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Static dielectric screening and exchange in the layered electron gas Glaus, Ulrich Walter 1980

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STATIC DIELECTRIC SCREENING AND EXCHANGE IN THE LAYERED ELECTRON GAS / by. ULRICH WALTER GLAUS D i p l . M a t h . , . S w i s s F e d e r a l I n s t i t u t e o f T e c h n o l o g y , Z u r i c h 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n \ I " » THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA J u n e , 1980 (_) Ulrich Walter Glaus, 1980 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t n f Physics  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e June 9 1980 ABSTRACT The s t a t i c s c r e e n i n g p r o p e r t i e s o f the l a y e r e d e l e c t r o n gas a r e s t u d i e d u s i n g a v e r y i d e a l i z e d m o d e l . Exchange and c o r r e l a t i o n have been i n c l u d e d t o f i r s t o r d e r . A g e n e r a l i z e d s u s c e p t i b i l i t y i s o b t a i n e d , w h i c h i s c o n s i s t e n t w i t h the r e s u l t s o f o t h e r a u t h o r s . The i n d u c e d c h a r g e d e n s i t y i n l i n e a r r e s p o n s e t o a p o s i t i v e p o i n t c h a r g e in one o f t he p l a n e s p r e d i c t e d by t h e new s u s c e p t i -b i l i t y e x h i b i t s s l o w l y d e c a y i n g l o n g range o s c i l l a t i o n s . F i n a l l y , a f o r m u l a f o r t h e s c r e e n e d p o t e n t i a l o f e x t e r n a l c h a r g e s between l a y e r s i s g i v e n , w h i c h m i g h t be o f some use in an a p p l i c a t i o n t o i n t e r c a l a t i o n compounds. T a b l e o f C o n t e n t s Page A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f f i g u r e s i v Acknow ledgemen ts v C h a p t e r 1 I n t r o d u c t i o n 1 II Random Phase A p p r o x i m a t i o n t o t he E f f e c t i v e I n t e r a c t i o n 5 I I I F i r s t O r d e r Exchange C o r r e c t i o n s t o t h e P o l a r i z a b i 1 i t y 16 IV The Response F u n c t i o n o r S u s c e p t i b i l i t y X 32 V S t a t i c S c r e e n i n g o f t he L E G 39 C o n c l u s i o n s R e f e r e n c e s 49 i v L i s t of F igures Fi gure Page 1 Diagram fo r n° 8 2 The n o n i n t e r a c t i n g p o l a r i z a b i 1 i ty 11° p l o t t e d as - n ° (q/k ) / n ° (0) versus q / k p 10 3 I t e r a t i o n diagram fo r the e f f e c t i v e i n t e r a c t i o n 11 A Diagrams fo r II1 17 5 Geometry fo r q < 2 23 6 Geometry fo r q > 2 28 7 The f i r s t order exchange c o r r e c t i o n n 1 30 8 The induced charge dens i t y p. . in c o n f i g u r a t i o n space 43 V Acknow ledgemen ts I am o b l i g e d t o D r . B i r g e r B e r g e r s e n who has i n i t i a t e d t h i s w o r k . He a l s o gave v a l u a b l e a d v i c e and c o n t r i b u t e d i d e a s t h r o u g h o u t i t s c o m p l e t i o n . The f i n a n c i a l s u p p o r t w h i c h I r e c e i v e d f r om t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a as an exchange s t u d e n t i s a l s o g r a t e f u l l y a c k n o w l e d g e d . 1 C h a p t e r One I n t r o d u c t i on The t h e o r y o f an i n t e r a c t i n g two d i m e n s i o n a 1 l a y e r e d e l e c t r o n gas (LEG) has r e c e n t l y been s t u d i e d and d e v e l o p e d by a c o n s i d e r a b l e number o f w o r k e r s and f rom a v a r i e t y o f d i f f e r e n t v i e w p o i n t s . The t h e o r i s t s i d e a l i z a t i o n o f ' j e l l i u m ' , w h i c h i n ou r c a s e r e p l a c e s t h e t w o - d i m e n s i o n a l p o s i t i v e - i o n l a t t i c e s by a u n i f o r m l y c h a r g e d and l a y e r e d j e l l y , s e r v e s as a u s e f u l s t a r t i n g p o i n t t o i n v e s t i g a t e t h e p h y s i c a l p r o p e r t i e s o f t h i s s t r o n g l y a n i s o t r o p i c s y s t e m . V i s s c h e r and F a l i c o v ( 1 9 7 0 ) f i r s t p roposed a m o d e l , w h i c h c o n s i s t s o f a s e r i e s o f e q u a l l y spaced p a r a l l e l p l a n e s . A f i n i t e t w o - d i m e n s i o n a l d e n s i t y o f e l e c t r o n s i s a l l o w e d t o move f r e e l y w i t h i n each l a y e r , but t u n n e l i n g between p l a n e s does no t t a k e p l a c e . T h i s model was c h o s e n f o r i t s r e s e m b l a n c e t o g r a p h i t e and g r a p h i t e -12. . i n t e r c a l a t i o n compounds. G r e c u ; ( 1 9 7 5 ) i n c l u d e d t u n n e l i n g i n an a t t e m p t t o s t u d y p lasma o s c i l l a t i o n s and s u r f a c e e f f e c t s . The c l a s s i c a l h y d r o d y n a m i c a 1 model o f a c h a r g e d f l u i d t r e a t e d by 13 F e t t e r " ( 1 9 7 4 ) d i d a l o t t o c l a r i f y how t h e g e o m e t r i c a l c o n f i g u r a t i o n o f t h e L E G , as an i n t e r m e d i a t e between t h e two - and t h r e e - d i m e n s i o n a l e l e c t r o n g a s e s , a f f e c t s t h e d y n a m i c a l p r o p e r t i e s o f t h i s s y s t e m ( i . e . , c o n d u c t i v i t y , r e s p o n s e t o an e x t e r n a l mov ing t e s t c h a r g e , b a c k g r o u n d d y n a m i c s ) . 2 The a p p e a r a n c e o f c h a r g e d e n s i t y waves (CDW) i n l a y e r e d t r a n s i t i o n m e t a l d i c h a 1 g e n o g i d e s a t low t e m p e r a t u r e m o t i v a t e d Ma ldegue (1978) t o c a l c u l a t e t h e f i r s t o r d e r exchange c o r r e c t i o n s t o t h e s t a t i c p o l a r i z a b i 1 i t y o f a homogeneous t w o - d i m e n s i o n a l e l e c t r o n g a s , w h i c h was found t o be s h a r p l y peaked a t 2 k p , where kp i s t h e u s u a l F e r m i , w a v e v e c t o r . T h i s i n d i c a t e s a s e n s i t i v e r e s p o n s i v e n e s s t o e x t e r n a l p e r t u r b a t i o n s a t t h i s w a v e v e c t o r and t hus a t e n d e n c y t owards a CDW. G e l d h a r d t and T a y l o r gave a v e r y u s e f u l method t o e v a l u a t e f i r s t o r d e r terms s u c h as t h o s e m e n t i o n e d a b o v e , a method w h i c h i s a l s o employed in t h i s t h e s i s . We w i l l f i r s t d e f i n e t h e VF-mode l and then d e t e r m i n e a s e l f - c o n s i s t e n t s c r e e n e d i n t e r a c t i o n between c h a r g e d p a r t i c l e s immersed in the LEG ( § 2 ) . S i n c e t h i s w i l l be done w i t h i n t he f r a m e -work o f t h e random phase a p p r o x i m a t i o n ( R P A ) , t h e r e i s no need to d i s t i n g u i s h between an e f f e c t i v e p o t e n t i a l seen by an e x t e r n a l t e s t c h a r g e and an e l e c t r o n , w h i c h t a k e s p a r t i n t he s c r e e n i n g . The f i r s t o r d e r c o r r e c t i o n t o t h e s t a t i c p o l a r i z a b i 1 i t y i s t hen d e t e r m i n e d ( § 3 ) . I n s t e a d o f u s i n g a b a r e Coulomb - o r a T h o m a s - F e r m i - p o t e n t i a l , we w i l l t a k e t h e s c r e e n e d one o b t a i n e d in § 2 , w h i c h i n c l u d e s i n t r a -and i n t e r l a y e r b a r e C o u l o m b - i n t e r a c t i o n s . In §k an e x p r e s s i o n f o r t h e s u s c e p t i b i l i t y o f t h e LEG i s o b t a i n e d , w h i c h i s j u s t i f i e d by c o m p a r i s o n t o t h e o n e s - o b t a i n e d by o t h e r a u t h o r s f o r the homogeneous e l e c t r o n g a s , u s i n g d i f f e r e n t a p p r o x i m a t i o n t e c h n i q u e s . The t h e s i s c o n c l u d e s w i t h a l o o k a t t h e s c r e e n i n g p r o p e r t i e s o f t he LEG p r e -d i c t e d by the new s u s c e p t i b i l i t y . In p a r t i c u l a r , t h e i nduced c h a r g e d e n s i t y i n l i n e a r r e s p o n s e t o an e x t e r n a l p o i n t c h a r g e w i t h i n one o f t h e l a y e r s and t h e s c r e e n e d p o t e n t i a l between two p o s i t i v e c h a r g e s 3 h a l f w a y between two l a y e r s a r e o b t a i n e d a n a l y t i c a l l y as f u n c t i o n s o f w a v e v e c t o r and d i s t a n c e p e r p e n d i c u l a r t o t h e p l a n e s . A n u m e r i c a l F o u r i e r t r a n s f o r m has been o b t a i n e d i n t h e f i r s t c a s e . A l l a p p r o x i m a t i o n s a r e c a r r i e d ou t a t z e r o t e m p e r a t u r e . 1. The V F - M o d e l [ 1 ] C o n s i d e r a c o n f i g u r a t i o n o f i n f i n i t e l y many p l a n e s s t a c k e d w i t h i n t e r p l a n e s p a c i n g c . E l e c t r o n s a r e a l l o w e d t o move f r e e l y i n t h e p l a n e s but t h e y a r e assumed t o have no e x t e n s i o n p e r p e n d i -c u l a r t o t he l a y e r s , s a y z . Fo r t h e p u r p o s e s o f n o r m a l i z a t i o n , l e t r=(x, jy) be c o n f i n e d t o a r e g i o n o f a r e a A . We can now d e f i n e t h e one e l e c t r o n wave f u n c t i o n s : 1 i t. ~r 1.1 t V = — a e X (z -mc) k , a , m / A - a a : s p i n s t a t e 1 ° 2 X (z - mc) = 1 <5(z - mc)) ,m : p l a n e l a b e l k = (k k ) 2 - d i m wave v e c t o r x , y H a v i n g d e t e r m i n e d a c o m p l e t e s e t o f one e l e c t r o n s t a t e s , we can g i v e t h e H a m i l t o n i a n , w h i c h w i l l be c o n s i d e r e d t h r o u g h o u t t h i s t h e s i s , i n second q u a n t i z e d n o t a t i o n . I t c o n t a i n s bo th i n t e r - and i n t r a -l a y e r C o u l o m b - i n t e r a c t i o n s between e l e c t r o n s . The s y s t e m i s e l e c t r i c a l l y n e u t r a l i z e d by a r i g i d homogeneous p o s i t i v e b a c k g r o u n d w i t h i n t he p l a n e s . 1.2 H = _ E ^ a * a-> + -> k k , m , a k , m , o k , m , a _, V^. a + a + a_^ , , a + + g ,m-m ' r5+^l»a»m T t - ( | » a V k , a , m $,otm k , P , q^O [ m,m\o,o where t he k i n e t i c ene rgy E-£ and t h e F o u r i e r - t r a n s f o r m e d Coulomb p o t e n t i a l V-* i a r e g i v e n by ^ q ,m-m 1.3 t ,2 .P 2m* m* b e i n g t h e e f f e c t i v e e l e c t r o n mass . 1.4 q,m-m' KA d2(f--r') tq.(r-r ) ( ( ? - ? ' ) 2 +(m-m' j z c z ) 1_ 'V 2_2\2 where K i s an e x t e r n a l d i e l e c t r i c c o n s t a n t . T h e q = o te rm in t h e i n t e r a c t i o n p a r t o f t h e H a m i l t o n i a n H j u s t s e r v e s t o c a n c e l t h e p o s i t i v e b a c k g r o u n d c h a r g e . The i n t e g r a l (1.4) can be e v a l u a t e d a n a l y t i c a l l y 1.5 V-*- t = q,m-m 2 i T e 2 1 KA a where q ; = q S i n c e t h e c o e f f i c i e n t s o f H a r e i n d e p e n d e n t o f s p i n , we w i l l f r om now on s u p p r e s s s p i n - l a b e l s , c a r r y i n g o u t t h e i r sums a t t h e a p p r o -p r i a t e p l a c e s w i t h o u t f u r t h e r m e n t i o n i n g them. I t seems c l e a r t h a t t h e p r e s e n c e o f many e l e c t r o n s i n ou r s y s t e m w i l l a l t e r t he i n f l u e n c e o f one p a r t i c l e upon a n o t h e r o n e , t h u s g i v i n g r i s e t o an e f f e c t i v e i n t e r a c t i o n . In t he n e x t s e c t i o n we w i l l o b t a i n t h e s o c a l l e d Random P h a s e A p p r o x i m a t i o n (RPA) t o t h e e f f e c t i v e i n t e r -a c t i o n . 5 C h a p t e r Two RANDOM PHASE APPROXIMATION TO THE EFFECTIVE INTERACTION L e t P be d e f i n e d by 2.1 P* (q) " I 4,SL a ^ , £ T h i s i s t h e F o u r i e r - t r a n s f o r m e d d e n s i t y o p e r a t o r i n t h e fc'th p l a n e . W i t h t h e H a m i l t o n i a n (1.2)-,' we can r e p r e s e n t i n t h e H e i s e n b e r g p i c t u r e 2.2 p j ( q , t ) = e I H f l P £ (*) e"'"* We can now d e f i n e t h e d e n s i t y - d e n s i t y c o r r e l a t i o n f u n c t i o n D n , m <<» u> 2.3 D (q,w) = - i n ,m ^ +oo . -io)t_- r H/->- _\ H/->' v, -a t dt e T < { p n ( q,t ) p _ ( q,0) } > e 1 1 T : t i m e o r d e r i n g o p e r a t o r < > : e x a c t g round s t a t e e x p e c t a t i o n v a l u e e a ! t l . a d i a t a d i c s w i t c h i n g on (a=0 +). 6 The e q u i v a l e n c e o f a l l t h e p l a n e s e n s u r e s t r a n s 1 a t i o n a1 i n -v a r i a n c e in the z - d i r e c t i o n , w h i c h i m p l i e s t h a t D = Di i . r nm | n - m | D|n _| i s a v e r y i m p o r t a n t q u a n t i t y , s i n c e i t d e s c r i b e s t he i n -t r i n s i c p r o p e r t i e s o f ou r s y s t e m . I t r e l a t e s e l e c t r o n d e n s i t y f l u c t u a t i o n s t o e l e c t r o n - e l e c t r o n i n t e r a c t i o n t h r o u g h t h e H a m i l t o n i a n (1.2). I t i s a l s o c o n n e c t e d t o t he d i e l e c t r i c s u s c e p t i b i l i t y X | n _ _ j t h r o u g h t h e e q u a t i o n 2.k X , | = i f 1 Di I | n - m | | n - m | w h i c h f o l l o w s f r om l i n e a r r esponse t h e o r y . X'|___m | d e t e r m i nes t h e i nduced c h a r g e d e n s i t y P | n c j c a u s e d by a s m a l l e x t e r n a l p o t e n t i a l V e x t a c t i n g on t h e LEG 2.5 P =1 X , I V ind^n L ' n-m e x t , m m ' where t h e l a b e l s as u s u a l r e f e r t o t h e p l a n e l o c a t i o n s . We w i l l t h e r e f o r e somet imes c a l l X< i t he r e s p o n s e f u n c t i o n . ' 'I n-m| r We w i l l o b t a i n an a p p r o x i m a t i o n t o X by u s i n g Feynman-Dyson p e r t u r b a t i o n t h e o r y and Greens f u n c t i o n s . The H a m i l t o n i a n (1.2) can n a t u r a l l y be s p l i t i n t o two p a r t s 2.6 H = H° + H 1 where H° =• I at a t J* k k,m k,r k,m i s t h e e n e r g y f o r n o n i n t e r a c t i n g e l e c t r o n s . A famous theorem o f 7 quantum f i e l d t h e o r y a s s e r t s t h a t t he e x a c t H e i s e n b e r g g round s t a t e e x p e c t a t i o n v a l u e < > e n t e r i n g ( 2 . 3 ) can be e x p r e s s e d i n te rms o f t h e w e l l - k n o w n n o n - i n t e r a c t i n g g round s t a t e o f ou r s y s t e m . T h i s i n v o l v e s an i n f i n i t e sum o f p r o d u c t s o f t h e i n t e r a c t i o n o p e r a t o r g i v e n in ( 1 . 2 ) combined w i t h t he d e n s i t y o p e r a t o r s P_ and p_ g i v e n in t h e i n t e r a c t i o n p i c t u r e I i H 0 t " i H 0 £ P = e n p e n ,m n ,m In a c c o r d a n c e w i t h c o n v e n t i o n , we w i l l name the so c a l l e d p r o p e r p a r t s o f t h e sum m e n t i o n e d above p o l a r i zab i 1 i t y o f n - t h o r d e r T I n . The s u s c e p t i b i l i t y and t h e s c r e e n e d i n t e r a c t i o n a r e then o b t a i n e d f r om the p o l a r i z a b i 1 i t y i n a way m e n t i o n e d l a t e r . A d e -t a i l e d a c c o u n t o f the method and i t s a p p l i c a t i o n t o t he d e g e n e r a t e e l e c t r o n gas can be f ound i n t h e e x c e l l e n t book o f F e t t e r and W a l e c k a [ 2 ] , S e c # 9 - 1 2 . L e t us f i r s t l o o k a t t h e n o n i n t e r a c t i n g s y s t e m w h i c h i s o b t a i n e d by s e t t i n g H 1 = 0 o r , " V _• = 0 i n ( 1 . 2 ) . T h r o u g h o u t t h i s t h e s i s , t h e t e m p e r a t u r e w i l l be t a k e n t o be z e r o . The g round s t a t e i n t h i s c a s e i s g i v e n by a s u c c e s s i o n o f t w o - d i m e n s i o n a l " F e r m i -d i s k s " f o r each p l a n e . L e t t h e i r r a d i i be k p . The e l e c t r o n d e n s i t y n i s r e l a t e d t o kp by k 2 N F 2 ' 7 n = A = 2 ? The non i n t e r a c t i ng p o l a r i z a b i 1 i t y 11^  i s g i v e n by 2.8 n ° ( q , o > ) = (2n) 2Ti d 2 k e k F ( k + q)6k F ( k ) e ^ ( k + q ) 9 ^ F ( k ) k q+k k q+k where = — x { k i n e t i c energy E^}, k Vk» - e l se 6 k ( k )  K F = 1 e ^ C k ) It has t h e f o l l o w i n g d i a g r a m m a t i c r e p r e s e n t a t i o n . F i g u r e .1: Diagram f o r I I k Y where >— d e n o t e s the f r e e F e r m i o n G r e e n s - f u n c t i o n 9 Figure 1 can be interpreted as an elementary electron-hole excitation in which momentum is conserved. It should be emphasized that electrons with wavevector close to the Fermi-surface give the dominant contri-bution to n ° . The integral in (2.8) has f i r s t been evaluated by Stern (1967) [3]. Since we are mainly interested in static screening properties in this thesis, we will only state the relevant real part of 11° below. The vector signs (-*-) are from now on omitted because i f is obvious that q, k, ... denote two-dimensional wavevectors. Let v = a n c' ^  k e 9' v e n ' n units of k_. Then 2.9 Re n ° (q,v) = irti 2q •sg 'r.2 I V 9 I' ( v " § ) 0 ( J  2- I-1 m { i q + e ( i — i - l - i)(£ + f)2 - i ) 2 q q 2 f ) - I ) 2 ) q 2 where sg x =,-| , 8(x) = 0(x) = 1 x>o 0 else for to = 0, this reduces to 2.10- Re n ° (q,0) = - m - m ] q<2 {1 - (1-Vq 2) 2} ' else a graph of n°(q,0) is plotted in Figure 2. 10 F i g u r e 2 : The non i n t e r a c t ing p o l a r i zab i 1 i t y 11° p l o t t e d as - n ° ( q / k p ) / I I 0 ( 0 ) / v e r s u s q / k p L n 2 . 0 Q/KF 3 . 0 4 . 0 Note the d i s c o n t i n u i t y o f the f i r s t d e r i v a t i v e a t q = 2 k p . The c u r v e has an i n f i n i t e s l o p e , w h i c h d i v e r g e s l i k e an i n v e r s e s q u a r e r o o t i x on the r i g h t - h a n d s i d e o f t h i s v a l u e . The t h r e e - d i m e n s i o n a l n o n i n t e r a c t i n g p o l a r i z a b i 1 i t y has a l o g a r i t h m i c s i n g u l a r i t y o f the f i r s t d e r i v a t i v e a t q = 2 k p . We can now get a f i r s t a p p r o x i m a t i o n t o the e f f e c t i v e i n t e r -a c t i o n by summing o v e r a l l p o s s i b l e c o n f i g u r a t i o n s o f d i a g r a m s such as in F i g u r e 1, l i n k e d by b a r e i n t e r a c t i o n s ( 1 5 ) . The f o l l o w i n g i t e r a t i o n p r o c e d u r e does t h a t . ( F i g u r e 3 ) . F i g u r e 3- I t e r a t i o n d i a g r a m f o r t he e f f e e t i v e : i n t e r a c t ion m n m n m m 1 Doub le z i g - z a q l i n e : e f f e c t i v e i n t e r a c t i o n U 3 3 mn S i n g l e z i g - z a g l i n e : b a r e Coulomb i n t e r a c t i o n g i v e n in (1.5) The m1 - m' c o n n e c t i o n in t h e second term i s a 0 o r d e r p o l a r i z a t i o n i n s e r t i o n 11° g i v e n in (2.9) 12 T h i s i s c a l l e d t he R i n g o r Random B h a s e a p p r o x i m a t i o n . The e f f e c t i v e i n t e r a c t i o n U i s o b t a i n e d f rom t h e b a r e Coulomb -mn p o t e n t i a l V „ n and t he n o n i n t e r a c t i n g p o l a r i z a b i 1 i t y by a D y s o n -t y p e e q u a t i o n i n v o l v i n g sums o v e r p l a n e l a b e l s . S i n c e a l l t h e p l a n e s a r e i d e n t i c a l , we have the a d d i t i o n a l r e q u i r e m e n t o f t r a n s -l a t i o n a l i n v a r i a n c e in t he z - d i r e c t i o n p e r p e n d i c u l a r t o them. I t f o l l o w s t h a t U and V can o n l y depend on | m - n [ . D iag ram (2 .11 ) then r e p r e s e n t s t h e f o l l o w i n g a l g e b r a i c e q u a t i o n : 2 . 1 2 Ui I (q,co) = Vi , (q) + 11° (q ,co) E V, . , (q) U, . i i ( q ,w) m-n m-n ^ i m-m ^ n-m I I I I m i i i i As u s u a l , we can d e c o u p l e t he sum in t he second term by F o u r i e r t r a n s f o r m i n g w i t h r e s p e c t t o t h e p l a n e l a b e l s . L e t I = |m - n | , t hen t he F o u r i e r t r a n s f o r m s o f t h e e f f e c t i v e -and b a r e Coulomb p o t e n t i a l s a r e g i v e n b y : + 0 ° 2 . 1 3 U(q , . y ,u ) = _ e 1 * Y U . ( q , u ) 2 , ] h V ( q y ) = e U y ' e - q c U | ^ ' Kq £=-°o = V Q ( q ) S ' (q .y ) E q u a t i o n ( 2 . 1 2 ) , e x p r e s s e d in te rms o f t h e s e f u n c t i o n s , becomes: 2 . 1 5 U(q,y,o>) = V Q ( q ) S ( q , y ) + n° ( Q , W ) V Q ( q ) S ( q , y ) U ( q , y , u ) I t i s now s t r a i g h t f o r w a r d t o s o l v e t h i s f o r U w i t h t h e r e s u l t : 13 V n ( q ) S ( q , y ) 2 . 1 6 U ( q . y . u ) = ^ - • = 1 - V Q ( q ) S ( q , y ) I T ( q , w ) I t i s i n t e r e s t i n g t t o compare t h i s , r e s u l t t o t he e q u i v a l e n t e x p r e s s i o n f o r a c o m p l e t e l y homogeneous e l e c t r o n g a s , where V (q) 2 . 1 7 U (q ,_ ) = - i i 1 - n U ( q , „ ) V Q ( q ) i s t h e R P A - s c r e e n e d p o t e n t i a l . No te t h a t t h e a d d i t i o n a 1 f a c t o r S ( q , y ) i n ( 2 . 1 6 ) f o r t he LEG e n t e r s f o r p u r e l y g e o m e t r i c a l r e a s o n s . We h a v e : 2 . 1 8 S ( q , y ) = f e U y e _ q C W - 2Re £ e U y - 1 £=-°° Z=0 s h q c c o s h q c - c o s y , where s h q c = s i n h q c t h i s te rm i s t h e m a t h e m a t i c a l t r a d e mark o f t h e L E G . The d e n o m i n a t o r o f ( 2 .16 ) can be i n t e r p r e t e d as the RPA - d i e l e c t r i c f u n c t ion 2 . 1 9 e R p A (q-,y,a))-.- ]•- V Q ( q ) S ( q , y ) l i ° ( q , u ) I t s h o u l d be m e n t i o n e d t h a t t h e z e r o s o f e (q , y, w) i m p l i c i t l y d e t e r m i n e t h e d i s p e r s i o n r e l a t i o n f o r t he p l asma o s c i l l a t i o n s w p ^ = f o r o u r s y s t e m . To o b t a i n the e f f e c t i v e i n t e r a c t i o n between two p l a n e s in te rms o f t h e i r d i s t a n c e t o each o t h e r , one has t o i n v e r t ( 2 .16 ) a c c o r d i n g t o t h e e q u a t i o n : 2TT 2.20 U (q,co)__L 2ir e iJLy U ( q , y , u ) dy 0 T h i s i s most e a s i l y c a r r i e d out w i t h a u n i t c i r c l e c o n t o u r i n t e r -g r a t i o n in the complex z - p l a n e , wh ich a f t e r the s u b s t i t u t i o n s z = e ' y dy = - - T — , becomes i z 2 . 2 1 u (q,_) = + 2 V n ( q ) . s n q c dz 2 a z - (z^+l) z = 1 where a = c o s h q c - n V Q s h q c . Note t h a t c o s y has been r e p l a c e d by -j (z+X) . For the s t a t i c c a s e , n^(q ,cu=0) has a n e g a t i v e r e a l p a r t f o r a l l q wh ich i m p l i e s a > 1 . 1 L e t Z Q = a - ( a 2 - l ) 2 ) t h i s i s p o s i t i v e r e a l and l e s s than 1 . The i n t e g r a l in ( 2 . 2 1 ) can then be e x p r e s s e d in terms o f z^ and i t s v a l u e i s equa l to the r e s i d u e o f the s i m p l e p o l e a t z = Z Q , w h i c h i s the o n l y one i n s i d e the u n i t c i r c l e . 2 . 2 2 2 T T i z0 = 1 we have t a c i t l y assumed t h a t £ 0 , wh ich can be done w i t h o u t l o s s o f g e n e r a l i t y , s i n c e = U ^ . 2.23 U„(q,0) = V n ( q ) s h q c — z n = ( c o s h q c - n° V n s h q c ) - ( ( c o s h q c - n ° V . s h q c ) 2 - 1 ) 2 15 This is the static screened or effective potential in the RPA expressed in terms of &=|m-n| and the wavevector q. Its properties are discussed in [1]. It depends on three basic input parameters. The interplane distance c, the electron density n or equivalently the Fermi wavevector kp and the effective electron mass m which enters through n^. This can be expressed in terms of an effective Bohr radius 2.2k a* 0 * 2 m e Then 2.25 n°(q) V Q(q) = - " V H 0 (3. ) V Q k p a 0 * F, where ft^ , VQ are now dimension 1 ess. Note also that V„(q) 2.26 Mm U t = Q (,,0) = , . ^(q.olvjq) which is the expected result for a two-dimensional homogeneous electron gas or the presence of only one layer. 16 Chapter Three FIRST ORDER EXCHANGE CORRECTIONS TO THE POLAR IZABILITY So far we have approximated the polarizabi1ity by its non interact ing part 11°. As seen from Figure 1, it neglects any exchange interaction between electron-hole pairs represented by solid lines. These effects enter however i f one takes into account the f i r s t order contribution n 1 to the polarizabi1ity. Calculations by Maldegue [k], who eva1uates .TI 1for the two-dimensional homogeneous electron gas as a function of wavevector q ( OJ=0) indicate a sharp peak at q = 2k p with IT1 (2kp) ^ 2.5 I 1 (0) . This is in contrast to its three dimensional analog for which Geldhardt and Taylor [5] found that n 1 (2kp) < I I 1 (0)Y . In the following, we will evaluate the static f i r s t order correction n 1 (q, UJ=0) in two dimensions, using the RPA- screened intralayer potential U Q (q) of the preceding chapter to take into account effects entirely due to the L E G . The next section is very mathematical, since care has to be taken in treating singularities, which might completely f a l s i f y numerical results. The excellent methods in [5] have been used and quoted subsequently. They greatly simplify this tricky problem. -The diagrams for I I 1 are depicted in Fig. k F i g u r e h: D iagrams f o r n e f f e c t i v e i n t e r a c t i o n U f rom 2.23 I I 1 is the sum o f t h e s e t h r e e c o n t r i b u t i o n s 18 According tothewell known rules of the Feynman-propagator formalism, those diagrams correspond to the following integrals: 3 . 1 (q, =£2§7F d 2k dk 0d 2pdp 0 (U(k-p) G°(p, p Q ) G°(p+q, p0+q0) G°(k, k_) G°(k+q, k0+q0)} 3 . 2 n_ ( q , q J = d 2k dk 0d 2pdp Q {U(k-p) (G°(p,p 0)) 2 G°( P +q, P o + qG) G°(k,k_)e I K ° n } ik nn. 3 . 3 (q,qQ) = ^ 2 f f ) B dk Qd2pdp 0{ U(k-p). G°(p, P o) (G°(p+q, P + q 0 ) ) 2 G°(k+q, k Q + % ) { k o + ^ } where the spin summation has given a factor of two and q Q, p Q, k Q denote the frequency parts (q Q = co). The evaluation of the frequency parts is straightforward with contour integration. Let q Q = 0, then: 3 . k nA ( q ' 0 ) = -tP" QK ( k)6 > (k+q) [0 < (p)9 > (p+q) -6> ( p)9 < (p+q) ] } K ~ \ + q ) (cop - U p + q ) Let I I * = I I * + I I * , then: 3 - 5 n £ ( q . o ) - * _ . f d2k d 2p v e <(k)[9 <(p)0 >(p+q)-0 >(p ) e <(p+q)3 ] (CJ - 0) , ) 2 p p+q 19 where 0 < ( k ) = 0(k -k) „ > ( k ) = 1 - 0"(k) We now want to e x p r e s s the i n t e g r a l s in d i m e n s i o n i e s s form and a l s o f i n d t h e e x p a n s i o n p a r a m e t e r f o r t h e p e r t u r b a t i o n s e r i e s o f the p o l a r i z a b i 1 i t y . The f o l l o w i n g change o f v a r i a b l e s w i l l do t h a t : P P k p , k •+ k k_, q + q k p Tik ^ 2 then O J = F P and s i m i l a r f o r q and k P — i i i * 2 ' t a k i n g a l l the c o n s t a n t f a c t o r s o u t o f the i n t e g r a n d , we g e t : 3 ' 6 n A , B ( q ' ° > - S h T • ' f A , B <<> where F D ( q ) i s now d i m e n s i o n 1 e s s . Compar ing t h i s w i t h 11° f rom (2.10), i t becomes c l e a r , t h a t 2 3-7a \ = k r a '* F o i s the e x p a n s i o n p a r a m e t e r , s i n c e 'i ' s n a t u r a l l y taken w i t h the i n t e g r a n d . The u s u a l e l e c t r o n i c d e n s i t y r i s c o n n e c t e d t o the Fermi wave-s v e c t o r kp in two d i m e n s i o n s th rough the f o l l o w i n g e q u a t i o n : i I2 20 X e x p r e s s e d i n . . r . and a * i s then s o x 2 2 r 3-7b X = a 5 o It f o l l o w s t h a t the e f f e c t i v e mass :"m* and the e x t e r n a l d i e l e c t r i c c o n s t a n t k in a * and the. e l e c t r o n i c d e n s i t y r d e t e r m i n e t h e o s exchange e f f e c t s . Note t h a t the i n t e r p l a n e s p a c i n g c a l s o e n t e r s i n t o n 1 t h r o u g h the e f f e c t i v e i n t e r a c t i o n (2.23) w i t h i n the i n t e g r a n d in F. . ( q ) . A , D We a r e now l e f t to e v a l u a t e the sum F^_, = F^(q)+Fg(q) to get TI1 (q , 0 ) . The idea i s to a p p l y a s e r i e s o f t r a n s f o r m a t i o n s to the i n t e g r a l s and thus r e n d e r them s u i t a b l e f o r a n u m e r i c a l e v a l u a t i o n . A t t e m p t s a t e v a l u a t i n g at l e a s t some o f the f o u r v a r i a b l e s o f i n t e g r a t i o n a n a l y t i c a l l y have t u r n e d out t o be not v e r y i l l u m i n a t i n g b e c a u s e one q u i c k l y runs i n t o h o r r e n d o u s :a 1 geb ra ix .iexp r e s s i o n s . From (3.4) and (3.5) we g e t : 3.8 F A ( q ) d 2 p d 2 k U ( k - p ) T 1, r x (U), -to. ) (OJ -to ) \ ~k+q' ^ p "'p+q' x [ e < ( p ) G > ( p + q ) 0 < ( k ) 0 > ( k + q ) - 0 > ( p ) 0 < ( p + q ) © < ( k ) 0 > ( k + q ) ] 3.9 F B ( q ) = d 2 p d 2 k U (k -p ) 1 ((o -to , J 2 X p p+q x [0 ( p ) 0 > ( p + q ) - 0 > ( p ) 0 < ( p + q ) ] 0 < ( k ) 21 where = ( j ) 2 , and 0 < ( p ) = 0(1 - | p | ) - , '.We want t o have the same s e t o f 0- f u n c t i o n s f o r F . and F D . A b D e f i n e Fg = d 2 p d 2 k ( . . . same as in F o ) 0 ( k + q ) , now change b v a r i ab 1 es p-H-'-p+q', k«->-k+q • i n . " ? , , ! i t t hen f o l lows ? D (q) = - ? D (q) .: b D o hence ? g ( q ) = 0 . So we can m u l t i p l y F g by 0 > ( k+q ) i n s i d e t h e i n t e g r a l w i t h o u t c h a n g i n g i t s ' v a l u e . In b o t h F. and F D we now have two t e r m s , one f o r p < 1, t h e A b o t h e r f o r p > 1 . In t h e l a t t e r , make t he t r a n s f o r m a t i o n p '-»- - p - q , t o ge t 3 . 10 F A ( q ) d ' p d ' k [ u ( k - p ) ^ ( k W ] e < 6 ' l 9 > ( p * q ) B < ( k ) ^ ( R ^ ) (uJ . - 0 ) ) (u3. , p+q p k+q 3.11 F B ( q ) - dWk [u (k -p ) -u ( k+p+ q ) ] e< j p ) e > (p+qj f (k)e> (k+q) P p+q The t r a n s f o r m a t i o n s p •»• p - j , k k - ^ w i l l f u r t h e r s y m m e t r i z e t h e e x p r e s s i o n s and we d e f i n e : 3 .12 F A B ( q ) - F A ( q ) + Fg (q) = (-1) [ ^ ( q J + F ^ ^ q ) ] wi th 22 3.13 F A r * ( , ) . { d W k U ( l p l k l ) j ^ l ^ x where we have now s u b s t i t u t e d f o r the w' s , i n t e r c h a n g e the dummy v a r i a b l e s k+->-p and t a k e t h e a v e r a g e , a l s o d e f i n e p 1 = k, then 3 .14 ^ ( q ) = ± 1 d 2 p d 2 p ' U ( | p ± p [ q - ( p ± p ' ) )Z (q p)2 (q p')2 x e ^ p - a j e ^ p + a j e ^ p ' - f - J e ^ p ' + f - ) w h i c h i s c o m p l e t e l y s y m m e t r i c in p and p ' . T h e r e a r e e s s e n t i a l l y 2 g e o m e t r i c a l l y d i f f e r e n t c a s e s d e p e n d i n g w h e t h e r q <-2 o r q > 2 . L e t . ' s l o o k a t q <. 2 f i r s t : 3 . 14 e x p r e s s e d in p o l a r c o o r d i n a t e s i s : <p,;<p' b e i n g the a n g l e s between q and p o r p 1 r e s p e c t i v e l y . The 0- f u n c t i o n s r e s t r i c t p (and p 1 ) t o the f o l l o w i n g a r e a : Figure 5 : geometry for q < 2 The shaded area is the region of integration For fixed <f>, we have 3.16 m( cf> ) < p < M( <j) ) where 3. 17 m( <f> ) = f cos* + ( l - ( f j 2 s i n 2 * ) 1 7 2 3.18 M(<j>) = 5- cos * + (1 - (^) 2 S i n 2 1 / 2 hence the range of p is then 3.19 A( <f> ) = M ((f)) - m( <f> ) = q cos <f> 2k I n t r o d u c e a new i n t e g r a t i o n v a r i a b l e s i n s t e a d o f p by 3.20 p = m( ([>;) + sA( (j) ) 0 < s i 1 dp = ds • A( cf> ) = ds q cos <f> The l i m i t s o f i n t e g r a t i o n a r e then f i x e d and 3.15 becomes , n o t i n g t h a t the same t r a n s f o r m a t i o n can be done w i t h p'><j>' 3.21 FM* (q) - ± | . dcfidcf. ds d s ' U ( p , p ' , cf), <))') x IT x ( p c o s ^ p ' c o s j ' ) 2 , p=p(s,*), p'=p'(s',V) p COScf) p' COS cf) 1 3.22 u " ( p , p ' , cf.,cf)') = u ( ( p 2 + p l 2 ± 2pp' ( c o s ; * cos cf.1 + s i n cf) s i n <j>'))'0 S i n c e we w i l l t a k e t h e s c r e e n e d p o t e n t i a l 2.23 f o r U , w i t h 1 = 0, t h i s w i l l n e v e r ge t s i n g u l a r and s o the o n l y s i n g u l a r i t i e s a p p e a r in t h e 3.23 (p cos <j> ± p' cos cf>')' P cos * p 1 cos cf>1 term Let <f)1 -*--(()1 in F AB then 3.2k U ( ( p 2 + p ' 2 + 2 p p ' (s in cf) s i n cf)1 ± costf. cos $*)Y i) M u l t i p l y 3.23 o u t and i n t e r c h a n g e cf. t^-cf.1 in the t h i r d term to g e t : 25 4 3.25 AB (q) = dtj) d<f>' dsds 1 ± / p c p s _ U " ( P ' coscj)1 J) "TT f i n a l l y use 3.12, then 3.26 F A B = (-1) d<}>d<}> d s d s ' [ ( U + + IT) + (u + - u")E^2l4 ] p 1 COS <j>' TT l e t t i n g <{>->-<}> and <J> 1 - <j> 1 , we see t h a t TT d<j>' ( ) = TT ~T d<j>' ( ) in 3.26, thus TT T 3.27 F A B - (-2) d<j>' IT T TT d<j> p cos * d s d s ' [(U +U ) + (U -U Jp'coscf . ' ] S i n c e q<2, p and p 1 w i l l n e v e r be z e r o . But t h e r e i s a s i n g u l a r i t y in t h e i n t e g r a n d f o r <j>'= f o r t u n a t e l y , i t t u r n s ou t t h a t t h i s s i n g u l a r i t y i s a removable one as shown below Lemma: <f>,= ,2" ' s a removable s i n g u l a r i t y in (3-27) P r o o f : Le t x = cos 4 , x 1 = cos <!> ' We a r e i n t e r e s t e d in .3.28.. l i m ( u ( t + ) - U ( t - ) ) P ( ( ^ j ^ x , = A ( p ( x ) , x ) x ' + 0 3.29 t ± = ( p 2 ( x ) + p l 2 ( x , ) + 2 p ( x ) p ( x ' ) (a (]-x2)h ( l - x 1 - 2 ) ^ x x ' ) ) The s ( s 1 ) - dependance o f p ( p 1 ) i s not i m p o r t a n t f o r t h i s and i s t h e r e f o r e s u p p r e s s e d . S i n c e x 1 = 0, we have t + = t and t i s maximal i f x = 1 o r <}> = 0, hence '3.30. t max = (( 1 + ( f - ) ) 2 + ( l - ( f - ) 2 ) ) ' = (2 + q)H < 2 So t i s c o n f i n e d to 0 < t < 2 . In t h i s r a n g e , U i s an a n a l y t i c f u n c t i o n o f t and t i s an a n a l y t f u n c t i o n o f x 1 by i n s p e c t i o n . T h e r e f o r e U can be expanded in a power s e r i e s a r o u n d x 1 = 0: 27 3.31 IT ( p . p ' . x . x ' H l T ( p , p ' , x , 0 ) + ^ r x 1 + 0 ( x * 2 ) x ' = 0 T h i s y i e l d s A ( p ( x ) , x ) = dU dU dx ' x ' = 0 p(x) . x 7 ¥ It remains t o e v a l u a t e dU dx" dU d x 1 x'=0 dU dt. , t(x'=0) ( d t •• dt" d x 1 : : . ' dx ' x'=0 ± V V - M E l ( 1 - X 2 J ' . dt = ( p ' + p ( . a ( l - x 2 ) 2 (1-x 1 2 ) 2 : + x x ' ) dx ' + p p 1 ( - a x ' T T ^ x 7 2jq ±x) d x 1 i± dt" dx' d p 1 = ( p ' + p a O - x 2 ) ^ x'=0 + P P 1 x x'=0 t(x'=0) 3.32 T h u s : A ( p ( x ) , x ) - ^ f f i p f ' § t ( x '=b ) wh ich i s f i n i t e q . e . d . II at ((,' = — , the integrand in (3.27) has to be replaced by 3-39 2U(t(x'=0)) + S j i d t 2p 2cos 2j> t(x'-O) t { x ' = 0 ) The preceding analysis can also be used for the removabl singularity at p' = 0, which occurs i f q = 2. Hence the integrand in F^(q) has a well defined value and stays f over the whole region of integration for q < 2. For q >2. the geometry is Figure 6 29 where <j> = a r c s i n ( 2 / q ) . A g a i n f o r f i x e d (j> we h a v e : m ( < ( . ) - p - M( • ) wi th 3.34 . m( <j>) = J cos <(>-(l /qx2 . 2 {p s i n <p K 3.35 M(<p) = | c o s ( f > + (1 - ( | ) 2 s i n 2 cf,)12 (3.36) A ;(<)>) = M (,<p ) - m(<p ) = 2(1 - ( | ) 2 s i n 2 <p)' which then l e a d s t o < '.0 .37-; F A B ( q ) -= -2 d<j> 1 d;<p - < i 0 d s d s ' ; - V M * ) A (*') q 2 [(u + + u " ) — L -r+ (u + : u" i , p ? A , ] c o s <p cos <p' p 1 cos2<p' T h e r e a r e no s i n g u l a r i t i e s in t h i s i n t e g r a n d (<f>0< y- ) F^g (q) can now be e v a l u a t e d n u m e r i c a l l y wh ich has been done by a m u l t i d i m e n s i o n a l Simpson - i n t e g r a t i o n on an Amdahl 470 V/6 Computer . A graph i s p l o t t e d . i n F i g u r e 7. The a n a l y t i c a l f e a t u r e s [4] o f our n 1 a r e the same as t h o s e o f Maldegues . The s i n g u l a r i t y o f t h e f i r s t d e r i v a t i v e a t q = 2kj_ i s e v i d e n t in b o t h c a s e s . F i g u r e ' 7: Thr* f i r s t , o r d e r exchange c o r r e c t i o n II1 to the po la r i zab i 1 i ty p l o t t e d as - I I 1 (q /k F > / " 1 (0) v e r s u s q / k p L O 0 . 0 1.0 2 . 0 3 .0 4 . 0 Q / K F 31 However , t h e peak i s c o n s i d e r a b l y damped n 1 (2kp) = 1.25 n'(0) compared t o 2.5n'(0) in [k] . T h i s shows t h a t the p r e s e n c e o f i n t e r a c t i n g e l e c t r o n s in n e i g h b o u r i n g l a y e r s t ends t o make exchange e f f e c t s more t h r e e - d i m e n s i o n a l . For c o m p a r i s o n we have c h o s e n the e l e c t r o n i c d e n s i t y r g and t h e e f f e c t i v e mass. :m* t o be the same as t h o s e in [k]. The i n t e r l a y e r s p a c i n g c was taken t o be 6 K. The peak n'(2kp) i s lowered in c o m p a r i s o n t o IT1 (0) , i f r s i s i n c r e a s e d , i . e . f o r an e l e c t r o n gas w i t h lower d e n s i t y , b e c a u s e l o n g - w a v e l e n g t h i n t e r a c t i o n s (sma l l q) can then a l s o . t ake p a r t in e x c h a n g e . 32 C h a p t e r Four THE RESPONSE FUNCTION OR SUSCEPTIBIL ITY X In t h e p r e s e n t a p p r o x i m a t i o n , in s p i t e o f the f a c t t h a t i n t e r l a y e r i n t e r a c t i o n s have been t a k e n : i n t o a c c o u n t t o o b t a i n I I 1 , bo th 11° and n 1 a r e d e f i n e d o n l y f o r v e r t e x -p a i r s b e g i n n i n g and e n d i n g in the same p l a n e . T h i s i s a c o n s e q u e n c e o f o u r a s s u m p t i o n t h a t no t u n n e l i n g s h a l l o c c u r between p l a n e s . In o r d e r t o be a b l e t o s t u d y s c r e e n i n g o r i n d u c e d c h a r g e d e n s i t i e s , we have t o o b t a i n an e x p r e s s i o n f o r the s u s c e p t i b i l i t y Xi I as a f u n c t i o n o f i n t e r l a y e r d i s t a n c e . T h e r e has been I n-m |.j a l o t o f c o n t r o v e r s y as t o how one can r e a s o n a b l y t a k e i n t o a c c o u n t exchange and c o r r e l a t i o n e f f e c t s in a g e n e r a l i z a t i o n o f the RPA - r e s p o n s e f u n c t i o n [ 6 , 7 , 8 ] . We w i l l p r o p o s e a s t r a i g h t f o r w a r d a p p r o a c h t o e x p r e s s the s u s c e p t i b i l i t y in terms o f t h e z e r o - a n d f i r s t o r d e r p o l a r i z a b i 1 i t i e s in a ) . The r e s u l t t h u s o b t a i n e d i s then compared in i t s e s s e n t i a l f e a t u r e s t o t h e a p p r o x i m a t i o n s o b t a i n e d by o t h e r a u t h o r s , u s i n g the more I n t u i t i v e - " t h e o r y o f l o c a l f i e l d c o r r e c t i o n s ( b ) ) . 33 To a v o i d t e d i o u s p l a n e l a b e l sums, we w i l l work w i t h F o u r i e r t r a n s f o r m s w i t h r e s p e c t t o them as d e f i n e d i n ( 2 . 1 3 ) , ( 2 . 1 4 ) . Then t h e e q u a t i o n : c = I a b n m n-m m can be t r a n s f o r m e d i n t o h ( y ) = f ( y ) g ( y ) where h = I c e ] n \ f = I a e m \ g = Eb e ' n y n n ' n n n n a) H a v i n g o b t a i n e d n 1 ( q , u>=0) , t he new p r o p e r p a r t o f the r e s p o n s e f u n c t i o n i s g i v e n b y : L i A o 1 n = n + n One can now r e s u b s t i t u t e t h i s i n t o t h e a p p r o x i m a t i o n ( 2 . 1 1 ) o f C h a p t e r 2 and o b t a i n a new e f f e c t i v e i n t e r a c t i o n V Q (q) S ( q , y ) k ' 2 = 1 - V o ( q ) S ( q , y ) II* (q) where 11° has been r e p l a c e d by n " . T h i s a p p r o x i m a t i o n i s s t i l l s e I f c o n s i s t e n t i n t he s e n s e t h a t t h e e f f e c t i v e i n t e r a c t i o n U i s g i v e n by t h e sum o f the b a r e Coulomb p o t e n t i a l and t h e p o t e n t i a l a r i s i n g f r om c h a r g e - d e n s i t y f l u c t u a t i o n s . T h i s can be e x p r e s s e d a s : 3h A . 3 U ( q , y ) = V Q S (1 + X ( q , y ) V Q S ) where X i s t h e s u s c e p t i b i l i t y . E q u a t i n g {k.2) and ( A . 3 ) and s o l v i n g f o r X , t h i s y i e l d s : H*(q) k.k X ( q , y ) = 1 - V o ( q ) S ( q , y ) n * ( q ) w h i c h can be t r a n s f o r m e d back t o p l a n e l a b e l s v e r y e a s i l y w i t h ( 2 . 2 3 ) : | m-n j k - S X | n - m | ( q ) = I I " ( c ' ) ( ' 6 m , n ' H*(q) V Q ( q ) 2shqc f ° ) Z ° " T o h.6 z Q = a - ( a 2 - l ) , a= cosh qc - H V D sh q c b) L e t s look a t e q u a t i o n 2.k a g a i n in i t s F o u r i e r t r a n s f o r m e d s t a t i c f o r m : k - 7 p i n d ( q , y ) = X ( q , y ) V e x t ( q , y ) The w e l l known RPA - r e s u l t f o r X i s o b t a i n e d by t a k i n g Tl" = 1 1 ° i n a ) , * - 8 x • It has been p o i n t e d o u t by Bohm and P i n e s [ 9 ] t h a t t h i s a p p r o x i m a t i o n n e g l e c t s any s o r t o f exchange and c o r r e l a t i o n e f f e c t s between e l e c t r o n s . Hubbard [ 1 0 ] e t a l . t h e n came up 35 w i t h a n a t u r a l g e n e r a l i z a t i o n o f the RPA, w h i c h i n c l u d e s l o c a l f i e l d c o r r e c t i o n s G in the f o r m : 4.9 x(q.y) = ^ M 1 - V 0 ( q ) S ( q , y ) n ° ( q ) ( l - G ( q , y ) ) G i s r e l a t e d t o t h e p a i r c o r r e l a t i o n f u n c t i o n g ( r ) [14] , w h i c h s u g g e s t s i t s i n t e r p r e t a t i o n as a l o c a l f i e l d c o r r e c t i o n . It i s i n t e r e s t i n g now t o compare our v a l u e s f o r G , w h i c h can be o b t a i n e d by r e a r r a n g i n g t h e r e s u l t in a ) , w i t h t h a t o f o t h e r a u t h o r s , who o b t a i n e d t h e same f u n c t i o n u s i n g d i f f e r e n t a p p r o x i m a t i o n t e c h n i q u e s , f o r a r e v i e w o f t h i s , see [ 6 ] . A s t r a i g h t f o r w a r d c a l c u l a t i o n y i e l d s f rom ( 4 . 5 ) . A . , 0 G ( q , y ) - . - n l ( q ) >n*(q), n ° ( q ) , V 0 ( q ) ^ S ( q , y ) T h i s has p r o b a b l y n e v e r been c o n t e m p l a t e d f o r a l a y e r e d s y s t e m , bu t t h e r e a r e n e v e r t h e l e s s f e a t u r e s o f G ( q , y ) w h i c h must be s a t i s f i e d i n any c a s e . l) f o r s m a l l q:, y o r l a r g e w a v e v e c t o r s , we e x p e c t t h e RPA t o be a good a p p r o x i m a t i o n and c o r r e l a t i o n e f f e c t s s h o u l d become n e g l i g i b l e , hence 4.11 '.I l i m G ( q , y ) = 0 q 0 y = 0 which i s e a s i l y v e r i f i e d 2) as p o i n t e d out by S i n g w i e t . a l . [11], in the s t a t i c c a s e , we s h o u l d have 4.12 l i m G ( q , y ) = c o n s t a n t q -y oo which i s indeed t r u e as we s h a l l show p r e s e n t l y : In t h e ' p r e v i o u s c h a p t e r , we have o b t a i n e d n ' ( q ) f o r q » 2 in t h e form n ' ( q ) = a F A g ( q ) where r" A B (q) i s g i v e n by (3-37) , 2 1 m a n d a = . T _ F _ f o r l a r g e q , one can take the f o l l o w i n g l i m i t s : l i m —, =1, l i m _ cosc|> = 1, l i m . .cos <p1 = 1 and U ( | p+p' | )'-& U(q) Hence: 37 2 % L j 4 . 1 3 | F A Q ( q ) I * 1 AB ^ 1 q -*• o= q .4 d<f>d<f>1 dsds 'A((t)) A(cf>') where A( (f. ) = 2 ( 1 - ( J ) 2 s i n 2 (j))^ and s i m i l a r l y f o r <{>1 and * 0 = a r c s i n l e t <}> = a r c s i n — , dd> = q 2 d x q ( l - ( ^ ) 2 ) i S +1 then d * A(<j>)' = -( l - x 2 ) ' q - i q The i n t e g r a l then c o n v e r g e s to | - f o r q -> oo > wh ich i m p l i e s 4 . 1 4 n ' ( q ) £ a k_ o F • U ( q ) TTT.2 as q a l s o f o r l a r g e q : 4 . 1 5 n ° ( q ) = f F T ( l v v q ' ' ^ q< m 38 and then, neglecting I I 1 in the denominator of G(q,y): k. 16 G ( q , y ) 2 q •*• OO note that lim q->- oo S(q,y) = 1 for a l l y Owing to the geometry of our system and the nature of the present approximation which is closely related to the one of Toigo and Woodruff :[:8] , our G (q,y) exhibits a sharp peak at q = 2kp with a singularity in its f i r s t derivative, not unlike the TW result for a 3 dimensional homogeneous electron gas. These results show that our approximation is probably not too unreasonable since other authors using different schemes have obtained similar results for homogeneous systems. 39 C h a p t e r F i v e STATIC SCREENING OF THE L E G The g e n e r a l i z e d r e s p o n s e f u n c t i o n ( 4 . 5 ) o f C h a p t e r k can now be used to s t u d y the s c r e e n i n g p r o p e r t i e s o f o u r s y s t e m . S i n c e , a c c o r d i n g to. " ( 4 . 7 ) » we have t o ge t the e x t e r n a l p o t e n t i a l o f some a r b i t r a r y c h a r g e d i s t r i b u t i o n i n t r o d u c e d i n t o the L E G , we w i l l g i v e the Green "s" f u n c t i o n s u i t e d t o t h i s p r o b l e m b e l o w : Gi ven ' d 2 x e " < x p e x t ( x ' z ) ' 5 - ' P e x t < « ' z > - S the e x t e r n a l c h a r g e d i s t r i b u t i o n , we want t o f i n d the c o r r e s p o n d i n g p o t e n t i a l V e x t ( q , z ) Let 5 . 2 g ( z , z ' ) = 0 I I e 2 q | - q z - z ' Then 5 . 3 (^r - q2) g ( z , z ' ) = - 6 ( z - z ' ) Hence c l w (q» z) _ ^ ^ e 2 e x t " ~ A ~ d z ' g ( z . z ' ) , p e x t ( q , z ' ) ko s o l v e s the c o r r e s p o n d i n g P o i s s o n e q u a t i o n and i s t h e r e f o r e the wanted p o t e n t i a l . We w i l l now c o n s i d e r two c a s e s wh ich a r e o f p h y s i c a l i n t e r e s t : a) e x t e r n a l p o s i t i v e p o i n t c h a r g e l o c a t e d in one o f t h e l a y e r s . b) two e x t e r n a l p o s i t i v e p o i n t c h a r g e s b e i n g sandwiched h a l f w a y between two l a y e r s . a ) L e t ° e x t ( x , z ) = 6 ( x ) 6 ( z ) 2 5 . 5 V (q z) = Zl"& e " q I2' n - Yt, P e x t V q , Z ; Aq 6 » q - |q hence in the p l a n e s z = mc e x t ^ 'm o We can now take the F o u r i e r t r a n s f o r m w i t h r e s p e c t t o the p l a n e l a b e l s and use (k.k) t o get 5 - 6 P ; n d ( q , y ) = - X (q ,y ) V Q ( q ) S ( q , y ) = - n"(q)Vn(q)S(q,y) 1 - n' v(q)V 0(q)S(q,y) where V 0 and S a r e d e f i n e d in (2.14) w h i c h can i m m e d i a t e l y be t r a n s f o r m e d back t o g i v e w i t h (2.23) 5 ' 7 p i n d ( q ) ? n - - n ' " ( q ) U n ( q ) We e x p e c t the e x t e r n a l c h a r g e t o be s c r e e n e d c o m p l e t e l y a t l a r g e d i s t a n c e , w h i c h i s p roven b e l o w : In c o n f i g u r a t i o n s p a c e we h a v e : 5 . 8 p. , ( x ) , = ind ' n fdfq T27]T e I C | X p . (q) , i nd ^ 'n I n t e g r a t i n g t h i s o v e r a l l a v a i l a b l e x - s p a c e and summing o v e r a l a y e r s , we s h o u l d ge t -1 f o r t he c o m p e n s a t i o n o f the e x t e r n a l c h a r g e . . 5 . 9 i nd , To t r A2 d z x d e i q . x T27T 2 6 E p . , (q) , n n i nd ^ ' = A d 2 q 6(q) E p . ^ (q)^ = A E p. ,(0) , n n K i n d ' U s i n g (2.23) and n o t i n g t h a t l i m q+o q 2.We we have w i t h a = ^ — c n '(0) , hi :.zo(q = 0) = 1 - a - ((1 - a ) 2 - ] ) h h e n c e 5.10 p. .. T n t = 2a — =— Z z J n l i n d » l o t 1 n ° z - — ° z o " 2 a (a + ((1 - a ) 2 - - l ) % ) 2 =-1 a s e x p e c t e d . S i n c e p , (q) o n l y d e p e n d s on q = | q | , we c a n i n t e g r a t e i n d w i t h r e s p e c t t o t h e a n g l e i n (5-8) u s i n g •"'2TT 5.11 Vqx)-^ J e i q X c o s * d^ o t h e ,o - o r d e r B e s s e l f u n c t i o n , t h i s y i e l d s 5.12 P ; n H ( x ) . = \ ~ ind n 2 T T q dq J 0 ( q x ) 9 i n d ( q ) , n ; x = I We h a v e c a l c u l a t e d 5.12 n u m e r i c a l l y and p l o t t e d i t i n F i g u r e 8. The p a r a m a t e r s h a v e b e e n t a k e n t o be t h o s e o f ' a ' t y p i c a 1 . . t r a n s i t i o n meta1 d i c h a 1 c o g e n i de - . 43 F i g u r e 8: The induced c h a r g e d e n s i t y p . , (x) o f e l e c t r o n s i nd , o in the p l a n e o f the i m p u r i t y g i v e n as a f u n c t i o n o f d i s t a n c e r f rom a p o s i t i v e p o i n t c h a r g e at the o r i g i n . x = r. k One o b s e r v e s t h a t t h e c h a r g e d i s t r i b u t i o n P j n c j i s a l m o s t z e r o f o r r . k p >2, m e a n i n g a few 8 . The o s c i l l a t i o n s o f v e r y s m a l l a m p l i t u d e w h i c h d o m i n a t e t h e b e h a v i o u r o f P j n c j a t g r e a t e r v a l u e s o f r , a r e commonly c a l l e d F r i e d e l : ' o s c i 1 l a t i o n s . Our r e s u l t s seem t o i n d i c a t e , t h a t t h e y d e c a y much l e s s r a p i d l y t h a n l i k e w h i c h w o u l d be t h e c a s e f o r a RPA - t r e a t m e n t o f r 3 s c r e e n i n g i n a t h r e e d i m e n s i o n a l homogeneous e l e c t r o n g a s . I t seems s a f e t o p r e d i c t a d e c a y f o r t h e F r i e d e l - o s c i l l a t i o n s l i k e -pj- w h e r e 2 - a ^ 1. H o w e v e r , t o be a b l e t o g i v e a more a c c u r a t e v a l u e f o r a, o n e w o u l d h a v e t o f i n d an a n a l y t i c a l t r e a t m e n t o f t h e s i n g u l a r i t y o f II 1 a t q = 2 k p w h i c h d e t e r m i n e s t h e b e h a v i o u r o f t h o s e o s c i l l a t i o n s . b ) L e t p e x t ( x ' z ) = ( 5 ( x " a) + 6 ( x + a ) ) 6(y) 6(z) a n d t h e o r i g i n o f t h e p l a n e s be a t (0,0, (ri + j) c ) n = 0, ± 1 ,, h e n c e q - (q x + q y ) 5.1* and V e x t ( q , („4)c) - v^W,,, = c o s ^ . a j a ^ l ^ l 5.15 V e x t ( q , y ) = - I e - | n * (q) 2 Aire + v ° ° - q | ( n + i s ) | — c o s q a I e M | 1 Aq x n = - 0 0 * V ~ 2 (q,y) A s i m p l e c a l c u l a t i o n y i e l d s 2 sh S | c o s V. 5.16 3 ( q , y ) = 2 2 c o s h qc - cos y and the induced c h a r g e d e n s i t y i s t h e r e f o r e g i v e n by 5.17 P i n d(q.v) - - X ( q . y ) V e x t ( q , y ) n " ( q ) 3(q,y) V D 2 c o s q x a 1 - I T S ( q , y ) V o wi th V o kq I) he We a r e now i n t e r e s t e d in the induced p o t e n t i a l V . ^ a r i s i n g f rom the induced c h a r g e d e n s i t y P j n c j d e t e r m i n e d by (5.17). Then the s c r e e n e d o r t o t a l p o t e n t i a l w i l l be g i v e n by 5 . 1 8 V \ = V + V . . t o t e x t ind which can be e a s i l y e v a l u a t e d a n a l y t i c a l l y a t z = 0, i . e . f o r the p l a n e , where the e x t e r n a l c h a r g e s a r e l o c a t e d . Wi th (S-h) we g e t : 2.7T 5.20 V . n d ( q , z = 0 ) Vo27 s(q,y) P j n d ^ ' y ) d y r V ( i + e - i y ) 2TT dy (cosh qc - cos y ) ( c o s h qc - V ^ - s h q c - c o s y) w i t h T = 2 cos q a sh a£ sh qc V (q) lT(q) - i x 2 o ^ Let z = e ' y , dy = -JJ , wh ich t r a n s f o r m s (5.20) i n t o : 47 5.21 _4r_ 2TT i z ( l + z ) dz . = 1 ( 2 c o s h q c - z z - l ) ( 2 ( c o s h q c - l T ^ s h q c ) - z z l e t „ = c o s h q c - n " v 0 s n c i c g = c o s h q c and z = a - ( a 2 - 1) 2 o Z ] = B - ( 3 2 - D ; then (5-21) can be e x p r e s s e d in z^ and Z Q : 5.22 4r 2TT i :( l+z) dz = 1 ( z - Z l ) ( z - i )(z z o n o t e t h a t z and z . a r e r e a l and < 1, hence i n s i d e t h e u n i t o I c i r c l e and (5.22) i s j u s t the sum o f the r e s i d u e s o f the two s i m p l e p o l e s a t z and z . 5.23 v.nd(q,z=o) = - 4r[ O O z,(l+r,) O 1 . 1 o ] = + 4r z z ] l ) ( z , - l ) 48 CONCLUSIONS In t h i s t h e s i s , we have p r o p o s e d a method f o l l o w i n g f rom Feynman-Dyson p e r t u r b a t i o n t h e o r y t o t r e a t s c r e e n i n g phenomena o f a d e g e n e r a t e l a y e r e d e l e c t r o n gas g o i n g beyond the random phase a p p r o x i m a t i o n . The r e s u l t s were found t o be c o n s i s t e n t w i t h p r e v i o u s g e n e r a l i z a t i o n s by o t h e r a u t h o r s . The method c o u l d thus be used t o i n c l u d e exchange and c o r r e l a t i o n e f f e c t s , when o t h e r p r o p e r t i e s o f an i n t e r a c t i n g e l e c t r o n gas a r e i n v e s t i g a t e d . The s c r e e n i n g examples t r e a t e d in C h a p t e r 5 seem to i n d i c a t e t h a t the major c o n s e q u e n c e o f the new a p p r o x i m a t i o n i s an enhancement o f the F r i e d e l . o s c i 1 l a t i o n s a t l a r g e d i s t a n c e s . O t h e r w i s e , the induced c h a r g e d e n s i t y i s v e r y w e l l d e s c r i b e d by the RPA a t l e a s t in the VF - model f o r t h e d e n s i t i e s c o n s i d e r e d . 49 REFERENCES 1) P . B . V i s s c h e r and L . M . F a l i c o v , P h y s . Rev . B3 3 (1971) 2541 2) A . L . F e t t e r and J . D . W a l e c k a : Quantum Theo ry o f many P a r t i c l e S y s t e m s , Mac G r a w - H i l l . 3) F. S t e r n , P h y s . Rev . L e t t e r s j M 1967) , 546 4) P . F . M a l d e g u e , S o l i d S t a t e Comm., 2G_ ( 1 9 7 8 ) , 133 5) D . J . W . G e l d h a r d t and R. T a y l o r , C a n . J . o f P h y s : , 4 8 ' ( 1 9 7 0 ) , 155 6) A . A . K u g l e r , J . o f S t a t . P h y s . , J2_ ( 1 9 7 5 ) , 35 7) L . J . Sham, P h y s . Rev . B3 7_ (1973), 4357 8) F. T o i g o and T . O . W o o d r u f f , P h y s . Rev . B 3 , 2_ (1970) 3958 9) D. Bohm:.. and D. P i n e s , Phys Rev . 9_2_ (1953), 609 10) J . H u b b a r d , P r o c . Roy. S o c . A243 (1957), 336 11) K . S . S i n g w i " . , A . S j o l a n d e r , M . P . T o s i and R . H . L a n d , P h y s . Rev . B3 7 ( 1 9 7 0 ) , 1044 12) D. G r e c u , J . P h y s . C , 8 (1975), 2627 13) A . L . F e t t e r , Ann. ,o f P h y s . , 818 (1974), 1 14) M. J o n s o n , J . P h y s . C , 9 (1976), 3055 

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