- Email: [email protected]
Surface plasmons in RPA
Volume 34A, number 7
PHYSICS LETTERS
SURFACE
PLASMONS
19 April 1971
IN RPA
Ch. HEGER and D. WAGNER
Institut Far Theoretische Physik der Ruhr-Universittit Bochum, Germany Received 16 March 1971
The high density limit for the dispersion of the surface plasmons is presented.
In c l a s s i c a l or s e m i - c l a s s i c a l a p p r o x i m a t i o n the p r o b l e m of the s u r f a c e p l a s m o n d i s p e r s i o n is solved f a i r l y well [1,2]. This does not apply to a c o n s i s t e n t quantum m e c h a n i c a l t r e a t m e n t ; so that some r e c e n t p a p e r s on this subject exhibit p a r t i ally c o n t r a d i c t o r y r e s u l t s [3,4]. We have solved this p r o b l e m in an approximate R P A. Using the equation of motion method for the density m a t r i x , we d e t e r m i n e d the e i g e n f r e q u e n c i e s of a Coulomb gas confined to a slab of t h i c k n e s s L by infinite potential walls. Because of the lacking t r a n s l a t i o n a l i n v a r i a n c e the H a r t r e e functions of the e l e c t r o n s are not given by plane or standing waves. The true H a r t r e e functions deviate f r o m this because of the F r i e d e l o s c i l l a t i o n s n e a r the s u r f a c e s . As approximate H a r t r e e functions we use the complete set of functions in the slab
where K 1 i s diagonal in the "P3 - r e p r e s e n t a t i o n " ; h 2~ -~- (~_~)2 K - l ( # 3 ) : (k 2 +p2)EL(k,P3; ¢0) e L being the Lindhard function; w h e r e a s K2 is non-diagonal: g2(P3, p'3) =
1 f
8n2
d2k ,
ei(kP'3;o~)
O(P~-¼(p+p') 2) _ O(p2F-¼(p-p') 2) p.p'/2m - w
p : = (k.P3) , p ' : = (k.P3)
with ~ = (~, 7/) ; ~ i s the coordinate n o r m a l to the surface,
The sum in (1) i s r e s t r i c t e d to the even (odd) c l a s s of the m o m e n t a P3.P'3 • O(x) = step function. The solution of (1) depends e s s e n t i a l l y on the s i n g u l a r i t i e s of E L in the complex p 3 - p l a n e . This shows i m m e d i a t e l y that high frequency app r o x i m a t i o n for E L a r e inadequate for the solution of this p r o b l e m [4], [6]. In the high density l i m i t we find for a s e m i infinite metal
0>~ > -L,
¢o = ½~f2 ~Op + (0.7888-0.0420i- 0.3918rsl/2)
( ~ ]kP3 ) : = 2(2#)- 3/2 exp (i k e) sin P3
P3: = n n / L ( n = 1 , 2 , 3 . . . ) .
T h e s e functions a r e the true H a r t r e e functions in the i n t e r i o r of the slab and satisfy the c o r r e c t boundary conditions. Deviations f r o m the true H a r t r e e functions can be substantial_ only within a s m a l l l a y e r of t h i c k n e s s k ~ ' , which d e c r e a s e s with i n c r e a s i n g density of the e l e c t r o n s (rs--" 0). L i n e a r i z i n g the equation of motion for the d e n s i t y m a t r i x we get a l i n e a r homogeneous i n t e g r a l equation for the deviation f r o m the equil i b r i u m d e n s i t y , which has been d e r i v e d p r e v i ously by other authors in p r i n c i p l e . T h i s i n t e g r a l equation can be solved exactly [5]. F r o m this we find for the d i s p e r s i o n r e l a t i o n for the surface plasmons:
1 = 448
~
P 3 ' P'3
(J~ I K 1(1-K2 )-1 [P'3 ,~
(1)
½kv F
The lowest o r d e r in r s the t e r m l i n e a r in k c o r r e s p o n d s to the s e m i - c l a s s i c a l solution, where the density of e l e c t r o n s i s taken as constant [2]. The next t e r m is new and shows that c l a s s i c a l a p p r o x i m a t i o n s deviate appreciably from the R 1~ A, c o n t r a r y to the case of volume p l a s m o n s .
References [1] R. H. Ritchle, Progr. Theor. Phys. 29 (1963) 607; R. H. Ritchie and A. L. Marusak, Surf. Sci. 4 (1966) 234. [2] D.Wagner, Z.f. Naturf. 21a (1966) 634. [3] P. A. Fedders, Phys. Rev. 153 (1967) 438. [4] P. J. Feibelmann, Phys. Rev. 176 (1968) 551. [5] Ch. Heger, Diplomarbeit, KC~ln1970; Ch. Heger, D.Wagner, to be published. [6] J. Harris, A. Griffin, Can. J. Phys. 48 (1970) 2592.
.
PHYSICS LETTERS
SURFACE
PLASMONS
19 April 1971
IN RPA
Ch. HEGER and D. WAGNER
Institut Far Theoretische Physik der Ruhr-Universittit Bochum, Germany Received 16 March 1971
The high density limit for the dispersion of the surface plasmons is presented.
In c l a s s i c a l or s e m i - c l a s s i c a l a p p r o x i m a t i o n the p r o b l e m of the s u r f a c e p l a s m o n d i s p e r s i o n is solved f a i r l y well [1,2]. This does not apply to a c o n s i s t e n t quantum m e c h a n i c a l t r e a t m e n t ; so that some r e c e n t p a p e r s on this subject exhibit p a r t i ally c o n t r a d i c t o r y r e s u l t s [3,4]. We have solved this p r o b l e m in an approximate R P A. Using the equation of motion method for the density m a t r i x , we d e t e r m i n e d the e i g e n f r e q u e n c i e s of a Coulomb gas confined to a slab of t h i c k n e s s L by infinite potential walls. Because of the lacking t r a n s l a t i o n a l i n v a r i a n c e the H a r t r e e functions of the e l e c t r o n s are not given by plane or standing waves. The true H a r t r e e functions deviate f r o m this because of the F r i e d e l o s c i l l a t i o n s n e a r the s u r f a c e s . As approximate H a r t r e e functions we use the complete set of functions in the slab
where K 1 i s diagonal in the "P3 - r e p r e s e n t a t i o n " ; h 2~ -~- (~_~)2 K - l ( # 3 ) : (k 2 +p2)EL(k,P3; ¢0) e L being the Lindhard function; w h e r e a s K2 is non-diagonal: g2(P3, p'3) =
1 f
8n2
d2k ,
ei(kP'3;o~)
O(P~-¼(p+p') 2) _ O(p2F-¼(p-p') 2) p.p'/2m - w
p : = (k.P3) , p ' : = (k.P3)
with ~ = (~, 7/) ; ~ i s the coordinate n o r m a l to the surface,
The sum in (1) i s r e s t r i c t e d to the even (odd) c l a s s of the m o m e n t a P3.P'3 • O(x) = step function. The solution of (1) depends e s s e n t i a l l y on the s i n g u l a r i t i e s of E L in the complex p 3 - p l a n e . This shows i m m e d i a t e l y that high frequency app r o x i m a t i o n for E L a r e inadequate for the solution of this p r o b l e m [4], [6]. In the high density l i m i t we find for a s e m i infinite metal
0>~ > -L,
¢o = ½~f2 ~Op + (0.7888-0.0420i- 0.3918rsl/2)
( ~ ]kP3 ) : = 2(2#)- 3/2 exp (i k e) sin P3
P3: = n n / L ( n = 1 , 2 , 3 . . . ) .
T h e s e functions a r e the true H a r t r e e functions in the i n t e r i o r of the slab and satisfy the c o r r e c t boundary conditions. Deviations f r o m the true H a r t r e e functions can be substantial_ only within a s m a l l l a y e r of t h i c k n e s s k ~ ' , which d e c r e a s e s with i n c r e a s i n g density of the e l e c t r o n s (rs--" 0). L i n e a r i z i n g the equation of motion for the d e n s i t y m a t r i x we get a l i n e a r homogeneous i n t e g r a l equation for the deviation f r o m the equil i b r i u m d e n s i t y , which has been d e r i v e d p r e v i ously by other authors in p r i n c i p l e . T h i s i n t e g r a l equation can be solved exactly [5]. F r o m this we find for the d i s p e r s i o n r e l a t i o n for the surface plasmons:
1 = 448
~
P 3 ' P'3
(J~ I K 1(1-K2 )-1 [P'3 ,~
(1)
½kv F
The lowest o r d e r in r s the t e r m l i n e a r in k c o r r e s p o n d s to the s e m i - c l a s s i c a l solution, where the density of e l e c t r o n s i s taken as constant [2]. The next t e r m is new and shows that c l a s s i c a l a p p r o x i m a t i o n s deviate appreciably from the R 1~ A, c o n t r a r y to the case of volume p l a s m o n s .
References [1] R. H. Ritchle, Progr. Theor. Phys. 29 (1963) 607; R. H. Ritchie and A. L. Marusak, Surf. Sci. 4 (1966) 234. [2] D.Wagner, Z.f. Naturf. 21a (1966) 634. [3] P. A. Fedders, Phys. Rev. 153 (1967) 438. [4] P. J. Feibelmann, Phys. Rev. 176 (1968) 551. [5] Ch. Heger, Diplomarbeit, KC~ln1970; Ch. Heger, D.Wagner, to be published. [6] J. Harris, A. Griffin, Can. J. Phys. 48 (1970) 2592.
.