UBC Theses and Dissertations

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UBC Theses and Dissertations

Electronic properties of NBS₂ single layers Li, Zhanming 1984

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ELECTRONIC PROPERTIES OF NBS 2 SINGLE LAYERS by ZHANMING £ 1 B.Sc.,Zhongshan U n i v e r s i t y , 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA Sept. 1984 © Zhanming L i , 1984 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e h e a d o f m y d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Plltj^ l^ S T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 1 9 5 6 M a i n M a l l V a n c o u v e r , C a n a d a V 6 T 1 Y 3 D a t e QcX. IS y 1 9%4 D E - 6 ( 3 / 8 1 ) i i A b s t r a c t A theory of o p t i c a l l y and m a g n e t i c a l l y a n i s o t r o p i c NbS 2 p l a t e l e t s suspended i n water i s developed, and a two dimensional t i g h t - b i n d i n g model i s used to d e s c r i b e the magnetic and o p t i c a l p r o p e r t i e s of s i n g l e l a y e r NbS 2 and NbSe 2. For the f i r s t time both p o l a r i z a t i o n and t r a n s i t i o n matrix e f f e c t s are in c l u d e d i n c a l c u l a t i n g the i n t e n s i t i e s of a b s o r p t i o n of the t r a n s i t i o n metal d i c h a l c o g e n i d e s . The r e s u l t s are i n q u a l i t a t i v e agreement with experiments and can be i n t e r p r e t e d i n terms of the two dimensional(2D) nature of the band s t r u c t u r e . Table of Contents A b s t r a c t i i L i s t of Tables , IV L i s t of F i g u r e s V Acknowledgement VI Chapter I INTRODUCTION 1 A. STRUCTURE AND PROPERTIES OF THE 2H-TMD 2 B. A RECENT EXPERIMENT 3 Chapter II THEORY OF ANISOTROPIC PLATELETS IN WATER 5 A. A MACROSCOPIC MODEL 5 B. FITTING THE MODEL TO THE EXPERIMENT 7 Chapter III FERMI SURFACES AND BAND STRUCTURES OF THE SINGLE LAYERS ..10 A. A TIGHT BINDING FIT TO THE BANDS OF THE 2H-TMD 10 B. BAND STRUCTURE OF SINGLE "LAYERS 12 Chapter IV MAGNETIC SUSCEPTIBILITY OF SINGLE LAYER COMPOUNDS 14 Chapter V OPTICAL PROPERTIES OF THE SINGLE LAYER COMPOUNDS 18 Chapter VI CONCLUSIONS 24 BIBLIOGRAPHY 25 APPENDIX A - FORMULATION OF THE MAGNETIC SUSCEPTBILITY ...27 APPENDIX B - EVALUATION OF MATRIX ELEMENT OF TNE ORBITAL ANGLAR MOMENTUM 28 APPENDIX C - TETRAHEDRON METHOD IN TWO DIMENSION 29 APPENDIX D - EVALUATION OF THE ATOMIC TRANSITION MATRIX ..31 i v L i s t of Tables I. C r y s t a i l o g r a p h i c data 2 I I . Data of the f i t t i n g f o r NbS 2 8 I I I . Experimental data of magnetic s u s c e p t i b i l i t y of TMD 8 IV. Atomic o r b i t a l s used i n the t i g h t b i n d i n g f i t 13 V. Magnetic s u s c e p t i b i l i t i e s from the theory 17 V L i s t of F i g u r e s 1. S t r u c t u r e of 2H-TMD 33 2. O p t i c a l Absorption Spectra of NbS 2 34 3. The Aligment of NbS 2 P l a t e l e t s i n Magnectic F i e l d ....35 4. F i t t i n g of the Model to Experiment 36 5. Coordinates of the S i n g l e Layer 37 6. Absorption s p e c t r a i n Magnetic F i e l d 38 7. R a t i o of I n t e n s i t i e s of D i f f e r e n t P o l a r i z t i o n s 39 8. Band S t r u c t u r e of 2H-NbS2 40 9. Band S t r u c t u r e of 2H-NbSe 2 41 10. D e n s i t y of S t a t e s of the 3D Compounds 42 11. Symmetry P o i n t s i n the B r i l l o u i n Zone 42 12. Band S t r u c t u r e of S i n g l e Layer NbS 2 43 13. Band S t r u c t u r e of S i n g l e Layer NbSe 2 44 14. Fermi Surface of S i n g l e Layer NbS 2 45 15. Fermi Surface of S i n g l e Layer NbSe 2 46 16. D e n s i t y of S t a t e s ( d z 2 ) of the S i n g l e Layers 47 17. Diagram I l l u s t r a t i n g the Tetrahedron Method 48 18. Se p a r a t i o n of Bands of the S i n g l e Layer NbS 2 49 19. Se p a r a t i o n of Bands of the S i n g l e Layer NbSe 2 50 20. A m p l i t i t u d e C o e f f i c i e n t s of the Wave Funtion f o r d z 2 Band 51 21. J o i n t D e n s i t y of S t a t e s of S i n g l e Layer NbS 2 52 22. J o i n t D e n s i t y of Stat e s of S i n g l e Layer NbSe 2 ., 53 v i Acknowledgement I am o b l i g e d to my a d v i s o r , P r o f e s s o r B. Bergersen who has i n i t i a t e d t h i s work and given v a l u a b l e a d v i c e throughout i t s complet i o n . Mr. C. L i u and Dr. R . F . F r i n d t have k i n d l y p r o v i d e d d e t a i l s and data of the experiment. I have b e n e f i t e d from the i n s p i r i n g c o n v e r s a t i o n s with Dr. P.Palffy-Muhoray on the r a t i o of the o b s o r p t i o n i n t e n s i t i e s . Thanks a l s o go to Dr. S. Sharma f o r her h e l p f u l i n f o r m a t i o n p r o v i d e d d u r i n g the computer programing of the band s t r u c t u r e c a l c u l a t i o n . The f i n a n c i a l support from the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1 I. INTRODUCTION The t r a n s i t i o n metal d i c h a l c o g e n i d e s (TMD's) have been the o b j e c t of a great d e a l of recent study due to the o b s e r v a t i o n of charge d e n s i t y waves (CDW's) with a s s o c i a t e d p e r i o d i c l a t t i c e s d i s t o r t i o n s [ 1 3 . Two models are proposed f o r the formation of CDW's[2,3], Since both of them are based on the quasi~2D nature of the compounds, i t w i l l be of experimental and t h e o r e t i c a l i n t e r e s t s to study the p r o p e r t i e s of s i n g l e l a y e r TMD's. The recent s u c c s s f u l p r e p a r a t i o n of s i n g l e l a y e r NbS 2 [4,17,19] motivate us to study the e l e c t r o n i c p r o p e r t i e s of the compounds using a 2D model. Although no CDW's are observed i n the s i n g l e l a y e r s , a t h e o r e t i c a l study w i l l h e l p one to understand to what extent the TMD's are two d i m e n s i o n a l . In both the experiment and t h i s work one f i n d s c o n s i d e r a b l e enhancement of magnetic s u s c e p t i b i l i t i e s i n the s i n g l e l a y e r compounds, which i s r e l a t e d to the recent o b s e r v a t i o n s of the magnetic p r o p e r t i e s of t h i n f i l m s , s u r f a c e s and i n t e r f a c e s [ 5 ] ; t h e r e f o r e the magnetic behavior of the compounds i s i n t e r e s t i n g i n i t s own r i g h t . A f t e r a b r i e f i n t r o d u c t i o n to the s t r u c t u r e and p r o p e r t i e s of the TMD's and the recent experiment r e s u l t s on s i n g l e l a y e r NbS 2 i n t h i s chapter ,we w i l l analyse the experiment r e s u l t s with a simple model in chapter 2, and-we present a t i g h t b i n d i n g c a l c u l a t i o n as m o d i f i e d from that of Doran e t . a l . [ 6 ] i n chapter 3. F i n a l l y we w i l l use the t i g h t b i n d i n g model to c a l c u l a t e the magnetic and o p t i c a l p r o p e r t i e s of the compounds. 2 A. STRUCTURE AND PROPERTIES OF THE 2H-TMD The m a t e r i a l s 2H-NbS2 and 2H-NbSe 2 belong to the f a m i l y of l a y e r t r a n s i t i o n metal d i c h a l c o g e n i d e s (from group IVB, V and VIB) as reviewed by Wilson and Y o f f e e [ 7 ] , The s t r u c t u r e of both compounds c o n s i s t s of molecular sandwiches with one sheet of t r a n s i t i o n metal atoms between two sheets of chalcogen atoms. The c r y s t a l has two sandwiches i n a u n i t c e l l with the atoms in each sandwich forming a t r i a n g u l a r l a t t i c e . To e s t a b l i s h our c o o r d i n a t e system and n o t a t i o n we i l l u s t r a t e the s t r u c t u r e s c h e m a t i c a l l y i n FIGURE 1. The c r y s t a l l o g r a p h i c data are l i s t e d i n TABLE 1. * C r y s t a l l o g r a p h i c data ( i n atomic uni t s )• [ 7 , 1 0 ] used i n our c a l c u l a t i o n . a, s e p a r a t i o n w i t h i n a monolayer; w, height of Van der Waals gap; s, height of sandwich. a s w 2H-NbS2 6.26 5.94 5.29 2H-NbSe 2 6.50 6.31 5.48 Table I - C r y s t a l l o g r a p h i c data Due to the weak i n t e r l a y e r i n t e r a t i o n , the a n i s o t r o p i e s in e l e c t r i c c o n d u c t i v i t y i s very l a r g e . For example, i n m e t a l l i c 3 compounds, the c o n d u c t i v i t y p a r a l l e l to the l a y e r can be as l a r g e as 50 times that p e r p e n d i c u l a r to the l a y e r . The most i n t e r e s t i n g f e a t u r e of TMD's, however, i s that many of them e x h i b i t a t r a n s i t i o n to both CDW s t a t e s and superconducting s t a t e s at low temperatures. There have been two popular models f o r the CDW t r a n s i t i o n s , the n e s t i n g model and the saddle p o i n t model. In the n e s t i n g model[2] , p a r a l l e l s e c t i o n s of Fermi s u r f a c e s that are spanned by a wave ve c t o r q, give r i s e to a d i v e r g e n t e l e c t r i c s u s c e p t i b i l i t y , xe. X = p n ( k ) ( l - n ( K ) ) / ( E ( k ) - E ( k + q ) ) (1 .1 ) where n(k) i s the Fermi f u n t i o n . In the saddle p o i n t model[3], two saddle p o i n t s on the Fermi s u r f a c e spanned by a wave vector q lead to the divergence of (1.1). C l e a r l y both of the models are based on the geometry of the quasi-2D band s t r u c t u r e . B. A RECENT EXPERIMENT We w i l l next d i s c u s s some experiment r e s u l t s [ 4 , 1 9 ] as the mo t i v a t i o n f o r t h i s work. In the experiment, s i n g l e l a y e r p l a t e l e t s of NbS 2 f r e e l y suspended i n water have been prepared using a technique i n v o l v i n g i n t e r c a l a t i o n of hydrogen and water, and the s i n g l e l a y e r s t r u c t u r e has been determined by X-ray d i f f r a c t i o n experiment. Since the sample i s homogeneous one can perform o p t i c a l a b s o r t i o n experiment on i t . For the convevience of l a t e r r e f e r e n c e , the o p t i c a l a b s o r p t i o n s p e c t r a ( p l o t t e d as ln(Io/I),where Io i s the i n t e n s i t y of the i n c i d e n t l i g h t ) of 4 s i n g l e l a y e r NbS 2 are reproduced in FIGURE 2[4]. Compared to the a b s o r p t i o n s p e c t r a of the 3D s i n g l e c r y s t a l , the s p e c t r a of the p l a t e l e t s have the same main a b s o r p t i o n peak at 2.7 eV, but a c o n s i d e r a b l e r e d u c t i o n at lower f r e q u e n c i e s i s observed. The r e d u c t i o n can be e x p l a i n e d i n terms of the i n c r e a s e i n band gap between p/d bands (bands a r i s i n g from p o r b i t a l s i n the language of t i g h t b i n d i n g method) and the d z 2 band as w i l l be shown i n chapter 3. Due to the a n i s o t r o p y i n magnetic s u s c e p t i b i l i e s , o n e can a l i g n the p l a t e l e t s with a strong magnetic f i e l d ( s e e FIGURE 3). The a n i s o t r o p y i n o p t i c a l p r o p e r t i e s then enables one to use p o l a r i z e d l i g h t to study the p o l a r i z a t i o n e f f e c t on the a b s o r p t i o n s p e c t r a . The p o l a r i z a t i o n dependent a b s o r p t i o n i n t e n s i t i e s at 4600 Angstrom are shown i n FIGURE 4. To understand the s p e c t r a and the p o l a r i z a t i o n e f f e c t , which i s the goal of t h i s work, one needs a theory of band s t r u c t u r e as w e l l as a macroscopic model f o r the s t a t i s t i c a l behavior of the p l a t e l e t s suspended i n water. 5 I I . THEORY OF ANISOTROPIC PLATELETS IN WATER A. A MACROSCOPIC MODEL For a continuous medium with o p t i c a l a b s o r p t i o n c o e f f i c i e n t , a, the i n t e n s i t y has the f o l l o w i n g forms: where I i s the i n t e n s i t y of l i g h t and t i s the l e n g t h of the path through the medium. Consider a d i s t r i b u t i o n of p l a t e l e t s with mean volume V 0 and l e t P 2 ( v ) be the p r o p a b i l i t y that a given p l a t e l e t has a volume V=vVg. We assume that 1) the whole system can be c o n s i d e r e d as a continuous medium. 2) macroscopic s c a t t e r i n g i s n e g l i g i b l e . 3) demagnetization and e l e c t r o s t a t i c e f f e c t s are n e g l i g i b l e . We can d e r i v e s i m i l a r r e l a t i o n s f o r the two cases when the p o l a r i z a t i o n v e c t e r , E , i s p a r a l l e l and p e r p e n d i c u l a r to the a p p l i e d magnetic f i e l d , , r e s p e c t i v e l y : Case 1. E p a r a l l e l to B d l = - a l . d t (2.1 ) 1=1 0exp(-at) (2.2) (2.3) 6 Here a, and a 3 are the a b s o r p t i o n c o e f f i c i e n t s c orresponding to p o l a r i z a t i o n s p a r a l l e l and p e r p e n d i c u l a r to the plane r e s p e c t i v e l y , a n d i i = (8$) are the p o l a r angles d e f i n e d i n FIGURE 5, and P^vJDdO- i s the c o n d i t i o n a l p r o b a b i l i t y of f i n d i n g a p l a t e l e t w i t h i n d£l when the volume i s v. Case 2. E p e r p e n d i c u l a r to B - d I / l d t = / / v [ a 3 c o s a f sin 2 Q+a, (sin^p + cos?p cos 29) ]P, (v,Q)P 2 (v)dQdv (2.4) The i n t e r a c t i o n energy can be w r i t t e n as H = - V B 2 [ X l + ( x 3 - X i ) c o s 2 0 ] / > U (2.5) Thus we have P, (vP)=exp(-H/kT)//exp(H/kT)dO. (2.6) where Xi and x3 are the magnetic s u s c e p t i b i l i t i e s p a r a l l e l and p e r p e n d i c u l a r to the plane r e s p e c t i v e l y . For the s i z e d i s t r i b u t i o n we choose P 2(v)=exp(-v) (v>0) (2.7) for the f o l l o w i n g reasons. I t t r u n c a t e s at zero and has a t a i l towards v=°°, and i t s standard d e v i a t i o n i s i n reasonable agreement with the experimental o b s e r v a t i o n s [ 4 ] that s i z e ranges from about 50nm to 300nm. I t i s simple mathematically. And most important of a l l i s that i t f i t s the experiment curve b e t t e r than the other d i s t r i b u t i o n s such as normal and vexp(-v) 7 d i s t r i b u t i o n s when we use a l e a s t square f i t method. I n t e g r a t i n g (2.3) and (2.4) over dQ-,dv and dt giv e s I 3 o / l 3 = e x p { [ a , - ( a , - a 3 ) <vcos 2B>]t} (2.8) I 1 0 / I i = e x p { 0 . 5 [ a 3 + a 1 + ( a 1 -a 3)<vcos 2 9>]t} (2.9) <vcos29>=J"vcos2Qp,P2dQdv (2.10) where I 3 and I, are the a b s o r p t i o n i n t e n s i t i e s with E p a r a l l e l and p e r p e n d i c u l a r to the magnetic f i e l d r e s p e c t i v e l y , and I 3 0 and I 1 0 are the corresponding i n t e n s i t i e s of the i n c i d e n t l i g h t . < > here means average over ft and v. (2.8) and (2,9) are the r e s u l t s of our model to be f i t t e d to the experiment data. B. FITTING THE MODEL TO THE EXPERIMENT If one chooses a,t,and a 3 t i n (2.8,9) and ( X 3 - X 1 ) as parameters to be f i t t e d to the experiment curve i n r e f e r e n c e 4, one i s able to p l o t 1 , 0 / 1 , and I 3 0 / l 3 as f u n t i o n s of B,as i s i l l u s t r a t e d i n FIGURE 4. The value of q u a n t i t i e s i n the f i t t i n g and the r e s u l t s of the three parameters are shown i n TABLE 2. Compared to the p u b l i s h e d data shown in TABLE 3, where we have converted the o r i g i n a l emu/g u n i t i n t o the cgs di m e n s i o n l e s s u n i t , one f i n d s that the a n i s o t r o p i e s of the s i n g l e l a y e r NbS 2 are a few times l a r g e r than those of many of the 3D TMD c r y s t a l s . T h i s value i s i n v e r s e l y p r o p o r t i o a l to the assumed mean value of the p l a t e l e t s . The value given i n TABLE 2 was suggested to us by C. L i u based on e l e c t r o m i c r o s c o p e o b s e r v a t i o n s [ 4 ] . 8 V ( M 3 / 1 0 2 3 ) 2 . 4 K T ( J / 1 0 2 1 ) 4 . 0 4 a,t 3 . 3 4 a 3 t 0 . 8 1 X 3 - X 1 ( c g s / 1 0 6 ) 2 6 Table II - Data of the f i t t i n g f o r NbS 2 NbSe 2 TaS 2 TaSe 2 X 3 * 8 . 8 8 . 4 9 X i * 4 . 8 4 . 2 3 . 6 * X ' s are a l l in c g s / 1 0 6 [ 1 4 ] [ 1 8 ] Table III - Experimental data of magnetic s u s c e p t i b i l i t y of TMD We comment here that probably because of l i g h t s c a t t e r i n g , there i s some descrepancy i n the r a t i o of the a b s o r p t i o n i n t e n s i t i e s ( s e e e q u a t i o n ( 2 . 1 2 ) ) . I f one f i t s each curve s e p a r a t e l y , one o b t a i n s s l i g h t l y d i f f e r e n t parameters, and the X 3 - X i i n TABLE 4 i s the average of the two f i t s . Although experiments are l a c k i n g f o r data of s i n g l e c r y s t a l NbS 2, we can s t i l l make the comparision because,from TABLE 3 . the a n i s o t r o p i c s u s c e p t i b i l i t i e s are about the same f o r a l l 2 H -9 TMD's and i n agreement with the c a l c u l a t i o n of McDonald and G e l d a r t [ l 4 ] . To see how the p l a t e l e t s order under the magnetic f i e l d we a l s o p l o t the order parameter[8],S S=0.5(3<cos28>-1) (2.11) vs the magnetic f i e l d i n FIGURE 4. From the p l o t one sees that the alignment of the p l a t e l e t s reaches i t s s a t u r a t i o n value at about 50 kGauss. To examine the v a l i d i t y of the theory, we p l o t R = l n ( I ( 0 ) / I , ) / l n ( I 3 / I ( 0 ) ) (2.12) in FIGURE 7 a c c o r d i n g to the d i g i t i z e d data from experiment curve i n FIGURE 6, where 1(0) i s the i n t e n s i t y when B=0. I t i s found that R i s c o l o u r dependent and not always equal to 2 as the theory p r e d i c t s (see (2.8,2.9)). The R value(=2.6) at the a b s o r p t i o n peak e x p l a i n s the d e v i a t i o n of theory from the experiment. We b e l i e v e that the l a r g e d e v i a t i o n of R from 2 i n f r e q u e n c i e s away from the a b s o r p t i o n peak i s due to l i g h t s c a t t e r i n g . In those f r e q u e n c i e s a b s o r p t i o n i n both p o l a r i z a t i o n s i s so small that the e x t i n t i o n of i n t e n s i t y i s dominated by s c a t t e r i n g l i g h t . At low f r e q u e n c i e s , s c a t t e r i n g i n the p o l a r i z t i o n p a r a l l e l to the plane of the p l a t e l e t s i s l a r g e r than that i n the other d i r e c t i o n , and the i n v e r s e s i t u a t i o n a p p l i e s to the high f r e q u e n c i e s . 10 I I I . FERMI SURFACES AND BAND STRUCTURES OF THE SINGLE LAYERS Among the 2H f a m i l y of TMD's, NbS 2 has r e c e i v e d l e s s a t t e n t i o n than the others p a r t l y because i t i s d i f f i c u l t to make good specimens, and p a r t l y because i t appears to be l e s s remarkable than the other member of the f a m i l y . CDW which has been observed i n the other compounds[l] at low temperatures i s i n h i b i t e d i n NbS 2 , perhaps t o t a l l y . As the f i r s t attempt to study the d i f f e r e n c e s in e l e c t r o n i c s t r u c t u r e between NbS 2 and the other compounds, Wexler and Woolly(1975)[9] c a l c u l a t e d the band s t r u c t u r e s of TaSe 2,TaS 2,NbSe 2 and NbS 2 using a l a y e r method and a s i m i l a r p o t e n t i a l to that of M a t h e i s s f 1 0 ] . They found some d i f f e r e n c e s which seemed to be determined by the chalcogen r a t h e r than the metal. A. A TIGHT BINDING FIT TO THE BANDS OF THE 2H-TMD To c a l c u l a t e n u m e r i c a l l y x ( q ) , the e l e c t r i c s u s c e p t i b i l i t y which play a d e c i s i v e r o l e i n the occurrence of CDW, Doran e t . a l . [6,11] used a S l a t e r - K o s t e r t i g h t b i n d i n g f i t t i n g scheme[l2] to f i t t h e i r Hamiltonian to the band s t r u c t u r e c a l c u l a t i o n of Wexsler and W o o l l y [ 9 ] . To e s t a b l i s h n o t a t i o n s and f a c i l i t a t e comparision we d e s c r i b e below t h e i r r e s u l t s b r i e f l y . 2H-NbS2 (and 2H-NbSe2 ) has D^ space group symmetry. A u n i t c e l l i s shown in FIGURE 1(b). F o l l o w i n g the n o t a t i o n s of r e f . 6 , a Bloch sum over N l a t t i c e s i t e s i n the c r y s t a l i s given by 11 |+ >=N^exp[ik.(R+d)]|a(r-R-d)> ( 3 . 1 ) d i s a v e c t o r from the o r i g i n to atoms i n the u n i t c e l l , and the |a>'s are the symmetric and antisymmetric combination of the atomic o r b i t a l s belonging to the same sandwich. That i s |X_> = | x ,Vt+ei )> - ) xCSi+e;) > | *-:> = | x'.s i-re-0>+ | xCsHe 6 ) > e t c . ( 3 . 2 ) In t h i s way the Bloch sums are d i v i d e d i n t o even and odd f u n t i o n s with respect to the plane of the sandwich. With t h i s c h o i c e of the b a s i s the Hamiltonian matrix can be w r i t t e n s c h e m a t i c a l l y as: Sandwich 1 2 Even Odd Even Odd Even A(6x6) 0 Ccos(Sz) E s i n ( S z ) 1 Odd 0 B(5x5) - E T s i n ( S z ) Dcos(Sz) Even C cos(Sz) -E s i n ( S z ) A 0 2 Odd E* Tsin(Sz) D*"sin(Sz) 0 B* where Sz=0.5cK (c=2(w+s)), and the matrix elements are expressed i n terms of the energy i n t e g r a l of S l a t e r and K o s t e r [ l 2 ] which 1 2 are i n t u r n w r i t t e n i n terms of the two-centre parameters. It i s these l a t t e r parameters which are used in the f i t t i n g p rocedure. The r e s u l t s of the f i t t i n g are given i n FIGURES 8 , 9 and 10, where the symmetry p o i n t s are i n d i c a t e d i n FIGURE 11. The band s t r u c t u r e i s c h a r a t e r i z e d by an i s o l a t e d d z 2 band between the upper d-bands and the lower p-bands and a Fermi s u r f a c e w e l l above the saddle p o i n t s [ 9 ] . B. BAND STRUCTURE OF SINGLE LAYERS To apply the t i g h t b i n d i n g f i t to the s i n g l e l a y e r compounds, we modify the f i t by s e t t i n g the i n t e r l a y e r i n t e r a c t i o n to z e r o , t h a t i s , by s e t t i n g C,D and E matrices, to zero. The m o d i f i e d bands and the Fermi s u r f a c e s are shown in FIGURES 12,13 ; 14,15 r e s p e c t i v e l y , Shown in FIGURE 16 are the d e n s i t i e s of s t a t e s of d z 2 bands which are c a l c u l a t e d by a 2D analog of the t e t r a h e d r o n method[13]. The d e t a i l e d formulas in the method are worked out i n Appendix C. In the c a l c u l a t i o n we have generated a g r i d with 1600 t r i a n g l e s i n an i r r e d u c i b l e B r i l l o u i n zone as shown i n FIGURE 17(a) . The atomic o r b i t a l s and t h e i r angular part i n one sandwich are numbered i n the way d e s c r i b e d i n TABLE 4. The main f e a t u r e s of the 2D and 3D bands are e s s e n t i a l l y the same except a 0.5 eV i n c r e a s e i n the band gap from p/d to d z 2 which w i l l a f f e c t the interband t r a n s i t i o n s as w i l l be shown in chapter 5. One a l s o notes that the energy d i s p e r s i o n s along MK and ALHA are e x a c t l y the same as they should. Moreover, FIGURE 16 shows that the r e d u c t i o n of the t h i r d dimension g i v e s 13 r i s e to high d e n s i t y of s t a t e s , e s p e c i a l l y i n NbS 2. The reason why the d e n s i t y of s t a t e s of s i n g l e l a y e r s of NbSe 2 i s lower than that of NbS 2 i s that the saddle p o i n t s are l e s s pronounced i n NbSe 2 as can be seen from FIGURES 14 and 15. From F i g u r e 14 , one sees that besides the w e l l known saddle p o i n t between TP, a new saddle p o i n t a r i s e s between QP as a consequence of the 2D model, which leads to the peak around the Fermi energy. Mathematically,- due to the saddle p o i n t s the d e n s i t y of s t a t e s w i l l d i v e r g e l o g a r i t h m i c a l l y , but i n computations one h a r d l y get any divergence due to the l i m i t set by the computational method . Th e r e f o r e one must be c a r e f u l i f the Fermi s u r f a c e l i e s at the saddle p o i n t . even odd o r b i t a l s n umber- ang. f n o r b i t a l s number ang. fn 3 z 2 - r 2 1 xz 1 (Yi+Y^)i2 x 2 -y 2 2 (t + ? 2 )/2 yz 2 (Yl-€! )i2v xy 3 x- 3 (Y,' +Y~' ) i 2 x + 4 ( Y| +y",! )/a y- 4 (Y| -Y",' )xf2V y + 5 (Y; -y-; JXE; z + 5 0 Yi z- 6 Y° Table IV - Atomic o r b i t a l s used i n the t i g h t b i n d i n g f i t 1 4 IV. MAGNETIC SUSCEPTIBILITY OF SINGLE LAYER COMPOUNDS Although energy band s t r u c t u r e s of m e t a l l i c TMD's have been c a l c u l a t e d and s t u d i e d by many authors[1,7,6,9,10], the magnetic s u s c e p t i b i 1 i t y , x , was seldom c a l c u l a t e d by theory due to the complicated e l e c t r o n i c s t r u c t u r e of the compounds[14]. To e s t i m a t e the a n i s o t r o p i c magnetic s u s c e p t i b i l i t y of the s i n g l e l a y e r s which i s measured i n d i r e c t l y from the o p t i c a l a b s o r p t i o n experiment (as was shown in chapter 2.), we adopt the 2D t i g h t -b i n d i n g Hamiltonian as was shown i n l a s t c h a p t e r . In c o n s i d e r i n g the s u s c e p t i b i l i t y based on the 2D model, we f o l l o w the development of Misra and K l e i m a n [ 1 5 ] ( f o r the a p p l i c a t i o n of the theory i n our case see Appendix A ) . Because of the high d e n s i t y of s t a t e s , we expect the P a u l i s u s c e p t i b i l i t y to dominate and ignore c o n t r i b u t i o n s from other sources, and w r i t e where E n i s the Bloch eigenvalue i n the absence of the f i e l d , OC i s i n the d i r e t i o n of the f i e l d , and a. (4.1 ) 15 (gn*J 2 = 4 ( | <n , k , + \JHd\n , k , +> | 2 + | <n , k , + \M*\n > "> I 2 > (4.2) where |n,k, + (-)> i s one of the Kramer degenerate s t a t e s with wave v e c t o r k and band l a b e l n and JU. i s the p e r i o d i c p a r t of the magnetic moment o p e r a t o r [ 1 5 ] . In the t i g h t b i n d i n g l i m i t , the s i m p l e s t appproximation i s to w r i t e ^ ^ s l ^ l n ^ s ' ^ n ^ l L s J n ' ,k><51,+ & <G| <J"J<5> (4.3) where <n,k||n',k> i s the matrix element of the o r b i t a l angular momentum operator between the angular p a r t s of the t i g h t - b i n d i n g b a s i s f u n t i o n s and <6\6<x\6> i s the P a u l i s p i n matrix element(see Appendix B) . In w r i t i n g (4.2), we n e g l e t e d the interband matrix element due to the l a r g e band gap between d z 2 band and other bands. The above aproximation method i s i n the same s p i r i t as McDonald and G e l d a r t [ l 4 ] . In e v a l u a t i n g (4.2), we have taken the advantage of the 2D nature of the e l e c t r o n i c s t r u c t u r e by using a 2D t e t r a h e d r o n method mentioned in chapter 3. The e i g e n v a l u e s and e i g e n v e t o r s of the t i g h t b i n d i n g Hamiltonian are obtained by using standard matrix techniques(by c a l l i n g a s u b r o u t i n e EIGZC from the IMSL l i b r a r y ) to d i a g o n a l i z e the Hamiltonian matrix. (4.2) i s then e v a l u a t e d at each k p o i n t i n our B r i l l o u i n zone g r i d (see FIGURE 17.) by u s i n g (4.3) and the wave f u n t i o n obtained i n the program. We f i n d t h a t <g x 2>/<g * 2 > ~ 4 ( 4 . 4 ) 16 where < > means average over the B r i l l o u n zone. We i n t e r p r e t t h i s r e s u l t i n a simple model as f o l l o w s . F i r s t c o n s i d e r the l i g a n d f i e l d model by s e t t i n g H , the p e r i o d i c p o t e n t i a l to zero. Then the s p l i t t i n g of d - o r b i t a l s under D 3j, p o i n t group i s s i m p l l y d z 2 , ( x 2 - y 2 , x y ) , and (zx,yz) i n ascending o r d e r f l O ] , Since the Fermi s u r f a c e l i e s i n the second l e v e l , we have a s t a t e |2,±2>. When a magnetic f i e l d i s a p p l i e d to the system, the s p l i t t i n g of |2,±2> g i v e s g =2<2 , t2 | Lz | 2 , -h2> + 2 = 6 g =2<2,+2|Lx|2,+2>+2=2 (4.5) When the p e r i o d i c p o t e n t i a l i s turned on, the int r a a t o m i c i n t e r a c t i o n s quench the o r b i t a l c o n t r i b u t i o n s to g and we get (4.4) The r e s u l t s obtained f o r x(T=0) and D(E), the d e n s i t y of s t a t e s are shown i n TABLE 5. C o n s i d e r i n g the u n c e r t a i n t y i n the p a r t i c a l s i z e d i s t r i b u t i o n , the agreement between theory and experiment i s q u i t e s a t i s f a c t o r y . One a l s o notes t h a t the d e n s i t y of s t a t e s i n the 2D model i s c o n s i d e r a b l l y l a r g e r than i n 3D (see FIGURE 16) mainly because i n 2D there i s no d i s p e r s i o n of energy i n the z d i r e c t i o n . 1 7 NbS 2 | NbSe 2 D* 35 | 22 X! 4 .9 | 3 .2 X 3 24 I 1 6 X 3 - X 1 19 X 3 - x 7 P 2 6 I 1 3 *In the t a b l e the u n i t of D i s 1 / ( R y d . N b . s p i n ) ; the u n i t of x i s c g s / 1 0 6 . T a b l e V - Magnet ic s u s c e p t i b i l i t i e s from the theory 18 V. OPTICAL PROPERTIES OF THE SINGLE LAYER COMPOUNDS To understand the a b s o r p t i o n s p e c t r a a r i s i n g from the int e r b a n d t r a n s i t i o n s one needs to c a l c u l a t e the j o i n t d e n s i t y of s t a t e s and the o s c i l l a t o r s t r e n g t h . In the case of s i n g l e l a y e r s of NbS 2 and NbSe 2, the t i g h t b i n d i n g matrix block d i a g o n a l i z e s and the energy bands separate i n t o 6 and TV bands as i n d i c a t e d i n FIGURE 18 and 19 r e s p e c t i v e l y . For t r a n s i t i o n s below 5 eV, we only c o n s i d e r the t r a n s i t i o n s from p/d bands to d z 2 band because t r a n s i t i o n s from d - o r b i t a l s to d - o r b i t a l s are fo r b i d e n and d/p bands are dominated by d - o r b i t a l s ( t h e evidence of which i s shown i n FIGURE 20). We b e l i e v e that the f o l l o w i n g c a l c u l a t i o n i s the f i r s t attempt to i n c l u d e both the p o l a r i z a t i o n and the matrix element e f f e c t itfhen c o n s i d e r i n g o p t i c a l t r a n s i t i o n s i n a s o l i d . The o p t i c a l j o i n t d e n s i t y of s t a t e s f u n t i o n can be w r i t t e n as J ( E ) ~ ; d S / V ^ ( E v - E c ) (5.1) where E c and Es, are the energy e i g e n v a l u e s of the conduction band and valence band r e s p e c t i v e l y . Since we are c o n s i d e r i n g d i r e c t i n t e r b a n d t r a n s i t i o n s , care has been taken to a v o i d counting t r a n s i t i o n s from p/d band to the occupied p o r t i o n of the d z 2 band. The r e s u l t s o b t a i n e d by the t e t r a h e d r o n method mentioned e a r l i e r f o r (v-to-cf and TT-to-lT t r a n s i t i o n s are shown i n FIGURES 21 and 22 r e s p e c t i v e l y . From FIGURE 21, one notes that there i s no interband 19 t r a n s i t i o n below 2.5 eV which acounts f o r the r e d u c t i o n of a b s o r p t i o n i n low frequency region (see FIGURE 2). One a l s o f i n d s t hat both ( r t o - 6 and TP-to-IT t r a n s i t i o n s have a peak at 2.7 eV as was observed i n the experiment(see FIGURE 2.) . But the magnitude of d e n s i t y f u n t i o n f o r TT-to-'J t r a n s i t i o n i s about twice as l a r g e as that f o r <5-to-e5 t r a n s i t i o n , while the experiment gaves a 1=4a 3(see r e f . 4 and TABLE 5). To account f o r the experiment r e s u l t s one needs to c o n s i d e r the o s c i l l a t o r s t r e n g t h or the t r a n s i t i o n matrix Fcv=2mtLv| r c v | 2/H=27\ \ v„J 2/mEy- (5.2) where ^• = <^|r|\|/.> v,c=<VvH>c> ( 5 ' 3 ) and |V, ,>=£v; | j, > l> c>=£c; | i > (5.4) i a re the wave f u n t i o n s of the valence and conduction bands r e s p e c t i v e l y with | i- > being the b l o c h sum f o r the i t h atomic o r b i t a l . We c o n s i d e r t r a n s i t i o n s from 6 to <S f i r s t . In order to evaluate <r>\/v\r |'Mt> i n the t i g h t b i n d i n g l i m i t , one needs to c a l c u l a t e terms l i k e <Y*($,hq( | r-R | ) |r|Y'J\9,f )f ( r ) > e x p { i P 1 ( k , R ) - i P 2 ( k , R ) } (5.5) 20 where {) and ^  are f u n t i o n s of "0 , and "r , and exp{iP, (it ,R)} and exp{iP 2(k,R)} are the phase f a c t o r s that give the c o r r e c t l i n e a r combinations of atomic o r b i t a l s when combined with Vt*'s and C£'s r e s p e c t i v e l y . Since a k and d dependent f a c t o r i s i n c l u d e d i n the o r i g i n a l Bloch sums[6], the f u n t i o n P(k,d)=P,-P 2 i s r a t h e r complicated. We make the f o l l o w i n g approximations f o r the z component. <YJ( e,f')g( |f-R| ) |cos0r |Y*(0/f )f (r)>exp{iP(k,R)} 7 * < Y " < ( 9 , V |cos0rO|Y*(fl/?)><g( |r-R| ) | f (r )>exp{ iP(k ,R)} **<Y"'( 0,f ) IcosOrVt^'f >><9< I r-R | ) | f ( r ) >exp{ iP( k ,R)} (5,6) and s i m i l a r l y f o r the x and y componet, where r s i s some constant of the order of the l a t t i c e spacing, and the second step of the approximation t r e a t s the a n g l a r p a r t s of the atomic o r b i t a l s as i f they were on the same s i t e . A f t e r s u b s t i t u t i n g (5.6) i n t o ( 5 . 3 ) , we make the f u r t h e r approximation: <g(|r-R|)| f(f)>exp{iP(k,R)} ^S,-(1c) = <Y^'((7\f)g( | r-R | ) |Y*(0,<f )f (r)>exp{iP(k,R)} (5.7) where Sjj's are the o v e r l a p matrix elements generated from the computer program. T h i s s t e p i s needed because S^-OO's take care of the phase f a c t o r . The above three steps of approximation enable one to make use of the V/'s.Cf's and S-:(k)'s from the 21 .program and of the well-known s e l e c t i o n r u l e s in atomic p h y s i c s . Then = v f ( k ) C : (k)Tr;S-;(k) (5.8) where T i j ' s a r e fc^e r matrix element between the angl a r part of the atomic o r b i t a l s ( t h e atomic t r a n s i t i o n m a t r i c e s ) as worked out in Appendix D. In the case of 7* to 6 t r a n s i t i o n S-'-s are zero, and one has 'J to modify the above approximation method. In Appendix D. we note that there are only three nonzero T,-j's and only one of them i s l a r g e , i . e . ,corresponding to t r a n s i t i o n from p to d-o r b i t a l s . Therefore the phase f a c t o r exp{iP} i s not important and one can s t i l l use the nonzero Sjj's corresponding to the (S-t o - 6 t r a n s i t i o n to estimate the TT-to- (T t r a n s i t i o n . Averaging (5.8) over the B r i l l o u i n zone g i v e s |<^|X|^ >|2/|<>v|i|1c>|2 = 6.9 (5.9) Again care has been taken when averaging (5.8) to a v o i d counting occupied s t a t e s i n d z 2 band. At the a b s o r p t i o n peak EVc a1/a3=|<^|y|^C>|2J(EVC)/|<S'k,|z|%>|2J(EvJ = 6.9/2. 2 ^ 3 ( 5 . 1 0 ) 22 which i s i n q u a l i t a t i v e agreement with experiment. Using the same program, we c a l c u l a t e the corresponding q u a n t i t i e s f o r NbSe 2: a,/a 3 = l 5 / l .8^8 (5.11) In order to compare the r e s u l t s with experiment, we p l o t the c a l c u l a t e d i n t e n s i t i e s as f u n c t i o n s of the wavelength i n FIGURE 6, assuming that the matrix elements are frequency-independent i n the region under c o n s i d e r a t i o n . To conclude we make the f o l l o w i n g p o i n t s . 1) A s s i g n i n g the o p t i c a l experiment data to interband t r a n s i t i o n s has been a c o n t r o v e r s i a l subject f o r the TMD's. D i f f e r e n t authors o b t a i n d i f f e r e n t r e s u l t s based on d i f f e r e n t experiment or model[20,21,22], and l i t t l e c o n s i s t e n c y has been achieved ,even w i t h i n the r e s u l t s of the same author. 2) Bechthold e t . a l . [ 2 0 ] have assigned the t r a n s i t i o n at 2.leV to d z 2 - t o - d / p t r a n s i t i o n . But we b e l i e v e that t h i s should be the p / d - t o - d z 2 t r a n s i t i o n based on our c a l c u l a t i o n and the f a c t that t h i s t r a n s i t i o n peak i s st r o n g i n low f r e q u e n c i e s and i t s p o s i t i o n remains approximately unchanged a f t e r i n t e r c a l a t i o n . 3) U s u a l l y people take i t f o r granted that a l l t r a n s i t i o n matrix elements are approximately equal [21]. I t turns out ,however,that when p o l a r i z a t i o n i s i n c l u d e d , the r e l a t i v e magnitude of d i f f e r e n t matrix elements becomes important. In the case of s i n g l e l a y e r NbS 2 the matrix element of 6-23 to-0' t r a n s i t i o n i s about 7 times that of *n"-to-<3* t r a n s i t i o n . P h y s i c a l l y t h i s means that the d i p o l e moment p a r a l l e l to l a y e r i s about 7 times that p e r p e n d i c u l a r to the l a y e r . If one ignores the matrix element e f f e c t , he w i l l not be a b l e to understand why the <5-to-<5 t r a n s i t i o n i s stronger than TT-to-<5 t r a n s i t i o n based only on the j o i n t d e n s i t y of s t a t e s . 4) People have been a p p l y i n g the atomic s e l e c t i o n r u l e s i n c o n s i d e r i n g the i n t e r b a n d t r a n s i t i o n s i n TMD's[4,22] which i s e q u i v a l e n t to the one center approximation we made i n (5.6). But we b e l i e v e that i t i s not w e l l j u s t i f i e d a p r i o r i . Thus the matrix elements we c a l c u l a t e d i s a crude e s t i m a t i o n and the r e l a t i v e magnitude of the t h e o r e c t i c a l curves i n FIGURE 6 i s not very r e l i a b l e . 24 V I . CONCLUSIONS The s t a t i s t i c a l behavior of the s i n g l e l a y e r p l a t e l e t s suspended in water i s understood in terms of a simple model of m a g n e t i c a l l y a n i s o t r o p i c s i n g l e l a y e r s . The q u a l i t a t i v e agreement of t h i s work with experiments supports the o r i g i n a l band c a l c u l a t i o n of Wexler and W o o l l e y [ 9 ] . Moreover the r e d u c t i o n of d i m e n s i o n a l i t y can be acounted f o r by a t i g h t b i n d i n g model with zero i n t e r l a y e r i n t e r a c t i o n . More work i s needed to study the s i z e d i s t r i b u t i o n and l i g h t s c a t t e r i n g and to c a l c u l a t e the i n t e r b a n d matrix elements more a c c u r a t e l y . We hope t h i s work w i l l motivate the p r e p a r a t i o n of more s i n g l e l a y e r TMD's and i n i t i a t e more s o p h i s t i c a t e d t h e o r i e s . 25 BIBLIOGRAPHY 1] J.A.Wilson, F . J . d i S a l v o , and S. Mahajan, Adv. Phys. 24 , 117(1975) 2] S.K. Chan, and V. Heine, J.Phys. F 3 , 795(1973) 3] J.M.Rice and G.K.Scott, Phys. Rev. L e t t . 35, 120(1975) 4] C . L i u , O.Singh, P.Joesen, A.E.Curzon and R . F . F r i n d t , Thin S o l i d F i l m s . JMJ3 , 165(1984); See a l s o C. L i u , M.Sc. T h e s i s , Simon F r a s e r U n i v e r s i t y (1982) 5] J . T e r s o f f and L.M.Falicov, Phys. Rev. B26, 6186(1982) 6] N.J.Doran, B.Ricco, D . J . T i t t e r i n g t o n and G.Wexler,J.Phys.C, 1 1 , 685(1978) 7] J.A.Wilson and A.D.Yoffee, Adv. Phys. J_8, 1 93(1969) 8] W.H.de Jeu, P h y s i c a l P r o p e r t i e s of L i q u i d C r y s t a l l i n e M a t e r i a l s , New York: Gordon and Breach(l980) 9] G.Wexler and A.M.Woolly, J . Phys.C, 9, 1185(1976) 10 1 1 12 1 3 1 4 1 5 16 1 7 18 19 20 21 L.F.Matheiss, Phys. Rev. B5, 290(1972) N.J.Doran, B.Ricco, M.Schreiber, D . J . T i t t e r i n g t o n and G.Wexler, J . Phys. C, J j , 699(1978) J . C . S l a t e r and G.F.Koster, Phys. Rev. 58, 1498(1954) A.H.McDonald, S.H.Vosko and P.T.Coleridge, J . Phys. C, J_2 , 2991(1979) A.H.McDonald and D.J.W.Geldart, Phys. Rev. B24 ,469(1981) P.k.Misra and L.Kleiman, Phys. Rev. B5 ,4581(1972) Y . Y a f e t , S o l i d S t a t e Phys. J_4, 1 (1963) D.W. Murphy and G.W.Hull, J r . , J.Chem. Phys. 62 , 173(1975). S . J . H i l l e n i u s and R.V.Coleman, Phys.Rev. B18 ,3790(1978) C . L i u and R.F. F r i n d t ( 1 9 8 4 ) , to be b u p l i s h e d . P.S.Bechthold, R.Manzke and R.Schollhorn, J.Physique,Colloque C6 , 159(1984) W.Y.Liang and A.R.Beal, J.Phys.C, 9 , 2823(1976) 26 [22] S . S . P . P a r k i n and A . R . B e a l , P h i l . Mag. 42^  , No.5, 627(1980) 27 APPENDIX A - FORMULATION OF THE MAGNETIC SUSCEPTBILITY The theory of Misra and Kleinman[l5] g i v e s X - 0 . 5 g 2 l J f ( E n ) (A. 1 ) where f(En) i s the Fermi f u n t i o n , and En the energy, and where i s the d i r e c t i o n c o s i n e of the magnetic f i e l d B and S a= ( 1/2) <5e i s the component of the P a u l i matrice which i s f i x e d with respect to r o t a t i o n of the c o o r d i n a t e system. In our problem, we choose the c o o r d i n a t e system such that the components of the magnetic moment can be expressed i n terms of the the corresponding c o e f f i c i e n t s of the P a u l i m a t r i c e s , and the G's are d e f i n e d as f o l l o w s [ l 6 ] T P (A.2) (A.3) - GfcX = 2Re<YnfcflyUj4.fk*> (A.4) - G*z = 2<V^|>u|4vif> g 2 ( k ) =GowGooc+G<xyGtxy+GoizGcxz = 4 | <n+ |XJ n+> | 2 + 4 | <n+ \M*\n + > | 2 (A.5) f o r B along the <X d i r e c t i o n ( A . 5 T i s j u s t the d e s i r e d e q u a t i o n . 28 APPENDIX B - EVALUATION OF MATRIX ELEMENT OF TNE ORBITAL ANGLAR MOMENTUM From the t i g h t b i n d i n g f i t , one has ,using the same n o t a t i o n as (5.4), |t>= C i | i > (B.2) and we need to c o n s i d e r d z 2 band only, as a f i r s t approximation, One e a s i l y shows that Lz L 2 Lz 3> = 4 5> = 2 i> = 0 3> 3> (B.3) (i^3,5) T h e r e f o r e <>|Lz|1'>/<Y!f>=Z(4Ct'C3Sj3+Cl-C5Sl-5)/Zcl-Cj Sg (B.4) where S { j i s the o v e r l a p i n t e r g r a l , OHLX[H->=O Because L^|i>=|odd> 'J And (B.5) (B.4) and (B.5), together with C i ' s and Sy's from the program, enable us to ev a l u a t e g 2 over the the B r i l l o u i n zone. 29 APPENDIX C - TETRAHEDRON METHOD IN TWO DIMENSION The purpose i s to ev a l u a t e the f o l l o w i n g i n two dimension!13]: J(E)=/dkF(k )0(E-E(k)) I(E)=fdkF(k)£(E-E(k)) (C.1) where I(E)=dJ(E)/dE i s the e l e c t r o n i c d e n s i t y of s t a t e s . We d i v i d e the B r i l l o u i n zone i n t o t r i a n g l e s ( m i c r o z o n e s ) as was shown in FIGURE 17(a), and we are ab l e to w r i t e J ( E ) « V H I . V | j;V I ( E ) = V M £ 3i"I I» ?i ( C 2 ) where i s the volume of the microzone, F K i s the F value at co r n e r K of the i t h microzone and E ^ s have been arranged i n ascending order. We have the folowing cases(see FIGURE 17(b) fo r one p a r t i c u l a r zone as an example) ( 1 ) E<Ef g = 0 (2) E | < E<El D e f i n e f m= ( E - E M ) / ( E n -Em) n' = f i ( f 31 J, =(1+f ( l +f , 3 ) / 3 J A = f i , /3 J 3 = f 3 » / 3 g l=2n l/(E-E,) i', = ( f a + f I5 )/2 I ^ f K l / 2 (K=2,3) (3) Ea<.E-cE3 nl = d - f t 5 f .23) J,-(1-£|3 f ^ ) / 3 n l J2.= d - & f 13 )/3n l J 3 = [ l - f ( 3 f 2 3 < 1 + f 3 2 f.3| )]/3n' 30 g l = 2 ( 1 - n l ) / ( E 3 - E ) 1 ^ 3 / 2 ( K - 1 , 2 ) (4) E 3<E g = 0 the above equations enable us to e v a l u a t e (4.1). 31 APPENDIX D - EVALUATION OF THE ATOMIC TRANSITION MATRIX From TABLE 4. and the f o l l o w i n g equations sin Y =j Jti-*«Kt*ito™-\ f j j t ^ . g t ^ Y " 1 ' ' } e (D . 1) we e a s i l y w r i t e o - 1 13b zY 2 = ( i T r Y + zY 1 ? Y )r t 0 7 ? Y 0 ) r etc . Th T (D .2 ) e r e f o r e there are three nonvanishing T;;'s ?,= i % T ^ r J ^ (D.3) S i m i l a r l y we have f o r nonvanishing T *j ' s{ T*= -CCl^ Cn I X j &*^-r»> J) / 4 T T 2 x r Vii? T,V- I/LJS" T3V Vii? (D .4 ) FIGURE CAPTIONS 33 (a) o ® o ® (b) (c) F i g u r e 1 - S t r u c t u r e of 2H-TMD (a) S t r u c t u r e of 2H-TMD(0 c h a l c o g e n s , © m e t a l ) ; (b) The Un i t C e l l of NbS 2; (c) The St r u c u r e of NbS 2 l o o k i n g down the c axis(® S atoms i n sandwich 1 ,0 S atoms i n sandwich 2 ) . 34 300 .00 • 00 WAVELENGTH (ntr.) F i g u r e 2 - O p t i c a l A b s o r p t i o n S p e c t r a of NbS 2" ( — ) S p e t r a of s i n g l e l a y e r s ; ( ) S p e c t r a of s i n g l e c r y s t a l . •Taken from r e f . 4 35 (L 1 £> Sample ceil F i g u r e 3 - The Aligment of NbS 2 P l a t e l e t s i n Magnectic F i e l d k i s the wave v e c t o r of the i n c i d e n t l i g h t ; B i s the a p p l i e d magnetic f i e l d 36 KGauss (b) F i g u r e 4 - F i t t i n g of the Model to Experiment (a) F i t i n g I 0 / l to experiment i n r e f . 4 . (b) The order parameter F i g u r e 5 - Coordinates of the S i n g l e Layer 38 F i g u r e 6 - A b s o r p t i o n s p e c t r a i n Magnetic F i e l d * A,X are d i g i t i z e d data p o i n t s from r e f . 4 , 1 9 39 F i g u r e 7 - R a t i o of I n t e n s i t i e s of D i f f e r e n t P o l a r i z t i o n s 40 F i g u r e 8 - Band S t r u c t u r e of 2H-NbS 2 41 F i g u r e 9 - Band S t r u c t u r e of 2H-NbSe 2 42 F i g u r e 11 - Symmetry P o i n t s i n the B r i l l o u i n Zone 43 44 45 F i g u r e 14 - Fermi Surface of S i n g l e Layer NbS 2 Fermi Energy=-0.64 Ryd 46 F i g u r e 15 - Fermi Surface of S i n g l e Layer NbSe 2 Fermi Energy=-0.635 Ryd 4 7 60-, - » 1 1 • — i 1 r-» a. -0.S8 -0.66 -0.€4 -0.62 -0.60 -0.58 -0.5S Ryd (b) F i g u r e 16 - D e n s i t y of S t a t e s ( d Z 2 ) of the S i n g l e Layers (a) DOS of NbS 2; (b) DOS of NbSe 2, i n u n i t of l/(Nb.Spin.Ryd) 4 8 (b) F i g u r e 17 - Diagram I l l u s t r a t i n g the Tetrahedron Method (a) B r i l l o u i n zone g r i d ; (b) Diagram of the four c ases. 49 F i g u r e 18 (*r) «S) - S e p a r a t i o n of Bands of the S i n g l e Layer NbS 2 50 n o iff) F i g u r e 19 - S e p a r a t i o n of Bands of the S i n g l e Layer NbSe 2 5 1 F i g u r e 20 - A m p l i t i t u d e C o e f f i c i e n t s of the Wave Fu n t i o n f o r d z 2 Band 52 F i g u r e 21 - J o i n t D e n s i t y of S t a t e s of S i n g l e Layer NbS 2 (a) Tr to 5 t r a n s t i o n s ; (b) <T t o <5 t r a n s t i o n s . 53 2 0 - , Ryd (b) F i g u r e 22 - J o i n t D e n s i t y of S t a t e s of S i n g l e Layer NbSe 2 (a) TT to t r a n s t i o n s ; (b) 6 to 6 t r a n s t i o n s . 

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