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The charge density wave instability in 2H-NbSe₂ Gallant, Michel Isidore 1977

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c.\ THE CHARGE DENSITY WAVE INSTABILITY MICHEL ISIDORE GALLANT B.Sc, U n i v e r s i t y of Toronto, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s thesis as conforming IN 2H-NbSe2 by to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1977 Michel Isidore Gallant. 1977 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements f o r an advanced degree at the Un ivers 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 t h i s t h e s i s for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It i s understood that copying or p u b l i c a t i o n of th is thes is fo r f inanc ia l gain s h a l l not be allowed without my writ ten permission. n r Physics Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date A P r i l 5> 1 9 7 7 Abstract Experimental and t h e o r e t i c a l r e s u l t s on the charge density wave (CDW) layered compound 2H-NbSe2 have been reviewed and d i s -cussed i n r e l a t i o n to the nature of the CDW i n s t a b i l i t y . A simple extension of the saddle point model to three dimensions has been given. The r e s u l t s are discussed i n connection with the a v a i l a b l e band c a l c u l a t i o n s . The a n i s o t r o p i c Knight s h i f t , based on a d 2, d . d 2 2 r z xy x -y Fermi surface, has been estimated and compared with the NMR ex-perimental value. The r e s u l t s i n d i c a t e that the Fermi surface contains a s i g n i f i c a n t mixture of the three d o r b i t a l s , but not enough to place the saddle point at the Fermi energy i n the f i t -ted t i g h t binding bands. It i s concluded that more work i s necessary to decide whether saddle point or nesting behavior i s responsible f o r CDW formation i n 2H-NbSe„. i i i Table of Contents Page List of Tables iv List of Figures v Acknowledgements v i Chapter 1 Introduction 1 2 Crystal Structure 2.1 Normal Phase 3 2.2 CDW Phase 5 3 Experimental Properties and Band Structure 3.1 Early Results 9 3.2 Tight Binding F i t to APW Results 14 3.3 More Recent Results 27 4 Nuclear Magnetic Resonance 4.1 Introduction 36 4.2 Knight Shift and the Fermi Surface 37 5 Summary and Conclusion 44 Bibliography 46 Appendix A Symmetry, Group Theory and Band 49 Calculations Appendix B Matrix Elements of d Bloch Sums 51 Appendis C Numerical Matrix Elements of d Orbitals 52 Appendix D Atomic Units 53 iv List of Tables Table Page 1 Real d atomic orbitals. 16(a) 2 Nesting versus saddle point models. 45 V L i s t of Figures Figure Page 1 C r y s t a l structure of la y e r s . 4 2 Deformation of Nb l a t t i c e i n the CDW phase. 6 3 B r i l l o u i n Sone f o r 2H-NbSe2 i n the normal phase. 8 4 Nearest neighbor coordination of Nb atoms i n 17 a layer. 5 Unhybridized t i g h t binding bands along f 7 K. 19 6a Hybridized t i g h t binding "d 2" band along saddle 20 point d i r e c t i o n . 6b Hybridized t i g h t binding "d z2" band along PM. 21 7 Two dimensional t i g h t binding Fermi surface 22 f o r 2H-NbSe2. 8 Nesting sections of Fermi surface i n p e r i o d i c 23 zone scheme. 9 NbSe„ two dimensional t i g h t binding energy 25 function, f o r "K i n the P KM basal plane. 10 Cross section of Fermi surface i n PKHA plane 33 with modifications. 11 Energy bands i n PKHA plane with modifications. 34 12 Normal phase B r i l l o u i n zone with superposed 35 commensurate CDW zones. 13 Niobium 4d r a d i a l wavefunction. 41 Acknowledgements I am obliged to Dr. Birger Bergersen f o r h i s eagerness to guide me i n t h i s research. In p a r t i c u l a r , h i s many hours spent i n computing are greatly appreciated. Thanks must also go to Dr. D.LI. Williams and Dr. J.A.R. S t i l e s f o r allowing me to partake i n some NMR experiments on 2H-NbSe2, and also f o r some f r u i t f u l discussions. Dr. Dan L i t v i n k i n d l y a s s i s t e d i n the preparation o f the Group Theory appendix. Part of t h i s research was f i n a n c i a l l y a s s i s t e d by a National Research Council of Canada Postgraduate Scholarship, which i s hereby g r a t e f u l l y acknowledged. 1 Chapter 1: Introduction Since about 1970, there has been a great deal of i n t e r e s t and a c t i v i t y i n the study of layered t r a n s i t i o n metal dichalcogenides. The t r a n s i t i o n metals included here are from the IVB, VB, and VIB columns of the p e r i o d i c t a b l e . The chalcogens include s u l f u r , t e l -lurium and selenium. A rather thorough review of the p h y s i c a l pro-p e r t i e s of these materials (Wilson and Yoffe 1969) i n d i c a t e s some general trends. A more recent review i s given by Wilson et a l . (1975). The IVB and VIB dichalcogenides tend to be semiconductors, whereas the VB compounds are narrow band metals. A l l are characterized by prominent d bands near o r at the Fermi energy and, depending on the extent of f i l l i n g of these bands, t h i s gives r i s e to various i n t r i -guing p r o p e r t i e s . Wilson and Yoffe (1969) have given a simple q u a l i -t a t i v e model of the e l e c t r o n i c energy bands i n these materials which helps to explain the e l e c t r i c a l and o p t i c a l p r o p e r t i e s . This i s a s t r i c t l y two dimensional model, but Bromley (1972) has suggested that the model may be extended to three dimensions in a simple manner. Att e n t i o n w i l l be focused on the group VB m e t a l l i c compound 2H-NbSe2 i n t h i s t h e s i s . 2H-NbSe2, as well as many other t r a n s i t i o n metal compounds, e x h i b i t s a low temperature phase t r a n s i t i o n to an v -incommensurate charge density wave (CDW) state (Williams et a l . 1975; Moncton et a l . 1975; Singh et a l . 1976). 2H-NbSe2 has the lowest CDW t r a n s i t i o n temperature ( T C D W = 33.5K) and the highest super-conducting c r i t i c a l temperature ( T r n w = 7.2K) of the t r a n s i t i o n 2 metal dichalcogenides. The nature of the charge density wave i n s t a b i l i t y , i n terms of e l e c t r o n i c band structure and the Fermi surface i s a matter of some controversy. Lomer (1962) showed, by examining the s u s c e p t i b i l i t y , that i f large parts of the Fermi surface are f l a t and nearly para-l l e l , so that they can be connected by a common wavevector, ("nes-t i n g " ) , then an i n s t a b i l i t y i n the l a t t i c e - e l e c t r o n system could develop. Chan and Heine (1973) discussed a microscopic model f o r t h i s i n s t a b i l i t y and gave c r i t e r i a f o r the occurrence of charge density waves, spin density waves, p e r i o d i c l a t t i c e d i s t o r t i o n s and combinations of these. More r e c e n t l y , Rice and Scott (1975) demon-strated that, f o r the case of a s i n g l e CDW i n a two dimensional system, a saddle point i n the energy bands at the Fermi surface y i e l d s a l o g a r i t h m i c a l l y diverging s u s c e p t i b i l i t y . They proposed that t h i s might be the o r i g i n of the CDW i n s t a b i l i t y . In t h i s t h e s i s , a v a i l a b l e d e t a i l e d band c a l c u l a t i o n s and t i g h t binding f i t s w i l l be used i n an attempt to determine which mechanism i s responsible f o r the CDW t r a n s i t i o n i n 2H-NbSe2- Trans-port, thermodynamic and o p t i c a l properties w i l l be examined i n re-l a t i o n to the e l e c t r o n i c energy bands to t r y and determine the best bands and thus the Fermi surface. Some estimates of quantites w i l l be given to determine whether the nesting or the saddle point model i s favored. 3 Chapter 2: C r y s t a l Structure r' 2.1 Normal Phase The t r a n s i t i o n metal dichalcogenides are composed of layers (or sandwiches). Each layer has three sheets of atoms, the two outer ones containing the chalcogens while the c e n t r a l sheet com-pr i s e s t r a n s i t i o n metal atoms as shown i n F i g . 1(c). These three sheet (X-T-X) layers are attracted by r e l a t i v e l y weak Van der Waals forces, and may be separated quite e a s i l y with i n t e r c a l a t e s , pro-v i d i n g t e s t s of the two dimensionality of these compounds (Yoffe 1973). Within a given layer, the binding i s strongly covalent with both the metal and chalcogen atoms forming hexagonal arrays. Two types of layers are p o s s i b l e . In one, each metal atom i s coordinated by a t r i g o n a l prism of chalcogens ( F i g . 1(a)), as i n 2H-NbSe2. The other form has octahedral coordination about the metal atoms, as i n lT-TaS 2 (Fig. 1(b)). The l a t t e r form w i l l not be discussed f u r t h e r here. A three dimensional c r y s t a l structure i s determined by the stacking sequence of these layers. One compound may form several d i f f e r e n t such sequences (polytypes). The notation used to d i s t i n -guish between the polytypes gives the number of layers per u n i t c e l l followed by the associated c r y s t a l system of the three dimensional structure. For example, 2H-NbSe2 i n d i c a t e s a two layer stacking sequence (2 layers per unit c e l l ) belonging to the hexagonal Figure 1. C r y s t a l structure of layers, (a) Trigon a l prism coordination u n i t . 0 metal; @ chalcogen. (b) Octahedral u n i t , (c) Side view of two layers with c/2 of 2H polytype. 5 4 c r y s t a l system. The space group of 2H-NbSe0 i s D,, , the same as 2. on f o r the HCP structure. NbSe 2 e x i s t s i n two other polytypes, but the 2H form i s the one of i n t e r e s t here. The polytype may be confirmed by X-ray a n a l y s i s . 2.2 CDW Phase Below T ^ D W = 33K, 2H-NbSe2 adopts an incommensurate super-l a t t i c e or p e r i o d i c l a t t i c e d i s t o r t i o n (Moncton et a l . 1975). Overhauser (1971) has shown how neutron d i f f r a c t i o n r e s u l t s may be analyzed i n terms of CDW formation. The s u p e r l a t t i c e remains incom-mensurate down to at least 5K, i n contrast to other t r a n s i t i o n metal dichalcogenides, such as 2H-TaSe 2 > which lock i n t o a commensurate s u p e r l a t t i c e at low temperatures. The atomic displacements are small compared to the nearest neighbor separation, and Berthier et a l . (1976) have estimated (from NMR r e s u l t s ) an upper l i m i t of•-^.09 f o r the corresponding f r a c t i o n a l e lectron r e d i s t r i b u t i o n within the un i t c e l l . F i g . 2 shows the atomic displacements i n the plane of a layer f o r the d i s t o r t e d phase. A t r i p l e incommensurate CDW i s coupled to t h i s s u p e r l a t t i c e . The CDW wavevectors are, where ficz. .02 and decreases somewhat as T i s lowered below T-...,,.; "a^ = 2PM i s one of three r e c i p r o c a l l a t t i c e vectors, oriented at (2/3)7\ r e l a t i v e to each other, d e f i n i n g the normal phase B r i l l o u i n (2) o> 0 «> e-».0*o e>0<o J? / Figure 2. Deformation of hexagonal(\v)Nb lattice in the plane of the layers in the CDW phase. The Nb deformations are assumed to be directed toward the maxima in the CDW o . The CDW phase is i n i t i a l l y anchored to a Nb atom The figure demonstrates how the incommensurate CDW gives rise to a distribution of inequivalent Nb sites (i.e. A, B, C). The distortions are greatly exaggerated 7 zone (Fig. 3). The CDW*s l i e i n the planes of the laye r s . Williams and Scruby (1975) interpreted the i n t e n s i t y v a r i a t i o n s of t h e i r transmission electron d i f f r a c t i o n observations on t h i n c r y s t a l s of 2H-NbSe2 i n terms of the r e l a t i v e phase of CDW*s on alternate layers. T h eir r e s u l t s suggest that the p e r i o d i c l a t t i c e d i s t o r t i o n s e i t h e r stack along the axis perpendicular to the layers i n phase with the 2 layer per u n i t c e l l l a t t i c e , or have no phase c o r r e l a t i o n between p a i r s of layer s . Figure 3. B r i l l o u i n zone f o r 2H-NbSe? i n the normal phase, (a) Three dimensions, (b) Basal plane. 9 Chapter 3: Experimental Properties and Band Structure 3.1 Early Results Since the CDW i n s t a b i l i t y i s be l i e v e d to have i t s o r i g i n i n c e r t a i n features of the Fermi surface (nesting or saddle p o i n t s ) , i t i s natural: to examine the various band structure c a l c u l a t i o n s (along with t h e i r corresponding Fermi surfaces), and, at the same time, analyze the various experimental r e s u l t s i n r e l a t i o n to these bands. S t r i c t chronological order w i l l not.be adhered to, as a n t i c i -pated r e s u l t s w i l l u n i f y the di s c u s s i o n . The e a r l i e s t attempts at understanding the e l e c t r o n i c structure of 2H-NbSe2 ( a n c* o t n e : r •" ayered compounds) involved schematic ligand f i e l d l e v e l s , which are e s s e n t i a l l y centres of g r a v i t y of the cor-responding d subbands. Goodenough (1968) and Huisman et a l . (1971) proposed somewhat d i f f e r e n t ligand l e v e l s . The l a t t e r r e s u l t s are i n good agreement with l a t e r band structure c a l c u l a t i o n s . Wilson and Yoffe (1969) proposed schematic models of the bands, making use of mainiy o p t i c a l r e f l e c t i v i t y spectra. Their bands derive mainly from niobium d z2 o r b i t a l s and selenium p o r b i t a l s . 2 2 The z band i n t h i s p i c t u r e i s higher i n energy and the z and p bands do not overlap. The same group estimated an occupied d con-duction bandwidth of about .5 eV. Lee et a l . (1970) showed that the maximum i n the magnetic s u s c e p t i b i l i t y at T^p^ and the r e v e r s a l i n sign of the H a l l c o e f f i -c i e n t which they observed, are c o r r e l a t e d . A d i f f e r e n t schematic model 10 of the bands was put forward i n an attempt to explain the H a l l c o e f f i c i e n t r e v e r s a l of sign (Pisharody et a l . 1972; Huisman et a l . 2 1971; V e l l i n g a et a l . 1970). The z and p bands were assumed to overlap. Photoemission (PE) data due to McMenamin and Spicer (1972) predicted an occupied d conduction bandwidth i n 2H-NbSe2 of about 1 eV, twice as large as Wilson and Yoffe's estimate. The lower selenium p valence bands and d conduction band were found to over-lap by .1 to .2 eV. An o v e r a l l bandwidth of 4 to 7 eV was found f o r the p bands. The extreme anisotropy i n the transport p r o p e r t i e s provides some evidence f o r the two dimensionality of 2H-NbSe2. The room tem-perature r e s i s t i v i t y r a c i o f o r conduction perpendicular and para-l l e l to the layers (Edwards and F r i n d t 1971) i s , Huntley and F r i n d t (1973) extended the previous transport pro-perty experimental work to include the temperature dependence of the thermoelectric power and magnetoresistance. A high temperature ( T ^ T ^ ^ ) H a l l c o e f f i c i e n t , consistent with a hole Fermi surface with one hole per niobium atom, was found. This group s t r i k i n g l y demonstrated that the H a l l c o e f f i c i e n t r e v e r s a l may be suppressed by i m p u r i t i e s . A two band model with temperature dependent hole amd electron mo-b i l i t i e s , i n v o l v i n g impurity s c a t t e r i n g , was put f o r t h i n an attempt to explain the H a l l anomaly, but the CDW phase t r a n s i t i o n i s now believed to be responsible for the H a l l behavior. 11 The f a c t that the H a l l c o e f f i c i e n t r e v e r s a l i s e a s i l y supp-ressed demonstrates that the CDW t r a n s i t i o n i n 2H-NbSe2 i s a tran-s i t i o n very s e n s i t i v e to d e t a i l s of the atomic environment. This i s evident from the very low T^,^ f o r t h i s compound (compared to other CDW t r a n s i t i o n metal layered compounds) and the fact that the CDW's are not observed to lock i n , or become commensurate, with the l a t t i c e . S t i l e s et a l . (1976) have studied the impurity dependence of T^p^, and f i n d that the CDW state i s completely subdued f o r a resi s t a n c e r a t i o lower than 12. Bromley (1972) discussed the two dimensional Fermi surface of NbSe2> based on the semiempirical t i g h t binding r e s u l t s of Bromley et a l . (1972). He showed that i f i n t e r l a y e r bonding i s greater than a c e r t a i n value (in terms of the bandwidth perpendicular to the l a y e r s ) , both electron and hole Fermi surfaces w i l l occur f o r any VB dichalcogeriide. A change i n the amount of e l e c t r o n and hole Fermi surface can lead to a change i n H a l l c o e f f i c i e n t . A change i n t h i s i n t e r l a y e r bonding could thus account f o r the H a l l r e v e r s a l . The smallness of the s u p e r l a t t i c e d i s t o r t i o n s (Moncton 1975) suggests that t h i s mechanism i s not the d e c i s i v e f a c t o r i n determining the H a l l behavior. Another point should be mentioned concerning Bromley's discussion. A completely d^2 band was assumed, the other d bands being s p l i t o f f . A t i g h t binding i n t e r p o l a t i o n f i t to a d e t a i l e d band c a l c u l a t i o n (Mattheiss 1973) shows that strong h y b r i d i z a t i o n between the d z2 and d^2 ^2, d Bloch sums produces a s i g n i f i c a n t band gap between these two x y 2 subbands. AtV1 , the lower band i s completely z i n character, but the lower band has d 2 2, d character at K ( F i g . 3). Thus, there x -y xy i s expected to be s i g n i f i c a n t admixture of these three o r b i t a l s i n 2 the conduction band which was previously described as "z ". This f a c t w i l l become important l a t e r when the Knight s h i f t i s discussed Ehrenfreund et a l . (1971) have done NMR on powdered samples of 2H-NbSe2, and have c l e a r l y demonstrated that the phase t r a n s i t i o n at T C D W does not r e s u l t from antiferromagnetic ordering (as was pre v i o u s l y suggested by Lee et a l . (1970)), but i s the r e s u l t of a s t r u c t u r a l change ( l a t t i c e d i s t o r t i o n ) accompanied by a r e d i s t r i -bution i n the valence e l e c t r o n i c charge density (CDW). Sobolev et a l . (1970) reported o p t i c a l r e f l e c t i v i t y data f o r the energy range 1 to 10 eV i n 2H-NbSe2> They proposed that the peaks above 2 eV probably r e s u l t from d i r e c t band to band t r a n s i t i o n s at s p e c i f i c points i n the B r i l l o u i n zone, although accurate assignments with respect to l a t e r d e t a i l e d band c a l c u l a t i o n s are d i f f i c u l t and ambiguous. The peaks below 2 eV were assumed to correspond to e x c i -tons, a view which disagrees with l a t e r views (Beal et a l . 1975). The f i r s t d e t a i l e d three dimensional band c a l c u l a t i o n i n 2H-NbSe 2 (and other layered compounds) was given by Mattheiss (1973). His augmented plane wave (APW) c a l c u l a t i o n p r e d i c t s a 1 eV h y b r i d i -zation gap between the lowest "d z2" conduction band and the higher d 2 2, d bands. An occupied conduction bandwidth of r^, .5 eV x -y xy • i s obtained which i s i n f a i r agreement with the .7 eV r e s u l t of Williams and Shephard (1973). Mattheiss finds strong Nb(5s), Se(4s, 4p) h y b r i d i z a t i o n , producing a large G^CT^bonding-antibonding) 13 gap in which a l l the d bands l i e . This means that, since the free 2 4 atom electronic configuration for selenium is 4s 4p and since the bands derived (mainly) from these orbitals are pushed well below t h e Fermi level, the selenium s and p bands w i l l be completely f i l -led implying significant charge transfer to the selenium atoms. Mattheiss assumes neutral atom charge densities in his calculation, and from the above remark, i t appears that this would lead to. some incorrect features in his bands, due to the lack of self consistency in his choice of crystal potential. The gap of .7 eV between the d^2 conduction band and the Se(p) bands is probably a consequence of this potential (PE results indicated that these bands overlap by ^-v^-.leV). One should therefore regard the finer details of such a calculation with caution. T h e Fermi surface derived from Mattheiss' bands is composed of a large hole pocket centered at A (Fig. 3), extending down close to P, and an open hole cylinder along KH, for the lower conduction band; the upper conduction band (corresponding to a second zone) yields open hole cylinders centered about both P A and KH, the up-per band surfaces enclosing the lower band ones. No electron pockets are found. Mattheiss' hole Fermi surface is consistent with the high temperature Hall coefficient (Huntley and Frindt 1973), R.. = 4X10" 4 cm3-Coulomb_1 H which corresponds to one hole per niobium atom, on a single carrier model. 14 Another feature of i n t e r e s t i n Mattheiss' bands i s the occurrence of a saddle point i n the energy bands along P K giving r i s e to a sharp peak i n the energy density of states at that point. In the de-t a i l e d APW r e s u l t s , t h i s saddle point l i e s well below the Fermi energy throughout the B r i l l o u i n zone (about .4eV below). Rice and Scott (1975) showed how a saddle point at the Fermi energy leads to a diverging response function ( s u s c e p t i b i l i t y ) . Mattheiss also pre-sents a t i g h t binding f i t which places the saddle point at (or very close to) the Fermi energy. It i s i n s t r u c t i v e to follow t h i s t i g h t binding f i t procedure i n some d e t a i l , as i t shows the e s s e n t i a l fea-tures of the d e t a i l e d band r e s u l t s . The following section i s devoted to t h i s study. 3.2 Tight Binding F i t to APW Results A two dimensional model for the layered compound 2H-NbSe2 i s used, neglecting i n t e r l a y e r i n t e r a c t i o n s . The extreme anisotropy i n the transport properties suggests that t h i s model could be f r u i t f u l . The chalcogen atoms are only taken i n t o account through the point symmetry (thus matrix elements w i l l have i n d i r e c t contributions, from the Se atoms). A purely two dimensional hexagonal l a t t i c e of Nb atoms would have point group symmetry, but the t r i g o n a l p r i s -matic coordination of Se atoms about each Nb atom reduces the sym-metry to i n t h i s purely 2^ case. The main content of the wave-functions at the Fermi surface i s d , d 2 2 and d 2 from the Nb xy x -y z atoms. Since i t i s the Fermi surface that i s of i n t e r e s t here, 15 Bloch states w i l l be formed from l i n e a r combinations of these three d o r b i t a l s . Also, f o r a s t r i c t l y two dimensional model, there i s r e f -l e c t i o n symmetry i n the basal plane so that matrix elements between d , d and the above three d states vanish, zx zy The electron wave function i s a l i n e a r combination of three Bloch wave functions, C3) (4) where, f o r example, ^^^Cr-R^) i s a d ^ o r b i t a l centered on hexagonal l a t t i c e s i t e R„. A S u b s t i t u t i n g t h i s electron wavefunction into the v a r i a t i o n a l c ondition, where H i s the si n g l e electron Hamiltonian, y i e l d s the secular equation determining the best energy eigenvalues and eigenfunctions, f o r the chosen basis of three d o r b i t a l s , det (H (k) - E(k)S..(k)) = 0 (6) where = ^ ^ j | H j i s a matrix element of the Hamiltonian, 16 and S „ = ^ j 4 ^ - j l ^ i s an overlap matrix element. N i s the number of p r i m i t i v e c e l l s i n the c r y s t a l (or the number of Nb atoms i n the layer f o r t h i s model). Taking nearest neighbor (nn) overlap only, and using the orthonormality of d o r b i t a l s (Table 1), one has, These matrix elements are given e x p l i c i t l y i n Appendix B. The s i m p l i f i e d method of Hiickel (Hoffman 1963) , which amounts to taking the H „ as proportional to the corresponding overlaps S „ with s c a l i n g constant k, was t r i e d , but f a i l e d to y i e l d a reasonable f i t to the d e t a i l e d APW bands. This i s due p r i m a r i l y to the non-vanishing of matrix elements such as, which would be zero i n the Hiickel method since, < d x y . ( r t * ^ | a x > y » - c n > > = 0 Here "t = a i i s one of the (nn) p r i m i t i v e t r a n s l a t i o n vectors (Fig. 4). It i s expected that i f the Hiickel method were applied including the Se o r b i t a l s e x p l i c i t l y , a reasonable f i t could be obtained. The Se atoms must be taken i n t o account, The S l a t e r and Koster (1954) l i n e a r combination of atomic o r b i t a l s i n t e r p o l a t i o n method was used to f i t the various overlap and exchange (Hamiltonian) matrix elements to Mattheiss' APW bands. The reduction Table 1. Real atomic o r b i t a l basis functions i n terms of angular momentum substates m. R_,(r) i s the r a d i a l part of the 4d wavefunction. m Magnetic State Real Wavefunction 1 i<p -2 A O fed 737 17 Figure J L . Nearest neighbor coordination of Nb atoms i n a layer. t 1 i s a p r i m i t i v e t r a n s l a t i o n vector. 18 of the matrix elements to a minimum independent set i s due to Miasek (1957). Appendix A discusses the use of Group Theory i n s i m p l i f y i n g the band problem. Matrix elements were f i t t e d to the unhybridized bandwidths, f o r a l l three bands, along P M, P K and the h y b r i d i -zation band gap between the d 2 and d , d 2 2 bands at K and M. z xy x -y Numerical values of these matrix elements are given i n Appendix C. I f one sets a l l matrix elements between d 2 and d 2 2, <1 z x -y xy equal to zero, one has the crossing bands i n F i g . 5. The strong over-lap i s evident. With h y b r i d i z a t i o n included, the lowest band i s shown i n F i g . 6 along c e r t a i n symmetry d i r e c t i o n s i n the B r i l l o u i n zone. There i s now a band gap between the d 2 and d , d 2 2 subbands. z xy x -y The o v e r a l l minimum i n energy f o r the bands i s at M. The feature of i n t e r e s t i s the occurrence of a saddle point along P K, which ap-parently i s a consequence of the strong h y b r i d i z a t i o n between the three d bands. The Fermi surface i s shown i n F i g . 7 demonstrating that t h i s two dimensional model y i e l d s the o v e r a l l correct hole Fermi surface, as i n the APW c a l c u l a t i o n . The saddle point seems to l i e close to the Fermi surface. Another feature of i n t e r e s t i s the r e l a t i v e f l a t n e s s of the Fermi surface sections near c. F i g . 8 shows how spanning vectors can connect e s s e n t i a l l y f l a t pieces of Fermi sur-face. Thus, the nesting appears to be favorable i n t h i s model. Of course, i n t e r l a y e r i n t e r a c t i o n s w i l l d i s t o r t the Fermi surface as one moves perpendicular to the B r i l l o u i n zone basal plane. The nes-t i n g wavevectors are i n the observed V M d i r e c t i o n s for the CDW. From these band r e s u l t s , the nesting model f o r the CDW i n s t a b i l i t y o Figure 6b. Hybridized t i g h t binding "d 2" band along f M . Figure 8. Nesting sections of Fermi surface i n period zone scheme. 24 seems to be favored, although the saddle point cannot be ru l e d out. In p a r t i c u l a r , Rice and Scott (1975) point out that a saddle point i n -s t a b i l i t y would only destroy the l o c a l part of the Fermi surface around the saddle point which acts as a s c a t t e r i n g sink (since the ele c t r o n group v e l o c i t y , "v* = "H * t ) E / ^ ~ k , i s small i n t h i s region) i n the high temperature phase, and thus could well account f o r the fa c t that 2H-NbSe2 i s a better conductor i n the CDW phase than i n the normal phase. A Fermi surface nesting condition would tend to anni-h i l a t e much Fermi surface producing a poorer conductor i n the CDW phase. The Mattheiss band c a l c u l a t i o n and the corresponding t i g h t binding f i t do not appear to c l a r i f y the issue. F i n a l l y , the basal plane energy surface i s portrayed i n F i g . 9. It i s u s e f u l to extend the saddle point model of Rice and Scott somewhat. The saddle point energy w i l l probably change as one moves perpendicular to the basal plane. I f i t i s assumed that the saddle point maintains i t s ( k x , k ) coordinates as k z v a r i e s , then i n Rice and Scott's model, t h i s i s equivalent to taking the chemical poten-t i a l (yU(= E.f) to be k z dependent so that /A =jU(k^). These authors suggest that three dimensionality might be taken into account by averaging t h e i r s u s c e p t i b i l i t y " ^ ( " c ^ ; jUtk^)) over k z. In the pre-sent study, such an estimate i s made. With the energy measured from the saddle p o i n t , jm (k z) i s assumed to have a cosine v a r i a t i o n along k as f o l l o w s : z C8) 25 26 where c i s the un i t c e l l height and k z ranges over the 3 B r i l l o u i n zone (Fig. 3 ) . Then, the average i s , (9) where "~^)L~ ^6 COS if A I. 4-ah (2i+e) k Q i s an upper c u t o f f of i n t e g r a t i o n , and i s approximately 2/3PM; N^, the nesting f a c t o r , i s taken to be constant i n the averaging. # -1 An estimate of m* , from the previous t i g h t binding band cur-2 2 vatures O E/£ k ), y i e l d s , where a i s the (nn) niobium distance; S Q ^ " 1 R y d b e r g s (Appendix D contains a discussion of atomic u n i t s ) . A rough estimate of b •—'1/3 causes nesting and N_. diverges. A meaningful numerical estimate of N_» i s therefore not possible i n t h i s d i s c u s s i o n . However, the average may be performed a n a l y t i c a l l y using, o (10) The f i n a l r e s u l t i s , L - — This i s s i m i l a r to ")£ (q* ; yU) f o r the s t r i c t l y two dimensional 27 case, and shows that the logarithmic s i n g u l a r i t y induced by the saddle point i n ' y . i s governed by the maximum dev i a t i o n , , of the saddle point energy from the Fermi energy. Thus s i g n i f i c a n t deviations of /A. from zero (the saddle point energy) w i l l tend to dampen the saddle point i n s t a b i l i t y . An extension of t h i s study might be to r e l a t e the const, term i n to some simple form, such as the free e lectron gas s u s c e p t i b i l i t y f o r a c y l i n d e r (Hodges et a l . 1971), and make a more c a r e f u l estimate of N_&. This could then be analyzed i n terms of the CDW i n s t a b i l i t y c r i t e r i a presented by Chan and Heine (1973). It should be mentioned here that Gupta et a l . (1976) have shown that the usual choice of taking a wavevector independent o s c i l l a t o r strength matrix element ^*+<?J e ^ ' r J k ^ in. (1), the general form f o r the s u s c e p t i b i l i t y , can give spurious r e s u l t s . It i s suggested that f o r nesting s i t u a t i o n s the v a r i a t i o n of these matrix elements can have s i g n i f i c a n t e f f e c t s on the s u s c e p t i b i l i t y . 3.3 More Recent Results Fong and Cohen (1974) t r i e d the empirical pseudopotential method. The upper chalcogen p valence band was found to l i e c l o s e r to the d conduction bands than i n Mattheiss' bands. Both e l e c t r o n and hole Fermi surfaces were found, but were not discussed i n d e t a i l . The saddle point which they obtained f l u c t u a t e s by almost h a l f the con-duction bandwidth of 1.5eV. The occupied width was*^.9eV, consistent with the PE r e s u l t s of McMenamin and Spicer (1972). Charge d e n s i t i e s were also calculated and discussed i n terms of bonding models. 28 Significant charge transfer to the Se atoms was v e r i f i e d , but charge was found to accumulate between the Nb and Se atoms as well. The pseudopotential method seemed to involve convergence d i f f i c u l t i e s , and does not appear to be considered as accurate as Mattheiss' results. The temperature dependence of the Bragg intensity in the neutron work of Moncton et a l . (1975) suggests that the observed superlattice. forms in a second order way. A strong Kohn anomaly was also observed, but, surprisingly, no soft phonon mode was found (Chan and Heine (1973) predicted that this mode should soften). The same group also observed that the f i n i t e discontinuity in the specific heat is characteristic of a second order phase transition (or a very weak f i r s t order one). This should be contrasted to the phenomenological Landau theory pro-posed by McMillan (1975) in which the electronic d band charge den-si t y is taken as the order parameter, and the CDW transition i s pre-dicted to be f i r s t order, for a t r i p l e CDW. Barmatz et a l . (1975) measured elastic constants and found be-havior near T_,mi. characteristic of a second order phase transition. LUW Harper and Geballe (1975) performed careful measurements of the specific heat. They found the expected relation, C = VT + fi, T 3 (12) with y = 16 mJ mole K , and 0 = .55 mJ mole K This value for ^ implies a high density of states N(E^) at E^, con-sistent with Mattheiss (1973). The important result that the CDW tran-sition does not have a large effect on the specific heat (a slight 29 d i s c o n t i n u i t y ) or r e s i s t i v i t y was also demonstrated. This group suggested that the CDW does not a l t e r the Fermi surface appreciably (although the H a l l r e v e r s a l implies some change of c o n n e c t i v i t y i n the Fermi surface), a r e s u l t which i s consistent with the Rice and Scott sad-dle point model. The e a r l i e r superconductivity data of Bachmann et a l . (1971) also suggests a large N(E^). Beal et a l . (1975) obtained normal (to layers) incidence, op-t i c a l r e f l e c t i v i t y spectra. The range investigated included a l l low energy e l e c t r o n i c t r a n s i t i o n s i n the range 1 to 5eV. Structure between 1 and 2eV was associated with e l e c t r o n i c t r a n s i t i o n s near the Fermi l e v e l , i n contrast to the e a r l i e r assignment (Sobolev et a l . 1970). Maxima i n the r e f l e c t a n c e near 2.5 to 3.5eV and 5 to 6eV were e s t i -mated to be t r a n s i t i o n s from two high d e n s i t i e s of states regions i n the p valence band to the upper d 2 conduction bands, although tran-z s i t i o n s from the Fermi l e v e l to the high metal 5s antibonding states cannot be r u l e d out. A Kramers-Kronig analysis was performed and the r e s u l t s were shown to be i n agreement with Mattheiss' band c a l c u l a -t i o n . Campagnoli et a l . (1976) have observed the thermoreflectance of 2H-NbSe2- The prominent features include a p o s i t i v e peak at 1.9eV and a negative peak at about 2.6eV. The f i r s t peak i s believed to be c o r r e l a t e d with the onset of d-d band t r a n s i t i o n s . Fong and Cohen (1974) have pointed out a s p e c i f i c feature i n t h e i r bands which could be the o r i g i n of t h i s strong absorption. The second peak i s i d e n t i -f i e d as a t r a n s i t i o n i n v o l v i n g a temperature broadened saddle point i n the j o i n t density of states (JDOS). This i s i n agreement with the 30 JDOS p r o f i l e s proposed by Mattheiss, based on h i s APW band stru c t u r e . Singh and Curzon (1976) observed a s u p e r l a t t i c e by e l e c t r o n d i f f r a c t i o n i n 2H-NbSe2» but could not resolve the incommensurabi-l i t y of the p e r i o d i c l a t t i c e d i s t o r t i o n s . Otherwise, t h e i r r e s u l t s agree with those of Moncton et a l . (1975). Wexler and Woolley (1976) have done a d e t a i l e d layer method band c a l c u l a t i o n . The method i s s i m i l a r to that of Mattheiss, but was c a r r i e d out i n greater d e t a i l so that l e s s i n t e r p o l a t i o n would be necessary to obtain the Fermi surface. The o v e r a l l shape of the d conduction bands obtained are l i k e Mattheiss 1 but the p valence bands are c l o s e r i n energy to the d bands. In a d d i t i o n , the absolute minimum i n energy of the d bands i s found to l i e along P M and not at M as Mattheiss f i n d s . The lowest d band at P i n the APW c a l c u l a -t i o n i s below the Fermi l e v e l but the layer method places these bands (at P ) well above the Fermi l e v e l . This feature i s presumably not very s e n s i t i v e to s l i g h t changes i n the layer method and determines the Fermi surface topology. The lower band forms an open hole c y l i n -der with axis P A, i n contrast to Mattheiss 1 large hole pocket about A, which does not "break through" the P KM basal plane. The remain-der of the Fermi surface resembles that of Mattheiss, except that the c y l i n d e r s about KH are more c i r c u l a r i n cross s e c t i o n than those of Mattheiss (weakening the nesting behavior). The saddle point again l i e s considerably below E^ throughout the zone; the saddle point model seems unfavorable again. Wexler and Woolley point out the important r o l e of the chalcogen 31 as compared to the metal atom by comparing 2H-NbSe2 with 2H-NbS2, which have very s i m i l a r band structures, but are quite d i f f e r e n t i n s t a b i l i t y against CDW formation (CDW formation i n 2H-NbS2 i s i n -h i b i t e d and i t s transport properties vary smoothly at low temperatures, i n contrast to the behavior of 2H-NbSe2 discussed p r e v i o u s l y ) . They examine the two respective Fermi surfaces and speculate that the d i f -ferences i n these two compounds might involve some f i n e d e t a i l i n the close p a i r of d^2 bands along P K . It seems u n l i k e l y that accurate f i n e d e t a i l could r e s u l t from a band c a l c u l a t i o n which does not i n -voke s e l f consistency i n the p o t e n t i a l (or charge t r a n s f e r ) . Another feature of the layer method c a l c u l a t i o n that deserves mention i s the p-d gap which i s less than Mattheiss' by about .5eV, which agrees better with PE r e s u l t s mentioned e a r l i e r . Up to date, the only d i r e c t experimental measurements ( i n CDW layered compounds) that gauge the Fermi surface are the Landau quantum o s c i l l a t i o n observations of Graebner and Robins (1976). These i n -cluded magnetization (de Haas-van Alphen) and temperature (magneto-thermal) o s c i l l a t i o n s . They are also the f i r s t observed o s c i l l a t i o n s i n a superconductor, which can be used to probe the superconducting f l u x l a t t i c e . Graebner and Robins f i n d that t h e i r o s c i l l a t i o n s can be ex-plained by proposing a s l i g h t change i n Mattheiss' conduction bands along the P M and P K d i r e c t i o n s , thereby creating a small pocket of electrons around P . This small pocket w i l l not contribute s i g n i f i -cantly to averages over the Fermi surface (which determines many 32 thermodynamic and transport p r o p e r t i e s ) , as i t s surface area i s much le s s than that of the remaining Fermi surface. It should be mentioned that the m o d i f i c a t i o n to Mattheiss, which Graebner and Robins show, i s also consistent with a hole torus i n the B r i l l o u i n zone basal plane with axis PA. The two i n t e r p r e t a t i o n s are shown i n F i g s . 10, 11. A greater s p e c i f i c a t i o n of the m o d i f i c a t i o n i s evidently required. The same group t r i e d f o l d i n g the unaltered Mattheiss band s t r u -cture i n t o a smaller ( s u p e r l a t t i c e ) B r i l l o u i n zone (Graebner and Mar-cus (1968) show the d e t a i l s of t h i s procedure f o r chromium), deter-mined by the CDW wavevector"q* C D W = . 9 8 ( 2 / 3 ) P M . F i g . 12 shows the high temperature B r i l l o u i n zone with the smaller hexagonal super-l a t t i c e zones f o r the commensurate case <1Q)W = (2/3) PM. For 2H-NbSe2, t h i s f o l d i n g - i n procedure i s very s e n s i t i v e to the Fermi surface topo-logy, as the hole c y l i n d e r s about KH have r a d i i close to the super-l a t t i c e zone r a d i i . Therefore, any statements as regards the change of c o n n e c t i v i t y i n the Fermi surface due to the i n t r o d u c t i o n of new zone boundaries leading to a change of H a l l c o e f f i c i e n t sign should be regarded with caution. Graebner and Robins found no electron poc-kets that matched t h e i r o s c i l l a t i o n s . In a d d i t i o n , they showed that the proposed high temperature e l e c t r o n pocket would survive the CDW energy gaps, as q„ n, does not connect the electron pocket to any other pieces of Fermi surface. The small electron pocket does not appear to play a large r o l e i n the CDW t r a n s i t i o n . It would be i n t e r e s t i n g to t r y and f i n d the o r i g i n of t h i s e l e c t r o n pocket i n terms of band matrix elements. Figure 10 . Cross section of Fermi surface i n TKHA plane with modifications. (a) Mattheiss. (b) Hole torus i n t e r p r e t a t i o n ^ (c) Electron pancake i n t e r p r e t a t i o n . Figure 11. Energy bands i n r KHA plane with modifications, (a) Hole torus, (b) Electron pancake (aa 1 as i n F i g . 10). Figure 12. Normal phase B r i l l o u i n zone (dark l i n e s ) with superposed commensurate CDW zones ( l i g h t l i n e s ) The wavevector f o r the CDW i s ~qrnw = (2/3)TM. 36 Chapter 4: Nuclear Magnetic Resonance 4.1 Int ro ducti on Nuclear magnetic resonance (NMR) i s a powerful t o o l f o r the i n v e s t i g a t i o n of CDW compounds. This i s due to the f a c t that the n u c l e i are perturbed by t h e i r c r y s t a l l o g r a p h i c environments. The on-set of a CDW should therefore manifest i t s e l f i n the NMR spectrum. In a d d i t i o n , NMR i n the normal phase gives information concerning the wavefunction content at the Fermi surface, which w i l l be d i s -cussed l a t e r i n some d e t a i l . V a l i c et a l . (1974) performed continuous wave (CW) NMR on si n g l e c r y s t a l s of 2H-NbSe2, measured the high temperature (T >' r C D W) Knight s h i f t , and interpreted t h e i r low temperature (T^ T^,^) lineshape broadening as a d i s t r i b u t i o n of Knight s h i f t . This i s evidence f o r a d i s t r i b u t i o n of inequivalent Nb s i t e s , induced by an incommensurate CDW. The same group observed a broadening i n the quadrupole resonance l i n e s as T ^ T ^ ^ from the normal phase. Since the quadrupole f r e -quencies depend d i r e c t l y on the e l e c t r i c f i e l d gradient tensor (EFG) at the Nb nucleus, t h i s behavior was in t e r p r e t e d as f l u c t u a t i o n s i n the EFG heralding a CDW phase t r a n s i t i o n . Berthier et a l . (1976) used pulsed-NMR on s i n g l e c r y s t a l s of 2H-NbSe2 to observe a l o c a l d i s t r i b u t i o n of Knight s h i f t and e l e c t r i c f i e l d gradient below T^,^. Th e i r r e s u l t s are compatible with the sym-metry of a t r i p l e incommensurate CDW. 37 S t i l e s and Williams (1976) have extended the CW-NMR work of V a l i c et a l . (1974). They measured the Knight s h i f t and EFG tensor f o r T>T^,p^, and observe a s i m i l a r "behavior of lineshape broadening f o r T ^ T ^ y . I t i s pointed out that the amplitude dependence of the CDW resembles that of a B.C.S. order parameter. They show that the Knight s h i f t d i s t r i b u t i o n can be corr e l a t e d with an incommensurate .CDW amp-l i t u d e . The Knight s h i f t r e s u l t s , i n the high temperature phase, w i l l now be discussed i n r e l a t i o n to the Fermi surface. 4.2 Knight S h i f t and the Fermi Surface I f the n u c l e i i n a solid.'have a nonzero spin, they tend to l i n e up with an external magnetic f i e l d g i v ing r i s e to the nuclear Zeeman energy l e v e l s . The number of l e v e l s i s determined by the number of magnetic sublevels (2S + 1) f o r a nucleus with spin S. The nuclear magnetic resonance frequency i s determined by the di f f e r e n c e i n energy between adjacent Zeeman l e v e l s , which i n turn i s determined by the external magnetic f i e l d strength and the nuclear magnetic moment. However, an external magnetic f i e l d also p o l a r i z e s the conduction electrons (Pauli paramagnetism), which y i e l d s an extra magnetic f i e l d at the nucleus. The f r a c t i o n a l s h i f t i n the NMR resonance frequency due to t h i s nuclear d i p o l e - e l e c t r o n dipole i n t e r a c t i o n i s c a l l e d the Knight s h i f t K, and i s of the form, K= K. + K (3cos 2e - 1) is o an^ (13) 38 f o r a nuclear s i t e with a x i a l symmetry (Rowland 1961). A t r i g o n a l or hexagonal axis s a t i s f i e s t h i s condition. The f i r s t term i s the i s o t r o p i c Knight s h i f t due to the so c a l l e d contact i n t e r a c t i o n and i s given by, h-,So = ^ ^ w ^ < l ^ l > £ • t l 4 ) -21 whereJL(o= 9.2741 10 erg/Gauss i s the Bohr magneton, the volume per Nb atom, N(E^) the density of states per unit volume per u n i t energy at the Fermi energy, and < J l P c c d * \ i s the Fermi surface average of the square of the electron wavefunction at the nucleus. S l i c h t e r (1963) gives an e x p l i c i t form f o r t h i s average. Since only atomic s o r b i t a l s have nonzero value at t h e i r n u c l e i , and Mattheiss points out that the Fermi surface from h i s band c a l c u l a t i o n i s pre-dominently d i n character, K ^ s q i s expected to be small and i n fa c t i s found to be s o . S t i l e s and Williams (1976) f i n d a value of K ^ s q = -.02 +_ .02% f o r one of t h e i r CDW samples. In the present work, a rough estimate of the electron wavefunction content at P was made, to v e r i f y Mattheiss' c a l c u l a t i o n . Using niobium 4d, 5s and selenium 4s, 4p atomic o r b i t a l s (Clementi and Roetti 1974; Basch and Gray 1966), the 4X4 secular determinant f o r mixing of these o r b i t a l s at f was computed. The "mainly" d^2 wavefunction at P was found to contain only a 1% admixture (amplitude squared) o f 5s o r b i t a l , a 5% admix-ture of 4p and a n e g l i g i b l e amount of 4s o r b i t a l . Since the d z2 band about the Fermi energy i s r e l a t i v e l y narrow compared to the 4d z2,5s bandgap, and since these two o r b i t a l s mix at a l l points i n the zone, 39 the 1% mixture of 5s estimated at P should also be a reasonable value at the Fermi surface. Thus, the i s o t r o p i c Knight s h i f t i s pr e d i c t e d to be small. A s t r i k i n g feature that was noticed i n t h i s c a l c u l a t i o n was the very strong h y b r i d i z a t i o n of the 4s, 4p and 5s o r b i t a l s i n the highest s t a t e at P. Thi s demonstrates the nature of t h e a - 0 - * gap which contains the d^2 conduction band i n 2H-NbSe2. The second term i n the Knight s h i f t i s the anis o t r o p i c Knight s h i f t ( f o r a x i a l symmetry), where © i s the angle between the axis of symmetry and the external magnetic f i e l d d i r e c t i o n , and, (15) PS The i n t e g r a l i n i s over the atomic c e l l with V() normalized to t h i s same c e l l , q^ i s a measure of the anisotropy i n the conduction band charge d i s t r i b u t i o n . I f Bloch states ti) based on d 2, d and d 2 2 T z xy x -y atomic o r b i t a l s are used, the i n t e g r a l i n (15) reads, . y a > e * ^ - * % ( 1 6 ) The i n t e g r a l s on the r i g h t are i n general three center i n t e g r a l s , but since the i n t e g r a t i o n i s over j u s t one unit c e l l , and the atomic or-b i t a l s f a l l o f f qu i c k l y with distance, i t i s a reasonable ap-_> _» proximation to take j u s t the terms R = R , = 0. The i n t e g r a l re-r n n 1 b duces to an average over one s i t e : 40 (17) 2 Now since ![3cos - 1) has the same angular dependence as the d z2 o r b i t a l , i t i s e a s i l y seen that the only nonzero terms are f o r j = j ' . Thus, (17) reduces to, 3 (18) This involves the quadrupole moment f o r each d o r b i t a l and the amount a., of each d o r b i t a l i n the Fermi surface wavefunction. By w r i t i n g d , d 2 2, d 2 (Table 1) as l i n e a r combinations of s p h e r i c a l harmonics times the same r a d i a l part R^(_r), and using the following i n t e g r a l s (Arfken 1970), (19) (18) becomes j=4(T%,r-isr-/vy»0<^> (20) where \ i ? - / - \— <r _3 The average o f r over the u n i t c e l l i s now replaced by the average over a l l space. This i s quite a good approximation as the square of the niobium r a d i a l 4d wavefunction (Herman and Skillman 1963) i s very small at the atomic c e l l boundary (Fig. 13) and the contributions from outside the u n i t c e l l have been found i n t h i s study to be of the order of 5%. Then performing the i n t e g r a t i o n numerically, the r e s u l t i s , 24 Figure 13. Niobium 4d r a d i a l wavefunction (Herman and Skillman 1963). The abscissa u n i t s are u n i v e r s a l . For Nb, X=3.89r with r i n atomic (Bohr) u n i t s . N.N. i s the nearest neighbor Nb distance i n a layer. 42 3.5. (21) where i s the Bohr radius f o r hydrogen. Thus (15) becomes, If the Fermi surface i s assumed to have wavefunctions which are completely d^2 i n character, then J az2J ^  = 1 and, q^ ..--^  2/ap3 or 25 -3 q f = 1.35X10 cm . Taking Mattheiss' (1973) value f o r the density of states at E f , N(E f) = 2X40 States(Ry.Atom.Spin)" 1 with the f a c t o r of 2 f o r spin, and converting to common u n i t s , an estimate of (15) f o r pure d 2 Fermi surface i s , K .oo43 = .43%. The value which S t i l e s and z an — Williams (1976) obtained i s , K & n = .18 +_ .01%. Comparison of the t h e o r e t i c a l and experimental values i n d i c a t e s that the Fermi surface can-not be considered to be completely d^2 i n character. The f a c t that the experimental value of K i s much less than the t h e o r e t i c a l d 2 value r an z obtained here indicates by (22) that roughly 25% of d or d^2 ^2 i s mixed i n . This i s consistent with the t i g h t binding f i t Fermi surface i f E^ does not l i e close to the saddle point. The t i g h t binding f i t shows that the saddle point (at least i n the PKM basal plane) has about 50% admixture of the d , d 2 2 o r b i t a l s . This i s expected, as the saddle xy x -y r » point comes about through h y b r i d i z a t i o n between these very o r b i t a l s . Since the saddle point region i s one of high density of sta t e s , i f i t occured at the Fermi surface, i t would tend to dominate any Fermi sur-face average. A saddle point at the Fermi surface would therefore 43 suggest a very small or even negative value for K . The saddle point mechanism appears to be inconsistent with the NMR results. 44 Chapter 5: Summary and Conclusion This thesis has been exploratory i n nature, with considerable discussion of the experimental and t h e o r e t i c a l work on the CDW layered compound 2H-NbSe2« Emphasis has been on properties r e l a t i n g to the CDW i n s t a b i l i t y mechanism. Table 2 gives a b r i e f summary of the saddle point versus nesting model prop e r t i e s . At present,. there appears to be more experimental evidence i n favour of the saddle point mechanism, but the NMR r e s u l t s c e r t a i n l y need to be explained. The various d e t a i l e d band structure c a l c u l a t i o n s have the same general features, but i t could be that some l o c a l feature (besides the saddle point) of the Fermi surface i s important. It would be use f u l to apply the very d e t a i l e d KKR method of band c a l c u l a t i o n (as has been done f o r lT-TaS^ by Myron et a l . (1975) showing very strong nesting) to 2H-NbSe2-Further quantum o s c i l l a t i o n experiments to map out the Fermi surface i n greater d e t a i l are also d e s i r a b l e . In conclusion, a d e f i n i t i v e choice between the saddle point and the nesting model f o r the CDW i n s t a b i l i t y i n 2H-NbSe2 cannot be made at present. More t h e o r e t i c a l and experimental i n v e s t i g a t i o n s are needed to s e t t l e t h i s unresolved problem. 45 Table 2. Nesting versus Saddle Point Models. NESTING SADDLE POINT Favorable Unfavorable Favorable Unfavorable • Nearly 2^ bands. • Approximately correct nesting wavevector. • Higher cr i n CDW phase. » Small change i n C p and f at • Higher (J- i n CDW phase. •Approximately correct connec-t i n g wavevector. •High N(E^) from superconduc-t i v i t y and C . P •Small discon-t i n u i t y i n ? a t W •Saddle point i n most band models not near E^. •K r e s u l t s , an 46 Bibliography Arfken, G. 1970. Mathematical Methods f o r P h y s i c i s t s sec. 12.9, (Academic Press, New York). • Bachmann, R., Kirsch, H.C., and Geballe, T.H. 1971. S o l i d State Commun. 9_, 57. Barmatz, M., T e s t a r d i , L.R., and DiSalvo, F.J. 1975. Phys. Rev. B 12, 4367. Basch, H., and Gray, H.B. 1966. Theoret. Chim. Acta. 4, 367. Beal, A.R., Hughes, H.P., and Liang, W.Y. 1975. J . Phys. C. 8^  4236. Berthier, C , Jerome, D., Molinie', P., and Rouxel, J . 1976. S o l i d State Commun. 19_, 131. Berthier, C , Jerome, D., Molinie"', P. 1976. S o l i d State Commun. 18, 1393. Bouckaert, L.P., Smoluchowski, R., and Wigner, E.P. 1936. Phys. Rev. 50, 58. Bromley, R.A., Murray, R.B., and Yoffe, A.D. 1972. J . Phys. C 5_, 759. Bromley, R.A. 1972. Phys. Rev. Let t . 29_, 357. Campagnoli, G., G u s t i n e t t i , A., and S t e l l a , A. 1976. S o l i d State Commun. 18, 973. Chan, S.K., and Heine, V. 1973. J . Phys. F 3_, 795. Clementi, E., and R o e t t i , C. 1974. Atomic Data and Nuclear Data Tables, 14_, 177. Cornwell, J.F. 1969. Group Theory and E l e c t r o n i c Energy Bands i n Solids (North Holland, Amsterdam). Coulson, C.A. 1961. Valence (Oxford U n i v e r s i t y press, New York). Edwards, J . , and F r i n d t , R.F. 1971. J . Phys. Chem. S o l i d s , 32, 2217. Egorov, R.F., Reser, B . I . , and S h i r o k o v s k i i , V.P. 1968. Phys. Status S o l i d i . 26, 391. 47 Ehrenfreund, E., Gossard, A.C., Gamble, F.R., and Geballe, T.H. 1971. J. Appl. Phys. 42, 1491. Fong, C.Y., and Cohen, M.L. 1974. Phys. Rev. Lett. 32_, 720. Goodenough, J.B. 1968. Phys. Rev. 171, 466. Graebner, J.E., and Marcus, J.A. 1968. Phys. Rev. 175, 659. Graebner, J.E., and Robbins, M. 1976. Phys. Rev. Lett. 36, 422. Gupta, R.P., and Freeman, A.J. 1976. Phys. Rev. B 13, 4376. Harper, J.M.E., Geballe, T.H., and DiSalvo, F.J. 1975. Phys. Lett. 54A, 27. Harrison, W.A. 1970. Solid State Theory sees. 152 (McGraw H i l l , New York). Herman, F., and Skillman, L. 1963. Atomic Structure Calculations (Prentice Hall, Englewood Cliffs, N.J.). Herring, C. 1942. J. Franklin Inst. 233, 525. Hodges, C, Smith, H. , and Wilkins, J.W. 1971. Phys. Rev. B 4_, 302. Hoffman, R. 1963. J. Chem. Phys. 39, 1397. Huisman, R. , DeJonge, R., Haas, C, and Jellinek, F. 1971. J. Solid State Chem. 3_, 56. Huntley, D.J., and Frindt, R.F. 1974. Can. J. Phys. 5_2, 861. Koster, G.F. 1964..Space Groups and their Representations (Academic Press, New York). Lax, M. 1974. Symmetry Principles in Solid State and Molecular Phy- sics Appendix A, ( J. Wiley and Sons, New York). Lee, H.N.S., Garcia, M., McKinzie, H., and Wold, A. 1970. J. Solid State Chem. 1_, 1970. Lomer, W.M. 1962. Proc. Phys. Soc. 80, 489. Mattheiss, L.F. 1973. Phys. Rev. B 8_, 3719. McMenamin, J.C, and Spicer, W.E. 1972. Phys. Rev. Lett. 29, 1501. McMillan, W.L. 1975. Phys. Rev. B U, 1187. Miasek, M. 1957. Phys. Rev. 107, 92. 48 Moncton, D.E., Axe, J.D., and DiSalvo, F.J. 1975. Phys. Rev. L e t t . 34, 734. Myron, H.W., and Freeman, A.J. 1975. Phys. Rev. B 11_, 2735. Overhauser, A.W. 1971. Phys. Rev. 167, 692. Pisharody, K.R., Thompson, A.H., and Koehler, R.F. 1972. B u l l . Amer. Phys. Soc. 17_, 315. Rice, T.M., and Scott, G.K. 1975. Phys. Rev. Le t t . 35_, 120. Rowland, T.J. 1961. Nuclear Magnetic Resonance i n Metals Progress i n Materials S c i . , 9_, No.1, 25 (Pergamon Press, New York). Singh, 0., and Curzon, A.E. 1976. Phys. Lett. 56_, 63. S l a t e r , J . C , and Koster, G.F. 1954. Phys. Rev. 9£, 1498. S l i c h t e r , C.P. 1963. P r i n c i p l e s of Magnetic Resonance (Harper and Row, New York). Sobolev, V.V., Donetskikh, V.I., Kalyuzhnaya, G.A., and Antonova, E.A. 1971. Sov. Phys. Semicond. S_, 844. S t i l e s , J.A.R., and Williams, D.LI. 1976. J . Phys. C £, 3941. S t i l e s , J.A.R., Williams, D.LI., and Zuckermann, M.J. 1976. J . Phys. C 9, L489. V a l i c , M.I., A b d o l a l l , K., and Williams, D.LI. 1974. Proc. 18th Ampere Meeting, Nottingham 2_, 369. V e l l i n g a , M.B., deJonge, R., and Haas, C. 1970. J . S o l i d State Chem. 2, 299. Wexler, G., and Woolley, A.M. 1976. J . Phys. C £, 1185. Williams, P.M., and Shephard, F.R. 1973. J . Phys. C 6_, L36. Williams, P.M. , and Scruby, C B . 1975. S o l i d State Commun. 17_, 1197. Wilson, J.A., DiSalvo, F.S., and Mahajan, S. 1975. Adv. i n Phys. 24, 117. Wilson, J.A., and Yoffe, A.D. 1969. Adv. i n Phys. 1_8, 193. Yoffe, A.D. 1973. Festkorperprobleme Vol. 13 (Vieweg, Braunschweig, Germany). 49 Appendix A: Symmetry, Group Theory  and Band Structure C a l c u l a t i o n s The symmetry of a c r y s t a l l i n e s o l i d i s the basis f o r s i m p l i f i c a t i o n of e l e c t r o n i c band structure c a l c u l a t i o n s . T r a n s l a t i o n a l symmetry of the l a t t i c e y i e l d s electron states of the Bloch form, labeled by wave-—* vectors, k i n the B r i l l o u i n Zone(B.Z.) (Harrison 1970). Possible a d d i t i o n a l symmetry operations include r o t a t i o n s , r e f l e c t i o n s and s p e c i a l combinations of r o t a t i o n s or r e f l e c t i o n s and t r a n s l a t i o n s ( i e . g l i d e planes and screw axes). The set of a l l symmetry operations of the c r y s t a l c o n s t i t u t e the c r y s t a l ' s SPACE GROUP. The mathematical methods of GROUP THEORY allow one to s y s t e m a t i c a l l y study the con-sequences of the c r y s t a l symmetry. Thus, electron states are indexed according to the IRREDUCIBLE REPRESENTATIONS (IRREPS) of the GROUP OF THE WAVEVECTOR*k (Cornwell 1969). Dimensionalities of the IRREPS correspond to the degeneracies of the electron wavefunctions indexed by these IRREPS. The l a b e l i n g of points i n the various B.Z.*s i s f a i r l y standard i (Koster 1964). The IRREPS of the group of the wavevector at k are X labeled e i t h e r by subscripts of the B.Z. l a b e l at k, or more commonly, by numbers augmented with a plus or minus sign. For example, the center of the B.Z. i s labeled by P and, f o r a cubic l a t t i c e , the IRREPS at P are P^ , J '25» (Bouchaert et a l . 1936); f o r a hexagonal l a t t i c e , the IRREPS at P are 1 +, 2~, etc. (Herring 1942). It should be noted, however, that the IRREP notation i s not standardized, and care must be 50 taken i n comparing s i m i l a r band c a l c u l a t i o n s by d i f f e r e n t authors (Lax 1974). In the LCAO method, each basis function i s a Bloch sum based on a p a r t i c u l a r atomic o r b i t a l at a s p e c i f i c l o c a t i o n i n the unit c e l l . Group t h e o r e t i c a l matrix element theorems are used to reduce the number of nonvanishing matrix elements between d i f f e r e n t b a s i s functions. Basis functions are u s u a l l y chosen at the outset so as to transform accor-ding to the IRREPS of the wavevector k. Often, the consequences of group theory can be determined by simple in s p e c t i o n , but. f o r complicated space groups, group theory i s needed to e x p l o i t the f u l l symmetry. Egorov et a l . (1968) have given a c l e a r treatment of the group t h e o r e t i c a l considerations as applied to the LCAO scheme, with electron spin taken i n t o account. They stress the minimum number of matrix elements of the Hamiltonian that need to be evaluated. This requires s p e c i f i c information of the i r r e d u c i b l e representation matrices, and not j u s t the character t a b l e s . In terms of the symmetry of the atomic o r b i t a l s i n the LCAO method, reduction of matrix elements by inspection i s valuable i n that i t pro-vides i n s i g h t into p o s s i b l e bonding i n t e r p r e t a t i o n s and charge density d i s t r i b u t i o n s (Coulson 1961). A table of matrix elements f o r s, p and d atomic o r b i t a l Bloch sums has been given by Miasek (1957) f o r the HCP st r u c t u r e . These have been adapted f o r the present s i n g l e layer LCAO c a l c u l a t i o n with appropriate modifications to the <^niobium|seleniun^ matrix elements. 51 Appendix B: Matrix Elements of d Bloch Sums Tight binding matrix elements H „ = ••C^-jH} 1^ f o r the planar hexagonal l a t t i c e i n the nearest neighbor approximation (p. 16) a f t e r Miasek (1957) are given. <$ = d , = d 2 2 <P = d 2; » 1 xy' 'a. x -y , »3 z ^ T) =JjLLc\ Kv * a '""s t h e n e a r e s t neighbor distance. + a<4 l Cn+-t l ) |H|<?/n \>^a5 + ^ < < ? ^ + ? ^ | H) rb> c o s A 1 H33 = <4^it>JH)4Uit>>^ H 1 2 = J T ^ C ^ I Hl^O>> - <<*?,C?Ut,)| H| <P,<fc>|S *" H | S J H I J + a I <<$>( H) a)> COS I Sih ^ S „ ='^ y,-|v{,j^  may be obtained d i r e c t l y from the Hamiltonian matrix elements by putti n g H = 1. H.. = H.*; Note: Miasek does not note that the matrix element. 52 Appendix C: Numerical Matrix Elements of d O r b i t a l s Numerical values of matrix elements f o r the two dimensional t i g h t binding f i t to APW bands i n 2H-NbSe2 (Mattheiss 1973) are presented h e r e . " ^ =ai i s shown i n F i g . 4. (f.are as in-app. B. Hamiltonian Matrix Elements (Rydbergs) <4>3<fo)H|<VS>> < ^ f i l 4 l M H l ^ . 1 \ > < < ^ 5 C S ^ | H | ^ > <^<Sfti|H|^(iij> Overlap Matrix Elements <<2i(*+t>|<V™> = -0.4478 = -0.5078 = 0, = +0.0433 = -0.0233 = -0.0505 = +0.0200 = -0.0200 = +0.0956 -0.0484 +0.0259 +0.0300 +0.1000 0.0000 = <f.. (normalization) Vi These are the only independent overlap i n t e g r a l s . 53 Appendix D: Atomic Units I f a s o l i d i s viewed as an assembly of atoms, i n i t i a l l y at very large separations which subsequently come together, the p i c t u r e of how e l e c t r o n i c energy bands a r i s e i s c l e a r . The overlapping atomic wavefunctions simply broaden the i n i t i a l l y degenerate o r b i t a l s i n t o a quasicontinuum(band). Thus, i t i s natural to measure band energies i n atomic u n i t s . The i o n i z a t i o n energy of a hydrogen atom, the Rydberg, i s a commonly chosen u n i t : 1 Rydberg =.*>C =jg- = 13.6058 eV — 8 where a^ = .52918 X 10 cm i s the Bohr radius f o r hydrogen, - 1 0 -01 e = 4.80325 X 10 esu i s the electron charge, and-If = 1.05459X v-erg-sec i s Planck's constant. -12 The e l e c t r o n - v o l t (1 eV = 1.60219X 1° e r § ) i s a l s 0 a commonly used band energy. The energy gap between the valence and conduction band i n many semiconductors i s of the order of 1 eV; at room temperature, K T ^ l e V = .025 eV. A s t i l l larger u n i t , which i s sometimes confused with 40 the Rydberg, i s the Hartree: a 1 Hartree = JL.= 2 Rydberg = 27.116 eV. *o The Hartree i s often c a l l e d the "atomic u n i t of energy". Atomic u n i t s of length are often taken to be Bohr u n i t s (a^), e s p e c i a l l y when atomic o r b i t a l s are being discussed. Reduced Bohr 54 u n i t s (depending on the atomic number Z, and the p r i n c i p a l quantum number n) are sometimes used (Herman and Skillman 1963) to produce un i v e r s a l forms which are u s e f u l i n general d i s c u s s i o n s . 

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