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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 U n i v e r s i t y , 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 t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October, 1975  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  further  for  freely  the  requirements  B r i t i s h Columbia, I agree  available  for  this  representatives. thesis for  It  financial  Department  of  p Af  that  this  thesis or  i s understood that c o p y i n g or p u b l i c a t i o n gain s h a l l  S /C-T  U n i v e r s i t y of B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  for  r e f e r e n c e and study.  t h a t p e r m i s s i o n for e x t e n s i v e c o p y i n g o f  w r i t ten pe rm i ss i o n .  The  of  s c h o l a r l y purposes may be granted by the Head of my Department  by h i s of  fulfilment  the U n i v e r s i t y of  s h a l l make it  agree  in p a r t i a l  not  be allowed without my  ii  Abstract  The e l e c t r o n i c energy band structures of two h i g h l y conducting, a n i s o t r o p i c s o l i d s have been calculated using the extended Hiickel method. One-,  two-,  and three-dimensional models of the organic charge t r a n s f e r  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 i n d i c a t e that the band s t r u c t u r e of TTF-TCNQ i s w e l l described by a t i g h t - b i n d i n g , one-dimensional model i n which i n t e r a c t i o n s between stacks of molecules are neglected. The Fermi surface i s seen to consist of extremely f l a t e l e c t r o n and hole surfaces, the nature of which i s i n c o n c l u s i v e i n p r e d i c t i n g a Fermi-surface-related  instability  leading to a P e i e r l s d i s t o r t i o n . A one-dimensional model of (SN)x p r e d i c t s 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  s i n g l e chain i s h i g h l y unstable against  a  symmetry-reducing d i s t o r t i o n ; however, three-dimensional i n t e r c h a i n i n t e r actions appear to s t a b i l i z e the s t r u c t u r e . Consequently, the Fermi i s that of a semimetal with electron and hole pockets. The  surface  essential  features of the band s t r u c t u r e can be explained by a simple t i g h t - b i n d i n g model i n v o l v i n g 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 c a l c u l a t i o n are seen to have no qualitative effect.  iii  Table of Contents  Page L i s t o f Tables L i s t of Figures Acknowledgements  iv v  v i i  Chapter 1 Quasi-One-Dimensional Metals 1.1 1.2 1.3 1.4 1.5  Introduction TTF-TCNQ: An Organic Metal (SN)x: A Polymeric Superconductor One-Dimensional Theories Energy Band C a l c u l a t i o n  1 2 3 4 10  2 TTF-TCNQ 2.1 C r y s t a l Structure 2.2 One-Dimensional TTF-TCNQ 2.3 Three-Dimensional TTF-TCNQ  13 13 19  3 (SN)x 3.1 C r y s t a l Structure 3.2 One-Dimensional Band Structure  31 31  3.3 Three-Dimensional Band Structure  38  4 Discussion  48  Bibliography  50  Appendix (SN)x Band Structure .for the Penn C r y s t a l Structure  52  iv  L i s t of Tables  Table  Page  1  Molecular coordinates i n TTF-TCNQ.  23  2  Overlap i n t e g r a l s between molecular o r b i t a l s i n TTF-TCNQ.  23  Atomic coordinates i n the (SN)x u n i t c e l l f o r the s t r u c t u r e determined by Boudeulle (1974).  34  Atomic coordinates i n the (SN)x u n i t c e l l f o r the structure determined by Cohen et a l . (1975).  56  3 Al  L i s t of Figures  Page  Figure 1  E f f e c t of a P e i e r l s d i s t o r t i o n on the e l e c t r o n i c energy bands of a l i n e a r atomic l a t t i c e .  5  S h i f t of the e l e c t r o n i c 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  C r y s t a l s t r u c t u r e of TTF-TCNQ.  14  4  One-dimensional energy bands f o r a charge t r a n s f e r o f 0.5 electrons per TTF molecule, (a) For the undistorted chain, (b) For the d i s t o r t e d chain.  17  5  Density of states f o r 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  2  D"-C" plane.  22  8  E f f e c t of the F-Q coupling on the bands i n F i g 6.  24  9  Three-dimensional band s t r u c t u r e of TTF-TCNQ.  25  10  Histogram o f the d e n s i t y of states f o r the 3-D band s t r u c t u r e . Energy bands near the Fermi energy i n the TZ direction.  11 12a  Superimposed c r o s s - s e c t i o n s of the e l e c t r o n and hole surfaces i n two d i f f e r e n t planes p a r a l l e l to the = 0 plane.  26 27  29  12b  Cross-section of the Fermi surface i n the k plane.  = 0  13a  P r o j e c t i o n of the c r y s t a l s t r u c t u r e o f (SN)x onto a plane perpendicular to the chain a x i s .  30 32  VI  Figure 13b 14  15 16  Page Perspective drawing of a side view of the crystal structure.  33  Energy bands f o r an i s o l a t e d (SN)x chain, (a) For the observed s t r u c t u r e , (b) For the d i s t o r t e d s t r u c t u r e shown i n F i g . 16.  36  Density of states f o r the two 1-D bands nearest the Fermi energy.  37  D i s t o r t i o n which breaks the screw a x i s 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 s t r u c t u r e s of (SN)x and TTF-TCNQ.  40  18  Three-dimensional (SN)x band s t r u c t u r e .  41  19  Tight-binding energy bands.  45  20  Histogram of the d e n s i t y of states f o r the t i g h t - b i n d i n g 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 s t r u c t u r e .  54  A3  Tight-binding analogue of F i g . A2.  57  vii  Acknowledgement s  I would l i k e to thank Dr. Birger Bergersen f o r introducing me to the world of one dimension, and f o r h i s subsequent p a t i e n t and knowledgeable s u p e r v i s i o n . The ubiquitous and s t i m u l a t i n g presence of Dr. John B e r l i n s k y l e d 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 f o r valuable discussions must also be expressed to the Friends of TCNQ, Dr. Jim Carolan, Dr. Dan L i t v i n , Tom T i e d j e , and Dr. Larry Weiler, and to Dr. T.M.  Rice. I thank Dr. G i l b e r t  f o r the use of h i s Fermi surface p l o t t i n g  Lonzarich  program.  The members of the West Penthouse also deserve a word of a p p r e c i a t i o n f o r t h e i r constant encouragement and moral  support.  F i n a l l y ^ am g r a t e f u l to the National Research Council f o r t h e i r f i n a n c i a l a s s i s t a n c e 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 s t a t e physics i n recent years has been the dicovery that c e r t a i n organic s o l i d s with a high degree of anisotropy e x h i b i t m e t a l l i c p r o p e r t i e s . The importance of these "quasi-one-dimensional"  m a t e r i a l s i s that they provide a d i r e c t  experimental check of the theory which has been worked out f o r various one-dimensional  (1-D) models.  Of p a r t i c u l a r i n t e r e s t are the charge t r a n s f e r s a l t s of tetracyanoquinodimethan (TCNQ) which have the highest e l e c t r i c a l c o n d u c t i v i t y of any known organic s a l t s . The large planar TCNQ molecule i s a good acceptor with the e x t r a e l e c t r o n occupying, unpaired, a Tr-orbital ( i . e . a wavef u n c t i o n which i s odd under r e f l e c t i o n through the molecular plane). This f e a t u r e , together with the arrangement of the molecules face-to-face i n stacks imparts a strong 1-D character to the e l e c t r o n i c p r o p e r t i e s . 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 t e t r a t h i o f u l v a l i n i u m tetracyanoquinodimethan  (TTF-TCNQ). They observed a m e t a l l i c c o n d u c t i v i t y  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 c o n d u c t i v i t y  i n d i c a t i n g a phase t r a n s i t i o n to an i n s u l a t i n g s t a t e was observed.  Further  work by various groups has confirmed the m e t a l l i c p r o p e r t i e s and the metali n s u l a t o r t r a n s i t i o n , but has not d u p l i c a t e d the extremely high conduct i v i t y . The work on TTF-TCNQ has spawned a whole host of organic s o l i d s with s i m i l a r p r o p e r t i e s ; however, TTF-TCNQ remains the prototype, and  2  consequently the most studied. A good deal of the i n t e r e s t 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 a t t e n t i o n has been focussed on the TCNQ s a l t s i n t h i s regard, other quasi-l-D systems have also been i n v e s t i g a t e d . One, the inorganic polymer polysulphur n i t r i d e ((SN)x) which was known t o have a high c o n d u c t i v i t y , has r e c e n t l y been shown by Greene et a l .  (1975) t o undergo a superconduc-  t i n g t r a n s i t i o n at about 0.3K. (SN)x i s , i n some respects, q u i t e 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 m e t a l - i n s u l a tor t r a n s i t i o n . Superconducting or not, quasi-l-D m a t e r i a l s are of fundamental i n t e r e s t as novel, r e l a t i v e l y unstudied systems. This t h e s i s concerns i t s e l f w i t h a study of the two d i f f e r e n t examples already mentioned, TTF-TCNQ o  and (SN)x. Some experimental r e s u l t s and several t h e o r e t i c a l 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 r e s u l t s of the c a l c u l a t i o n s are discussed, and a comparison o f 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 i n t e r a c t weakly i n comparison to the i n t r a c h a i n coupling, r e s u l t i n g i n conduction p r i m a r i l y along the chain a x i s . Because TTF i s a good donor, a charge of ^ one e l e c t r o n per TTF i s t r a n s f e r r e d to the TCNQ. Each chain thus c a r r i e s a net charge so that i n a simple 1-D band p i c t u r e TTF-TCNQ i s expected t o be a metal. T y p i c a l room temperature c o n d u c t i v i t i e s p a r a l l e l to the chain a x i s  are a'' ^300 R1  - 1000 (n-cm) " -  1  (Tiedje, 1975; Schafer et a l . , 1974;  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 a'Jp. At 53K, a sudden drop occurs, R  s i g n i f y 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 c o n d u c t i v i t y , a ^ , i s only about 1.4  (ft-cm) * with a 58K peak of three times t h i s f i g u r e .  The anisotropy a " / o  i s thus greater than 300.  - L  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 c o n d u c t i v i t y (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 t o be h i g h l y conducting since the experiments of Goehring (1956). I t i s only w i t h the recent i n t e r e s t i n the quasi-l-D s o l i d s that (SN)x has again become subject to experiment. The s t r u c t u r e i s s i m i l a r to that of TTF-TCNQ, c o n s i s t i n g of p a r a l l e l chains of atoms. The room temperature c o n d u c t i v i t y 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 c o n d u c t i v i t y , a^~, i s much lower, the anisotropy a "/a-** ranging from 50 - 1000. I t should be stressed that no metali n s u l a t o r 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 s t r u c t u r e relevant to any d i s c u s s i o n 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 e f f e c t of t h i s f i b r o u s s t r u c t u r e on the transverse p r o p e r t i e s i s not c l e a r . 1.4 One-Dimensional Theories The phrase "one-dimensional metal" i s i n one sense a paradox, since a 1-D non-interacting e l e c t r o n gas i n a p e r i o d i c p o t e n t i a l cannot be a conductor. This was shown by P e i e r l s (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 F i g . 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 e l e c t r o n to the conduct i o n band, the band i s e x a c t l y 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 dimeri 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 e l e c t r o n i c energy and making the m e t a l l i c r chain unstable against such a p e r i o d i c 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 f o r any degree of b a n d - f i l l i n g . When electron-phonon i n t e r a c t i o n s are included, the same symmetryreducing t r a n s i t i o n may occur as the  manifestation of an e f f e c t f i r s t  predicted by Kohn (1959). P h y s i c a l l y t h i s e f f e c t a r i s e s from the f a c t 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 d e r i v a t i v e of the d i e l e c t r i c f u n c t i o n 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 e l e c t r o n gas to screen the l a t t i c e  5  o  o  o o  —^ a if— o o o  s\  o o  2ft  o  o o  o  o  Figure 1. E f f e c t of a P e i e r l s d i s t o r t i o n , (a) Tight-binding energy band f o r the l i n e a r chain with interatomic spacing a. (b) The bands f o r the chain i n which the atoms have dimerized to double the l a t t i c e parameter to 2a.  v i b r a t i o n . The r e s u l t a n t 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 ( P e i e r l s ) d i s t o r t i o n w i t h wavevector 2k^,. The strength of the e f f e c t depends on how much of the Fermi surface can be connected by the wavevector 2kL (or how w e l l the Fermi surface nests) r  In 2- or 3-D, the e f f e c t i s a weak one; i n 1-D however, the anomaly i s  6  pronounced since the Fermi surface c o n s i s t s of two s i n g u l a r points at kp, and the nesting i s p e r f e c t . A d i s c u s s i o n of the Fermi surface i n s t a b i l i t y  w i t h i n mean f i e l d  theory has been given by Rice and S t r a s s l e r (1973). They consider a l i n e a r chain with a h a l f - f i l l e d t i g h t - b i n d i n g band. The ( F r 6 h l i c h ) Hamiltonian i s w r i t t e n as,  H - Xe 2*Vt+ P  ^*WW  faM*  Here the b's and c's are the usual operators f o r the unperturbed phonons of energy hu^ and Bloch electrons of energy  r e s p e c t i v e l y ; N i s the  number of ions i n the chain, and g(q) i s the electron-phonon constant. The screened phonon frequencies, o,  coupling  of the coupled e l e c t r o n -  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  P  i s the Fermi d i s t r i b u t i o n f u n c t i o n . 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 t o be,  ^. = where k T  c  * Z . Z % £  f  t  - ' -  x  C5  j  7  Thus ft approaches zero l o g a r i t h m i c a l l y as T approaches the c r i t i c a l temperature T . Below T , Rice and S t r & s s l e r view the l i n e a r chain as a condensed c phonon s t a t e of wavevector q + <b >= (Nu/2)6  q  q>qo  Q  such that the expectation values <b > =  , where u i s a dimensionless amplitude. In t h i s case an  energy gap o f magnitude 2A(T) appears i n the e l e c t r o n i c energy spectrum -2A at p = k„. From a zero temperature value A(0) = 4e e c  creases, vanishing f i n a l l y a t the T  c  , the gap de-  given by (5). In the framework o f  t h i s model, then, the high temperature metal undergoes a second order phase t r a n s i t i o n t o a low temperature i n s u l a t o r . It has been argued that the Kohn anomaly does not always lead t o an i n s u l a t i n g s t a t e . F r o h l i c h (1954) considered a system w i t h the Hamiltonian given by (1) and argued that superconductivity might r e s u l t i n the f o l lowing manner. When an e x t e r n a l f i e l d i s applied the electrons are d i s placed i n k-space w i t h a v e l o c i t y v . I f 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 e l e c t r o n s , a superr  l a t t i c e of p e r i o d i c i t y 2?r/2kp e x i s t s i n the frame o f the e l e c t r o n s . Consequently,  energy gaps w i l l appear at the d i s p l a c e d Fermi surface as  i n 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 s c a t t e r i n g 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 e l e c t r o n i c energy i s E(k) + hkpV . When hkpV becomes greater than h a l f the gap, A, the f r e e g  g  energy i s lowered by electrons being scattered i n t o the conduction band, thereby i n t r o d u c i n g a r e s i s t a n c e and decreasing the supercurrent. The coupled electron-phonon  mode described here i s known as a F r o h l i c h mode.  Bardeen (1973) has suggested that the F r o h l i c h mode might be a mechanism  8  J  \  ,/  1  \ 1  A  Ti*\ 1  1  i -or  -jr  1  \% T  >  /1  I  \  7  \  v n  i i• ii i i  m .  "£  Figure 2. S h i f t of the e l e c t r o n i c 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 c o n d u c t i v i t y i n TTF-TCNQ. Unfortunately, mean f i e l d theory i s inaccurate i n d e s c r i b i n g systems due to f l u c t u a t i o n s of the order parameter. Landau and  1-D  Lifshitz  (1969) have shown that f o r a system with short range f o r c e s , f l u c t u a t i o n s p r o h i b i t a phase t r a n s i t i o n except at zero temperature. Lee, R i c e , and Anderson (1973) have included f l u c t u a t i o n s 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 general i z e d Landau theory. They f i n d that f o r T £ T t r a n s i t i o n temperature),  c  (T  being the mean f i e l d  £(T) becomes l a r g e , i n c r e a s i n g e x p o n e n t i a l l y  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 e f f e c t of the f l u c t u a t i o n s on the e l e c t r o n i c d e n s i t y of c states i s to change the well-defined gap which e x i s t s 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 r e s u l t would then have some i n f l u e n c e on the Kohn anomaly.  9 One-dimensional models a r e ' u n r e a l i s t i c i n the sense that they ignore the f i n i t e , i f s m a l l , i n t e r c h a i n i n t e r a c t i o n s that are present systems. H o r o v i t z , Gutfreund, and Weger  i n real  (HGW,1975) have included i n t e r -  chain e f f e c t s i n the model described by (1) with nearest-neighbour t i g h t - b i n d i n g of the form,  = - £ p Cos  fyt (6)  p a r a l l e l t o the c h a i n , and  £(p ,p ) = ttpj ~~ q€ (t°s tap*. + tos ape) x  (  0  transverse to the chain. Here £p, pp are the Fermi energy and wavevector for a s i n g l e chain, e the i n t e r c h a i n  i s an energy parameter - £p, n i s the r a t i o of  Q  t o the i n t r a c h a i n 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 t o the wavevector q  = ( /aj2pp, /a); that i s 7r  Q  7r  to say, the Fermi wavevector nests p e r f e c t l y f o r 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,  r e s u l t i n g i n a smearing of the Kohn anomaly. There e x i s t s a c r i t i c a l value n of the i n t e r c h a i n coupling strength such that f o r n > n the £  c  P e i e r l s t r a n s i t i o n i s suppressed; f o r example, at T = 0, n c  3.5 ^T (n=0)/e |a| , T c  Q  £  again denoting the mean f i e l d t r a n s i t i o n temper-  ature. I t i s i n t e r e s t i n g to note that f o r a = 0, the i n t e r c h a i n coupling does not. a f f e c t 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 e x i s t at any  for which the e l e c t r o n -  phonon coupling parameter A(q) i s a maximum. I f there i s no strong dependence on q  , however, the q" =  i n s t a b i l i t y w i l l be favoured f o r T  those values of the i n t e r c h a i n  coupling f o r 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 e f f e c t s are small. When f l u c t u a t i o n s  are included i n a treatment of the i n s t a b i l i t y  at q , HGW f i n d a temperature c h a r a c t e r i s t i c o f the system, Q  = ne /4; Q  f o r 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 t o be appropriate i n the r e g i o n ,  VIP  1  (8)  o -> -> where T i s the mean f i e l d T f o r q = q . c c o n  1.5 Energy Band  n  Calculation  In t h i s section the method used i n c a l c u l a t i n g the energy bands i s outlined.  A l i n e a r combination of atomic o r b i t a l s (LCAO) method i s used  f o r (SN)x  because of the covalent bonding nature of the c o n s t i t u e n t s . 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 f o r TTF-TCNQ. The  e l e c t r o n i c wavefunction, ^ ( r ) i s expanded as a l i n e a r combin-  a t i o n of Bloch o r b i t a l s ,  where  11 ->  -y  -V-  Here d>. fr - R - r.) i s the i ' t h atomic o r b i t a l centered at p o s i t i o n i m 1 y  v  r . i n the u n i t c e l l with postion vector i r  m  , and N i s the number of u n i t  c e l l s i n the c r y s t a l . S u b s t i t u t i o n of (9) into the Schrodinger leads eventually to the set of coupled l i n e a r  a  equation  equations,  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*,  ^  ue  ha  V  (12)  S  V  The i n t e g r a l s h™.. were c a l c u l a t e d 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 s ?. as f o l l o w s : 1  F  ki\  il  -i  (13)  K i s the phenomenological Wolfsberg-Helmholtz parameter which i s v a r i e d to give the best f i t to experiment. U s u a l l y K i s . t a k e n to be 1.75, value which was used i n our c a l c u l a t i o n ,  a  i s the experimentally deter-  mined i o n i z a t i o n p o t e n t i a l f o r o r b i t a l i . I f n o r b i t a l s per u n i t c e l l are i n c l u d e d , the energy bands are obtained by s o l v i n g the n x n eigenvalue equation  (11) f o r each wavevector.  The EHM allows the f u l l symmetry of the c r y s t a l to be incorporated c o r r e c t l y . While the method i s somewhat crude, i t i s expected t o give energy d i f f e r e n c e s such as bandwidths q u i t e w e l l . In a d d i t i o n , i t s simp l i c i t y leads t o an easy i n t e r p r e t a t i o n of the energy eigenvectors. The EHM has been used i n s i m i l a r c a l c u l a t i o n s by several authors: M Cubbin and Manne (1968) and Fleming and Falk (1973) f o r polyethylene, K o r t e l a and Manne (1974) f o r graphite, and B e r l i n s k y et a l . (1974) f o r TTF-TCNQ.  13  Chapter 2: TTF-TCNQ  2.1 C r y s t a l Structure The c r y s t a l s t r u c t u r e as reported by Kistenmacher  et a l . (1974) i s  shown i n F i g . 3; molecular coordinates are given i n t a b l e 1. The dominant feature i s the segregated chains o f TTF and TCNQ molecules stacked i n a monoclinic l a t t i c e . Neighbouring chains along the c-axis are r e l a t e d by a screw a x i s symmetry operation: a r o t a t i o n o f 180° about an a x i s midway between the- chains, together with a t r a n s l a t i o n of b/2 along the a x i s takes one chain i n t o the other. A u n i t c e l l thus contains two TTF and two TCNQ molecules. The space group o f the s t r u c t u r e i s P2^/c, the symmetry elements of which are a center o f i n v e r s i o n a t each molecular s i t e and the screw a x i s .  2.2 One-Dimensional TTF-TCNQ The stacking arrangement o f the s t r u c t u r e , together with the observed anisotropy i n the c o n d u c t i v i t y point to a model i n which there are i n t e r a c t i o n s 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 i n t e r c h a i n coupling i s taken i n t o account. A part o f t h i s s e c t i o n i s based on the work of B e r l i n s k y , Carolan and Weiler (BCW, 1974) since i t i s a necessary prelude to what f o l l o w s . BCW have used the EHM to c a l c u l a t e the molecular o r b i t a l s (MO's) f o r both TTF and TCNQ molecules, with the r e s u l t that the highest occupied MO +  on each molecule i s a i r - o r b i t a l . The symmetry o f these MO's i s such that the TTF wavefunction i s even and the TCNQ +  wavefunction odd under  14  (b) Figure 3. C r y s t a l s t r u c t u r e of TTF-TCNQ. TTF and TCNQ are l a b e l l e d F and Q r e s p e c t i v e l y , while the numbers i n d i c a t e those molecules whose coordinates are given i n Table 1; 1,3,4,6 are i n the same u n i t c e l l and 2 and 5 are i n the c e l l displaced by ft. (a) View perpend i c u l a r t o the chain a x i s . The s o l i d c i r c l e s i n d i c a t e those atoms p o s i t i o n e d 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 b i s e c t i n g 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 g r e a t l y d i m i n i s h i n g i n t e r c h a i n e f f e c t s . 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 b a s i s f u n c t i o n s for our energy band c a l c u l a t i o n . I t should be pointed out that the a-axis overlap i s zero only f o r those wavefunctions c a l c u l a t e d f o r i s o l a t e d molecules. I n c l u s i o n of the c r y s t a l f i e l d reduces the MO  symmetry,resul-  t i n g i n a f i n i t e overlap. This can be seen i n F i g . 3 where the atoms marked  and S£ are s i t u a t e d i n d i f f e r e n t environments. The s i z e 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 f o r the l i n e a r atomic chain of l a t t i c e constant b and one o r b i t a l per u n i t c e l l . We w r i t e the wavefunction as, (14) 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 s u b s c r i p t i = F or Q, which w i l l herea f t e r denote TTF and TCNQ r e s p e c t i v e l y . The energy f o r band i i s then given by,  2X 22*  (15)  16 Including only nearest-neighbour i n t e r a c t i o n s g i v e s ,  l  ~  l-Htoskk<fi(?+t)lfifr)>  Since <<J>^(r + b) |<jK(r)> E  <<  ( 1 6 )  1, the denominator can be expanded to  get,  £; = { < < f t ( r ] / t f / < £ ^  . (17)  Making use of the Huckel approximation and r e t a i n i n g 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 i o n i z a t i o n p o t e n t i a l of molecule i , and t . = (K - l ) e V a . . l I I i r  V  The Fermi l e v e l i n t e r s e c t s both bands, w i t h the consequence that the Fermi wavevector determines the amount of f i l l i n g of each band, or e q u i v a l e n t l y , the degree of charge t r a n s f e r . Grobman et a l . (1974) i n t e r preted t h e i r photoemissidn data as i n d i c a t i n g a charge t r a n s f e r of about one e l e c t r o n 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 c a l c u l a t i o n s were therefore i n i t i a l l y performed f o r a h a l f - f i l l e d band model. Using the BCW estimate that e° = -7.5 eV, t o r gether with the c a l c u l a t e d overlap i n t e g r a l s 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 f o r the F and Q bands. The f a c t that t„ and t„ are opp o s i t e 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 i n t e r a c t i n the presence of F-Q coupling. More r e c e n t l y , evidence f o r a P e i e r l s d i s t o r t i o n has been provided by an X-ray a n a l y s i s of the c r y s t a l s t r u c t u r e by Denoyer et a l . (1975). Below 40K they observe a 3-D s u p e r l a t t i c e 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 nearl y q u a r t e r - f i l l e d Q band. The i o n i z a t i o n p o t e n t i a l s were accordingly changed to E ° = -7.58 eV and E Q = -7.35 eV. The proposed band s t r u c t u r e i s presented i n F i g . 4a f o r an undistorted chain and i n F i g . 4b f o r a chain with a d i s t o r t i o n wavelength of 4b.  Figure 4. (a) One-dimensional energy bands f o r a charge t r a n s f e r of 0.5 electrons per TTF molecule, (b) The e f f e c t of a P e i e r l s d i s t o r t i o n of wavelength 4b on the bands of (a).  18 The density of states f o r 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 P i g . 5.  D(€)  Figure 5. Density of states f o r the two non-interacting bands i n F i g . 4a. E„ i s the Fermi energy.  19 2.3 Three-Dimensional TTF-TCNQ Before considering the three-dimensional band s t r u c t u r e i t might be i n s t r u c t i v e to look at the 2-D case i n which we include i n t e r a c t i o n s 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 l i n e s o f equal energy normal to the b- and c-axes. Each o f the doubly degenerate bands i n F i g . 4a i s s p l i t by the i n t e r c h a i n coupling. The 4 x 4 secular determinant  H  i n (11) assumes a block diagonal form,  11  '21  S  ll  H  12  £S  21  H  22  e  eS eS  12 22  = 0 *33 " 3 3 e S  Ul  " 43 £ S  H  34 "  e S  34  H  44 ~  £ S  44  where, i f only nearest neighbour overlaps, C K , on neighbouring chains are included,  -y  =  S  r  S  S  S  S  12 34  =  S  21  *  = a (l + e  c  *  = a (l + e  C  p  ) ( l + e  ) ( l + e 43 0 O t> = H„„ = e„ + 2Ke a„cosk t 22 r F F =  Q  S  :  H  ll  H  33  H  12  =  H  21  34  =  H  43  H  -y  ll 2 2 - - 1 + 2a ,cosk-b F -y -y = 44 = 1 + 2aQCOsk*b 33  s  E Q ° +  :  44  2K£Q°O-QCOS1C"  *  = Ke °S  *  =  F  KG  12  Q° 34 S  t  (20)  The r e s u l t i n g energy bands are approximately given by the expressions,  S°£i±  where,  All  ;  Z>L-£i Z(K-l)£i<r +  i  coctt  In the region of the band crossing the bands w i l l appear as i n F i g . 6  t  CD  cr UJ  2:  UJ  FERMI LEVEL  Figure 6 . Energy bands along k 2-D case.  i n the region of the band crossing i n the  y  The points at which the Fermi l e v e l i n t e r s e c t s 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 d e v i a t i o n of the Fermi wavevector from k^ = ft/4b w i l l then be s m a l l , wo we may w r i t e i t a s ,  21  (23) Using equations  (22) g i v e s ,  7» -» T  (24)  (Jos  A c r o s s - s e c t i o n of the Fermi surface i n the b-c plane i s sketched i n F i g . 7. The surfaces have been l a b e l l e d F or Q depending on which band the Fermi surface crosses. I t i s r e a d i l y apparent as i n d i c a t e d by the arrows that the p o r t i o n o f the Fermi surface i n the upper h a l f of the zone nests p e r f e c t l y with the p o r t i o n i n the lower h a l f . One would thus expect the 2-D model of TTF-TCNQ as o u t l i n e d here to be unstable against a d i s t o r t i o n of wavevector \ = (q ,q ,q ) = (iT/aa ,ir/2b, 0) , with a h  3.  D  1 deter- •  C  mining the p e r i o d i c i t y i n the a - d i r e c t i o n . From the d i s c u s s i o n of HGW, a would be that value maximizing the electron-phonon  coupling constant X(q).  A 2-D model f o r TTF-TCNQ i s not r e a l l y v a l i d since the i n t e r a c t i o n energy between F and Q chains i s roughly of the same magnitude as that f o r the F-F and Q-Q couplings. When F-Q i n t e r a c t i o n s are added the degeneracies at the crossover points are l i f t e d and some small curvature i n -y  -y  the a x e d i r e c t i o n i s added to the equal energy l i n e s . F i g . 8 i n d i c a t e s what happens to the bands i n the crossover region i n a d i 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 i n t e r c h a i n overlaps (from BCW) are presented i n Table 2. With these overlaps, the terms i n the secular determinant not given by (21) a r e ,  S  = S * = S * 31 " 42 24 13 S  ->-  -ik-b  ik-b ik • a )(1 + e - 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 d e v i a t i o n ( i n u n i t s 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 f o r those molecules shown i n Fig.3. In the coordinate system used,the l a t t i c e vectors a r e , i n A u n 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  9.992  1.910  8.942  6 .  Overlap i n t e g r a l s s . between MO's centered on the molecules l i s t e d i n Table 1 aAd shown i n F i g . 3.(taken from B e r l i n s k y et a l . , 1974).  Table 2.  m  1  2  1  1  2.0xl0"  2  2.0xl0"  2  3  -3.0xl0"  4  4  0  5  1.4xl0"  4  6  1.8xl0"  7  3 2  -4  -3.0xl0~  4  -1.4xl0"  4  0  1 6  -4 4x10 0  -1  -4 6  -4 4x10 0  4x10  1 7xl0"  1 7xl0" 3  9  4  1  0  0x10  -3. OxlO"  1  3.3xl0"  -3.  6.  5  4  3  3 -4  4x10  -9 3 x l 0 "  1  1 8xl0"  -9 3 x l 0 "  3  1  4 8xl0"  5  4 8xl0"  3  5  ?  -4 ' 3x10 0  4 8xl0"  5  4 8xl0"  5  1  24  > o  L±J UJ  Figure 8. E f f e c t of the F-Q coupling on the bands i n F i g . 6.  '32 5  41  H.  11  = S S  2 , - i t - (S+c)  ilc-a,  23  14* ^ F Q ^ ' ^  K(,° °)S../2 +  e  "  ^  C25)  •  The energy bands i n various d i r e c t i o n s i n the B r i l l o u i n Zone ( F i g . 17) are shown i n F i g . 9. The most s t r i k i n g f r a t u r e of the bands i s that they are very nearly f l a t , demonstrating the 1-D nature of TTF-TCNQ. A f u r t h e r corroboration of t h i s observation  i s the appearance o f the density of  states which has been c a l c u l a t e d f o r 5246 wavevectors i n the Zone. The histogram p l o t t e d i n F i g . 10 i s nearly i d e n t i c a l t o the sketch i n F i g . 5. The i n s e t of F i g . 10 shows the density of states c a l c u l a t e d f o r a f i n e r mesh of wavevectors near the band c r o s s i n g . The small peaks on e i t h e r side of the d i p a r i s e from the mutual r e p u l s i o n of the bands while the d i p i t s e l f i s due to the gap opened up i n most d i r e c t i o n s .  25  "Figure 9. Three-dimensional energy bands along several d i r e c t i o n s i n the B r i l l o u i n Zone which i s shown i n F i g . 17.  26  Figure 10. Histogram of the density of states f o r the 3-D band s t r u c t u r e . 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 F i g . 11. To p i c t u r e 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 f l a t n e s s 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 s m a l l , the Fermi l e v e l l i e s very close to and e , i n t e r s e c t i n g both bands, only one band, or n e i t h e r band, crea t i n g hole and e l e c t r o n surfaces. I t must be pointed out that because the overlap of the bands i s small the shape of the Fermi surface i s q u i t e s e n s i t i v e to even small changes i n the bands. In a d d i t i o n a p r e c i s e determination 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  r  = -7.51  eV) using the c a l c u l a t e d band s t r u c t u r e ,  Cross-sections of the Fermi surface i n several planes are shown i n F i g . 12; as expected the surface c o n s i s t s of a very f l a t e l e c t r o n and hole pocket i n each h a l 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 d i f f e r e n c e , decreasing the s i z e of the e l e c t r o n surface somewhat and i n c r e a s i n g the hole surface. In any case, the extremely small degree of curvature i n the surfaces makes i t p o s s i b l e f o r d i f f e r e n t nesting vectors to connect various combinations  of e l e c t r o n 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 e l e c t r o n 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 e l e c t r o n and hole surfaces i n 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 e l e c t r o n and hole contours i n the'upper h a l f of the zone. The v e r t i c a l scale i s i n u n i t s of ir/b  31  Chapter 3:  (SN)x  3.1 C r y s t a l S t r u c t u r e Two determinations of the (SN)x c r y s t a l s t r u c t u r e have r e c e n t l y become a v a i l a b l e . The r e s u l t s of the f i r s t , an e l e c t r o n d i f f r a c t i o n study by Boudeulle (1974), are shown i n F i g . 13 and are used i n the c a l c u l a t i o n s 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 d i f f e r e n t s t r u c t u r e (see Appendix) i n which the l a t t i c e constants are n e a r l y the same but the atomic  bond angles are d i f f e r e n t from those of Boudeulle.  C a l c u l a t i o n s i n c o r p o r a t i n g the Cohen data are presented i n the Appendix.  Despite the d i f f e r e n c e i n c r y s t a l s t r u c t u r e the r e s u l t s are essen-  t i a l l y the same. Thus, the f a c t that the e l e c t r o n 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 s t r u c t u r e 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 n i t r o g e n atoms stack along the b-axis forming a l t e r n a t i n g short (1.58 A) and long (1.72 X) SN bonds such that four atoms per chain are included i n the u n i t c e l l .  Nearest-  neighbouring chains i n the c - d i r e c t i o n are r e l a t e d by i n v e r s i o n and are thus i n e q u i v a l e n t , g i v i n g a t o t a l of eight atoms per u n i t c e l l . The  atomic  coordinates are l i s t e d i n Table 3.  3.2 One-Dimensional Band S t r u c t u r e As i n the preceding d i s c u s s i o n of TTF-TCNQ, we f i r s t consider the case of i s o l a t e d chains. The wavefunction \ ( r ) i s w r i t t e n as,  32  Figure 13. (a) P r o j e c t i o n of the c r y s t a l s t r u c t u r e of (SN)x onto a plane perpendicular to the chain a x i s . The sulphur and nitrogen atoms have been l a b e l l e d S. and N. , i = 1,2,3,4 depending on the p o s i t i o n i n the u n i t c e l l .  33  Figure 13. (b) Perspective drawing of a side view of the c r y s t a l s t r u c t u r e . The s o l i d rods i n d i c a t e the bonds along the chain and the open rods represent the most important i n t e r c h a i n overlaps discussed i n §3.3.  Table 3. Atomic coordinates i n the (SN)x u n i t c e l l f o r the Boudeulle s t r u c t u r e . In the coordinate system used, the l a t t i c e vectors are, i n A u n i t s : £ = (4.12,0,0), ft = (0,4.43,0), and  t = (-2.55,0,7.20). The b-axis i s the chain a x i s .  X  Y  Z  Si  -1.20  1.74  1.20  s2  -0.08  -0.48  2.40  s3  1.20  -1.74  -1.20  S  0.08  0.48  -2.40  Ni  -0.90  0.20  1.05  N2  -0.38  -2.02  2.55  N3  0.90  -0.21  -1.05  0.38  2.02  -2.55  H  (26)  ^-'U^^'/^?)  Here N i s the number of u n i t c e l l s , R i s the coordinate o f the center of m the m'th c e l l , ?. i s the coordinate with respect to the center o f the c e l l 3  of the j ' t h o r b i t a l , and . (r) i s an atomic S l a t e r o r b i t a l . ^rK?) s t i t u t e d i n t o the Hiickel formula using the constants e^g -26.0, =  w  a  sub-  s  =  -13.4  eV, with S l a t e r exponent 1.95 f o r n i t r o g e n , and e^g -20.0, e^p  -11.0  eV with S l a t e r exponents 2.122  =  =  and 1.827 f o r sulphur.  I n c l u s i o n o f the sulphur 3d o r b i t a l s i n the b a s i s set was found to r e s u l t i n a narrowing of the bands near the Fermi l e v e l w i t h no other q u a l i t a t i v e changes i n the band s t r u c t u r e . In some of the Huckel c a l c u l a tions mentioned i n §1.5 the value K = 2.0 f o r the  Wolfsberg-Helmholtz  constant was found to lead to b e t t e r r e s u l t s that the more popular value K = 1.75.  An increased K produces broader bands, an e f f e c t opposite t o  the i n c l u s i o n o f the 3d s t a t e s . Thus i t was f e l t that we were safe i n taking  K = 1.75 and n e g l e c t i n g 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 s i n g l e chain u n i t c e l l c o n s i s t s o f four atoms, (SN)^, the  energies f o r each k-value are obtained by s o l v i n g a 16 x 16 eigenvalue equation. F i g . 14a shows the energy bands which r e s u l t . Since the (SN)^ u n i t has 22 valence e l e c t r o n s , the 11 lowest bands w i l l be e x a c t l y f i l l e d at T = 0. O r d i n a r i l y one would then expect (SN)x to be an i n s u l a t o r . However, the screw a x i s symmetry o f the c r y s t a l s t r u c t u r e r e q u i r e s p a i r s o f 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 d e n s i t y o f s t a t e s f o r the highest occupied and lowest unoccupied bands i s shown i n F i g . 15 ( c f . Fig. 5 ) .  36  Figure 14. Energy bands f o r an i s o l a t e d (SN)x chain, (a) For the observed s t r u c t u r e . The symmetry l a b e l s are defined i n the t e x t while the dashed l i n e represents the Fermi l e v e l , (b) For the d i s t o r t e d s t r u c t u r e shown i n F i g . 16.  37  CO UJ I—  < I—  CO  u. o > I—  CO  z  UJ Q  -10  -9.5  E  -9  F  E N E R G Y (eV)  Figure 15. Density of states f o r the two bands s t r a d d l i n g the Fermi energy i n F i g . 14a.  Because the chains are n e a r l y planar, the l a b e l s s or IT may be used to i n d i c a t e 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 l a b e l l e d by *, and  1  labels a  k = 0 o r b i t a l with a node on the long SN bond. These r e s u l t s agree w i t h 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. I t 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 F i g . 16 (note that the s i z e of the u n i t c e l l remains unchanged).  undistorted  T  o  distorted  T  Figure 16. D i s t o r t i o n which breaks the screw a x i s symmetry.  The e f f e c t 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 F i g . 14b. We conclude, t h e r e f o r e , 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 d i f f e r e n t chains are l e s s than twice the Van der Waals r a d i i f o r sulphur and nitrogen. The i n t e r c h a i n coupling should then be r e l a t i v e l y l a r g e , implying that the 1-D  approximation  i s not very good.  Since there are two chains per u n i t c e l l , the 1-D bands are doubly degene r a t e , the degeneracy being removed by the i n t e r c h a i n coupling. The  split-  t i n g of the bands s h i f t s the Fermi l e v e l away from the Zone boundary, while  39  h y b r i d i z a t i o n 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 i n t o account an (SN)^ u n i t on one chain i n a unit c e l l with 20 neighbouring ( S N ^ u n i t s (two on the same chain and three on each of s i x surrounding chains) has been c a l c u l a t e d and i s shown along several symmetry d i r e c t i o n s of the B r i l l o u i n Zone ( F i g . 17) i n F i g . 18. The l i n e s and planes of degeneracies which a r i s e are the r e s u l t of the screw axis and time r e v e r s a l symmetry. The important feature i n F i g . 18 i s the crossing of the bands i n TZ and YC d i r e c t i o n s while there are gaps i n a l l the other d i r e c t i o n s . The crossings are a c c i d e n t a l degeneracies and are allowed because the wavefunctions of the c r o s s i n g 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 l e 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 d i f f e r e n t energies so that the Fermi l e v e l i n t e r s e c t s the bands at p o i n t s other than the degenerate p o i n t s . One thus expects a Fermi surface c o n s i s t i n g of an e l e c t r o n pocket near the c r o s s i n g along YC and a hole pocket near the crossing on TZ, leading us to describe (SN)x as a semimetal. I t i s g r a t i f y i n g to note that our r e s u l t s agree q u a l i t a t i v e l y with the more s o p h i s t i c a t e d r e l a t i v i s t i c OPW c a l c u l a t i o n s of Rudge (1975). The most important feature of Rudge's that i s not present i n our r e s u l t s i s the lower band i n the ZE d i r e c t i o n r i s i n g across the Fermi l e v e l , c r e a t i n g a hole pocket near point B. His conclusion r e i n f o r c e s ours that (SN)x i s a semimetal. The 3-D band s t r u c t u r e 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 w i t h the c r y s t a l s t r u c t u r e s of (SN)x and TTF-TCNQ. Symmetry d i c t a t e s two-fold degeneracies i n the band s t r u c t u r e throughout the top and bottom zone faces and along the l i n e s AE and BD. CY i s not a l i n e of degeneracy.  r  Z  IE  C C  Y Y  Figure 18. Three-dimensional band structure i n those d i r e c t i o n s indicated i n F i g . 17 as obtained from the extended Huckel c a l c u l a t i o n . Only the four bands c l o s e s t to the Fermi l e v e l are shown. The dashed l i n e marks the approximate p o s i t i o n of the Fermi energy.  P  42  noting that the highest occupied and lowest unoccupied bands i n the 1-D c a l c u l a t i o n ( F i g . 14a) look very much l i k e simple t i g h t - b i n d i n g (cosine) bands. The k = 0 wavefunction symmetries i n d i c a t e d i n F i g . 14a suggest that a s u i t a b l e b a s i s f o r such a c a l c u l a t i o n might be a IT* SN molecular o r b i t a l . There are four such SN molecules per u n i t 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 c l o s e r examination of the c r y s t a l s t r u c t u r e (Table 2 and F i g . 13). The molecule l a b e l l e d S^N^ i s considerably f a r t h e r away from molecules l a b e l l e d S^N^ i n i t s own and i n neighbouring c e l l s than from those l a b e l l e d  S2N2  and S^N^. Taking symmetry i n t o account, i t  should be a good approximation t o 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 c a l c u l a t i o n of the overlaps. The Hamiltonian then takes the form ( i f the energy of the SN MO i s taken t o be 0 ) ,  H  0  y*  6  y  0  0  0*  h  0  0  y  y*  0  0  6  2  0  1  C27)  Expansion of the corresponding secular determinant gives a c h a r a c t e r i s t i c equation of the form, E  4  - 2ae  2  + 6 = 0  (28)  with r o o t s ,  ± Jo~±  Ja  2  -8  (29)  43  where, 2a  = 2 i u 12 + \Si\  + |6 |  2  2  2  Quantitative r e s u l t s f o r t h i s model can be obtained by examination of the wavefunctions coming out o f the band s t r u c t u r e c a l c u l a t i o n . The TT* SN MO which i s suggested has the form,  * = +  - 2<f>  s  (30)  N  The most important i n t e r a c t i o n s i n v o l v i n g these MO's are then: ( i ) t : the i r - i n t e r a c t i o n between nearest-neighbouring molecules on the same chain. o ( i i ) t ^ : a ^ - i n t e r a c t i o n with a bond length o f 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 - i n t e r a c t i o n with bond length 2.81 A between atoms N^ and N^ i n the same u n i t c e l l . (iv)  t ^ : a a - i n t e r a c t i o n between S^ and S^ i n the c e l l d i s p l a c e d by t).  Including symmetry-equivalent i n t e r a c t i o n s gives f o r the matrix elements the expressions,  . ,., -ik-b. y = t(l+ e ) ^ ik-(a-b) . . -ik'b = t e + t + t e  S  r  l  6  2  2  -it.-(a+c)+  = t e x  v  ^ i^-(D"-C) + t o e ~ t e  J  v  J  l k  (31) 0  3  2  The parameters t , t , t , t 1  ^  3  2  3  can be c a l c u l a t e d 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 b e t t e r reproduce the band s t r u c t u r e of F i g . 18; these t i g h t - b i n d i n g curves are presented i n F i g . 19. Two important features of the band s t r u c t u r e are r e a d i l y apparent from equation (29) : ( i ) When a  2  = 6 , e = ± Ja and the band s t r u c t u r e e x h i b i t s a two-fold  degeneracy. The d i r e c t i o n s i n lt-space f o r which t h i s occurs are e x a c t l y those f o r which symmetry d i c t a t e s degeneracies, ( i i ) An a c c i d e n t a l degeneracy occurs when 8 = 0 . Then e = 0 and the bands cross at the Fermi l e v e l . 8 = 0 when P2  =  (32)  Taking the r e a l and imaginary parts of the equation gives two equations, the s o l u t i o n of which defines a curve i n it-space, which, w i t h the parameters 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 o f t h i s o v e r s i m p l i f i e d model therefore i s that (SN)x i s a zero-gap semiconductor. A s o p h i s t i c a t i o n of t h i s model to include non-zero terms along the anti-diagonal and i n t e r a c t i o n s with other bands would remove the accident a l degeneracy at t h t Fermi l e v e l . One would expect the small s p l i t t i n g s to cause i n d i r e c t overlaps o f the bands, thereby c r e a t i n g a Fermi surface of a semimetal w i t h e l e c t r o n and hole pockets. This conclusion agrees with that o f the more complex Huckel c a l c u l a t i o n . Since a density of states c a l c u l a t i o n incorporating the f u l l Huckel treatment proved to be u n f e a s i b l e , the t i g h t - b i n d i n g bands were used as  45  Figure 19. Band structure analogous to that of F i g . 18 r e s u l t i n g from the t i g h t - b i n d i n g c a l c u l a t i o n .  46 they represent the energy bands reasonably w e l l . F i g . 20 shows the r e s u l t s , and while the d e t a i l s of the p l o t  are not b e l i e v a b l e , the o v e r a l l features  should be c o r r e c t . The lack of resemblance to the 1-D  d e n s i t y of states  i n F i g . 15 i s an i n d i c a t i o n of the d i s p e r s i o n of the bands i n d i r e c t i o n s transverse to the chain a x i s and c l e a r l y demonstrates the need f o r a  3-D  model of (SN)x. Other authors who have c a l c u l a t e d the band s t r u c t u r e and the methods they have used are: ( i ) Rajan and F a l i c o v (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 c a l c u l a t i o n s 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 p o s s i b l e 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 l o w - l y i n g 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  f o r (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 e l e c t r o n i c energy spectra of two outwardly s i m i l a r systems. What we can conclude i s that both TTF-TCNQ are semimetals, i n agreement w i t h experiment. Both materials also e x h i b i t s i m i l a r anisotropy i n e l e c t r i c a l c o n d u c t i v i t y and o p t i c a l p r o p e r t i e s . Why then i s i t that TTF-TCNQ undergoes a P e i e r l s t r a n s i t i o n to a semiconducting s t a t e while (SN)x not only does not show t h i s feature but a c 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 c l o s e r look at the (SN)x c r y s t a l s t r u c t u r e i n the framework of our c a l c u l a t i o n . I f one b e l i e v e s the simple t i g h t - b i n d i n g MO treatment, then a s i n g l e (SN)x chain (type-^I) can be looked upon as c o n s i s t i n g of two inequivalent (type-II) chains, each c o n t r i b u t i n g one SN "molecule" to the u n i t 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 l a b e l l e d F and Q since they give r i s e to separate bands ( the two w i t h the degeneracy at the Fermi l e v e l i n F i g . 14a). Regarding the second type-II chain i n the same manner, one has two F and Q "molecules" per u n i t 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 f o r d i f f e r e n c e s i n the intermolecular distances. The d i f f e r e n c e between the two s o l i d s i s immediately apparent when the i n t e r c h a i n i n t e r a c t i o n s are turned on. The e f f e c t i s seen d r a m a t i c a l l y 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 ( i n terms of the bandwidth along TZ) i n d i r e c t i o n s transverse to TZ range up to 0.1 f o r (SN)x, but are less than 0.01 f o r TTF-TCNQ. I t seems, t h e r e f o r e , that the suppression of the P e i e r l s t r a n s i t i o n i n (SN)x must be a t t r i b u t e d to the strong i n t e r c h a i n coupling. HGW e x p l a i n the occurrence of a P e i e r l s 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 cond u c t i v i t y data they obtain a value of the i n t e r c h a i n coupling ri - 0.1, which f a l l s i n t o the region given by (8) i n which a 1-D mean f i e l d approach to the P e i e r l s t r a n s i t i o n i s v a l i d . Whether or not t h i s  theory explana-  t i o n i s c o r r e c t i s not c l e a r since our MO c a l c u l a t i o n gives a maximum n = 0.015 f o r two neighbouring TCNQ chains. Further, the HGW model f a i l s to describe TTF-TCNQ as a two-component system with c r o s s i n g bands i n t e r acting at the Fermi l e v e l to create a Fermi surface which d i f f e r s from the one r e s u l t i n g from equations  (6) and (7). However, the a c t u a l shape  of the surface i s l i k e l y not very important at temperatures near the observed T  c  - 60K, i n l i g h t of the small d i s p e r s i o n of the energy bands  transverse to the chain d i r e c t i o n . Thus our c a l c u l a t i o n does not preclude the P e i e r l s 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 s t r u c t u r e i n F i g . 9 shows, TTF-TCNQ i s w e l l explained by a 1-D model and consequently  the observed a n i s o t r o p i e s i n the e l e c t r o n i c  p r o p e r t i e s are not s u r p r i s i n g . S i m i l a r observations f o r (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 m a t e r i a l are i n c o n s i s t e n t with the band s t r u c t u r e i n which 3-D e f f e c t s are important. One must conclude, t h e r e f o r e , that the observed anisotropy i s due i n most part to the f i b r o u s nature of the c r y s t a l .  50  Bibliography  Bardeen, J . 1973. S o l i d State Comm. 13, 357. B e r l i n s k y , A.J., Carolan, J.F., and Weiler, L. 1974. S o l i d State Comm. 15_, 795. Boudeulle, M. 1974. Thesis, U n i v e r s i t e de Lyon. C h a i k i n , P.M., Kwak, J.F., Jones, T.E., G a r i t o , A.F., and Heeger, A.J. 1973. Phys. Rev. Lett. 31_, 601. Coleman, L.B., Cohen, M.J., Sandman, D.J., Yamagishi, F.G., G a r i t o , A.F., and Heeger, A.J. 1973. S o l i d State Comm. 12_, 1125. Cohen, M.J., G a r i t o , 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. L e t t . 3_2, 769. Denoyer, F., Comes, R., G a r i t o , A.F., and Heeger, A.J. 1975. Phys. Rev. L e t t . 35, 445. Etemad, S., Penney, T., Engler, E.M., S c o t t , B.A., and Seiden, E. 1975. Phys. Rev L e t t . 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., S t r e e t , G.B., and Suter, L.J. 1975. Phys. Rev. L e t t . 34_, 577. Grobman, W.D., P o l l a k , R.A., Eastman, D.E., Maas, E.T., J r . , and S c o t t , B.A. 1974. Phys. Rev. L e t t . 32_, 534. Hoffman, R. 1963. J . Chem. Phys. 39_, 1397. H o r o v i t z , 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 Y o f f e , A.D. 1975. S o l i d State Comm. 17_, 49. Kistenmacher, T.J., P h i l l i p s , T.E., and Cowan, D.O. 1974. Acta C r y s t . B30, 763.  51 Kohn, W. 1959. Phys. Rev. L e t t . 2_, 393. K o r t e l a , 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 , O n t a r i o ) . Lee, P.A., R i c e , T.M., and Anderson, P.W. 1973. Phys. Rev. L e t t . 31_, 462. L i t t l e , W.A. 1964. Phys. Rev. A 134, 1416. M Cubbin, W.L., and Manne, R. 1968. Chem. Phys. L e t t . 2_, 230. C  Parry, D.E., and Thomas, J.M. 1975. J . Phys. C 8_, L45. P e i e r l s , R.E. 1955. Quantum Theory o f S o l i d s (Oxford U n i v e r s i t y Press, London). Rajan, V.T., and F a l i c o v , L.M. 1975. To be published. Rice, M.J., and S t r a s s l e r , S. 1973. S o l i d State Comm. 1_3, 125. Rudge, W. 1975. B u l l . Am. Phys. Soc. 20_, 359, and p r i v a t e 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. S o l i d State Comm. 14, 342. Schluter, M., Chelikowsky, J.R., and Cohen, M.L. 1975. To be published. T i e d j e , T., Carolan, J.F., B e r l i n s k y , A.J., and Weiler, L. 1975. Can. J . Phys. 53_ (to be published). T i e d j e , T. 1975. M.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia. Tomkiewicz, Y., S c o t t , B.A., Tao, L . J . , and T i t l e , R.S. 1974. Phys. Rev. L e t t . 32, 1363. Weiler, L. P r i v a t e Communication.  52  Appendix: (SN)x Band Structure f o r the Perm C r y s t a l Structure The s t r u c t u r e of (SN)x reported by Cohen et a l . (1975) of the U n i v e r s i t y 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 ) .  Firstly,  the i n t r a c h a i n 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 n e a r l y planar as shown i n Fig.  A l . Secondly, i n e q u i v a l e n t chains are t r a n s l a t e d with respect to each o  other by about 1 A along the b-axis as compared to those i n the Boudeulle s t r u c t u r e . Table A l l i s t s the coordinates of the atoms i n the u n i t c e l l . The c a l c u l a t i o n s of Chapter 3 have been repeated using the Penn data. Fig A2 shows the 3-D band s t r u c t u r e a r i s i n g from the f u l l Hiickel c a l c u l a t i o n . O v e r a l l , the bands resemble those of F i g . 18; the d i f f e r e n c e s i n d e t a i l which are evident are an increased bandwidth along TZ and a small e r band s p l i t t i n g which pushes the Fermi wavevector back out to the zone boundary. The i m p l i c a t i o n , t h e r e f o r e , i s that i n the Penn s t r u c t u r e the i n t r a c h a i n coupling i s greater than i n the Boudeulle s t r u c t u r e while the i n t e r c h a i n coupling decreases. The t i g h t - b i n d i n g approximation to the band s t r u c t u r e introduced i n §3.3 has a l s o been applied i n t h i s case. Because of the change i n the r e l a t i v e p o s i t i o n of the i n e q u i v a l e n t chains, d i f f e r e n t i n t e r a c t i o n s become important and the g r e a t l y s i m p l i f y i n g feature that elements on the a n t i diagonal are zero does not appear. Thus no expression such as (28) can be w r i t t e n down and the 4 x 4 Hamiltonian must be d i a g o n a l i z e d numerically. I f 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 s t r u c t u r e of (SN)x from the data of Cohen et a l . (1975).  54  Figure A2. Three-dimensional band s t r u c t u r e obtained by the extended Hitckel method incorporating the Penn c r y s t a l s t r u c t u r e .  55  is, *  6*  a  6*  * V  u  6  a  3*  6  V  e  a  a  3  e  y  (Al)  The matrix elements are given by,  a  = 2t cosk'.a 1  e = t (l 2  e" ^- ") 1  +  y = tae- ^  0  1  v =t e-  i k  3  « = t (l 5  +  +  ^ e  +  i]t  t | | e  t t f 6  i t (a-?)  (A2)  - i l t . Ca+c")  ' ) &  where the t ^ ' s represent the f o l l o w i n g i n t e r a c t i o n s : ( i ) ti'. a a - i n t e r a c t i o n between two  atoms i n c e l l s displaced by a.  ( i i ) t : the i n t e r a c t i o n between molecules on the same chain, 2  ( i i i ) t : a a - i n t e r a c t i o n between an Sj and the S 3  3  i n the c e l l  t r a n s l a t e d by 1). (iv) t ^ : a u - i n t e r a c t i o n between an S i and the S  3  i n the c e l l d i s -  placed by D" - a. (v) t s : a a - i n t e r a c t i o n between an  and the  i n the c e l l  trans-  lated by c. With the parameters t  1  = 0.02 eV, t  2  = -0.57 eV, t  eV, and t$ = 0.04 eV, the energy bands depicted  3  = 0.25 eV, t  k  = -0.06  i n F i g . A3 again bear a  56 strong s i m i l a r i t y to those i n F i g . A2. The conclusion to be drawn from t h i s c a l c u l a t i o n i n c o r p o r a t i n g the Penn s t r u c t u r e 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 u n i t c e l l f o r the Penn s t r u c t u r e . 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).  X  Y  Si  -1.143  1.275  1.119  s2  -0.143  -0.945  2.475  s3  1.143  -1.275  -1.119  s4  0.143  0.945  -2.475  Ni .  -1.043  -0.307  1.279  N2  -0.243  -2.526.  2.315  N3  1.043  0.307  -1.279  N  0.243  2.526  -2.315  I  k  Z  57  r  Z  P  Z Z  Z  E E  A A  P  C C  Y Y  P  ' Figure A3. Tight-binding analogue of F i g . A2.  

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