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Phonon broadening of spectral lines in impurity doped solids Rystephanick, Raymond Gary 1965

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The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY B. Sc. (Hons.), University of Manitoba, 1962 M-.Sc.j University of Manitoba^ 1963 FRIDAY, APRIL 23;, 1965 AT 10:30 A.M. IN ROOM 301,, HENNINGS (PHYSICS) of RAYMOND GARY RYSTE PHANICK COMMITTEE IN CHARGE Chairman,' Dean W.H. Gage R. Barr.ie L. G. de Sobrino R. A. Restrepo H. Schmidt C. F. Schwerdtfeger R. F. Snider External Examiner: A. A. Maradudin Westinghouse Research Laboratories Pittsburgh, Pennsylvania PHONON BROADENING OF SPECTRAL LINES IN IMPURITY".DOPED SOLIDS ABSTRACT The theory of the phonon broadening of o p t i c a l absorption l i n e s i n impurity"doped semiconductors i s discussed s t a r t i n g with Kubo's formulation of the adiabatic d i e l e c t r i c , s u s c e p t i b i l i t y . The absorption constant i s r e l a t e d to : a two p a r t i c l e temperature Green's function using Matsubafa's method. The i n t e r -action of the electrons (or holes) with the. l a t t i c e v i b r a t i o n s i s assumed to be. small so that, i t can be treated as a perturbation to the independent electron and l a t t i c e v i b r a t i o n systems. Using, t h i s approxima-t i o n , Dyson's equations; are obtained,- using the" technique of Feynman diagrams, f o r the one p a r t i c l e electron and phonon Green's functions and for the two part:dele Green's function and these are solved within the framework of the, approximation. The l i n e shape function.is c a l c u l a t e d e x p l i c i t l y and i t i s found that the l i n e consists of a sharp peak whose width depends on temperature and i s due to the f i n i t e l i f e t i m e of the bound c a r r i e r states, and of a continuous back-ground due to:multi™phonon processes which accompany the o p t i c a l absorption. • The technique used here i s compared with previous work on t h i s problem, employing temperature dependent doublestime advanced and retarded Green's functions. • The r e s u l t s obtained are compared with those obtained by previous" authors. GRADUATE STUDIES F i e l d of Study: Physics E l ementary Quantum Mechanic.s Electromagmetic Theory Special Theory of R e l a t i v i t y Statistical Mechanics Advanced Quantum Mechanics F.A. Kaempffer - G. M. ..Vol'koff H. Schmidt : R. B a m e. F. A,. Kaempffer PHONON BROADENING OF SPECTRAL LINES, IN IMPURITY DOPED SOLIDS by RAYMOND GARY RYSTEPHAN I CKr B.Sc.(Hons.), U n i v e r s i t y o f M a n i t o b a , 1962 M.Sc., U n i v e r s i t y of M a n i t o b a , 1 9 6 3 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the department of PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1965 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely, available for reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia, Vancouver 8, Canada. Date i i " ABSTRACT The t h e o r y of the phonon b r o a d e n i n g of o p t i c a l a b s o r p t i o n l i n e s i n i m p u r i t y doped s e m i c o n d u c t o r s i s d i s c u s s e d s t a r t i n g w i t h Kubo's f o r m u l a t i o n of the a d i a b a t i c d i e l e c t r i c s u s c e p t i b i l i t y . The a b s o r p t i o n c o n s t a n t i s r e l a t e d t o a two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n u s i n g Matsubara's method. The i n t e r a c t i o n of t h e e l e c t r o n s (or h o l e s ) w i t h the l a t t i c e v i b -r a t i o n s i s assumed t o be s m a l l so t h a t i t can be t r e a t e d as a p e r t u r b a t i o n t o t h e independent e l e c t r o n and l a t t i c e v i b r a t i o n systems. U s i n g t h i s app-r o x i m a t i o n , Dyson e q u a t i o n s a r e o b t a i n e d , u s i n g the t e c h n i q u e of Feynman di a g r a m s , f o r t h e one p a r t i c l e e l e c t r o n and phonon Green's f u n c t i o n s and f o r the two p a r t i c l e Green's f u n c t i o n and t h e s e a r e s o l v e d w i t h i n the framework of the a p p r o x i m a t i o n . The l i n e shape f u n c t i o n i s c a l c u l a t e d e x p l i c i t l y and i t i s found t h a t the l i n e c o n s i s t s of a sharp peak whose w i d t h depends on t e m p e r a t u r e and i s due t o the f i n i t e l i f e t i m e of the bound c a r r i e r s t a t e s , and of a c o n t i n u o u s background due t o mu1ti-phonon p r o c e s s e s which accompany the o p t i c a l a b s o r p t i o n . The t e c h n i q u e used here i s compared w i t h p r e v i o u s work on t h i s p roblem e m p l o y i n g t e m p e r a t u r e dependent d o u b l e - t i m e advanced and r e t a r d e d Green's f u n c t i o n s . The r e s u l t s o b t a i n e d a r e compared w i t h t h o s e o b t a i n e d by p r e v i o u s a u t h o r s . - i i i -TABLE OF CONTENTS Page A b s t r a c t i i T a b l e o f Contents i i i Acknowledgements ... i v Chapter I: I n t r o d u c t i o n ... ... ... I Chapter 1 I : Genera 1 Forma 1i sm ... ... 5 Chapter I I ! : L i n e Shape C a l c u l a t i o n i n Z e r o t h Order A p p r o x i m a t i o n ... ... 18 A. C a l c u l a t i o n of two p a r t i c l e Green's f u n c t i o n ... 18 B. C a l c u l a t i o n of one p a r t i c l e e l e c t r o n Green 1 s f u n c t ion ... ... 2 1 C. C a l c u l a t i o n of l i n e shape f u n c t i o n 23 Chapter !V: L i n e Shape C a l c u l a t i o n w i t h i n t e r a c t i o n ... 2 7 A. I n t r o d u c t i o n ... 2 7 B. The diagrams f o r n=2 2 9 C. Higher o r d e r diagrams ... ... 35 D. Dysons e q u a t i o n s ... 37 E. C a l c u l a t i o n of one p a r t i c l e Green's f u n c t i o n s ... 4-3 F. C a l c u l a t i o n of l i n e shape f u n c t i o n ... ... ... 51 G. D i s c u s s i o n o f h i g h e r o r d e r c a l c u l a t i o n ... ... 65 Chapter V: Compari son. of Methods 68 Chapter V I : Summary ••• ••• •-• 77 A p p e n d i x I: J u s t i f i c a t i o n o f (4.E.29) ... 7 9 Bi b l i o g r a p h y 82 - i v -ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e g r a t i t u d e t o Dr. R. B a r r i e f o r s u g g e s t i n g the problem and f o r c o n t i n u o u s g u i d a n c e and a s s i s t a n c e t h r o u g h o u t the c o u r s e of t h i s work. The a u t h o r a l s o g r a t e f u 1 1 y acknpwledges t h e encouragement o f h i s w i f e and her h e l p i n the p r e p a r a t i o n of the t h e s i s . T h i s r e s e a r c h was s u p p o r t e d by the N a t i o n a l Research C o u n c i l of Canada i n t h e form o f two S t u d e n t s h i p s awarded t o t h e a u t h o r . ! CHAPTER I : INTRODUCTION The i n t r o d u c t i o n o f a s m a l l c o n c e n t r a t i o n o f Group H i or Group V i m p u r i t i e s i n t o a s e m i c o n d u c t o r such as germanium or s i l i c o n l e a d s t o a set of energy l e v e l s l o c a l i z e d a t the i m p u r i t y s i t e s . The r e s u l t i n g bound e l e c t r o n s and h o l e s , which l i e i n s t a t e s j u s t below the c o n d u c t i o n band and j u s t above the v a l e n c e band r e s p e c t i v e l y , a c t as c a r r i e r s o f e l e c t r i c charge when they a r e e x c i t e d from the i m p u r i t y s t a t e s t o the c o n d u c t i o n or v a l e n c e band. It has been shown (Kohn, 1957) t h a t the bound c a r r i e r s t a t e s can be r o u g h l y d e s c r i b e d by a h y d r o g e n i c model, b e i n g d e s c r i b e d by the same S c h r o d i n g e r e q u a t i o n as t h e s t a t e s of an e l e c t r o n i n a hydrogen atom, but w i t h the coulomb p o t e n t i a l m o d i f i e d by t h e d i e l e c t r i c medium. At low t e m p e r a t u r e s t h e bound c a r r i e r s a r e m o s t l y in the ground s t a t e and can be e x c i t e d t o one of t h e e x c i t e d s t a t e s by a b s o r p t i o n of photons. The l i n e spectrum of t h e s e o p t i c a l a b s o r p t i o n s o c c u r s i n the i n f r a r e d r e g i o n and has been o b s e r v e d by s e v e r a l a u t h o r s . The bound c a r r i e r s i n t e r a c t w i t h t h e l a t t i c e v i b r a t i o n s c a u s i n g a b r o a d e n i n g o f the l i n e s as w e l l as a s h i f t o f the peak p o s i t i o n o f the s p e c t r a l l i n e . Our aim i n t h i s work i s t o s t u d y t h e shape o f t h e s e s p e c t r a l l i n e s . . We assume t h a t the o n l y s o u r c e o f b r o a d e n i n g i s the i n t e r a c t i o n of the bound c a r r i e r s w i t h l o n g w a v e l e n g t h a c o u s t i c phonons. There a r e two mechanisms by which t h e e1ectron-phonon i n t e r a c t i o n can cause b r o a d e n i n g . One i s t h e " l i f e -t i m e " e f f e c t (Kane, I960) which i s due t o t h e f a c t t h a t because of the e1ectr o n - p h o n o n i n t e r a c t i o n the bound c a r r i e r s t a t e s have a f i n i t e l i f e t i m e and can decay i n t o o t h e r i m p u r i t y s t a t e s . Because of the quantum m e c h a n i c a l u n c e r t a i n t y r e l a t i o n t h i s f i n i t e l i f e t i m e g i v e s r i s e t o a sp r e a d i n energy. If one assumes the l i f e t i m e e f f e c t t o be the mechanism of b r o a d e n i n g , one 2. can c a l c u l a t e by means of t i m e dependent p e r t u r b a t i o n t h e o r y (Weisskopf and Wigner, 1930) t h e - l i n e shape i n the v i c i n i t y of t h e peak. One f i n d s i n t h i s c ase t h a t the w i d t h of t h e l i n e s i s d e t e r m i n e d p r i m a r i l y by the w i d t h o f the e x c i t e d l e v e l and hence d i f f e r e n t l i n e s have d i f f e r e n t w i d t h s . A n o t h e r p o s s i b l e mechanism of b r o a d e n i n g i s the f a c t t h a t the e x c i t a t i o n o f a bound c a r r i e r can be accompanied by the s i m u l t a n e o u s emmission or a b s o r p t i o n of phonons and i s c a l l e d a ,"mu11i-phonon 1 1 p r o c e s s . For m u l t i - p h o n o n p r o c e s s e s t h e w j d t h s a r e d e t e r m i n e d m a i n l y by the w i d t h of. the ground s t a t e and hence a l l l i n e s have n e a r l y the same w i d t h (Lax and B u r s t e i n , . 1955)• E a r l i e r work on t h e t h e o r y of phonon b r o a d e n i n g (Lax and B u r s t e i n , Kubo and Toyazawa,. 1955) was based on a model i n which t h e bound c a r r i e r s i n t e r a c t o n l y w i t h the long w a v e l e n g t h a c o u s t i c phonons and the i n t e r a c t i o n between bound c a r r i e r s i s i g n o r e d . Both s e t s of a u t h o r s worked w i t h i n the framework of t h e a d i a b a t i c a p p r o x i m a t i o n and used the Bethe-Sommerfeld f o r m u l a f o r the a b s o r p t i o n c o n s t a n t as a s t a r t i n g p o i n t (Spmmerfeld and Bethe, 1933)- They were not a b l e t o o b t a i n a l i n e shape d i r e c t l y but i n s t e a d c a l c u l a t e d t h e moments. The r e s u l t s of t h e s e t r e a t m e n t s a t t r i b u t e d the broad-e n i n g t o m u l t i - p h o n o n p r o c e s s e s . These methods a r e not s a t i s f a c t o r y s i n c e Grant (1964) has shown t h a t a knowledge of the moments, even-of an a r b i t r a r y l a r g e but f i n i t e number of moments y i e l d s no i n f o r m a t i o n about the f u n c t i o n w i t h i n any f i n i t e i n t e r v a l . Sampson and Margenau (1956), u s i n g the same model, have d i s c u s s e d the problem w i t h i n the framework of t h e o r i g i n a l L o r e n t z b r o a d e n i n g t r e a t m e n t and o b t a i n e d a L o r e n t z i a n c u r v e whose w i d t h i s r e l a t e d t o the i n v e r s e of t h e t i m e between c o l l i s i o n s of the bound c a r r i e r s and t h e l a t t i c e v i b r a t i o n s . Kane has shown t h a t when the e1ectron-phonon i n t e r a c t i o n i s weak the o p t i c a l a b s o r p t i o n unaccompanied by phonon e m i s s i o n o r a b s o r p t i o n i s much 3. s t r o n g e r t h a n t h e one phonon p r o c e s s . To account f o r t h e l i n e b r o a d e n i n g he su g g e s t e d t h e l i f e t i m e e f f e c t . L a t e r work by N i s h i k a w a and B a r r i e (1963), based on t h e same model, was an e x t e n s i o n o f Kane's work a l t h o u g h u s i n g a d i f f e r e n t t e c h n i q u e . These a u t h o r s s t a r t e d from Kubo's f o r m u l a {Kubo, 1957) f o r t h e a d i a b a t i c d i e l e c t r i c s u s c e p t i b i l i t y and r e l a t e d t h i s f u n c t i o n t o a t e m p e r a t u r e dependent d o u b l e - t i m e Green's f u n c t i o n . The e l e c t r o n - p h o n o n i n t e r a c t i o n was t r e a t e d as a s m a l l p e r t u r b a t i o n . The a p p r o p r i a t e Green's f u n c t i o n was t h e lowest o r d e r Green's f u n c t i o n i n a h i e r a r c h y o f e q u a t i o n s f o r t h i s and h i g h e r o r d e r Green's f u n c t i o n s . A f t e r s o l v i n g t h e s e e q u a t i o n s f o r t h e lowest o r d e r f u n c t i o n t h e s e a u t h o r s were a b l e t o c a l c u l a t e t he l i n e shape f u n c t i o n d i r e c t l y . They found a L o r e n t z i a n l i n e shape i n the v i c i n i t y o f t h e peak, the w i d t h b e i n g due t o a l i f e t i m e e f f e c t i n agreement w i t h Kane's s u g g e s t i o n . The background was, however, due t o mu1ti-phonon p r o c e s s e s . The main d i f f i c u l t y i n t h i s work was t o d e c o u p l e the i n f i n i t e h i e r a r c h y o f c o u p l e d e q u a t i o n s . The purpose... of the p r e s e n t work i s t o st u d y the same problem u s i n g the same s t a r t i n g p o i n t as N i s h i k a w a and B a r r i e but a d i f f e r e n t t e c h n i q u e . Maradudin {196*0 a p p l i e d the method o f t e m p e r a t u r e Green's f u n c t i o n s w i t h a s s o c i a t e d Feynman diagrams t o a st u d y of t h e f r e q u e n c y s h i f t , a n d l i f e t i m e o f a l o c a l i z e d v i b r a t i o n mode due t o a l i g h t i m p u r i t y atom i n a c r y s t a l . !t was f e l t t h a t a s i m i l a r t e c h n i q u e a p p l i e d here would shed some l i g h t on the n a t u r e o f t h e d e c o u p l i n g i n t h e p r e v i o u s method s i n c e , i n g e n e r a l , w i t h the diagram t e c h n i q u e i t i s much e a s i e r t o see what p h y s i c a l a p p r o x i m a t i o n s a r e b e i n g made. The e x t e n s i o n o f t h e d e c o u p l i n g p r o c e d u r e t o l a r g e c o u p l i n g c o n s t a n t s has never been j u s t i f i e d and the diagram t e c h n i q u e may be needed f o r such c a s e s . The p r o c e d u r e i s t h e r e f o r e t o d e f i n e a two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n u s i n g Matsubara's method (Matsubara, 1955) as o u t l i n e d i n the book by A b r i k o s o v , Gorkov and D z y a 1 o s h i n s k i ( 1 9 6 3 ) and t o r e l a t e t h i s Green's f u n c t i o n t o t h e a b s o r p t i o n c o n s t a n t . We r e p r e s e n t the Green's f u n c t i o n as a sum of Feynman diagrams and 'Slim the c o n t r i b u t i o n s from t h e s e diagrams t o o b t a i n the l i n e shape f u n c t i o n . The r e s u l t s which a r e o b t a i n e d a g r e e w i t h t h o s e of N i s h i k a w a and B a r r i e . The hope t h a t t h e t e c h n i q u e would s h i n e l i g h t on the d e c o u p l i n g p r o c e d u r e i n t h e i r method i s not r e a l i z e d , however. In c h a p t e r II we s p e c i f y the system t o be s t u d i e d and d e r i v e the r e l a t i o n between t h e a p p r o p r i a t e Green's f u n c t i o n and' t h e l i n e shape f u n c t i o n . In c h a p t e r I 9 I we c a l c u l a t e t h e l i n e shape f u n c t i o n i n the absence of e l e c t r o n -phonon i n t e r a c t i o n i n o r d e r t o i n d i c a t e t h e method f o r a s i m p l e case. !n c h a p t e r IV we s t u d y the more c o m p l i c a t e d case where i n t e r a c t i o n i s p r e s e n t and d i s c u s s some of t h e d i f f i c u l t i e s o f t h e method. Chapter V i s a comparison of our method w i t h the method of N i s h i k a w a and B a r r i e and c h a p t e r VI g i v e s a b r i e f summary and a d i s c u s s i o n of t h e r e s u l t s . 5 CHAPTER ]1 : GENERAL FORMAL 3 SM In o r d e r t o stu d y t h e shape o f a b s o r p t i o n l i n e s i n i m p u r i t y s o l i d s we r e q u i r e t he a b s o r p t i o n c o n s t a n t f o r l i n e a r l y p o l a r i z e d r a d i a t i o n which can be e x p r e s s e d as (2.1) (T(LS)— C f r w s v t . u 3 < J n w . X - ( ^ where X_(^0) i s t h e a d i a b a t i c d i e l e c t r i c s u s c e p t i b i l i t y . Kubo (1957) has shown t h a t t h e l i n e a r r esponse o f a p h y s i c a l q u a n t i t y IB t o an e x t e r n a l f o r c e F i t ) may be wr i t t e n as ( 2 . 2 ) ^ 6 f l ( t ) = : - ^ - / t A , B(t)] where j 3 i s the s t a t i s t i c a l m e c h a n i c a l d e n s i t y m a t r i x and the p e r t u r b a t i o n t o t he n a t u r a l m o t i o f i o f the system i s g i v e n by < 2 - 3 > J ¥ ' f t ) - " ft F ( 0 . The e x p r e s s i o n i n b r a c k e t s i n (2.2) i s j u s t t h e usu a l commutator ( 2- 4 ) [ A.BltYj - ABCtf-BltfA . In d e r i v i n g (2.2) i t i s assumed t h a t t he e x t e r n a l f i e l d F " ( " t V n a s been a p p l i e d a d i a b a t i c a l l y s t a r t i n g from t h e d i s t a n t p a s t such t h a t (2.5) o & w x \~ it) - O t-^-oo and t h a t t he system i s i s o t r o p i c and i n thermal e q u i l i b r i u m a t l " — -OO . The a d m i t t a n c e o f the system i s then g i v e n by ( 2 . 6 ) X B f t M = - U f i ^ . J ' ! t t e - i w t ' f i t J a . f [ f l , B l t l l ] . In o r d e r t o o b t a i n t h e a d i a b a t i c d i e l e c t r i c s u s c e p t i b i l i t y , we t a k e t h e e x t e r n a l p e r t u r b i n g H a m i l t o n i a n t o be ( 2 . 7 ) j ^ ' i t ) = -rt .Rt) 6. where r 1 i s t h e e l e c t r i c d i p o l e moment o p e r a t o r o f an e l e c t r o n bound t o an i m p u r i t y s i t e . In t h i s form, t h e l i g h t quanta can be a b s o r b e d o n l y by e l e c t r o n s t r a p p e d a t i m p u r i t y s i t e s and the i n t e r a c t i o n between t h e s e e l e c t r o n s i s i g n o r e d . Making use o f ( 2 . 7 ) , we can w r i t e f o r the a d i a b a t i c d i e l e c t r i c s u s c e p t i b i l i t y (2.8) t(uS) = i£i^ fit e 1 ^ <•.[ pi(t)) Pi]} £ - 7 0 + Jo where (2.9) = e U H - M N l ) t ^ e - U H - ^ ) t } j — | b e i n g t h e H a m i l t o n i a n o f the system which we a r e c o n s i d e r i n g , jJi, i s the c h e m i c a l p o t e n t i a l and |\1 i s the, number o p e r a t o r o f t h e e l e c t r o n system. The ther m a l average ^ } i s g i v e n by ( 2 . , o , < - _ - - ) = (e-^-^\--) s i n c e f o r t h e t e c h n i q u e we w i s h t o adopt i t i s more c o n v e n i e n t t o work i n the grand c a n o n i c a l ensemble. T h e r e f o r e , i s the grand p a r t i t i o n f u n c t i o n and (2 = where R a i s t h e Boltzmann c o n s t a n t and T i s t h e t e m p e r a t u r e o f the system. From ( 2 . 8 ) , we see t h a t We make use of t h e p r o p e r t y o f t h e c y c l i c i n v a r i a n c e o f t h e t r a c e t o w r i t e t h i s as which becomes on s u b s t i t u t i n g t ' ^ r - t i n t h e second i n t e g r a l on the r i g h t hand s i d e o f t h i s e x p r e s s i o n , 7. From ( 2 . 1 ) , we see t h a t i n o r d e r t o c a l c u l a t e t h e f r e q u e n c y dependent a b s o r p t i o n c o n s t a n t we r e q u i r e t h e i m a g i n a r y p a r t o f *X-.(LS) which i s g i v e n by (2.12) d U . X M S i n c e < K ) ( t > l M > V = < f t ' t ) ' ? l > . w e 9et t h a t so t h a t we can w r i t e f i n a l l y We d e f i n e now a f u n c t i o n F(t*A by t h e r e l a t ion (2-15) F M = Qte^<r\(t)$\> - fit ^ u t <r1(0 h>. From (2.14) and (2.1) we see i m m e d i a t e l y the r e l a t i o n between and the f r e q u e n c y dependent a b s o r p t i o n c o n s t a n t (2.16) <r(u^ oc ^ F M « S i n c e we s h a l l be i n t e r e s t e d i n l o o k i n g a t the shape o f a s i n g l e l i n e r a t h e r than i n l o o k i n g at the whole spectrum o f l i n e s as g i v e n by p"(c«J^  , we d e f i n e a new n o r m a l i z e d l i n e shape f u n c t i o n F by t h e c o n d i t i o n < 2 I 7> F W M - C [ F M ] i where by £ ^ . we mean t h a t o n l y the l i n e peaked a t CO - ^ [ and p a r t o f ^ I—(i) t h e background s h o u l d be p i c k e d o ut. The f u n c t i o n [~ (uf) i s the one which we s h a l l c a l c u l a t e . The c o n s t a n t £ i s d e t e r m i n e d by t h e n o r m a l i z a t i o n c o n d i t i o n (2.18) ntfi We now i n t r o d u c e e l e c t r o n f i e l d o p e r a t o r s ^ ( 7 f ) and W ) i n t he 8. S c h r o d i n g e r r e p r e s e n t a t i o n . We note t h a t t h e s e o p e r a t o r s a r e not f r e e e l e c t r o n o p e r a t o r s but a r e r a t h e r f i e l d o p e r a t o r s f o r an e l e c t r o n i n t h e f i e l d o f an i m p u r i t y i o n . In terms o f t h e s e f i e l d o p e r a t o r s we can w r i t e f o r the d i p o l e moment o p e r a t o r i n t h e u s u a l way (2.19) ^ = - e so t h a t (2-20) p\tt) = - e - j ^ ^ t ^ where "Y* an^ j(f^t\ a r e o p e r a t o r s i n the He i senberg r e p r e s e n t a t i o n g i v e n by and (2.22) ^(fiti* e M - ^ t ^ f l e j U H - * * ! * In terms o f t h e s e o p e r a t o r s we can w r i t e , t h e r e f o r e , (2-23) <M(tm> - £\Ui?£t%jZl<$\^ We d e f i n e e i g e n s t a t e s and e i g e n v a l u e s o f the H a m i l t o n i a n , |—| , of our system by t h e e q u a t i o n (2.2k) H i w > = E ^ l ^ > We can then w r i t e f o r the aver a g e o f t h e p r o d u c t i n (2.23) (2.25) From (2.15) and (2.23) we f i n d t h a t t t I 9. ( 2 . 2 6 ) We now use Matsubara's method (Matsubara,, 1955) t o i n t r o d u c e a two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n by the r e l a t i o n where T i s a n , " i m a g i n a r y t i m e " v a r y i n g i n t h e i n t e r v a l from O t o |S and has the p r o p e r t y t h a t ( 2 . 2 8 ) H ^ C f i K i A i T U - - - - F i f r ^ J = ^ n ) f l ^ ) — f o r f.>TJ> ; V| =. •+• I i f t h e c o n v e r s i o n o f the sequence T i i f , » ^ ' N T O ^ J TJ , --^b r e q u i r e s an even number o f f e r m i o n o p e r a t o r t r a n s p o s i t i o n s and V? - -| f o r an odd number o f f e r m i o n o p e r a t o r t r a n s p o s i t i o n s . In ( 2 . 2 7 ) , ^(ffjV) and < P^(^ 1)f^  a r e o p e r a t o r s i n a He i s e n b e r g l i k e r e p r e s e n t a t i o n and a r e g i v e n by ( 2 . 2 9 ) f (x» r ) = e ( H _ / t m r - x f o e i e - l H - w l * and ( 2 . 3 0 , f ^ r l = e . " * - ^ ^ ( S ) e - ^ " ^ . We note t h a t ^ ( N 0 , ^ and ^(T\?)T') a r e no l o n g e r h e r m i t i a n c o n j u g a t e s o f each o t h e r . We now d e f i n e a new two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n jkhfalTP'ofir,/?'*?) by'the c o n d i t i o n 10. From (2.27) we see t h a t t h i s can be w r i t t e n as (2.32) ^ ^ f f ' f l j ^ t f ' o ) ~ < T ^ ( f ( ^ « ^ ^ / > ° ) W l V ) / ? ( ^ < : > ^ > which on making use of (2.28) becomes (2.33) /Hfaft'o'tfr^c) - < ^ A " , l r j f d / f J « - ' $ W , , J o ^ ^ , J o ^ . By making use of t h e p r o p e r t y o f t h e c y c l i c i n v a r i a n c e o f t h e t r a c e , we can ex t e n d t h e d e f i n i t i o n of ^ ^ f t f j C ' o J /Ctt A?'o^ t o f ^ o and we f i n d t h a t (2.3*0 ^ Y ^ ' O ; ^ , ^ Jfov r We w r i t e t he F o u r i e r t r a n s f o r m o f our Green's f u n c t i o n i n t h e s t a n d a r d way (2. 3 5 ) . l(c\-U)^ = i [ W ^ ' " ^ * ^ W ' * S fr,*'o) where c TllX a n d t O n ^ *1^F , and "r)^. b e i n g i n t e g e r s . T h e i n t e g r a l on t h e r i g h t hand s i d e . o f (2.35) can be w r i t t e n i n t h e form J- tv ^ - ^ ^ ^ 0 3 ^ j B ' o ) 4- JL f i r e £ ( ^ r t J B j r ^ ^ 0 ; A ^ ' q ) • S u b s t i t u t i n g f o r JOT (X <o) f rom (2.3*0 i n t o t h e second i n t e g r a l and making th e change o f v a r i a b l e t^ '-="C+jQ, we f i n d We note t h a t t h i s e x p r e s s i o n v a n i s h e s u n l e s s f ^ Y ^ - ^ X ' W e t h e r e f o r e wr i t e (2-37) ffifi'liw*) - ( e1^*JkhftpOitxfio)** ( t J n - 1 - ^ ) ^ The i n v e r s e t r a n s f o r m i s then g i v e n by ( 2 ' 3 8 ) ^ ^ f o - ^ t ' o ) = - t E - e ^ ^ i V ' ; ' ^ • II. If we make use o f (2.33) we can w r i t e it . (2.39) y b 1 ^ ' ) ^ ) = - j e ^ < ^ ( * > V ^ ( A V ) fyfto) $(Ao))^« In terms o f t h e e i g e n v a l u e s and e i g e n f u n c t i o n s (2.24) o f t h e Ham.iltonian t h i s becomes Q ( 2 , Z f 0 ) * ( w l ^ ^ l U X w I f o P 1 ) ^ ^ e f ^ " ^ L i m S i n c e £ ^ and a r e n e c e s s a r i l y r e a l , we see t h a t t h i s f u n c t i o n i s f i n i t e a t a l l p o i n t s i u i y ^ l i ^ T e x c e p t f o r the p o i n t a t 6J^=<0 . We c o n s t r u c t a f unct * on jtf(^'> £ ^ , of t h e complex v a r i a b l e F , which i s a n a l y t i c f o r a l l n o n r e a l F by an a n a l y t i c c o n t i n u a t i o n i n which we r e p l a c e iuXy^ by £ f o r C J * ^ O . ^ For our p u r p o s e s , we s h a l l n e g l e c t ^(iu\^) when iJ^^C s i n c e we s h a l l be i n t e r e s t e d o n l y i n t h e f u n c t i o n which i s a n a l y t i c i n the upper and. lower h a l f p l a n e s t o w i t h i n an i n f i n i t e s i m a T d i s t a n c e £ of the r e a l a x i s . P e r f o r m i n g t h i s c o n t i n u a t i o n we o b t a i n , w i t h t h e u n d e r s t a n d i n g t h a t <HfW#)l*>WV)l|W')|M> 1^ 1^ ^ I . We can r e l a t e t h e d i s c o n t i n u i t y o f ^/^^fyS 1 J E^  a c r o s s t he r e a l a x i s t o the f u n c t i o n F U d e f i n e d i n (2.26). We d e f i n e (2,42) /Sfyfi'l u>) = rf^|[^%,^i^€)-^r0Ey;W-L^]. On making use of (2.41) and t h e r e l a t i o n 7J1 •• Baym and Mermin (1961) g i v e a d i s c u s s i o n o f t h e m a t h e m a t i c a l j u s t i f i c a t i o n f o r t h i s c o n t i n u a t i o n o f t h e F o u r i e r c o e f f i c i e n t t o a l l n o n r e a l £ . 12. (2.43) o / i v w - = ^ i l c ) + t^Z*-*) t h i s becomes (2.M0 We def i ne a I so and i n t h e same way as above t h i s becomes Comparing (2.46) and (2.44) w i t h (2.26) we note t h a t (2.47) FdA= pyffrinjLwafi'f M w x ' _ A K C A £ \ If we d e f i n e a new Green's f u n c t i o n by t h e c o n d i t i o n (2.48) feF(E). e-we see from (2.47) t h a t { 2 49) = { £ h * \ -I \-erP" i - e ^ With t h e a i d of (2.41) we see a l s o t h a t (2.50) / f i ^ u H t e } = ^ r ( - u 3 - t O and ( 2 . 5 0 ^ ( o J - i t . ) - JPi-uHrit") and s i n c e i we get f i na11y 13. (2.52) h M = ^^^-^^i^+iz) - ^ ( ^ - i L ) We now pr o c e e d t o s p e c i f y our p h y s i c a l system and i t s H a m i l t o n i a n . We c o n s i d e r e s s e n t i a l l y the same system as t h a t c o n s i d e r e d by N i s h i k a w a and B a r r i e (1963). We assume t h a t we have an e l e c t r o n (or h o l e ) bound t o an i m p u r i t y atom, l a t t i c e v i b r a t i o n s , and the i n t e r a c t i o n o f the e l e c t r o n w i t h the l a t t i c e v i b r a t i o n s . We w r i t e t h e H a m i l t o n i a n i n the second q u a n t i z e d forma 1i sm as (2.53) H = Ho+" H e f , H o = H e . * H f (2-54). H e = L.TK(LtdK <2-55) Hp - \ btf ^ (2.56) Hep = X I 11 ^ ^ * ^  kg ] ^ • \—\ £ i s the H a m i l t o n i a n o f t h e bound e l e c t r o n when <|jhe l a t t i c e i s f i x e d a t the p o s i t i o n which c o r r e s p o n d s t o t h e e q u i l i b r i u m p o s i t i o n i n t h e absence of the bound e l e c t r o n and |—(p i s the H a m i l t o n i a n of the l a t t i c e v i b r a t i o n s c e n t e r e d a t the e q u i l i b r i u m p o s i t i o n s . Hep ' s t n e e l e c t ron-phonon i n t e r -a c t i o n which i s assumed t o be v e r y weak so t h a t t he c o u p l i n g c o n s t a n t k which measures t h e s t r e n g t h of the i n t e r a c t i o n s a t i s f i e s K « \ It i s assumed here t h a t t he e l e c t r o n i n t e r a c t s o n l y w i t h l o n g i t u d i n a l long w a v e l e n g t h a c o u s t i c phonons and t h a t t he i n t e r a c t i o n can be t r e a t e d i n the i s o t r o p i c d e f o r m a t i o n p o t e n t i a l a p p r o x i m a t i o n of Bardeeh and S h o c k l e y (1950). The s p i n o f the e l e c t r o n i s i g n o r e d and and t h e s u f f i x e s \ and s p e c i f y the complete e l e c t r o n i c and phonon s t a t e s , r e s p e c t i v e l y . The and fl.^ a r e the usua1 c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r the bbund e l e c t r o n and the b>-f? and b>-g> a r e the c o r r e s p o n d i n g o p e r a t o r s f o r the phonons. These o p e r a t o r s s a t i s f y the u s u a l r e l a t i o n s O x , * * ] * = O W ] v = ° and the e l e c t r o n o p e r a t o r s a l l commute w i t h t h e phonon o p e r a t o r s . The (L^ and d\ a r e r e l a t e d t o the f i e l d o p e r a t o r s 1 p + ( ^ ) a n d ^(rV) . wh i ch we i n t r o d u c e d p r e v i o u s l y , by t h e r e l a t i o n s and (2.59) where ([ ^ ( f z l ) ' s a h y d r o g e n i c e i genf unct i on of the H a m i l t o n i a n | —| g ( f o r an e l e c t r o n i n t h e coulomb f i e l d of the i m p u r i t y ion) b e l o n g i n g t o the e i g e n -I ue TA • va In o r d e r t o c a l c u l a t e t h e Green's f u n c t i o n g i v e n by ( 2 . 3 2 ) i t w i l l be c o n v e n i e n t t o e x p r e s s the i n t e r a c t i o n H a m i l t o n i a n M e - p ' n terms of f i e l d o p e r a t o r s . We i n t r o d u c e phonon f i e l d o p e r a t o r s '^(Tp) such t h a t where \ / i s the volume of the system. In terms of t h e e l e c t r o n and phonon f i e l d o p e r a t o r s we can wrhte (2.61) H where we now have t h e r e l a t i o n s , by comparing ( 2.61) w i t h ( 2 . 5 6 ) (2.62 a) K \ / A x . g - ^ and (2.62.b) i < y (£>>) e k ' n ^ ( H U t i 15. We c o n s i d e r a g a i n the Green's f u n c t i o n g i v e n by (2.32), The H a m i l t o n i a n o f our system i s from (2.53) g i v e n by H = H ^ H , r where f |ep i s the e l e c t r o n - p h o n o n i n t e r a c t i o n . We t r a n s f o r m t o a s p e c i a l i n t e r a c t i o n r e p r e s e n t a t i o n , l i k e t h e i n t e r a c t i o n r e p r e s e n t a t i o n of quantum f i e l d t h e o r y , i n which we w r i t e (2.63) - e ( " - -^ , r 4^ie- l H - -^ v and In terms o f t h e s e o p e r a t o r s i n t h e i n t e r a c t i o n r e p r e s e n t a t i o n we can w r i t e ( A b r i k o s o v , Gorkov and D z y a 1 o s h i n s k i , 1963) ( 2 - 6 5 ) Jkh^wo'-fofi'cb < r t ( < y f ^ 1 D \ t f r d o ) i t p ) > 0 where ^(j5>) c o r r e s p o n d s t o the S m a t r i x of quantum f i e l d t h e o r y and i s g i v e n by a (2-66) f^ja) = Tr e K p | - J H ^ t r O ^ ' J • and <^  = " ^ " ^ . J ' e " ^ 6 " ^ • S i n c e H e p i s a P r o d u c t o f e l e c t r o n and phonon f i e l d o p e r a t o r s , and from the form of t h e d e n s i t y m a t r i x , we see t h a t i n (2.65) we can average e l e c t r o n and phonon o p e r a t o r s s e p a r a t e l y . (1) We also.make use of Wick's theorem^ , which i n our f o r m u l a t i o n s t a t e s t h a t ^ Wick's theorem as used i n our f o r m u l a t i o n i s proved i n t h e book by A b r i k o s o v , Gorkov and D z y a I o s h i n s k i . 16. the average o f a ( ^  p r o d u c t of a c e r t a i n number o f ^  - o p e r a t o r s decomposes i n t o a sum of p r o d u c t s o f a l l p o s s i b l e 1^- av e r a g e s i n v o l v i n g p a i r s o f o p e r a t o r s ^ and ty . The same k i n d of r e l a t i o n h o l d s as w e l l f o r a pro d u c t o f tp o p e r a t o r s . It t u r n s o u t , t h e r e f o r e , t h a t we can w r i t e (2.65) as ( 2 - 6 7 ) ^ < T r ( ^ u y ) W j ^ r ^ rL (ot£ where j&ie s u b s c r i p t C means t h a t i n a c e r t a i n term o f the e x p a n s i o n of (2.67) where we have o p e r a t o r s H e p l v O We^M we c o n s i d e r o n l y t h o s e terms f o r which ^ ( f ) i s p a i r e d w i t h a ijJ i n 1-1^ Cr^) , ty i n WepCTpl w i t h a 'Hp i n {) and so on u n t i l ty i n HepOt^ i s p a i r e d w i t h tyCr1) (or ^(o) ) , and ^(o) i s p a i r e d w i t h a i n r l e j = f t ^ and 1p i n 'Hept^fi) w i t h a ^ i n r l e p O f y ^ and so on u n t i l we re a c h ^Lo) ( o r ^ ( r ) ) w i t h o u t a s i n g l e "Me^ > , i n the term o f the e x p a n s i o n we a r e d e a l i n g w i t h , b e i n g l e f t out of t h i s p r o c e s s . ^ Any terms i n which t h e r e a r e o p e r a t o r s f"| €^, which a r e not c o n n e c t e d by p a i r i n g s in t h i s f a s h i o n a f t e r Wick's theorem has been a p p l i e d a r e t o be n e g l e c t e d s i n c e t h e sum of t h e s e terms cancel.s t h e \ denominator o f (2.65). In s u c c e e d i n g c h a p t e r s we s h a l l be c o n s i d e r i n g diagrams f o r the Green's f u n c t i o n JQ . The s u b s c r i p t C means, t h e r e f o r e , t h a t of th e s e diagrams we must c o n s i d e r o n l y t h o s e which a r e c o n n e c t e d . It w i l l become e v i d e n t a t t h i s l a t e r s t a g e what i s meant by c o n n e c t e d diagrams. ^ We do not here r u l e out t h e p o s s i b i l i t y o f the " s e t " R ep(*£) ^ 4 f ^ be i n g a " n u l l s e t " p r o v i d e d the o t h e r " s e t " c o n t a i n s a l l t he f - j ^ i n the p a r t i c u l a r term of the e x p a n s i o n . 17. From (2.60) we see t h a t (^(7^) i s l i n e a r i n phonon c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . From t h e form of th e d e n s i t y m a t r i x used i n t a k i n g t h e t r p c e i n (2.67) and from (2.55) we see t h a t (2-68) <<{W)> = 0 -T h i s means t h a t i n t h g e x p a n s i o n o f (2.67) we c o n s i d e r o n l y the terms f o r which V l i s even s i n c e the terms f o r which VL i$ odd a r e z e r o . In c h a p t e r I I I we s h a l l c a l c u l a t e t h e two p a r t i c l e Green's f u n c t i o n and t h e l i n e shape f u n c t i o n f o r the case when Yl— O i n (2.67) . 18. CHAPTER M l : LINE SHAPE CALCULATION IN ZEROTH ORDER APPROXIMATION A. C a l c u l a t i o n o f two p a r t i c l e Green's f u n c t i o n . We s h a l l now pr o c e e d t o c a l c u l a t e the two p a r t i c l e t e m p e r a t u r e Green 1 f u n c t i o n ^(TLV, 7?;O j ^ C , r t v o ) , g i v e n i n (2.67),. i n the z e r o t h o r d e r ( n = o a p p r o x i m a t i o n . T h i s i s the e x a c t Green's f u n c t i o n i n the absence of e l e c t r o n phonon i n t e r a c t i o n s . From (2.67) we have i n t h i s a p p r o x i m a t i o n (3.A. I) ^(ff^&orft.hVo) = < T r ( 4 i ( a > ^ ( 7 C ^ ? ( ^ ^ ) i>tf'Jo))y where we have dropped the s u b s c r i p t C and s h a l l do t h e same i n the remainder of t h i s work remembering t h a t when c o n s i d e r i n g diagrams f o r the Green's f u n c t i o n s , we t a k e i n t o account o n l y c o n n e c t e d diagrams. Making use of Wick's theorem, s t a t e d i n c h a p t e r I I , we can w r i t e (3 - A.1) as (3-A.2) __ We now d e f i n e a one p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n by the r e l a t i o n (3.A.3) £ M ( z , * ' \ * - * ) = - < i ; ( ^ a » ^ ( ^ f ) ) > . The f a c t t h a t we a r e a b l e t o w r i t e as a f u n c t i o n o f the d i f f e r e n c e of the f v a r i a b l e s o n l y i s an immediate consequence o f t h e d e f i n i t i o n o f t h i s f u n c t i o n as can be r e a d i l y seen by making use of the p r o p e r t y o f t h e c y c l i c i n v a r i a n c e o f t h e t r a c e . In terms of one p a r t i c l e Green's f u n c t i o n s , (3-A.2) becomes (3-A.4) filfafl'o'^lPo) 19. We r e c a l l t h a t because of t h e l i m i t s which we imposed i n ( 2 . 3 0 t h a t jb^i/C,??'\O) i s i n f a c t t o be c o n s i d e r e d ascZtXvi $la\/C,rt'] -c) . We e x p r e s s ^ ^ ^ f f j ^ f ' - r - V ' ) d i agramat i ca 1 1 y as f o l l o w s ; (3.A.5) ' / ' V ^ j r - r ' , - ^ > ^ . in terms of di a g r a m s , (^ >^ Ix10 - f i f , 7?'o) can.be w r i t t e n as a sum of two diagrams rc,r Note t h a t the a p p r o p r i a t e a l g e b r a i c s i g n s a r e a s s o c i a t e d w i t h the a n a l y t i c e x p r e s s i o n s f o r t h e diagrams so t h a t we c o n s i d e r o n l y a d d i t i o n of diagrams. In our case a d i s c o n n e c t e d d i a g r a m f o r t h e two p a r t i c l e Green's f u n c t i o n i s a d i a g r a m w h i c h c o n t a i n s some p a r t s which a r e not c o n n e c t e d i n any way t o one or both of t h e " e x t e r n a l p o i n t s " (7? ,"€ ) and {}\J, o ). S i n c e i n the diagrams g i v e n i n (3.A.6) t h e r e a r e no p a r t s which a r e not c o n n e c t e d t o e i t h e r (ff , V ) o r ( f t ' , 0 ),' both o f the diagrams a r e c o n n e c t e d and hence both must be c o n s i d e r e d . It i s im p o r t a n t t o note here the d i s t i n c t i o n between c o n n e c t e d and d i s c o n n e c t e d diagrams when d e a l i n g w i t h two p a r t i c l e Green's f u n c t i o n s and w i t h t h e more f a m i l i a r s i n g l e p a r t i c l e Green's f u n c t i o n s From (2.37) we see t h a t We w r i t e , t h e r e f o r e , w i t h the a i d o f (3-A.4) r/1 \ v k l t ) ( t ^ ( ) t \ f t ' ] 6 \ . 20. We n o t i c e i m m e d i a t e l y t h a t t he second term on t h e r i g h t hand s i d e o f t h i s e q u a t i o n when i n t e g r a t e d w i l l g i v e r i s e t o a f a c t o r & • The term i s t h e r e -f o r e z e r o u n l e s s O d ^ - O .We can, t h e r e f o r e , n e g l e c t t h i s term s i n c e we saw i n c h a p t e r II t h a t we were not i n t e r e s t e d i n t h e v a l u e of A^irt^\ Li*J>C) when • Hence, t h e o n l y d i a g r a m which we sha 1 l(,cons i der as c o n t r i b u t i n g t o A ^  (n?^') iWu) ' s t h e f i r s t d i a g r a m on the r i g h t hand s i d e o f (3-A.6). We now expand ^ ^ ( J f) i n a F o u r i e r s e r i e s We f i n d t h a t rH^LWn)-* O u n l e s s tOw has t h e form g i v e n above from th e f a c t t h a t /&(o) ffijp'* ?<Lo) - -/&(*>t ^  ]& ] ?+f) f o r f e r m i o n s i n g l e p a r t i c l e Green's f u n c t i o n s . We d i s t i n g u i s h between "even" and "odd" f r e q u e n c i e s i n t h i s work by w r i t i n g f o r " e v e n " f r e q u e n c i e s and f o r "odd" f r e q u e n c i e s . S u b s t i t u t i n g from (3.A.8) i n t o (3-A.7) and n e g l e c t i n g t h e second term on t h e r i g h t h a n d , s i d e - o f t h i s e q u a t i o n *rfe get t h a t (3-A. 9) A i f / _ 7 _ 7 / , , u I r r f ! U f l ^ A + ^ f c f « ) ^ ^ "? i >K \ P &, u\J6 which becomes on i n t e g r a t i n g o ver x (3.A.10) , \ - I T T ? ~ - i A ' ° W F ? / . ' ^ \ A , l ) / ! " s ' . ' " f >\ On summing over ^ f i , t h i s becomes ( 3 - A - M ) ' A f o ) ( ^ A ^ ) = /^^l&J / i / 0 , ^ / f ; i ^ , - ^ . r e l , 21 B. C a l c u l a t i o n of one p a r t i c l e e l e c t r o n Green's f u n c t i o n . We now p r o c e e d t o c a l c u l a t e t h e one p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n /Q (f?,lt''] i&*C\ which we need t o know i n o r d e r t o c a l c u l a t e /hfo)Ln'lj€/)iu)^) . From (3-A.3) we have t h a t Making use of t h e f a c t t h a t t h i s f u n c t i o n depends o n l y on the v a r i a b l e s t h r o u g h the d i f f e r e n c e we w r i t e ( 3 ' B ' , ) i ^ t f ' j t O - - <rv(^(fQ,v) $(&3o))j . From (2.58) and (2.&9) we have t h a t and S i n c e from (2.63) f ( a » = e f M . - ^ ^ ^ e - i « . - ^ ) v where \\j i s t h e number o p e r a t o r d e f i n e d by ( 3 - B - 2 ) N/-- ^ . ^ ^ we o b t a i n on making use of (2.53), (2. 5^ +) and (2.55) a s - w e l l as the ant commutation r e l a t i o n s (2.57)/ The one p a r t i c l e Green's f u n c t i o n (3-B.1) can be w r i t t e n as 22. C o n s i d e r i n g the case i n which V > o we get on s u b s t i t u t i n g from above (3.B.5) Because o f the form of the d e n s i t y m a t r i x g i v e n i n c h a p t e r II f o l l o w i n g (2.66) and t h e form of \~\a , the H a m i l t o n i a n of our system i n the absence o f i n t e r a c t i o n s , we have where ( 3 . B . 7 ) n A = [ e p w . J I i s t h e e x p e c t a t i o n v a l u e o f t h e number of e l e c t r o n s i n t h e e l e c t r o n i c s t a t e /V • T h e r e f o r e , (3-B.5) becomes (3.B.8) ^ O ^ ' J W ) - - Z e " ^ - ^ & C f f ) § * W ) ( i - ^ . T a k i n g the i n v e r s e t r a n s f o r m of (3-A.8) we have (3'B'9) '^V.^ iitt.) - (jvftiS.*/4"'of1S";rt A,-'2-^ . S u b s t i t u t i n g from (3.B.8) i n t o t h i s e x p r e s s i o n we get ( 3 - B - ' 0 ) M T ^ Making use of (3.B.7) we o b t a i n f i n a l l y (3.B.I1) ^ ' V , 7 ? ' / I - ^ W g f ( f f ' ) . T h i s i s the one p a r t i c l e Green's f u n c t i o n f o r an e l e c t r o n i n the coulomb f i e l d o f a n u c l e u s . When d e a l i n g w i t h f r e e p a r t i c l e Green's f u n c t i o n s one u s u a l l y at t h i s p o i n t t r a n s f o r m s t o t h e momentum r e p r e s e n t a t i o n s i n c e i n t h i s r e p r e s e n t a t i o n d e l t a f u n c t i o n s r e p r e s e n t i n g c o n s e r v a t i o n of momentum tend t o s i m p l i f y t h e r e s u l t i n g s t r u c t u r e . However, s i n c e we a r e d e a l i n g w i t h e l e c t r o n s 23. i n t h e f i e l d o f a n u c l e u s no such s i m p l i f i c a t i o n i s o b t a i n e d and., t h e r e f o r e , f o r c o n v e n i e n c e we s h a l l c o n t i n u e t o work i n t h e c o o r d i n a t e r e p r e s e n t a t i o n . C. C a l c u l a t i o n o f l i n e shape f u n c t i o n . We now pr o c e e d t o c a l c u l a t e t he l i n e shape f u n c t i o n , u s i n g our form-u l a t i o n , i n the absence o f e l e c t r o n - p h o n o n i n t e r a c t i o n . In o r d e r t o do t h i s we must f i r s t e v a l u a t e e x p l i c i t l y t h e Green's f u n c t i o n g i v e n i n (3.A.II). S u b s t i t u t i n g from (3.B.11) i n t o t h i s e q u a t i o n we have (3-C.1) P uu, In o r d e r t o e v a l u a t e t h i s Green's f u n c t i o n we must p e r f o r m f i r s t t h e sum-mat ion over t U ? ^ - 7 ^ •• We do t h i s by r e p r e s e n t i n g the sum as a c o n t o u r i n t e g r a l i n t h e complex p l a n e -as o u t l i n e d below ( L u t t i n g e r and Ward, I 9 6 0 ) . 1 = - - | 4§: - f C ^ h L ^ We c o n s i d e r t h e c o n t o u r i n t e g r a l (3.C.2) where ( 3 . C 3 , f W = has p o l e s a t £ — Z- ) i s an a r b i t r a r y f u n c t i o n o f exce p t f o r p o s s i b l e p o l e s . We assume t h a t t h e p o l e s o f n(2r) do not c o i n c i de .wi t h t h e p o l e s o f - f ( i ) and we t a k e t h e c o n t o u r C i n (3-C.2) t o e n c i r c l e (1) T h i s a s s u m p t i o n i s made f o r convenience' i n o b t a i n i n g t he d e s i r e d r e s u l t but does not l i m i t t h e f i n a l r e s u l t t o t h e p a r t i c u l a r c a s e i n which t h e p o l e s do not c o i n c i d e . 2k. a l l t h e p o l e s o f -fit) i n the n e g a t i v e sense but none o f the p o l e s o f . The r e s i d u e of - f ( 0 a t 2r =• £ ^ ,s-"p s o t h a t (3.C.4) I - ' T o - I - h ( 2 ^ . !f Zr-f(i) h ( i ) O as (£\-=?©o we can r e p l a c e the c o n t o u r C by the c o n t o u r C ' t h a t e n c i r c l e s a l l t h e p o l e s of j") ( i t ) i n t h e p o s i t i v e sense. Comparing t h e s e two e v a l u a t i o n s of we get (3.C.5) - J r - x i k z : ^ = £ - f ( * r h i ^ . We can t h e r e f o r e w r i t e (3.C.6) J C _ We know t h a t ^ ^ = 1 and u s i n g (3-B.7) we f i n d t h a t f (T*-/uJ) . T h e r e f o r e (3.C.6) becomes (3.C.7) ~ L _ H L — - = M x > ~ I > and from (3.C.1) (3.C.8) = % ^ & W ® W ^ C i S ' ) ® [ f f l - P ^ = . P e r f o r m i n g t h e a n a l y t i c c o n t i n u a t i o n t o a f u n c t i o n of t h e complex v a r i a b l e £. w h i c h ; i s a n a l y t i c everywhere o f f the r e a l a x i s we have From (2.48) we have t h a t (3-C.io) jb^LE) - E ) which on s u b s t i t u t i n g from (3-C.$) becomes ( 3 . c m i f : c e ) = L L R v K . 3 3 , 25-where i s t h e m a t r i x element o f the d i p o l e moment o p e r a t o r o f t h e e l e c t r o n between th e two e l e c t r o n i c s t a t e s K and \ / . From (3.C.I2), we see t h a t (3. C. 13) R A * - R „ so t h a t we can r e w r i t e ( 3 - G . l l ) as From (2.52) we see t h a t t he l i n e shape f u n c t i o n can be w r i t t e n as which on s u b s t i t u t i n g from (3.C. 14) and making use of (2-.,43) becomes ( 3 - C " , 6 ) K 0 ) M - - z X ^ ^ I F f J 1 ^ . ^ S C L O + I ^ - T ; ) . From (2.17) we have t h a t where from (2.18) we f i n d t h a t t he n o r m a l i z i n g c o n s t a n t C ,has the v a l u e (3.C.18) We see t h e r e f o r e t h a t i f the e l e c t r o n i c s t a t e s a r e nondegenerate and the e l e c t r o n i c energy l e v e l s a r e w e l l s e p e r a t e d , t h e l i n e shape f u n c t i o n i s a s e r i e s o f d i s t i n c t d e l t a f u n c t i o n s . There i s no b r o a d e n i n g o f the a b s o r p -t i o n l i n e s due t o t h e l a t t i c e v i b r a t i o n s i n t h e absence of e I e c t r o n - p h o n o n i n t e r a c t i o n . T h i s means t h a t i n t h e absence o f i n t e r a c t i o n s between t h e e l e c t r o n s and phonons the o b s e r v e d w i d t h i n an a c t u a l measurement i s j u s t t h e i n s t r u m e n t a l w i d t h r e l a t e d t o t h e power o f r e s o l u t i o n o f t h e ex p e r i m e n t . 26. The r e s u l t which we have o b t a i n e d c o u l d have been e a s i l y o b t a i n e d from (2.8) and (2.1) w i t h o u t making use of Green's f u n c t i o n s . The r e s u l t was d e r i v e d u s i n g t h i s t e c h n i q u e , however, i n o r d e r t o g i v e a s i m p l e example of the use of t h i s method b e f o r e p r o c e e d i n g t o t h e more c o m p l i c a t e d case i n which i n t e r -a c t i o n s a r e p r e s e n t . It i s f e l t t h a t t h i s c a l c u l a t i o n w i l l make the m a t e r i a l which f o l l o w s . m o r e e a s i l y d i g e s t i b l e f o r t h o s e r e a d e r s u n f a m i l i a r w i t h the t e c h n i q u e employed. A number of the r e s u l t s d e r i v e d here w i l l be needed i n succeeding, c h a p t e r s . 27. CHAPTER IV : LINE SHAPE CALCULATION WITH INTERACTION A. I n t r o d u c t i o n . We saw i n t h e l a s t c h a p t e r t h a t the a b s o r p t i o n spectrum i s a s e r i e s of d i s t i n c t d e l t a . f u n c t i o n s , i n t h e absence of degeneracy, when t h e r e i s no e I e c t r o n - p h o n o n i n t e r a c t i o n so t h a t under t h e s e c i r c u m s t a n c e s the w i d t h of a l i n e o b s e r v e d i n an e x p e r i m e n t i s s i m p l y r e l a t e d t o t h e power of r e s o l u t i o m , of t h e measuring a p p a r a t u s . In o r d e r t o a ccount f o r t h e a d d i t i o n a l broad-e n i n g o b s e r v e d i n a c t u a l measurements, we have t o i n c l u d e e l e c t r o n - p h o n o n i n t e r a c t i o n s i n our c a l c u l a t i o n and t h i s means c o n s i d e r a t i o n of h i g h e r o r d e r terms i n the e x p a n s i o n (2.67) f o r the two p a r t i c l e Green's f u n c t i o n . It t u r n s out t h a t one cannot get any s a t i s f a c t o r y r e s u l t by c o n s i d e r i n g o n l y t h e terms i n t h e p e r t u r b a t i o n e x p a n s i o n f o r which y\ =. £L s i n c e t h i s l e a d s o n l y t o a p e r t u r b a t i o n t h e o r y a p p r o x i m a t i o n f o r the l i n e shape f u n c t i o n . We a r e more p a r t i c u l a r l y i n t e r e s t e d i n an a p p r o x i m a t i o n f o r t h e l i n e w i d t h and f o r the s h i f t o f t h e a b s o r p t i o n I i n e . In o r d e r t o o b t a i n t h i s i n f o r m a t i o n , i t i s n e c e s s a r y , t o c o n s i d e r d i a g r a m s . a r i s i n g from a l l o r d e r s i n the e x p a n s i o n . In g e n e r a l , i n the p r e s e n c e of i n t e r a c t i o n s , j u s t as i n quantum e l e c t r o d y n a m i c s , i t i s i m p o s s i b l e w i t h our p r e s e n t knowledge t o sum a l l of the diagrams e x a c t l y and we a r e f o r c e d t o r e s o r t t o a p p r o x i m a t e methods. The p r o c e d u r e i s t o s e l e c t c e r t a i n t y p e s of diagrams which w i l l g i v e r i s e t o i m p o r t a n t c o n t r i b u t i o n s and t o sum t h e c o n t r i b u t i o n s from a l l of t h e s e diagrams. We have assumed t h a t the c o u p l i n g c o n s t a n t K f o r t h e e 1ectron-phonon i n t e r -a c t i o n i s s m a l l . We s h a l l , t h e r e f o r e , c o n c e r n o u r s e l v e s w i t h c a l c u l a t i n g t h e w i d t h and the s h i f t of the a b s o r p t i o n 1ine c o r r e c t t o o r d e r K 2 i n t h i s c h a p t e r . We s h a l l a l s o d i s c u s s the p o s s i b i l i t y o f p e r f o r m i n g the c a l c u l a t i o n 28 t o h i g h e r o r d e r s i n K which may i n c e r t a i n problems become n e c e s s a r y s h a l l see t h a t i n t h i s c a s e t h e p r o c e d u r e i s e x t r e m e l y l a b o r i o u s i f not p r e s e n t i m p o s s i b l e . 29. B. The diagrams f o r y\ . We s h a l l now pr o c e e d t o enumerate t h e diagrams w h i c h one o b t a i n s i n the e x p a n s i o n (2.67) f o r the case r \ — £_. The form o f t h e s e diagrams w i l l e n a b l e us t o f o r e c a s t t he t y p e o f diagrams which o c c u r upon g o i n g t o h i g h e r o r d e r s i n the e x p a n s i o n . We s h a l l see t h a t t he v a r i o u s diagrams t h a t a r e ob-t a i n e d can be c l a s s i f i e d i n t o d e f i n i t e t y p e s which can, a f t e r making s u i t -a b l e a p p r o x i m a t i o n s , be summed. where we have made use of (2.61) f o r t h e i n t e r a c t i o n H a m i l t o n i a n . We saw i n c h a p t e r 1 I t h a t when c o n s i d e r i n g an aver a g e o f p r o d u c t s of e l e c t r o n and phonon o p e r a t o r s we can average the e l e c t r o n and phonon p r o d u c t s s e p a r a t e l y . We can t h e r e f o r e r e w r i t e (k. B.I) as C o n s i d e r i n g t h e term of (2.67) f o r Ti-iJL, we have (k.B.\) (k.B.2) We i n t r o d u c e now a phonon Green 1 s f u n c t i o n d e f i n e d by (4.B.3) The second r e l a t i o n f o l l o w s from t h e f i r s t by making use of the usual commut-a t i o n r e l at i onsiirf or t he phonon c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . !n diagram form we wr i t e 30. rip It',*' We need no a r r o w on t h i s dashed l i n e s i n c e t he phonon Green's f u n c t i o n i s an even f u n c t i o n o f the c o o r d i n a t e and " t i m e " d i f f e r e n c e s . Making use of (k.B.k) and (3-A.3) as w e l l as Wick's theorem we can r e w r i t e (k.B.2) as . | ^ ^ j O ^ ' V i ^ S o ) / * ' 0 ^ ^ ^ ^ fi^^o) (a.) + ^ °W)AV^°^7^^^^ (c) - fc (0)&, fC)o)M0)C^i °) J * 0 (e) -^6,^^o) ftn^y^^t^'W ^ ° K n l ^ ) (h) o)^%V^-^ ^ V^!^ ( i ) +• ^ a ^ ^ / H ^ (j) - fi'Hs,fP^!*'**) £ i t } ( A t f J«>) M 4 * V ^ ) (k) 31 We now w i s h t o e x p r e s s /S^Oft^ifoJ at^^'o) i n terms o f diagrams. To do t h i s we make use o f (3-A.5) and (k.B.h). We f i r s t c o n s i d e r the f i r s t f o u r terms o f (4.B.5). L a b e l l i n g t h e diagrams i n the same way as the terms i n ( 4 . B . 5 ) , t h e s e diagrams l o o k as f o l l o w s . 3 2 . ( c ) . (d) FIGURE k.I We note t h a t each of the f o u r diagrams shown i n F i g u r e k.\ has p a r t s which a r e not j o i n e d i n any way t o e i t h e r o f t h e e x t e r n a l p o i n t s (jC-,f) o r (fU, O ). Hence, a l l f o u r of t h e s e diagrams a r e d i s c o n n e c t e d and we must i g n o r e them. The r e m a i n i n g twenty terms of (k.B.5) g i v e r i s e t o t h r e e d i f f e r e n t t y p e s of diagrams. The terms l a b e l l e d from (e) t o ( I ) i n t h i s e x p r e s s i o n g i v e r i s e t o diagrams which we r e f e r t o as t y p e I. These a r e g i v e n i n F i g u r e k.2. FIGURE k.2 . ' The terms l a b e l l e d from (m) t o ( t ) i n (4.B.5) g i v e r i s e t o what we r e f e r t o c as t y p e I I diagrams which a r e shown i n F i g u r e k.3-33. (m) (n) (o) (p) (q) (r) (s) (t) FIGURE 4.3 The last four terms of (4.B.5) give r i s e to type III diagrams which are given in Figure 4.4. FIGURE 4.4 \t is readily seen from the above diagrams that each connected diagram that we have drawn has a topological equivalent which may be obtained from i t by interchanging the internal coordinates; i.e. in Figure 4.2, diagrams (e) and (h) are topol ogicaI Iy equivalent while in Figure 4.4 diagrams (v) and (w) 3*+. a r e e q u i v a l e n t . S i n c e from (4.B.5) we see t h a t we have t o i n t e g r a t e over the i n t e r n a l c o o r d i n a t e s o f the di a g r a m s , i n ca 1 c u l a t ing /^^(aVj/C'o ri^/C'o) we can drop t h e f a c t o r -—- a p p e a r i n g i n (4. B.5) and sum o n l y the c o n t r i b u t i o n s from a l l t o p o l o g i c a I 1y d i s t i n c t diagrams. In f a c t , i n the term" o f the expan-s i o n (2.67) f o r n = Y[ , we can drop the f a c t o r and sum o n l y t h e c o n t -r i b u t i o n s from t h e t o p o l o g i c a 1 1 y d i s t i n c t diagrams. T h e r e f o r e , i n o r d e r t o c a l c u l a t e / J ^ f o - ^ ^ k ^/ttf^'o) we would sum the c o n t r i b u t i o n s from ( e ) , ( f ) , ( i ) and ( j ) i n F i g u r e 4.2, from (m), ( n ) , (q) and ( r ) i n F i g u r e 4.3 and from (u) and (v) i n F i g u r e 4.4, r e c a l l i n g a t a l l t i m e s t h a t we must i n t e g r a t e over the i n t e r n a l c o o r d i n a t e s . We s h a l l not ca I cu I a t e j&^-f&^fi'o ; ^ f,^''o) e x p l i c i t l y s i n c e we a r e i n t e r e s t e d i n an a p p r o x i m a t i o n t o ^ ^(fSf,fl'OJJLV^'O) c o r r e c t t o o r d e r K which w i l l a l l o w us t o o b t a i n an a p p r o x i m a t i o n t o t h e w i d t h and the s h i f t o f a l i n e c o r r e c t t o o r d e r K z • In o r d e r t o do t h i s we have t o c o n s i d e r diagrams of h i g h e r o r d e r i n K1" than t h e ones we have a l r e a d y drawn. A l t h o u g h / ^ ( ^ ( f f f ^ ' o j r??V? /o) can be c a l c u l a t e d e x a c t l y and i s t h e o n l y term o f o r d e r K.1" i n the s e r i e s e x p a n s i o n (2.67) t h i s term g i v e s o n l y a p e r t u r b a t i o n t h e o r y a p p r o x i m a t i o n t o t h e l i n e shape f u n c t i o n i t s e l f and i s o f l i t t l e use in o b t a i n i n g a s a t i s f a c t o r y v a l u e f o r t h e l i n e w i d t h and l i n e s h i f t . We s h a l l , t h e r e f o r e , p r o c e e d t o l o o k at the form o f some of t h e h i g h e r o r d e r diagrams f o r fe^V^'o'^V^'o) . 35-C. High e r o r d e r diagrams. One f i n d s on g o i n g t o highfer v a l u e s o f Y) i n t h e e x p a n s i o n (2.67) t h a t t h e r e s u l t i n g diagrams can s t i l l be c l a s s e d i n t o t h e t h r e e t y p e s which were d i s c u s s e d i n the p r e v i o u s s e c t i o n f o r t h e case T) - A. . The g e n e r a l r u l e t o f o l l o w i n o b t a i n i n g t h e diagrams f or /3(M\fj\k'{nl'o'ifir) n!'o) i s t h a t one forms a l l c o n n e c t e d , t o p o l o g i c a 1 1 y n o n e q u i v a I e n t diagrams w i t h two e x t e r n a l c o o r d i n a t e s (/? , f ) and (n', O ) and ~V) i n t e r n a l c o o r d i n a t e s where two s o l i d l i n e s and one dashed l i n e meet at each i n t e r n a l c o o r d i n a t e . We r e c a l l t h a t i n the a n a l y t i c e x p r e s s i o n s f o r the' d i a g r a m s , an e l e c t r o n Green's f u n c t i o n i s a s s o c i a t e d w i t h a s o l i d l i n e and a phonon Green's f u n c t i o n i s a s s o c i a t e d w i t h a dashed l i n e . Some o f the t y p e \ \ diagrams which one o b t a i n s when Y) = <z\- a r e g i v e n i n F i g u r e 4.5 where we have o m i t t e d the l a b e l s o f the i n t e r n a l p o i n t s . (a) (b) (c) (d) (e) ( f ) (g) (h) FIGURE k.5 It i s not d i f f i c u l t t o see the form which diagrams o f t h i s t y p e t a k e f o r 3 6 . higher y] . As can be rea d i l y seen also, there are an i n f i n i t e number of diagrams of th i s type when we consider a l l "JO . The higher order type J diagrams can be deduced from the corresponding T) =• 3~ diagrams in the same fashion. Some of the type I f ! diagrams, which are s t r u c t u r a l l y d i f f e r e n t from the type! and type If diagrams, are given in Figure k.6 for the case T l - 4 - . (a) (b) (c) (d) FIGURE k.6 Again i t Js not d i f f i c u l t to v i s u a l i z e the form of higher order diagrams of th i s type. The problem before us now is to f i n d a method of summing the c o n t r i -butions from the i n f i n i t y of diagrams which we have for the two p a r t i c l e Green's function. Jt is a straightforward if somewhat cumbersome procedure to take any given diagram and f i n d i t s contribution but it is obvious that to sum a l l the contributions in th i s way is impossible. In the next section we shall indicate how one can sum the diagrams, after making suitable approxi-mations. 37. D. Dysons equations. Before attempting to sum the various diagrams we shall f i r s t make a simpl i f y i n g approximation. Salpeter and Bethe (1951) in considering the problem of Fermi-Dirac p a r t i c l e s interacting through v i r t u a l emission and absorption of quanta assumed that, i f the coupling constant was s u f f i c i e n t l y small, they could neglect processes in which two or more quanta are in the f i e l d simultaneously. K i t t e ! (1963) has shown that in the deformation poten-t i a l approximation, where the coupling constant is small, the number of v i r t u a l acoustic phonons accompanying a slow electron is of the order of .02. In t h i s instance the SaI peter-Bethe approximation should be rather good. Since we have assumed that our interaction could be treated in the deformation potential approximation and that the coupling constant K is small, we shall make a SaI peter-Bethe type approximation in which we neglect processes in which two v i r t u a l acoustic phonons accompany an electron at the same "imagi-nary time" X in analogy with the real time processes.^'^ (St should be re-membered that our problem does not. involve real time and the word process should not be taken too l i t e r a l l y . ) This means that we shall neglect diagrams l i k e (g) and (h) of Figure k.5 and (e) and (f) of Figure k.6. We consider now a f u l l one p a r t i c l e Green's function ^(/T^ 7£/* f - f ' ) which we express diagramaticaI 1y as In quantum electrodynamics an approximation of t h i s sort would correspond to replacing the vertex part by unity and summing only ladder diagrams. 38. and which i s d e f i n e d t o be (k.0.2) 4 ^ s e •+- •> s. j S K U .+ a l l h i g h e r o r d e r diagrams. We c o n s i d e r a l s o a f u l l phonon Green's f u n c t i o n fytsi-fV \Y-H1') which e x p r e s s d i a g r a m a t i c a l l y as (^•0.3) Q C h - J ? ' = we and which i s d e f i n e d t o be " ffjt" ._o -f- a l l h i g h e r o r d e r diagrams. ty From (if.D.2) and (k.D.k),. we see ^r-ead i 1 y t h a t we can w r i t e (^.D.5) 39. i and (k.0.6) tfv> Xfri fir E q u a t i o n (4.D.5) i s t h e Dyson e q u a t i o n f o r t h e one p a r t i c l e e l e c t r o n Green's f u n c t i o n and (4.D.6), i s t h e Dyson e q u a t i o n f o r t h e phonon Green's f u n c t i o n . From (3-A.5), (4.B.4), (4.D.1) and (4.D.3), we see t h a t (4.D.5) can be exp-r e s s e d a n a l y t i c a l l y as (4.D.7) J i ^ ^ v - v ) = j b i o \ g , g ' i * - * * ' ) where t h e a l g e b r a i c s i g n s have been chosen t o agree w i t h t h e s i g n s i n (k.B.5). S i m i l a r l y (4.D.6) can be w r i t t e n a n a l y t i c a l l y as (k. D.8) JBfc-tf'jf-fO = ^ ' ( t f - ^ 7 ; f - v ) If we now lo o k a t t h e ty p e ! i diagrams g i v e n i n F i g u r e 4.3 and F i g u r e k.S as w e l l as the f i r s t d iagram of (3-A.6) which i s a l s o a t y p e 11 diagram, we see t h a t i n the S a l p e t e r - Bethe t y p e a p p r o x i m a t i o n which we have made, the sum of a l l t y p e II diagrams can be w r i t t e n as We a l s o f i n d t h a t t he sum of a l l diagrams o f t y p e ! can be e x p r e s s e d diagrams at i ca11y as ho. A l s o f r o m F i g u r e h.h and F i g u r e 4.6, we see t h a t t h e sum of the t y p e 111 diagrams can be w r i t t e n i n the Sa1 p e t e r - B e t h e t y p e a p p r o x i m a t i o n as S i n c e t h e two p a r t i c l e t e m p e r a t u r e Green's f unct i on wh i'ch we a r e i n t e r e s t e d i n i s j u s t the sum o f a l l t h e s e d i a g r a m s , we see t h a t we can e x p r e s s C o n s i d e r now t h e f o l l o w i n g f u n c t i o n s (k.D. 10) • t * — n /> —° arid (4.D.11) >Qi 1 >i go • II and d e f i n e (if.D. 12) fihZV^'o'ftvfto) *Xl)($VlZ'o]%t>$'o) + / ^ ( £ t ; £ ' o ; ] ^ ' 0 ) t On comparing (4.D.9) w i t h (4.D.12) we see t h a t (h.D. 13) We c o n s i d e r a g a i n (4.D. 10) and r e w r i t e . i t as (k.p.\k) •f-* > w Ii We see t h a t the term i n b r a c k e t s i s a g a i n a two p a r t i c l e Green's f u n c t i o n o f the same form as (4.D.10) e x c e p t w i t h d i f f e r e n t e x t e r n a l c o o r d i n a t e s . We can, t h e r e f o r e , w r i t e (k.D.\k) a n a l y t i c a l l y as TLtO > n > ii ii ii 4-(^•0.15) if 2. In t h e same way we can w r i t e (if. D. 16) ^ttttf'e-falto) = -JkiKfi^Mfrrf-o) A f t e r t a k i n g t h e s u i t a b l e 1 i m i t s d e f i n e d i n (4.D.13), t h e s e e q u a t i o n s g i v e t h e Dyson e q u a t i o n f o r t h e two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n . We s h a l l c o n s i d e r t h e s o l u t i o n of t h i s e q u a t i o n a f t e r we have c o n s i d e r e d t h e s o l u t i o n o f the Dyson e q u a t i o n s f o r t h e one p a r t i c l e e l e c t r o n and phonon Green's f u n c t i o n s . E. C a l c u l a t i o n o f one p a r t i c l e Green's f u n c t i o n s . B e f o r e p r o c e e d i n g t o c o n s i d e r t h e s o l u t i o n s o f t h e Dyson e q u a t i o n s (k.D.7) and (k.0.8) f o r the one p a r t i c l e e l e c t r o n and phonon Green's f u n c t i o n s we s h a l l f i r s t o b t a i n the e x p l i c i t r e s u l t f o r t h e f r e e phonon Green's f u n c t i o n O^W-^C'j^-?') w h i c n w a s d e f i n e d i n (k. B. 3) • From (2.63) we see t h a t we can w r i t e where (+0(rt) i s t h e phonon f i e l d o p e r a t o r d e f i n e d i n (2.60), f - j ^ i s d e f i n e d i n (2.53) and |\| , the number o p e r a t o r , i s d e f i n e d i n (3-B.2). Making use o f (2.60), (2.5*+) and (2.55) as w e l l as t h e commutation r e l a t i o n s (2.57) we f i n d t h a t If we c o n s i d e r t h e phonon Green's f u n c t i o n d e f i n e d i n (4.B.3) f o r the case when £'>'V / we have (4.E.3) fe^CA-x'^f-t") = - < f ( ^ r ) ^ p ( / f > 0 > which by making use of (k.E.2) can be w r i t t e n s i n c e from the comments f o l l o w i n g (2.66) we know t h a t We have a l s o t h a t (k.E.S) <Jr£ i r ^ - •&•£> ^ a 7^ ^ / kk. and from t h e commutation r e l a t i o n s (2.57) (4.E.6) <^ l^ > = Sftf + (Ir&hy = WO^j?) ' i s s i m p l y t h e e x p e c t a t i o n v a l u e o f t h e phonon o c c u p a t i o n number i n the s t a t e and from B o s e - E i n s t e i n s t a t i s t i c s i s g i v e n by (4-E.7) l ) t - ^4 ' We can w r i t e t h e n , f o r f •> t " 1 (4.E.8) The f u n c t i o n w i t h which we s h a l l e v e n t u a l l y w i s h t o work i s the F o u r i e r t r a n s f o r m of t h e Green's f u n c t i o n (4.().8). With t h i s i n mind we s h a l l t h e r e -f o r e c a l c u l a t e t h e F o u r i e r t r a n s f o r m of (4.E.8), If we make use of the f a c t t h a t we f ind T T h e r e f o r e , t a k i n g t h e t r a n s f o r m o f (k. E.8) and making use of (k.E.7) we f i n d t h a t For a c o u s t i c phonons we have t h a t (4.E.11) cJg - CO_t Making use o f t h i s we o b t a i n f i n a l l y 45. B e f o r e p r o c e e d i n g t o s o l v e (4.0.7) and (4.4^8) f o r t h e one e l e c t r o n and the phonon Green's f u n c t i o n s , we s h a l l f i r s t t a k e t h e F o u r i e r t r a n s f o r m o f t h e s e e q u a t i o n s f o r i t i s t h e t r a n s f o r m s w i t h which we s h a l l e v e n t u a l l y want t o work. Making use o f (3.A.8) and which comes from (4.E.9), we can w r i t e t h e F o u r i e r t r a n s f o r m o f (4.D.7) as The F o u r i e r t r a n s f o r m of (4.D.8) i s w r i t t e n as We see t h a t we have t o s o l v e two c o u p l e d i n t e g r a l e q u a t i o n s i n o r d e r t o c a l c u l a t e feoS^'ji a n d h(ft-r2'j • W e s h a l I attempt t o s o l v e t h e s e e q u a t i o n s by an i t e r a t i v e p r o c e d u r e . We t h e r e f o r e f i r s t r e p l a c e ^(nr^jO) 'bin^i^-C^,) ' a n d ^ X ' , ^ 0 i n (4.E.14) by t h e i r z e r o t h o r d e r a p p r o x i m a t i o n s g i v e n i n (4.E.12) and ( 3 - B . l l ) r e s p e c t i v e l y t o o b t a i n (4.E. 16) $($,72'; i & J ^ ^ t f ^ l i^) (4.E.15) cr •k 46. In c a l c u l a t i n g the l i n e shape f u n c t i o n we s h a l l , i n f a c t , employ t h e s o l u t i o n of t h i s i n t e g r a l e q u a t i o n r a t h e r than the s o l u t i o n of (4.E.14). We s h a l l see t h a t t h e c o r r e c t i o n s t o t h e one p a r t i c l e e l e c t r o n Green's f u n c t i o n , o b t a i n e d by s u b s t i t u t i n g t h e s o l u t i o n o f (4.E.16) as w e l l as t h e lowest o r d e r i t e r a t e d s o l u t i o n of (4.E.15) i n t o (4.E.14) and s o l v i n g t h e r e s u l t i n g e q u a t i o n s , w i l l be of o r d e r |(^* and t h e r e f o r e can be n e g l e c t e d s i n c e we a r e i n t e r e s t e d o n l y i n t h e w i d t h and s h i f t o f an a b s o r p t i o n l i n e c o r r e c t t o o r d e r \t?~ . We now pr o c e e d t o s o l v e f o r ito^) i n (4.E.16) by i t e r a t i o n . We r e p l a c e / f e l ; r t 7 / 5 icOm) a s a f i r s t a p p r o x i m a t i o n by Jb^L&^Tl1) ic3^\ and make use of (3-B.11) and (4.E.12) t o w r i t e 0\t tf^-Z x x» f j We sum over the f r e q u e n c i e s by making use o f (3-C.5)• We f i n d t h a t the sum-ma t i o n over tJv,, i n t h e second term on t h e r i g h t hand s i d e o f (4.E.I7) g i v e s on u s i n g ( 3 . C 3 ) and (3. B. 7) (4.E.18) _ L _ 2 L * - / * * - £ & L - = . The convergence of the i n t e g r a l at i n f i n i t y i s a s s u r e d s i n c e i n f a c t t h e r e s h o u l d be a f a c t o r *H a s s o c i a t e d w i t h the above e x p r e s s i o n where £ - 7 0 s i n c e i n c h a p t e r I I I we saw t h a t J)^\??)r2,'i o) i n f a c t means ^(^ /& <°£rC^? /' - £ ) . Summing over i n t h e t h i r d term on the r i g h t hand s i d e oif.: (4. E. 17), t h i s becomes hi. C^-E.19) | on making use o f (3- C . 3 ) , (3-B.7) and (k.E.l). On making use o f t h e above two r e l a t i o n s as w e l l as (2.61) and (2.62) we can w r i t e f o r (4.E.17) (lf.E.20) - T & f l e ) & * ( / 8 0 If we now s u b s t i t u t e (4.E.20) back i n t o (if. E. 16) arid s o l v e t h e r e s u l t i n g e q u a t i o n we get (4. E. 2]) where (4.E.22) F^U&) ^ i ? tw*,-un-A«-u)tf We now d e f i n e (*KE.23) and (4.E.24) 48. The s o l u t i o n o f ( 4 . E . I 6 ) i s seen t o be ( 4 . E . 2 5 ) ^ ( / e ^ ' j i & O ~ % - ± ( ^ s K where (4.E.26) £K= » - ^ l L f M ^ ^ C - f ^ I ^ f ( t . ^ + ! From (4.E.26) we see t h a t Ct.E.27) S x = i - ^ "T-so t h a t by i t e r a t i o n we can w r i t e We s h a l l t a k e S x t o be so t h a t we can w r i t e f o r the Green's f u n c t i o n (4.E.25) <<f.E.30) i ^ T ^ j i ^ ) = H & t g > _ where In Appendix I we show t h a t the terms which we have n e g l e c t e d i n w r i t i n g (4.E.29) do not c o n t r i b u t e s i g n i f i c a n t l y t o the l i n e shape f u n c t i o n . We a l s o n o t e here t h a t f e e d i n g (4.E.30) back i n t o (4.E.I4) and s o l v i n g the e q u a t i o n w i l l have no s i g n i f i c a n t e f f e c t s i n c e t h i s would o n l y change the denominators i n (4.E.3I) by an amount of t h e o r d e r of (C 2 - and s i n c e i n (4.E.30) t h i s term i s a l r e a d y of o r d e r K the c o r r e c t i o n t o the one p a r t i c l e Green's f u n c t i o n i s of o r d e r ^ 49. In o r d e r t o s o l v e (4.E.I5) f o r t h e phonon Green's f u n c t i o n we go thr o u g h t h e same s o r t o f p r o c e d u r e as above. We r e p l a c e /^CK)UJCL. \ i-'uu^ and /Sift^T?, • i(u3^ *Sl*S) t h e i r z e r o t h o r d e r a p p r o x i m a t i o n s and o b t a i n t h e i n t e g r a l equat ion (4.E.32) ktf2-&')iSU) &• ft^tlZ-jZ'iiA*) Note t h a t we c o u l d have r e p l a c e d t h e e I e c t r o n r G r e e n ' s f u n c t i o n s i n (4.E.15) by t he Green's f u n c t i o n s we o b t a i n e d i n (4.E.30) but t h i s would o n l y mean a c o r r e c t i o n of o r d e r t o the phonon Green's f u n c t i o n . We f i n d on i t e r a t i n g (4.E.32) t h a t t h e s o l u t i o n can be w r i t t e n i n t h e form i (4.E.33) B(jZ-2')l.a~) = f Lt)fy where (4.E.34) (4.E.35) ^ = I +vc* ^ ^ £ , ^ ^ 3 ^ ^ +k* (4.E.36) -k(t,t') - . e . L t , t - l t ' ' * ' £ * & Making t h e same s o r t o f a p p r o x i m a t i o n s which we made f o r t h e case o f the e l e c t r o n Green's f u n c t i o n we o b t a i n t he s o l u t i o n (4.E.38) k?l!-&ii<u) = tyT. co^ where i 50. (if. E. 39) V g & f l J = 1121 ^ I V A X ' ^ K T ^ X ) ,. * iSU +T7; - T s A p a r t f r o m a term of o r d e r K** i n t h e denominator ,. (4. E. 38) can be w r i t t e n (4.E.40) $(£-}?'-iiu) = _ L T. e ^ ^ W . We see t h a t s u b s t i t u t i n g (4.E.40) i n t o (4.E.14) and s o l v i n g t h e r e s u l t i n g e q u a t i o n would o n l y i n t r o d u c e a c o r r e c t i o n o f o r d e r i n t o t he one. p a r t i c l e e l e c t r o n Green's f u n c t i o n . The s o l u t i o n s (4.E.30).and (4.E.40) a r e t h e r e f o r e s u f f i c i e n t f o r our purposes which i s t o o b t a i n a r e s u l t c o r r e c t t o o r d e r K*" • In t h e next s e c t i o n we s h a l l make use of t h e s e r e s u l t s t o c a l c u l a t e t h e l i n e shape f u n c t i o n w i t h t h e h e l p of t h e two p a r t i c l e Green's f u n c t i o n . 51 F.Ca1cu1 a t i o n of l i n e shape f u n c t i o n . We s h a l l now pr o c e e d t o c a l c u l a t e t h e l i n e shape f u n c t i o n c o r r e c t t o o r d e r K2" . R a t h e r t h a n c a l c u l a t e t he two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n j^fofi'o'fl^fl?o) e x p l i c i t l y , we s h a l l c a l c u l a t e the- 1 i n e shape f u n c t i o n , which i s r e l a t e d t o the d i s c o n t i n u i t y o f the a n a l y t i c a l l y - c o n t i n u e d Green's f u n c t i o n a c r o s s the r e a l a x i s , d i r e c t l y . From .(4. D- I 3) we see t h a t where F.2) / ^ ( E ^ C ' K ^ C ) ^ A ^ ' > r ) $ ( T L < f f ^ ) and We s h a l l c a l c u l a t e (4. F. 1) by i t e r a t i o n . R e p l a c i n g $^(h~l?{ji'o ;nlyi^'°) • ' n .(if.F.2) b y A ( ^ y r > ) A ( ^ ' A " - ^ ) a n d 4 ^ ^ ^ j ^ ) ^ ° > i n ( Z f F - 3 ) b y -b^tj^-t') ^ (tjill) -tV) w e o b t a i n f o r t h e f i r s t i t e r a t e d s o l u t i o n o f (if . F . I) - c|r[. • UrfM^ tetfafa vx^iZLfi; n-r) fce^f. .vjj&tx, \V{ )M^A; O 52. The f u n c t i o n which we a c t u a l l y r e q u i r e i s the F o u r i e r t r a n s f o r m of •ffijSt&O'ttivfi'o) • Making use of (2.37), (3-A.8) and (4, E.I 3) w e get on t a k i n g t h e t r a n s f o r m o f (4.F.4) (4.F.5) P a;, « v We note t h a t on t a k i n g the F o u r i e r t r a n s f o r m of the second term on the r i g h t hand s i d e of (4.F.4) t h a t t h i s term: i s z e r o u n l e s s -6^ = 0 , and from our remarks i n c h a p t e r ii we n e g l e c t t h i s term. We s u b s t i t u t e from (4.E.30) and (k.E.kO) i n t o (4.F.5) t o o b t a i n [i&.-r^-bu.-vKL LxWi& i l l [jJ&r Li^-T)! <*• to <&, -t!uu)] (4.F.6) 1-1 Making use of (2.48) and (3.C.12) as w e l l as (2.62 a) and (2.62 b) we w r i t e 53. ffJLU) | __ ^ I jQ (iav) = ~& i— i—L- r ; ~ n ~ V / r x * o», L ^ - ^ ^ ^ L x f r ^ (4.F.7) _ _ |(vT f _ ^ r ^ ^ « Z w « M > ! K % i V x ^ t « k -If we f o l l o w t h e same s o r t o f p r o c e d u r e which we f o l l o w e d i n o b t a i n i n g the one p a r t i c l e e l e c t r o n Green's f u n c t i o n i n (4.E.30) we o b t a i n (4.F.8) where (4.F.9) " i . ^ -JL 1 «- J.J 5k. The f i r s t term on t h e r i g h t hand s i d e of (if.F.8) c o r r e s p o n d s t o what N i s h i k a w a and B a r r i e c a l l e d the " d i r e c t t e r m " i n t h e i r work. T h i s i s the o n l y term which we s h a l l c o n s i d e r s i n c e the o t h e r term can be shown t o have a n e g l i g i b l e c o n t r i b u t i o n t o t h e l i n e shape f u n c t i o n i n t h e same way t h a t we show i n A p p e n d i x I t h a t t h e p a r t s which we have n e g l e c t e d i n w r i t i n g the one p a r t i c l e Green's f u n c t i o n can have no s i g n i f i c a n t e f f e c t . In summing over i n (4.F.9) we s h a l l n e g l e c t a l l the terms of o r d e r K2" i n t h e denominator s i n c e the f u n c t i o n we a r e c o n s i d e r i n g i s a l r e a d y of o r d e r K2" . On n e g l e c t i n g t h e s e terms and on summing over by means of (3-C.5) we get t h a t (4.F. 10) A'X'k +-4-We now have t o sum over 6Jy,| i n t h e f i r s t term on the r i g h t hand s i d e of (if.F.8). We make use of (3-C.5) t o w r i t e t h i s e x p r e s s i o n as where the c o n t o u r C/ i s shown i n F i g u r e if.7 where the c r o s s e s i n d i c a t e t h e p o l e s o f -|-(fc) . I-i 55-_: c' 2: - p l a n e FIGURE hj S i n c e L-xfe") i s r e g u l a r everywhere e x c e p t on the r e a l £ a x i s , and s i n c e t h e n a t u r e of t h e d i s c o n t i n u i t y a c r o s s t h e r e a l a x i s of t h i s f u n c t i o n i s g i v e n by (k.F. 12) LX(X±Lo)=[JACK)Tl!70c) where L \ 6 0 and (\(x) a r e r e a l , we see t h a t a p a r t from t h e p o l e s o f the i n t e g r a n d o f (4.F.11) i s r e g u l a r everywhere e x c e p t at the l i n e s and <JWI(_~UJK')=0 • We can t h e r e f o r e w r i t e (4.F.11) as the sum of f o u r i n t e g r a l s a l o n g t h e l i n e s H r = X — i o and 2: -X ±Lo + as shown i n t h e diagram where X i s r e a l . We a l s o make use o f the f a c t t h a t (4.F.13) t o w r i t e (k.F.\k) [ X - l*+/x_4-iclLxCX-ior|[K-iu^ 4 ! 56. From (4.F.10) we see that we can write (^.F. 1 5 ) <^Ux>j(xt£o ; i w ^ ) = K l H M i (x ; i u x ^ ^ - U ^ C x j - i ^ where (4. F. 16) J x y a;{u)M) *TE. WVx^ k . , J(Mf- ^ftxq >4- (a-^)+hte^£&-Tx^ug) l Focussing our attention now on the f i r s t two integrals on the right hand side of (if.F..14), we make use of (4.F.12) and (4.F.15) to write [ X - V A + ^ L x l X - i ' o U O + ^Rxxl (*-^W] C^+A+-K1'Lx^+^)]Q+-K2-^ls^ Ut-do \ i t o « j ] (4. F. 17) - | [ X . T ^ M f £L xUK^P x(*0(W H ^ ^ w O - d ^ J y (xjJuo] [ C * - T x + ^ L X ^ We see that t h i s function is peaked at X w"Tx- /u. • Since J?fo) , |_x(x.) ' {^ X^ iCX j and are a l l slowly varying functions we can replace the variable X in these functions b y T ^ - ^ . Making use of ( 4 . F . 1 6 ) , we find that 3^ x./(K j tux,) can be : neglected on making use as well of the fact that (4.F.18) 5 - V A A k \ 4 i t e * ° which follows from the fact that the density of phonon states vanishes at CJjg = o . Si nee L^^-i'^X^ ' s a ' s o a slowly varying function we can write the integral of the f i r s t two terms on the right hand side of (4.F.14) as F.19) r n . _ 57-where (4.F.20) K^CX-i'uJ.O - L _ x / ( K 4 W ^ 4- (x-ita,-TJj-t-/uL) WA>I (X J i u ^ . S i m i l a r l y , f o r the i n t e g r a l o f the l a s t two terms on the r i g h t hand s i d e of (k.F.\k) we can w r i t e w i t h t h e a i d of (4.F.22) - ? 5 ^ ^ T f ] 3 K where (k. F. 23) |<^ , (x+i'tO^ » L A U+tuv) + CX + t'uJv,-17.+ ^  Wx>i (xKu>^ •iu)^N) . In (2.52) we saw t h a t t he f u n c t i o n which we want i n o r d e r t o c a l c u l a t e t h e l i n e shape f u n c t i o n i s t h e d i s c o n t i n u i t y a c r o s s t h e r e a l a x i s of the Green's f u n c t i o n ^ ^ C B ) which i s t h e a n a l y t i c c o n t i n u a t i o n o f ( k . F . ) k ) . With t h i s i n mind we combine (4.F.19) and (4.F.22) t o w r i t e . f i l T x - ^ ) -^ iZcrv-^  + In t he same way we can w r i t e 58. ( i f . F . 25) 4- fei-^ ^ l ^ f l C W ,  We make use of and Cf. F. 27) ^ K x x, ( X t it) = K ^ t o +1 OO as w e l l as (2.52) and (3.B.7) t o w r i t e ( i f . F . 28) r — . . _ _ — , _ , / • " -&0 Cx+ to - r X 4 - ^ t < ? KX>/(T;.+ui^)]Vx-O^fafw-^J Q < K M ^ O i - u » \ T + t f (Iffta-A We now make use of the w e l l known r e s u l t , d e r i v a b l e by means of>the f a l t u n g theorem, t h a t the c o n v o l u t i o n of two L o r e n t z i a n f u n c t i o n s i s i t s e l f a L o r e n t z i a n f u n c t i o n t o w r i t e (if. F. 28) as (i f . F . 29) 59. We see that our. line shape function is the sum of Lorentzian curves. Before proceeding to look at a p a r t i c u l a r lin e and c a l c u l a t i n g i t s width and peak p o s i t i o n , it is necessary f i r s t to evaluate e x p l i c i t l y the various functions appearing in (4.F.29). We make use of (4.E.31), (2.43), (4.F. 12) and (4.F.21) to f i n d (4.F.33) fJ-Q-^ - i T L l [ l V x i ^ [ ^ . From (4.F.20), (4.F.12), (4.F.26), (4.E.30, (2.43) and (4.F.10) we f i n d a lso that (4.F.34) <»l (TK-^) = ^  5 - f f _ _ ± _ _ - 1 4- ^ w M k . 7 ~ L I I V x x , VI and (4.F.35) 60. In the same way from (4.F.23) and (k.F.2j) we f i n d + and (^•F-37) . We now specia Iize our result to the system in which we are a c t u a l l y interested. We are interested in the system of a. single electron bound to an impurity ion. Since in actual experiments what one usually measures is ab-sorption from the ground state, we assume that the electron is in the ground state. |f we label the ground state by Q(_ , assuming t h i s state to be non-degenerate, t h i s means that we can take C+.F.38) 1 ^ = 1 > f ^ " * ^ We mentioned also in chapter II that rather than look at the whole spectrum of lines we would concentrate instead on a p a r t i c u l a r line of the spectrum. We therefore focus our attention on a l i n e peaked say at t o ^ l ^ - T ^ . Making use of (2 .17) and ( 4 . F . 2 9 ) and remembering that the excited state may be degenerate we write (k. F. 39) f L % ) 'fluT^u ^[Q^-^^Ur^u-y.)] Z 7 T X [ w ^ T V > « ^ ^ 61. where t h e summation over \ e x t e n d s over s t a t e s A such t h a t (k.F.kO) \-TL We now d i s t i n g u i s h two c a s e s . Case I: The e x c i t e d l e v e l i s nondegenerate. In t h e case f o r which t h e e x c i t e d l e v e l ^ i s n ondegenerate, the l i n e shape f u n c t i o n f o r the- l i n e peaked at d J i i T ^ - T ^ i s a m o d i f i e d L o r e n t z i a n c u r v e g i v e n by T h i s e x p r e s s i o n i s v a l i d f o r a l l v a l u e s of CO t o lowest o r d e r i n . in p a r t i c u l a r , i n t h e wings where t h e l i n e shape f u n c t i o n can be w r i t t e n At t h e peak p o s i t i o n , (k.F.kl) can be a p p r o x i m a t e d by a L o r e n t z i a n c u r v e s i nee ^C^C^+^-p) and K^T^*- W -M^ a r e s l o w l y v a r y i n g f u n c t i o n s so t h a t where t h e f u l l w i d t h a t h a l f power tsu> i s g i v e n by. 62. From (4.F.43) we see that the peak po s i t i o n is (k.F.k5) Cd0 ~ ^ K ^ c^-At) + <^U^C - ^ ' Making use of (4.F.32) and (4.F.36) as well as (4.F.38) we can write ^ 0 (4.F.46) In general, the diagonal terms |V\\K.) a r e m u c h greater than the nondiagonal ones I V^i^l^CX^/W-) • If we ignore the nondiagonal terms we get for-the peak pos it ion < ^ 7 > C±~"t-X 'I s where We note that the s h i f t of the line as given here is temperature independent. The nondiagonal elements which we have neglected, however, have a temperature dependence through the phonon occupation numbers . The result (4.F.47) agrees with the line s h i f t obtained by Nishikawa and Barrie. From (k.F.kk) we obtain, with the aid of (4.F.37) and (4.F.33), that the f u l l width at half power is given by (k.F.kS) * where we have made use of the r e l a t i o n (4.F.18). The line width given by (k.F.kS) is exactly the same as that calculated by Nishikawa and Barrie as well. We see, therefore, that the broadening and the s h i f t of the line are 63. due t o a l i f e t i m e e f f e c t as sugg e s t e d by Kane s i n c e t h e w i d t h depends c r i t i c a l l y on t h e p o s i t i o n o f the peak. In f a c t , i f one b e g i n s by assuming a l i f e t i m e e f f e c t , one can o b t a i n (4.F.48) and (k.F.kS) from t h e We i s s k o p f f and Wigner (1930) t h e o r y o f n a t u r a l l i n e w i d t h s . One f i n d s u s i n g t h i s t h e o r y t h a t i s j u s t t he t r a n s i t i o n p r o b a b i l i t y f r o m the s t a t e p t o a l l o t h e r s t a t e s . We see, a l s o , from (k.F.kS) t h a t t h e w i d t h of t h e l i n e i s t e m p e r a t u r e dependent, depending on the t e m p e r a t u r e t h r o u g h t h e phonon o c c u p a t i o n numbers. At z e r o t e m p e r a t u r e we have t h a t l)g—b and t h e l i n e w i d t h i s g i v e n by (4.F.50) A £ > ( T - - O ) - - . T T ^ Z I Z I ( W r^cn-T^-^) s i n c e oC i s t h e ground s t a t e . The l i n e w i d t h i s t h e r e f o r e n o n - v a n i s h i n g a t z e r o t e m p e r a t u r e and i s due e n t i r e l y t o t h e b r o a d e n i n g o f t h e e x c i t e d s t a t e . Case I I : E x c i t e d s t a t e a p p r o x i m a t e l y d o u b l y d e g e n e r a t e . We l a b e l t he n e i g h b o u r i n g e x c i t e d s t a t e s by ^  and S* . From (4.F.39) we have t h a t near t h e peak t h e 1ine shape f u n c t i o n i s g i v e n by . We see t h a t i n t h i s case the l i n e i s a s u p e r p o s i t i o n o f two L o r e n t z i a n c u r v e s . The s e p a r a t i o n of t h e two peaks i s g i v e n by (4- F. 52) R2- K'^CT^) - K l K ^ C T r ^ ^ " " V 6k. and the r e l a t i v e h e i g h t s of t h e peaks i s g i v e n by 1 i ^ D ^ c r ^ + fLcw] 6 5 . G. D i s c u s s i o n o f h i g h e r o r d e r c a l c u l a t i o n . A l l o f t h e r e s u l t s which we have o b t a i n e d so f a r a r e based on t h e as s u m p t i o n t h a t the c o u p l i n g c o n s t a n t ( < « [ , and as such a r e c o r r e c t o n l y t o o r d e r !<.*" • The f a c t t h a t t h e c o u p l i n g c o n s t a n t i s s m a l l has p e r m i t t e d us t o make a number of s i m p l i f y i n g a s s u m p t i o n s and a p p r o x i m a t i o n s . In t h e o r y i t s h o u l d be p o s s i b l e t o o b t a i n a s o l u t i o n u s i n g t h i s method f o r t h e case when ifC i s not s m a l l , but i n p r a c t i c e t h i s i s as y e t i m p o s s i b l e . Perhaps the most s e r i o u s a p p r o x i m a t i o n which was made i n t h e p r e c e e d i work i n o r d e r t o s i m p l i f y t h e summation of t h e diagrams was t h e S a l p e t e r -Bethe t y p e a p p r o x i m a t i o n d i s c u s s e d i n s e c t i o n D of t h i s c h a p t e r . T h i s app-r o x i m a t i o n i s v a l i d i n t h e case which we have d i s c u s s e d s i n c e t he diagrams which were n e g l e c t e d by u s i n g t h i s a p p r o x i m a t i o n would not have c o n t r i b u t e d t o t h e s h i f t and t o t h e w i d t h o f the l i n e t o o r d e r fc2" . These d i a g r a m s , however, do c o n t r i b u t e t o o r d e r . T h e r e f o r e , i f the c o u p l i n g c o n s t a n t were not s m a l l t h e a p p r o x i m a t i o n would no l o n g e r be v a l i d and i n s t e a d o f the Dyson e q u a t i o n (k.0.7) f o r t h e one p a r t i c l e e l e c t r o n Green's f u n c t i o n we would have t o w r i t e where | OfVfc^j jt'o j /?, Yx ) i s a v e r t e x p a r t and i s t h e sum of a l l diagrams w i t h two e x t e r n a l e 1 e c t r o n I i r i e s and one e x t e r n a l phonon l i n e such t h a t d i a g r a m a t i c a l l y f l o o k s l i k e 66. U n f o r t u n a t e l y , j u s t as i n quantum e l e c t r o d y n a m i c s , i t i s as yet i m p o s s i b l e t o w r i t e an e q u a t i o n f o r I""7 i n c l o s e d form. Landau e t a l . (1956) were a b l e t o o b t a i n a c l o s e d e x p r e s s i o n f o r t h e v e r t e x p a r t i n quantum e l e c t r o d y n a m i c s by n e g l e c t i n g diagrams of t h e ty p e i i i but i t t u r n s out t h a t t h i s a p p r o x i m a t i o n i s c o n s i s t e n t o n l y i f t h e c o u p l i n g c o n s t a n t ( e l e c t r o n i c charge) i s i d e n t i c a l l y z e r o . We a l s o have t o w r i t e f o r the phonon Green's f u n c t i o n ; i n s t e a d o f (4.D .8) (4. G. 2) W / J - Jb^OC.fi'; t-^ j--jW> J t f ^ . ^ b'0)tf-ft j r - r ^ y f e - . In a d d i t i o n t o t h e c o m p l e x i t i e s i n t r o d u c e d by t h e v e r t e x p a r t , i f the c o u p l i n g c o n s t a n t i s not s m a l l so t h a t a - s o l u t i o n t o o r d e r K2" i s not s u f f i c i e n t we cannot make the same a p p r o x i m a t i o n s which we made i n s o l v i n g the i n t e g r a l e q u a t i o n s but we must i n s t e a d s o l v e t h e e q u a t i o n s as a set of c o u p l e d e q u a t i o n s . We c o u l d a l s o not n e g l e c t t h e terms which have been n e g l e c t e d i n w r i - t i n g (k.E.2S) and ( 4.E . 3 8 ) , i f t h e c o u p l i n g c o n s t a n t i s not s m a l l but i n s t e a d we would have t o t a k e account o f some of the h i g h e r o r d e r terms. The Dyson e q u a t i o n which we w r o t e f o r t h e two p a r t i c l e t e m p e r a t u r e Green's f u n c t i o n i s n o . l o n g e r c o r r e c t even w i t h t h e i n c l u s i o n o f a v e r t e x p a r t s i n c e we have 67-n e g l e c t e d diagrams o f t h e t y p e > s •> (4.G.3) < — ^ As y e t t h e r e has been found no s a t i s f a c t o r y method f o r t a k i n g such diagrams i n t o a c c o u n t . If one wanted t o o b t a i n a s o l u t i o n t o o r d e r k.t one c o u l d perhaps a p p r o x i m a t e t h e v e r t e x p a r t by t a k i n g o n l y the f i r s t few diagrams and t a k i n g a c c o u n t of diagrams l i k e (U.G.3) t o lowest o r d e r . With a c o n s i d e r a b l e amount of l a b o u r one c o u l d i n t h i s way o b t a i n a s o l u t i o n of t h e i n t e g r a l e q u a t i o n s t o o r d e r . For t h e case where K i s so l a r g e t h a t a c a l c u l a t i o n t o o r d e r K4" i s not s u f f i c i e n t , t h e problem as y e t seems t o be h o p e l e s s . 68. CHAPTER V: COMPARISON OF METHODS In t h i s c h a p t e r we s h a l l compare the method employed by N i s h i k a w a and v B a r r i e f o r i n v e s t i g a t i n g t h i s p roblem w i t h t h e method employed here. In s p i t e of t h e f a c t t h a t t h e s t a r t i n g p o i n t s i n t h e two c ases a r e t h e same, both s t a r t i n g w i t h K ubo 1s f o r m u l a t i o n of t h e s u s c e p t i b i l i t y t e n s o r , the methods of c a l c u l a t i o n a r e c o n s i d e r a b l y d i f f e r e n t . The p r e v i o u s a u t h o r s r e l a t e the f u n c t i o n ^)C(co) g i v e n i n (2.8) t o a t e m p e r a t u r e dependent doub 1 e-t ime Green 1 s f u n c t i o n , ^ f \ ( S ^ , f o r two o p e r a t o r s p\ and (3 , which i s a two branch a n a l y t i c f u n c t i o n o f E~ de-f i n e d everywhere o u t s i d e t h e r e a l a x i s by <3TTJ o The e x p r e s s i o n f o r J w c E ^ . O i s u s u a l l y c a l l e d the advanced Green's f u n c t i o n and *>0 i s c a l l e d the r e t a r d e d Green's f u n c t i o n . In terms of t h e Green' f u n c t i o n (5.1), X M was w r i t t e n (5.2) Making use of t h e e x p r e s s i o n f o r t h e d i p o l e moment o p e r a t o r , . the a b s o r p t i o n c o n s t a n t was w r i t t e n (5.3) c r M - covwi:. to ZL 7L Zj M**. fV \^X^MW ^ ^ a where A'A 69-We see, t h e r e f o r e , t h a t i n both methods the problem i s reduced t o c a l c u l a t i n g t h e a p p r o p r i a t e Green's f u n c t i o n . N i s h i k a w a and B a r r i e made use of the f a c t t h a t t h e Green's f u n c t i o n r^VlB/^ : s a t i s f i e s t h e e q u a t i o n o f motion ( 5 . 5 ) E C A I s ^ - j = < C M ) > +• <Cft,H3\B^ Making use of t h i s e q u a t i o n , they o b t a i n e d an i n f i n i t e h i e r a r c h y o f c o u p l e d e q u a t i o n s . The method o f s o l u t i o n c o n s i s t s i n d e c o u p l i n g t h e h i e r a r c h y o f e q u a t i o n s . The main d i f f i c u l t y i n t h i s approach c o n s i s t s o f j u s t i f y i n g t he d e c o u p l i n g . These a u t h o r s p e rformed the d e c o u p l i n g by r e l a t i n g t h e Green's f u n c t i o n (zr^LS) > which, i s j u s t ( 5 - 4 ) w i t h /J«x' and jU.*-\ , t o Green's f u n c t i o n s I i ke frtfQ. ^ (k | l^G-K^^ by n e g l e c t i n g terms o f the o r d e r o f K,3(^\.x'££) a n C ' ^ Gr^/j.'anc' h i g h e r o r d e r i n the c o u p l i n g c o n s t a n t as w e l l as n e g l e c t i n g terms o f o r d e r -L } where J\f* i s the number o f phonon s t a t e s f o r which thfe. m a t r i x e l e m e n t s l/^'t? ' n the i n t e r a c t i o n H a m i l t o n i a n a r e not equal t o z e r o . T h i s l a t t e r a p p r o x i m a t i o n a l l o w e d them t o w r i t e ^ k & t ^ K ^ ^ a s y ^ a ^ l d ^ ^ - S i n c e 4^a^\&^K>B i s of o r d e r K^Gs^iS.) f o r and the e q u a t i o n s i n which t h e s e terms appear a r e a l r e a d y of o r d e r < , the terms f o r which £ sjfc x/ c o u l d be n e g l e c t e d and an e x p r e s -s i o n f o r _r>^i (S) w a s o b t a i n e d i n c l o s e d form. The n e g l e c t of terms o f o r d e r was j u s t i f i e d by u t i l i z i n g t h e c o n t r a c t i o n theorem of Blo c h and D o m i n i c i s (1958) and the r e m a i n i n g d e c o u p l i n g was j u s t i f i e d by the s m a l l n e s s of t h e c o u p l i n g c o n s t a n t . It i s i n t e r e s t i n g t o see how the Green's f u n c t i o n i n t r o d u c e d by th e s e a u t h o r s can be e x p r e s s e d i n terms o f diagrams. The Green's f u n c t i o n G r ^ ( B \ can be r e l a t e d t o a c a u s a l Green's f unct i on ^($^fc$^tjt)^%^ where ~T~ i s a ti m e o r d e r i n g o p e r a t o r which has the same p r o p e r t y as the o p e r a t o r d e f i n e d 70. i n (2.28) and the o p e r a t o r s a r e d e f i n e d i n t h e H e i s e n b e r g r e p r e s e n t a t i o n . On t r a n s f o r m i n g t o the i n t e r a c t i o n r e p r e s e n t a t i o n we can w r i t e (5.6) < T ^ + W £ v t t ) « i + A x ) > - < T ( f i . x + l t l & v ftS&»fl>c-where (5.7) S(D O) -~T^.^ - [Vp(tUt-| If we e x p r e s s the f u n c t i o n b y (5.8) < T ( 0 | l t ) ^ ( o ] ) > = ^ t h e f u n c t i o n (5-6) can be e x p r e s s e d d i a g r a m a t i c a 1 Iy as t h e sum of a l l diagrams o f the form shown i n F i g u r e 5-1 where the d o u b l e l i n e s s i g n i f y t h a t s e l f energy p a r t s have been i n c l u d e d . +-FIGURE 5. 1 We see t h a t t he e v a l u a t i o n o f the Green's f u n c t i o n (5-6) i n v o l v e s summation of diagrams o f t h e same t o p o l o g i c a l form as t h e ones which we were r e q u i r e d t o e v a l u a t e . The Green's f u n c t i o n ^X(l^[t)^(±)A^Lt)^Lt)^K)^y c a n b e w r i t t e n as the sum of a l l diagrams o f the form shown i n F i g u r e 5-2.where diagrams 1 i ke 71 and are neglected, since they are of order — — which can be seen by using the contraction theorem of Bloch and Dominicis. FIGURE 5.2 4 -+ 4-From Figure 5-2 we see that the sum of diagrams for the Green's function < T ^ ) ^ t W ^ ) % W ^ f ^ > i s J " u s t e £> u a I t o the"bubble" ^ V times the sum of diagrams i.n . Figure 5.1. The funct ion ^ (£^fel^fct)«jfk)2^(*)&.jJ&xY> can therefore be approximated by We see, therefore, that the process of evaluating the o r i g i n a l Green's function (5.6) is equivalent to. performing a Dyson type summation of the diagrams in Figure 5.1. Our method of handling the problem was quite d i f f e r e n t . We defined a d i f f e r e n t Green's function from that defined by the previous authors by means of (2.27) and calculated t h i s function by methods borrowed from quantum f i e l d \ 72. t h e o r y . We t r a n s f o r m e d t o the i n t e r a c t i o n r e p r e s e n t a t i o n , made use of Wick's theorem and a v a i l e d o u r s e l v e s o f t h e Feynman dia g r a m t e c h n i q u e t o r e p r e s e n t the v a r i o u s terms of t h e p e r t u r b a t i o n s e r i e s . The problem i n t h i s case i s t o p i c k out from the i n f i n i t y of diagrams a sequence o f " p r i n c i p a l t e r m s " which exceed t h e o t h e r terms i n magnitude•and t o sum t h e c o n t r i b u t i o n s from t h e s e diagrams. A f u r t h e r problem i s t h e s o l u t i o n o f the v a r i o u s c o u p l e d i n t e g r a l equat i ons. We have seen t h a t t h e a b s o r p t i o n c o n s t a n t o~(u)') can be r e l a t e d t o the F o u r i e r t r a n s f o r m o f the f u n c t i o n I A A1 p- hi ' N i s h i k a w a and B a r r i e s t a r t e d w i t h t h e F o u r i e r t r a n s f o r m of t h e f u n c t i o n <^(K^(t)v^xi(t)&^(o)^M/{o))y and made use of an e q u a t i o n of mo t i o n f o r t h e t r a n s f o r m of t h i s f u n c t i o n d e r i v e d by t a k i n g t he f i r s t t i m e d e r i v a t i v e . T h i s p r o c e d u r e g i v e s a s a t i s f a c t o r y e q u a t i o n o f motion s i n c e t he p r o d u c t o f o p e r a t o r s • uSj^(-t) &)|(-tj does not commute w i t h j-j^p • The f u n c t i o n which we chose t o work w i t h was the Green's f unct ion 'PfffytyOPfafy- , t : s ^ o u ' d b e poss-i b l e t o t r e a t t h i s f u n c t i o n by some e q u a t i o n of motion as w e l l . However, i f we c o n s i d e r t h e f i r s t V d e r i v a t i v e o f t h e f u n c t i o n , we f i n d t h a t we need t o know J^npfoytO $OC^)) which i s r e l a t e d t o [^fytffi'tfU?^ r t f r V ] • • However, -t^Ov^f) ipOCj'^ n commutes w i t h e v e r y t h i n g i n the H a m i l t o n i a n e x c e p t . We, t h e r e f o r e , cannot o b t a i n a u s e f u l e q u a t i o n of motion f o r t h i s Green's f u n c t i o n by c o n s i d e r i n g o n l y t h e f i r s t <f d e r i v a t i v e . In o r d e r t o o b t a i n a s a t i s f a c -t o r y e q u a t i o n of. motion we would have t o c o n s i d e r h i g h e r d e r i v a t i v e s which i s a n . e x t r e m e l y awkward p r o c e d u r e . 73-We now i n v e s t i g a t e t h e r e l a t i o n s h i p between t h e Green's f u n c t i o n which we have i n t r o d u c e d and t h e one employed by t h e p r e v i o u s a u t h o r s . We c o n s i d e r t h e d o u b l e - t i m e r e t a r d e d Green's f u n c t i o n (5-9) - -_L_ J3T where the o p e r a t o r s a r e here d e f i n e d i n the H e i s e n b e r g r e p r e s e n t a t i o n so t h a t (5.10) The second e x p r e s s i o n i n (5-9) f o l l o w s from the f a c t t h a t t h e - p r e v i o u s a u t h o r s worked i n the c a n o n i c a l ensemble. In terms of t h e e i g e n v a l u e s and t h e e i g e n f u n c t i o n s of the H a m i l t o n i a n (2.2k), (5-9) can be w r i t t e n (5.11) w h i c h on making use of t h e r e l a t i o n i (5.12) becomes (5-13) 11 On r e l a b e l l i n g v a r i a b l e s t h i s can be w r i t t e n as S (5-14) ( E ) - ^ - f c £ T T | where 7k. ( 5 . 1 5 ) f(E) - ^ Z. 2__ ^ ^ < ^ l < ^ | M > < M t a X ^ > [ e ^ " ^ - L J ^ E ^ E J and ^ -=? From (2.40) we see t h a t t he Green's f u n c t i o n which we used i n t h e pres-ent work can be e x p r e s s e d as it* where , (5.17) . [ £ ^ " ^ - i ] ^ a ' ^ ^ E ^ - £ j . Making use of (2 . 4 8 ) , ( 2 . 5 8 ) , (2.59) and (3.C.12) we have where TXJZ.TL M^ ,MoK L-I L e ^ ^ y (5.19) ^ I ^ V ' ^ ^ I ^ A ^ ^ ^ ' ^ l ] & ( W E»rE^ . On comparing (5-19) w i t h (5-15) we see t h a t we cannot d e f i n i t e l y r e l a t e the two Green's f u n c t i o n s s i n c e one i s d e f i n e d u s i n g t h e c a n o n i c a l ensemble and t h e o t h e r u s i n g t h e grand c a n o n i c a l ensemble. If we had chosen t o work i n the c a n o n i c a l ensemble r a t h e r t h a n i n t h e more c o n v e n i e n t grand c a n o n i c a l -.0) ensemble, however, we would have been a b l e t o w r i t e t h e r e l a t i o n L. A A' 'U- Mi The r e l a t i o n s h i p between r e t a r d e d Green's f u n c t i o n s and t e m p e r a t u r e Green's f u n c t i o n s i s d e r i v e d on page 148 of the book by A b r i k o s o v e t a! 75. where ^^CS) i s the a n a l y t i c c o n t i n u a t i o n o f j^Lii^^) i n t o t he upper h a l f o f the complex p l a n e . From (5.20) we see t h a t t h e r e i s some c o n n e c t i o n between t h e temper-a t u r e Green's f u n c t i o n and t h e d o u b l e - t i m e t e m p e r a t u r e dependent Green's f u n c t i o n . From t h e d i s c u s s i o n f o l l o w i n g (5-6) i t i s e v i d e n t t h a t a p r o p e r d e c o u p l i n g i n t h e d o u b l e - t i m e Green's f u n c t i o n method by t h e e q u a t i o n of motion approach i s e q u i v a l e n t t o summing the most i m p o r t a n t Feynman diagrams i n our t e c h n i q u e . in t he case where r e a l t i m e Green's f u n c t i o n s a r e b e i n g c o n s i d e r e d ( i . e . t h e work of S a l p e t e r and B e t h e ) , the diagrams which one o b t a i n s can be i n t e r p r e t e d i n terms of a c t u a l l y o c c u r r i n g p h y s i c a l p r o c e s s e s . !n t h i s case t h e a p p r o x i m a t i o n s which a r e made i n o r d e r t o s i m p l i f y t h e r e s u l t i n g s t r u c -t u r e can be j u s t i f i e d on the b a s i s o f p h y s i c a l c o n s i d e r a t i o n s - n e g l e c t of c e r t a i n diagrams c o r r e s p o n d s t o the n e g l e c t o f t h e a s s o c i a t e d p h y s i c a l p r o c e s s e s . S i n c e i nCour.L t e c h n i que employ i n g " i m a g i n a r y t i m e " Green's f u n c t i o n s the r e s u l t i n g diagrams and m a t h e m a t i c a l f o r m a l i s m a r e i d e n t i c a l w i t h the r e a l t i m e f o r m a l i s m e x c e p t t h a t ~fc i s r e p l a c e d by i f , we s h o u l d be a b l e , t o n e g l e c t the same k i n d s o f diagrams as i n the case o f r e a l t i m e s . The hope was t h a t t h e p r e s e n t t e c h n i q u e i n which the n e g l e c t o f c e r t a i n diagrams can be j u s t i f i e d , by a n a l o g y w i t h the n e g l e c t o f c e r t a i n p h y s i c a l p r o c e s s e s i n the r e a l t i m e c a s e , would s h i n e some l i g h t on the d e c o u p l i n g p r o c e d u r e - i n o t h e r problems of t h i s t y p e which c o u l d be h a n d l e d i n ' t e r m s of d o u b l e - t i m e r e t a r d e d and advanced Green's f u n c t i o n s . T h i s hope i s not r e a l i z e d because of the d i f f i c u l t y of r e l a t i n g d i f f e r e n t terms i n the h i e r a r c h y of c o u p l e d e q u a t i o n s t o diagrams i n our t e c h n i q u e . T h i s d i f f i c u l t y i s f u r t h e r compounded by t h e f a c t t h a t i n our t e c h n i q u e t h e sums of s e r i e s of diagrams a r e r e p r e s -76. e n t e d as i n t e g r a l e q u a t i o n s which we were not a b l e t o s o l v e e x a c t l y . The app-r o x i m a t i o n s made i n s o l v i n g t h e s e e q u a t i o n s do not r e a d i l y l e n d t h e m s e l v e s t o a p h y s i c a l i n t e r p r e t a t i o n but r a t h e r a r e a n e g l e c t o f h i g h e r o r d e r terms i n the c o u p l i n g c o n s t a n t . The r e l a t i v e m e r i t s o f t h e two methods employed f o r t h i s problem a r e open t o q u e s t i o n . !n t h e method which we employed, i t' 4 ijs much e a s i e r t o see t h e p h y s i c a l s i g n i f i c a n c e of some of the a p p r o x i m a t i o n s which a r e made s i n c e by l o o k i n g at a diag r a m which has been n e g l e c t e d one can see what s o r t o f " p h y s i c a l p r o c e s s " has been n e g l e c t e d . On the o t h e r hand, i n t'he e q u a t i o n of motion approach the p h y s i c a l a p p r o x i m a t i o n s a r e a s s u m p t i o n s about the n e g l e c t of c e r t a i n c o r r e l a t i o n s and a r e , i n g e n e r a l , o b s c u r e d i n a c i o u d o f mathem= a t i c a l f o r m a l i s m . We f e e l t h a t f o r s m a l l c o u p l i n g c o n s t a n t t h e Matsubara method of t e m p e r a t u r e Green's f u n c t i o n s w i t h the a i d of the Feynman diagram t e c h n i q u e i s a s u p e r i o r method of h a n d l i n g a problem o f t h i s t y p e . However, in t h e case where t h e c o u p l i n g c o n s t a n t i s not s m a l l a n f one has t o o b t a i n a s o l u t i o n t o h i g h e r o r d e r i n the c o n s t a n t i t i s much e a s i e r t o o b t a i n a s o l u t i o n u s i n g the e q u a t i o n o f motion approach a l t h o u g h the j u s t i f i c a t i o n of t h e s o l u t i o n may be e x t r e m e l y d i f f i c u l t . 77. CHAPTER V i : SUMMARY Starting with K U D O ' S formula for the adiabatic d i e l e c t r i c s u s c e p t i b i l i t y tensor, we have used the Matsubara technique of temperature dependent two p a r t i c l e Green's functions to obtain the shape of absorption lines due to electrons bound to impurity ions in semiconductors. Assuming the interaction between the electrons and the l a t t i c e v ibrations to be weak, we have been able to obtain a solution to the Dyson equation for the two p a r t i c l e Green's function correct to second order in the coupling constant. In t h i s way we have calculated the l i n e shape function. We have also compared our method of solution with the method employed by Nishikawa and Barrie to study the same problem and we have indicated the r e l a t i o n between the two methods. The results which we obtained agree with those obtained by Nishikawa and Barrie. The line shape function obtained was a modified Lorentzian curve (or a superposition of such) which near the peak could be approximated by a pure Lorentzian curve. We found a s h i f t of the absorption;1ine which was independent of temperature, and a temperature dependent line width which depends c r i t i c a l l y on the position of the peak. This indicates that the broadening of the line at the peak is due primarily to a l i f e t i m e e f f e c t rather than to multi-phonon processes. Away from the peak, however, multi-phonon processes predominate as can be seen from (k.F.kl). These results then are contrary to those of Lax and Burstein and Kubo and Toyazawa and indicate that Kane:',yS interpretation that the broadening at the peak is due to a l i f e t i m e e f f e c t is correct. 78. Our method of h a n d l i n g the problem has t h e same advantages as the method of N i s h i k a w a and B a r r i e over t h e methods of p r e v i o u s a u t h o r s who merely c a l c u l a t e d the moments o f t h e a b s o r p t i o n l i n e s . R a t h e r , we have been a b l e t o o b t a i n the l i n e shape f u n c t i o n i t s e l f . Jn f a c t , N i s h i k a w a and B a r r i e showed t h a t t h e wings r a t h e r than the peak g i v e a g r e a t e r c o n t r i b u t i o n t o the moments so t h a t the moments a r e of l i t t l e v a l u e i n d e t e r m i n i n g t h e l i n e w i d t h . T h i s i s i n agreement w i t h G r a n t ' s r e s u l t t h a t a knowledge of the moments y i e l d s no i n f o r m a t i o n about t h e f u n c t i o n w i t h i n any f i n i t e i n t e r v a l . S i n c e we have not needed t o f i n d e x a c t e i g e n s t a t e s o r even a p p r o x i m a t e e i g e n s t a t e s of the H a m i l t o n i a n o f t h e system, we have been a b l e t o a v o i d a p p r o x i m a t i o n s such as the a d i a b a t i c a p p r o x i m a t i o n or the Condon a p p r o x i m a t i o n which were n e a r l y a lways used p r e v i o u s l y . N i s h i k a w a and B a r r i e have f o r m u l a t e d t h e s e a p p r o x i m -a t i o n s w i t h i n t h e framework of the p r e s e n t t h e o r y and have d i s c u s s e d t h e i r n a t u r e . 79-APPENDIX !: J U S T I F I C A T I O N OF (4.E.29) In t h i s a p p e n d i x we s h a l l j u s t i f y our n e g l e c t o f h i g h e r o r d e r terms i n w r i t i n g (4.E.29) by showing t h a t t h e terms which we have n e g l e c t e d w i l l g i v e a c o n t r i b u t i o n t o the l i n e shape f u n c t i o n which i s much s m a l l e r than t h a t o b t a i n e d i n (4 . F . 2 9 ) - The same p r o c e d u r e which we o u t l i n e here can be used t o j u s t i f y t he n e g l e c t o f the second term on t h e r i g h t hand s i d e o f (4 . F . 8 ) . The lowest o r d e r term which we n e g l e c t e d i n w r i t i n g (4.E.29) i s from (4.E.28) (A. I ) J&uffi%'ii&) = - ^ I L < & C ? ) f f i V ) F K , ( i ! u U  where L\(('ui\0 i s g i v e n by (4.E.31) and ^ ( L ' i X J ) i s g i v e n by (4.E.22). From <4.F.5), we see t h a t t h e lowest o r d e r c o n t r i b u t i o n t o t h e two p a r t i c l e Green's f u n c t i o n * i s g i v e n by In o r d e r t o f i n d t h e lowest o r d e r c o n t r i b u t i o n o f (A..T) t o t h e two p a r t i c l e Green's f u n c t i o n we need s u b s t i t u t e (A.'i) f o r one of the Green's f u n c t i o n s in t h e product; on t h e r i g h t hand s i d e of ( A .2) and s u b s t i t u t e from (4.E.30) f o r t he o t h e r Green's f u n c t i o n . P e r f o r m i n g t h i s o p e r a t i o n we get ( A . 3 ) ^ -'T^51^ ^w^w^m^Lt^) S i n c e L ^ i ' t X , ^ i s s l o w l y v a r y i n g we can w r i t e ( A .4) jfta^ ^ -£zzz. z. HI'^A' RIUS^) ' i 0 ] P> A A' - s i x « ^ -, r ... * 80. + b, ? (^A _ _ _ _ _ _ where we have made use of (2.48) as well. We shall make use only of the f i r s t term on the right hand side of (A,k). The second term has the identical form and the argument would proceed in the same way. We f i r s t transform the sum-mat ion over to an integration in the 2r -plane by means of (3 • C 5) so that (A.5) £ ^ * d £ _ . i ; _ M^>V where the contour C/ is the same as that shown in Figure 4.7••"We therefore use the same procedure we used in writing (h.F.Zk) to write F , ^ - ^ ) [x~Tx-t-/U 4 - | c * L - x j X+aDf[ [ x - u J - T ^ + M + ^ L x i U - u w ^ Y ] - r _ ^ * ± ^ ) 7 . [x +• iO-TX+-M -t- K^LxCx+ u)+-C£)"} fx-T^i +• M + L > | i i Making use of (4.F.12) and ( 4 . F . 2 I ) as well as (A-7) F ^ ; ( X . ^ 6 ) - { ^ ( x V T c ^ M we have that ( A " 8 ) F ^ X w D ) . F x ^ O C + M ~ T X 4-M.+-K?" L x ( X - l C X - T x t M + X x L x t X > i v>\} 81. T h i s f u n c t i o n i s s t r o n g l y peaked at X — "T^-M-K-1 L-xCx) • T h e r f o r e , f o r purposes o f i n t e g r a t i o n i n (A.6) we can w r i t e t h i s as ( A • 9 ) -zc^rLcrx-M^F^CTy^  T h e r e f o r e , f o l l o w i n g t he same p r o c e d u r e as we used i n o b t a i n i n g (k.F.ZS), we f i n d t h e lowest o r d e r c o n t r i b u t i o n t o the l i n e shape f u n c t i o n o f t h e Green's f u n c t ion (A.1) i s (A. 10) - j — . — (—. r _ " ? * X Ox-Tf') C W + T X . ~ T X 4 - £ U . x ( T X I + ^ ^ ^ +-J 2T x x1 On comparing t h i s w i t h ( 4 . F . 2 9 ) , we see t h a t (A.10) i s of o r d e r £ z as l a r g e as h(<^l c a l c u l a t e d p r e v i o u s l y . In a d d i t i o n t h e f u n c t i o n s £Ty -juS) and h^ ^ (Tx'A.^ ' n (A. 10) a r e s m a l l s i n c e t h e y c o n t a i n terms, l i k e V^i^ a n ^ "^A'X'fe^ "t1 k w n e r e \ t X i n the summation 7__ Vl'Aik. l/|x.k. t h e s e terms a r e much l e s s than u n i t y . From (A.10) i t can be seen t h a t t h e peak p o s i t i o n i s the same as t h a t g i v e n by (k.F.k7). S i n c e t he a m p l i t u d e i s l e s s than H 1 t i m e s the a m p l i t u d e which we had b e f o r e , (A.10) cannot c o n t r i b u t e t o t h e w i d t h o f the peak t o o r d e r |< . 82. BIBLIOGRAPHY Abrikosov, A.A., Gorkov, L.P. and Dzya1oshinski, I.E. 1963. Methods of Quantum F i e l d Theory in S t a t i s t i c a l Physics. (Prentice-Hall, Inc., Englewood C l i f f s , N.J.). Bardeen, J. and Shockley, W. 1950. Phys. Rev. 80, 72. Baym, G. and Mermin, N.D. 1961 . J. Math. Phys. 2, 232. Bloch, C. and Dominicis, C. 1958. Nuclear Phys. 7, 459. Grant, W. J. C. 1964. Physica, 30,. 1433-Kane, E.O. I960. Phys. Rev. JJ_9, 40. K i t t e l , C. 1963. Quantum Theory of Solids. (John Wiley and Sons, Inc., New York) p. 130. Kohn, W. 1957. Solid State Phys. (Seitz and Turnbull) v o l . 5, P- 257-Kubo, R. 1957- J- Phys. Soc. (Japan), J_2, 570 Kubo, R. and Toyazawa, Y. 1955- Progr. Theoret. Phys. (Kyoto) ,, J_3, 160. Landau, L.D., Abrikosov, A.A. and Khalatnikov, I.M. 1956. Nuovo Cimento 13 "Suppl . 3, 80. Lax, M. and Burstein, E. 1955- Phys. Rev. J_00, 592. Luttinger, J..M. and Ward, J.C. I960. Phys. Rev. J_18, 1417-Maradudin, A.A. 1964. Ann. Phys. _3_0, 371-Matsubara, T. 1955- Progr. Theoret. Phys. (Kyoto), 14, 351. 83-Nishikawa, K. and Barrie, R. 1963- Can. J. Phys. .1135-Sa1 peter, E.E. and Bethe, H.A. 1951. Phys. Rev. 84, 1232. Sampson, D. and Margenau, H. 1956. Phys. Rev. J_03, 879. Sommerfeld, A. and Bethe, H. 1933- Hanbuch der Physik, Vol. 24, Part 2, p.333-Weisskopf, V. and Wigner, E. 1930. Zs. f. Phys. 63, 54. 

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