t i n g of the bands s h i f t s the Fermi level away from the Zone boundary, while 39 hybridization between the bands causes a s p l i t t i n g were the bands should cross. A f u l l 3-D band structure taking into account an (SN)^ unit on one chain i n a unit c e l l with 20 neighbouring (SN^ units (two on the same chain and three on each of six surrounding chains) has been calculated and i s shown along several symmetry directions of the B r i l l o u i n Zone (Fig. 17) i n Fig. 18. The lines and planes of degeneracies which arise are the resul t of the screw axis and time reversal symmetry. The important feature i n Fig. 18 i s the crossing of the bands i n TZ and YC directions while there are gaps i n a l l the other directions. The crossings are accidental degeneracies and are allowed because the wavefunctions of the crossing bands transform d i f f e r e n t l y under the screw axis symmetry operation. An accurate determination of the Fermi le v e l i s not e a s i l y obtained. Q u a l i t a t i v e l y , however, i t i s not d i f f i c u l t to v i s u a l i z e the appearance of the Fermi surface. The crossings along TZ and YC occur at di f f e r e n t energies so that the Fermi le v e l intersects the bands at points other than the degenerate points. One thus expects a Fermi surface consisting of an electron pocket near the crossing along YC and a hole pocket near the cros-sing on TZ, leading us to describe (SN)x as a semimetal. It i s g r a t i f y i n g to note that our results agree q u a l i t a t i v e l y with the more sophisticated r e l a t i v i s t i c OPW calculations of Rudge (1975). The most important feature of Rudge's that i s not present i n our res u l t s i s the lower band i n the ZE di r e c t i o n r i s i n g across the Fermi l e v e l , creating a hole pocket near point B. His conclusion reinforces ours that (SN)x i s a semimetal. The 3-D band structure i s obtained from a 32 x 32 eigenvalue equation that i s cumbersome to work with. A s i m p l i f i e d treatment can be made by 40 Figure 17. B r i l l o u i n zone associated with the c r y s t a l structures of (SN)x and TTF-TCNQ. Symmetry dictates two-fold degeneracies i n the band structure throughout the top and bottom zone faces and along the lines AE and BD. CY i s not a l i n e of degeneracy. r Z IE C C Y Y P Figure 18. Three-dimensional band structure i n those directions indicated i n Fig. 17 as obtained from the extended Huckel c a l c u l a t i o n . Only the four bands closest to the Fermi level are shown. The dashed l i n e marks the approximate position of the Fermi energy. 42 noting that the highest occupied and lowest unoccupied bands i n the 1-D calculation (Fig. 14a) look very much l i k e simple tight-binding (cosine) bands. The k = 0 wavefunction symmetries indicated i n Fig. 14a suggest that a suitable basis for such a calculation might be a IT* SN molecular o r b i t a l . There are four such SN molecules per unit c e l l leading us to consider a 4 x 4 model Hamiltonian. An important s i m p l i f i c a t i o n of the matrix can be obtained by a closer examination of the cr y s t a l structure (Table 2 and Fig. 13). The molecule labelled S^N^ i s considerably farther away from molecules labelled S^N^ i n i t s own and i n neighbouring c e l l s than from those labelled S 2 N 2 and S^N^. Taking symmetry into account, i t should be a good approximation to neglect matrix elements along the a n t i -diagonal; the v a l i d i t y of t h i s approximation i s confirmed by a calculation of the overlaps. The Hamiltonian then takes the form ( i f the energy of the SN MO i s taken to be 0 ) , H 0 y* 61 0 y 0 0 0* h 0 0 y 0 62 y* 0 C27) Expansion of the corresponding secular determinant gives a ch a r a c t e r i s t i c equation of the form, 4 2 E - 2ae + 6 = 0 (28) with roots, ± Jo~± Ja2 - 8 (29) 43 where, 2a = 2 i u 12 + \Si\2 + |6 2| 2 Quantitative results for t h i s model can be obtained by examination of the wavefunctions coming out of the band structure c a l c u l a t i o n . The TT* SN MO which i s suggested has the form, * = + s - 2<f>N (30) The most important interactions involving these MO's are then: (i) t : the ir-interaction between nearest-neighbouring molecules on the same chain. o ( i i ) t ^ : a ^-in t e r a c t i o n with a bond length of 3.10 A between an atom S^ and the atom S^ i n the c e l l displaced by (B" - a). o ( i i i ) t^'- a a-interaction with bond length 2.81 A between atoms N^ and N^ i n the same unit c e l l . (iv) t ^ : a a-interaction between S^ and S^ i n the c e l l displaced by t). Including symmetry-equivalent interactions gives for the matrix elements the expressions, . ,., -ik-b. y = t ( l + e ) ^ ik-(a-b) . . -ik'b Sl = t r e + t 2 + t 3 e ^ (31) -it.-(a+c) ^ i ^ - (D " - C ) + t o e ~ l k 0 6 2 = t x e v J + t 2 e v J 3 The parameters t , t 1 , t 2 , t 3 can be calculated using the Hiickel formula (13) and the SN MO. One then finds that t = 0.45 eV, t j = -0.22 eV, 44 - -0.06 eV, and t ^ = 0.56 eV. Since these are only approximate values, we have chosen a set of parameters (t = 0.45 eV, t ^ = -0.19 eV, t ^ = -0.05 eV, and t ^ = 0.42 eV) which better reproduce the band structure of Fig. 18; these tight-binding curves are presented i n Fig. 19. Two important features of the band structure are r e a d i l y apparent from equation (29) : (i) When a2 =6, e = ± Ja and the band structure exhibits a two-fold degeneracy. The directions i n lt-space for which t h i s occurs are exactly those for which symmetry dictates degeneracies, ( i i ) An accidental degeneracy occurs when 8 = 0 . Then e = 0 and the bands cross at the Fermi l e v e l . 8 = 0 when P 2 = (32) Taking the r e a l and imaginary parts of the equation gives two equations, the solution of which defines a curve i n it-space, which, with the para-meters chosen extends from (0,0. 8fr/b,0) to (Tr/a,0. 5-rr/b. 0. 33-rr/a) . The simple form of the energy bands described by (29) means that a l l band crossings occur at the Fermi l e v e l . The conclusion of th i s oversimplified model therefore i s that (SN)x i s a zero-gap semiconductor. A sophistication of t h i s model to include non-zero terms along the anti-diagonal and interactions with other bands would remove the acciden-t a l degeneracy at tht Fermi l e v e l . One would expect the small s p l i t t i n g s to cause in d i r e c t overlaps of the bands, thereby creating a Fermi surface of a semimetal with electron and hole pockets. This conclusion agrees with that of the more complex Huckel calculation. Since a density of states calculation incorporating the f u l l Huckel treatment proved to be unfeasible, the tight-binding bands were used as 4 5 Figure 19. Band structure analogous to that of Fig. 18 r e s u l t i n g from the tight-binding c a l c u l a t i o n . 46 they represent the energy bands reasonably well. Fig. 20 shows the r e s u l t s , and while the d e t a i l s of the plot are not believable, the overall features should be correct. The lack of resemblance to the 1-D density of states i n Fig. 15 i s an indication of the dispersion of the bands i n directions transverse to the chain axis and c l e a r l y demonstrates the need for a 3-D model of (SN)x. Other authors who have calculated the band structure and the methods they have used are: (i) Rajan and Falicov (1975), ab i n i t i o . ( i i ) Kamimura et a l . (1975), semi-empirical. ( i i i ) Schluter et a l . (1975), pseudopotential. The feature these calculations have i n common i s a Fermi l e v e l which i n t e r -sects overlapping o and IT bands, d i f f e r i n g i n t h i s regard from our r e s u l t s . A possible explanation of the s i t u a t i o n (Weiler) i s that these methods overestimate the s p l i t t i n g of low-lying a bands i n the s o l i d , pushing them up i n energy and causing them to mix with the TT bands. The use of the EHM for (SN)x thus seems to be p a r t l y j u s t i f i e d . 48 Chapter 4: Discussion In the preceding two chapters we have considered the electronic energy spectra of two outwardly si m i l a r systems. What we can conclude i s that both TTF-TCNQ are semimetals, i n agreement with experiment. Both materi-als also exhibit s i m i l a r anisotropy i n e l e c t r i c a l conductivity and optic-al properties. Why then i s i t that TTF-TCNQ undergoes a Peierls t r a n s i t i o n to a semiconducting state while (SN)x not only does not show th i s feature but ac t u a l l y becomes superconducting? To underscore the s i m i l a r i t y l e t us take a closer look at the (SN)x cr y s t a l structure i n the framework of our ca l c u l a t i o n . I f one believes the simple tight-binding MO treatment, then a single (SN)x chain (type-^I) can be looked upon as consisting of two inequivalent (type-II) chains, each contributing one SN "molecule" to the unit c e l l . The two "molecules", although s t r u c t u r a l l y i d e n t i c a l , might be labelled F and Q since they give r i s e to separate bands ( the two with the degeneracy at the Fermi le v e l i n Fig. 14a). Regarding the second type-II chain i n the same manner, one has two F and Q "molecules" per unit c e l l . In the 1-D approximation of §3.2, the two bands are doubly degenerate. Thus, i n t h i s p i c t u r e , the (SN)x model i s completely analogous to TTF-TCNQ except for differences i n the intermolecular distances. The difference between the two solids i s immediately apparent when the interchain interactions are turned on. The effect i s seen dramatically i n the magnitude of the s p l i t t i n g s of the degenerate bands. In (SN)x, the r a t i o of the s p l i t t i n g to the 1-D bandwidth i s about 0.3 as compared to 49 a r a t i o of < 0.03 i n TTF-TCNQ. Further, bandwidths (in terms of the band-width along TZ) i n directions transverse to TZ range up to 0.1 for (SN)x, but are less than 0.01 for TTF-TCNQ. It seems, therefore, that the sup-pression of the Peierls t r a n s i t i o n i n (SN)x must be attributed to the strong interchain coupling. HGW explain the occurrence of a Peierls t r a n s i t i o n i n TTF-TCNQ i n terms of t h e i r r e s u l t s (see §1.4). From s p e c i f i c heat and e l e c t r i c a l con-d u c t i v i t y data they obtain a value of the interchain coupling ri - 0.1, which f a l l s into the region given by (8) i n which a 1-D mean f i e l d theory approach to the Peierls t r a n s i t i o n i s v a l i d . Whether or not th i s explana-tion i s correct i s not clear since our MO calculation gives a maximum n = 0.015 for two neighbouring TCNQ chains. Further, the HGW model f a i l s to describe TTF-TCNQ as a two-component system with crossing bands i n t e r -acting at the Fermi le v e l to create a Fermi surface which d i f f e r s from the one re s u l t i n g from equations (6) and (7). However, the actual shape of the surface i s l i k e l y not very important at temperatures near the ob-served T c - 60K, i n l i g h t of the small dispersion of the energy bands transverse to the chain d i r e c t i o n . Thus our calculation does not preclude the Peierls i n s t a b i l i t y with wavevector q = (Tr/a,2p„ ,Tr/c) which HGW con-r elude i s favoured to occur. As the band structure i n Fig. 9 shows, TTF-TCNQ i s well explained by a 1-D model and consequently the observed anisotropies i n the electronic properties are not surprising. Similar observations for (SN)x which i n i t i a l l y led to i t s being c l a s s i f i e d as a quasi-l-D material are inconsistent with the band structure i n which 3-D effects are important. One must conclude, therefore, that the observed anisotropy i s due i n most part to the fibrous nature of the c r y s t a l . 50 Bibliography Bardeen, J. 1973. Solid State Comm. 13, 357. Berlinsky, A.J., Carolan, J.F., and Weiler, L. 1974. Solid State Comm. 15_, 795. Boudeulle, M. 1974. Thesis, Universite de Lyon. Chaikin, P.M., Kwak, J.F., Jones, T.E., Garito, A.F., and Heeger, A.J. 1973. Phys. Rev. Lett. 31_, 601. Coleman, L.B., Cohen, M.J., Sandman, D.J., Yamagishi, F.G., Garito, A.F., and Heeger, A.J. 1973. Solid State Comm. 12_, 1125. Cohen, M.J., Garito, A.F., Heeger, A.J., MacDiarmid, A.G., and Saran, M. 1975. To be published. Craven, R.A., Salamon, M.B., DePasquali, G., Herman, R.M., Stucky, G., and Schwertz, A. 1974. Phys. Rev. Lett. 3_2, 769. Denoyer, F., Comes, R., Garito, A.F., and Heeger, A.J. 1975. Phys. Rev. Lett. 35, 445. Etemad, S., Penney, T., Engler, E.M., Scott, B.A., and Seiden, E. 1975. Phys. Rev Lett. 34_, 741. Fleming, J.E., and Falk, R.J. 1973. J. Phys. C 6_, 2954. Frb'hlich, H. 1954. Proc. Roy. Soc. A223, 296. Goehring, M. 1956. Q. Rev. Chem. Soc. 1_0, 437. Greene, R.L., Street, G.B., and Suter, L.J. 1975. Phys. Rev. Lett. 34_, 577. Grobman, W.D., Pollak, R.A., Eastman, D.E., Maas, E.T., J r . , and Scott, B.A. 1974. Phys. Rev. Lett. 32_, 534. Hoffman, R. 1963. J. Chem. Phys. 39_, 1397. Horovitz, B., Gutfreund, H., and Weger, M. 1975. To be published: Hsu, C , and Labes, M.M. 1974. J. Chem. Phys. 61_, 4640. Kamimura, H., Grant, A.J., Levy, F., and Yoffe, A.D. 1975. Solid State Comm. 17_, 49. Kistenmacher, T.J., P h i l l i p s , T.E., and Cowan, D.O. 1974. Acta Cryst. B30, 763. 51 Kohn, W. 1959. Phys. Rev. Lett. 2_, 393. Kortela, E.K., and Manne, R. 1974. J.Phys C 7, 1749. Landau, L.D., and L i f s c h i t z , E.M. 1969. S t a t i s t i c a l Physics §152, (Addison-Wesley, Don M i l l s , Ontario). Lee, P.A., Rice, T.M., and Anderson, P.W. 1973. Phys. Rev. Lett. 31_, 462. L i t t l e , W.A. 1964. Phys. Rev. A 134, 1416. M CCubbin, W.L., and Manne, R. 1968. Chem. Phys. Lett. 2_, 230. Parry, D.E., and Thomas, J.M. 1975. J. Phys. C 8_, L45. Pe i e r l s , R.E. 1955. Quantum Theory of Solids (Oxford University Press, London). Rajan, V.T., and Falicov, L.M. 1975. To be published. Rice, M.J., and Stra s s l e r , S. 1973. Solid State Comm. 1_3, 125. Rudge, W. 1975. B u l l . Am. Phys. Soc. 20_, 359, and private communication. Salamon, M.B., Bray, J.W., DePasquali, G., and Craven, R.A. 1975. Phys. Rev. B 11_, 619. Schafer, D.E., Wudl, F., Thomas, G.A., F e r r a r i s , J.P., and Cowan, D.O. 1974. Solid State Comm. 14, 342. Schluter, M., Chelikowsky, J.R., and Cohen, M.L. 1975. To be published. Tiedje, T., Carolan, J.F., Berlinsky, A.J., and Weiler, L. 1975. Can. J . Phys. 53_ (to be published). Tiedje, T. 1975. M.Sc. Thesis, University of B r i t i s h Columbia. Tomkiewicz, Y., Scott, B.A., Tao, L.J., and T i t l e , R.S. 1974. Phys. Rev. Lett. 32, 1363. Weiler, L. Private Communication. 52 Appendix: (SN)x Band Structure for the Perm  Crystal Structure The structure of (SN)x reported by Cohen et a l . (1975) of the University of Pennsylvania group d i f f e r s from that of Boudeulle i n two ways (although the l a t t i c e parameters are v i r t u a l l y i d e n t i c a l ) . F i r s t l y , the intrachain bonding angles are d i f f e r e n t , r e s u l t i n g i n S-N bond d i s -o tances which are almost equal (1.59 and 1.63 A as compared to Boudeulle's o 1.58 and 1.7.2 A) and i n chains which are very nearly planar as shown i n Fig. A l . Secondly, inequivalent chains are translated with respect to each o other by about 1 A along the b-axis as compared to those i n the Boudeulle structure. Table A l l i s t s the coordinates of the atoms i n the unit c e l l . The calculations of Chapter 3 have been repeated using the Penn data. Fig A2 shows the 3-D band structure a r i s i n g from the f u l l Hiickel calcula-t i o n . Overall, the bands resemble those of Fig. 18; the differences i n d e t a i l which are evident are an increased bandwidth along TZ and a smal-l e r band s p l i t t i n g which pushes the Fermi wavevector back out to the zone boundary. The implication, therefore, i s that i n the Penn structure the intrachain coupling i s greater than i n the Boudeulle structure while the interchain coupling decreases. The tight-binding approximation to the band structure introduced i n §3.3 has also been applied i n t h i s case. Because of the change i n the r e l a t i v e position of the inequivalent chains, di f f e r e n t interactions become important and the greatly simplifying feature that elements on the a n t i -diagonal are zero does not appear. Thus no expression such as (28) can be written down and the 4 x 4 Hamiltonian must be diagonalized numerically. If the energy of the SN MO's i s taken to be zero, then the Hamiltonian 53 F.igure A l . View down the b-axis of the c r y s t a l structure of (SN)x from the data of Cohen et a l . (1975). 54 Figure A2. Three-dimensional band structure obtained by the extended Hitckel method incorporating the Penn cr y s t a l structure. 55 is, a 3 * y 6* e a 6* * V u 6 a 3 * 6 V e a (Al) The matrix elements are given by, a = 2t1cosk'.a e = t 2 ( l + e" 1^- 0") y = t a e - 1 ^ + t | | e i t (a-?) v = t 3 e - i k ^ + t t f 6 - i l t . Ca+c") « = t 5 ( l + e i ] t' &) (A2) where the t^'s represent the following interactions: (i) ti'. a a-interaction between two atoms i n c e l l s displaced by a. ( i i ) t 2 : the interaction between molecules on the same chain, ( i i i ) t 3 : a a-interaction between an Sj and the S 3 i n the c e l l translated by 1). (iv) t ^ : a u-interaction between an Si and the S 3 i n the c e l l d i s -placed by D" - a. (v) t s : a a-interaction between an and the i n the c e l l trans-lated by c. With the parameters t1 = 0.02 eV, t 2 = -0.57 eV, t 3 = 0.25 eV, tk = -0.06 eV, and t$ = 0.04 eV, the energy bands depicted i n Fig. A3 again bear a 56 strong s i m i l a r i t y to those i n Fig. A2. The conclusion to be drawn from t h i s c a l c u l a t i o n incorporating the Penn structure i s the same as that of § 3 . 3 : (SN)x 1 S a semimetal. Table A l . Atomic coordinates i n the (SN)x unit c e l l for the Penn structure. In the coordinate system employed, the l a t t i c e vectors are, i n R u n i t s : £ = (4.15,0,0), t = (0,4.44,0), and t = (-2.57,0,7.19). I X Y Z Si -1.143 1.275 1.119 s 2 -0.143 -0.945 2.475 s 3 1.143 -1.275 -1.119 s 4 0.143 0.945 -2.475 Ni . -1.043 -0.307 1.279 N 2 -0.243 -2.526. 2.315 N 3 1.043 0.307 -1.279 Nk 0.243 2.526 -2.315 57 r Z Z E E A A P P Z Z C C Y Y P ' Figure A3. Tight-binding analogue of Fig. A2. 

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