@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Gallant, Michel Isidore"@en ; dcterms:issued "2010-02-16T02:14:12Z"@en, "1977"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "Experimental and theoretical results on the charge density wave (CDW) layered compound 2H-NbSe₂ have been reviewed and discussed in relation to the nature of the CDW instability. A simple extension of the saddle point model to three dimensions has been given. The results are discussed in connection with the available band calculations. The anisotropic Knight shift, based on a [sup d]z², [sup d]xy, [sup d]x²-y² Fermi surface, has been estimated and compared with the NMR experimental value. The results indicate that the Fermi surface contains a significant mixture of the three \"d\" orbitals, but not enough to place the saddle point at the Fermi energy in the fitted tight binding bands. It is concluded that more work is necessary to decide whether saddle point or nesting behavior is responsible for CDW formation in 2H-NbSe₂."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/20268?expand=metadata"@en ; skos:note "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 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

' 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 ? - / - \\— ^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| |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> < ^ f i l 4 l M H l ^ . 1 \\ > < < ^ 5 C S ^ | H | ^ > <^ Overlap Matrix Elements <<2i(*+t>| = -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 = 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 . "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0093977"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Physics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "The charge density wave instability in 2H-NbSe₂"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/20268"@en .