THEORY OF CHARGES NEAR A JELLIUM SURFACE

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THEORY OF CHARGES NEAR A JELLIUM SURFACE

M. Babiker

To cite this version:

M. Babiker. THEORY OF CHARGES NEAR A JELLIUM SURFACE. Journal de Physique Collo-

ques, 1984, 45 (C5), pp.C5-465-C5-469. �10.1051/jphyscol:1984570�. �jpa-00224190�

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THEORY OF CHARGES NEAR A J E L L I U M SURFACE

M. Babiker

Physics Depar,bent, University of Essex, CoZchester, Essex C04 3SQ, U.K.

~6sumB - Nous dgrivons l e p o t e n t i e l coulombien d'une charge en dehors d e l a s u r f a c e d l u n mgtal en employant l a t h e o r i e descharnpsquantiques pour l e c a s dumodPlede j e l l i u m a v e c d i s p e r s i o n s p a t i a l e . Nous montrons que l a d i s p e r s i o n s p a t i a l e n ' a p a s d'importance pourlecomportement des p a r t i c u l e s en dehors du m6ta1, q u o i q u ' e l l e s o i t

t r e s

importante pour c e l l e s B l ' i n t e r i e u r . Le nouveau p o t e n t i e l e s t d i s c u t g e t cornpar& avec ceux dans l a l i t t g r a t u r e r g c e n t e

.

A b s t r a c t

-

The Coulomb p o t e n t i a l a c t i n g on a charge o u t s i d e a metal s u r f a c e i s d e r i v e d using quantum f i e l d t h e o r y f o r t h e c a s e of t h e s p a t i a l l y d i s p e r - s i v e j e l l i u m model. I t i s shown t h a t s p a t i a l d i s p e r s i o n i s unimportant f o r charges o u t s i d e t h e m e t a l although it i s e s s e n t i a l f o r p a r t i c l e s i n s i d e . The new p o t e n t i a l i s d i s c u s s e d and compared w i t h t h o s e i n t h e r e c e n t l i t e r a t u r e .

The t h e o r y of q u a n t i s e d f i e l d s e x h i b i t s some fundamental a t t r i b u t e s which a r e c h a r a c t e r i s t i c of Coulomb i n t e r a c t i o n s . The q u a n t i s a t i o n of t h e s c a l a r (Coulomb) f i e l d r e q u i r e s a d e s c r i p t i o n i n terms of abnormal Boson o s c i l l a t o r s / I / . Indeed without such a s i g n a t u r e t h e b a s i c Coulomb p r o p e r t i e s , such a s t h e i n t e r p a r t i c l e Coulomb p o t e n t i a l energy, would emerge w i t h t h e wrong s i g n from quantum f i e l d t h e o r y . I t h a s r e c e n t l y been emphasised t h a t t h e s e Coulomb a s p e c t s a r e e x h i b i t e d n o t o n l y by r e l a t i v i s t i c quantum f i e l d t h e o r y b u t by t h e n o n - r e l a t i v i s t i c t h e o r y a s w e l l / 2 / .

These p e c u l i a r i t i e s i n t h e q u a n t i s a t i o n of t h e Coulomb f i e l d appear t o have e i t h e r been ignored o r overlooked by r e s e a r c h e r s i n t h e f i e l d of condensed m a t t e r t h e o r y . W e emphasise h e r e t h a t t h e s e a s p e c t s a r e a l s o of importance i n condensed m a t t e r t h e o r y e s p e c i a l l y when t h e q u a n t i s a t i o n of t h e Coulomb f i e l d i s an e s s e n t i a l

f e a t u r e of t h e p h y s i c a l e f f e c t s under c o n s i d e r a t i o n .

The p o t e n t i a l a c t i n g on a charged p a r t i c l e i n vacuum o u t s i d e a m e t a l l i c s u r f a c e i s a p h y s i c a l e f f e c t whose p r o p e r e v a l u a t i o n r e q u i r e s t h e q u a n t i s a t i o n of t h e allowed e l e c t r o m a g n e t i c modes. The r e q u i r e d p o t e n t i a l i s d e f i n e d a s t h e change i n t h e p a r t i c l e s e l f - e n e r g y ( r e l a t i v e t o f r e e space) due t o t h e p r o c e s s e s of emission and r e a b s o r p t i o n by t h e p a r t i c l e of t h e allowed quanta of t h e f i e l d s . The presence of t h e m a t e r i a l i s experienced by t h e p a r t i c l e v i a t h e s e quanta s i n c e t h e i r ampli- t u d e s a r e modified by t h e boundary c o n d i t i o n s a t t h e metal-vacuum i n t e r f a c e . By following t h e s t e p s of conventional quantum f i e l d t h e o r y one g u a r a n t e e s t h a t t h e l e a d i n g Coulomb p a r t of t h e p o t e n t i a l w i l l emerge w i t h t h e Coulomb a s p e c t s mentioned e a r l i e r f u l l y taken account o f .

I n t h i s paper we o u t l i n e t h e s t e p s needed f o r t h e e v a l u a t i o n of t h e p o t e n t i a l a c t i n g on a charge o u t s i d e t h e metal. From t h e o u t s e t we s h a l l i n c l u d e s p a t i a l d i s p e r s i o n b u t o u r i n i t i a l t a s k w i l l b e t o i n v e s t i g a t e whether s p a t i a l d i s p e r s i o n i s a s important ( f o r p a r t i c l e s o u t s i d e ) a s h a s p r e v i o u s l y been expected. The l e a d i n g Coulomb p o t e n t i a l emerging from t h e c a l c u l a t i o n s based on t h i s formalism i s t h e n d i s p l a y e d , b u t we s h a l l r e l e g a t e t o t h e end of t h e paper any d i s c u s s i o n s on t h e comparison w i t h p r e v i o u s approaches and r e s u l t s .

The system c o n s i s t s of t h e metal occupying t h e h a l f - s p a c e z

<

0 and t h e charged p a r t i c l e (of mass M and charge e ) i n t h e vacuum r e g i o n z > 0 o u t s i d e t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984570

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C5-466 JOURNAL DE PHYSIQUE

metal. We d e s c r i b e t h e metal by t h e s p a t i a l l y d i s p e r s i v e continuum j e l l i u m model whose e s s e n t i a l i n g r e d i e n t s a r e r e p r e s e n t e d by a c o l l e c t i v e p o l a r i s a t i o n f i e l d

5.

Associated with

5

a r e t h e conser ed charge and c u r r e n t d e n s i t i e s

S

- ne V . < and ^0-.- n e s , gnd a d e v ' a t i o n A n = - nm6

(1.5)

of t h e p r e s s u r e from i t s equilibrium v a l u e . Here (3 = 315 vi with vF t h e Fermi v e l o c i t y . I n t h e presence of t h e charged p a r t i c l e we have F

where a l l t h e terms i n v o l v i n g

5

a r e e f f e c t i v e only i n s i d e t h e j e l l i u m . The p a r t i c l e charge and c u r r e n t d e n s i t i e s a r e d e f i n e d by p = e6

(g-5)

and J = ex6

(G)

with

5

and X t h e p a r t i c l e p o s i t i o n and v e l o c i t y v e c t o r s . The a p p r o p r i a t e Lagrangian which -

produces t h e e q u a t i o n s of motion (1) t o ( 4 ) i s

The q u a n t i s a t i o n of t h e f i e l d s i s done i n terms of F o u r i e r modes with t h e time- dependence e-int i n t h e absence of t h e p a r t i c l e . For a mode of frequency R e q s . (1) and ( 3 ) g i v e

where

% W

w i t h E = e c z ) + ~ e c - z ) ; ^E= 1 - P

-

w = (4rneo /m) ²

.

n2 ' P ( 8 )

Equation ( 6 ) i s s a t i s f i e d by t h e following two c r i t e r i a

% %

(x.g(Q)

⁼^{0 with}

EG

#

2 ) ; (EE

=

2

with

x.E

# 0 ) . ( 9 )

The f i e l d s s a t i s f y i n g t h e second c r i t e r i o n i n e q . ( 9 ) a r e t h e w e l l known l o n g i t u d i n a l bulk f i e l d s which a r e c h a r a c t e r i s e d by t h e vanishing displacement

D

i n t h e e n t i r e space / 3 / . I t i s e a s y t o s e e t h a 5 a va i s h i n g displacement can be achieved only i f

2

9

we have ~ ' 1 =

2

f o r z

>

0 and ( E + B

/n

V

)Ell

= _O f o r z < 0. These o s c i l l a t i o n s a r e t h e r e f o r e confined e n t i r e l y t o t h e metal i . e . t h e y a r e n o t coupled t o t h e e x t e r i o r even i n t h e presence of s p a t i a l d i s p e r s i o n .

The f i e l d s which have non-vanishing amplitudes o u t s i d e t h e m e t a l a r e t h o s e s a t i s f y i n g t h e f i r s t c r i t e r i o n i n e q . ( 9 ) . For a t h e o r y aiming a t t h e q u a n t i s a t i o n of a l l t h e p o t e n t i a l s

A

and 4 , t h e Lagrangian i s w r i t t e n i n t h e form

.-

where we have t a k e n account of t h e requirement t h a t

) A . ! (

^and⁴a r e connected by a gauge c o n d i t i o n of t h e type $ = f ( x . f i ) . I n t h e absence of t h e p a r t i c l e

(e=

O,g=?) we o b t a i n from e q s . ( 2 ) and ( 3 ) f o r t h e mode fi

I f we now f i x t h e gauge by s e t t i n g

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Using t h e s e f i e l d e q u a t i o n s t h e gauge c o n d i t i o n (12) w i l l reduce t o t h e s i m p l e r form

' L .

V .

:-

+ 5

$ = 0. ⁽¹⁴⁾

The c a n o n i c a l procedure, with

5,

A_, $ and

5

a s v a r i a b l e s w i l l l e a d from t h e Lagrangian i n e q . ( 1 0 ) t o t h e Hamiltonian

f o r t h e " j e l l i u m p l u s f i e l d s "

2. 2 2

+ -) (1.i)

² d r

^- -

J- ⁸ⁿ

1

(l$.x$+c$ / c ) d ~

The n e x t s t e p i s t o e l i m i n a t e

5

i n favour of and $ u s i n g t h e r e l a t i o n s (17)

C

where we have a l s o made use of e q s . ( l 3 ) . The second e q u a t i o n i n ( 1 7 ) d e y o n s t r a t e s 2 2

t h a t s p a t i a l d i s p e r s i o n o n l y i n t r o d u c e s terms o f t h e o r d e r ( B / c ) 2 10-

.

S p a t i a l d i s p e r s i o n a f f e c t s a l s o t h e boundary c o n d i t i o n s which demand t h a t (?$I) be con- t i n u o u s a c r o s s t h e boundary. This i s e q u i v a l e n t t o t h e c o n d i t i o n E (1- B2/c 2 ) ~ ' l (0-) = E1'(O+) an9

is

t h e r e f o r e d i f f e r s from t h e d i s p e r s i o n l e s s case by t h e appearance gf t h e

£ & t o r ( B /c ) =

loe4.

Thus we can deduce t h a t s p a t i a l d i s p e r s i o n can l e a d t o only v e r y small c o r r e c t i o n s t o a l l t h e p r o p e r t i e s i n v o l v i n g t h e medium-modified Maxwell f i e l d s . By c o n t r a s t t h e bulk f i e l d s depend c r u c i a l l y on t h e f i n i t e n e s s of t h e parameter 6 b u t t h e y a r e i r r e l e v a n t f o r p a r t i c l e s o u t s i d e t h e j e l l i u m .

The q u a n t i s a t i o n of t h e f i e l d s i s done by s u b s t i t u t i n g from e q . ( 1 7 ) i n e q . ( 1 6 ) and by following t h e subsequent s t e p s of t h e d i s p e r s i o n l e s s case /4/. T h i s procedure w i l l l e a d f i r s t t o t r a n s v e r s e modes a s s- and p- space modes and t r a n s v e r s e s u r f a c e plasmons a l l of which couple only t o t h e t r a n s v e r s e p a r t i c l e c u r r e n t

3

^and^{w i l l}

t h e r e f o r e n o t c o n t r i b u t e t o t h e Coulomb p o t e n t i a l . The q u a n t i s a t i o n l e a d s secondly t o s c a l a r - l o n g i t u d i n a l space modes ( a ) and s c a l a r - l o n g i t u d i n a l s u r f a c e modes ( 6 ) . The a and 6 modes couple only t o t h e l o n g i t u d i n a l p a r t i c l e c u r r e n t J " which i s connected t o t h e p a r t i c l e charge d e n s i t y by t h e e q u a t i o n of c o n t i n u i t y . Exact c a n c e l l a t i o n s e x i s t between t h e c o n t r i b u t i o n s a r i s i n g s e p a r a t e l y from t h e longitud- i n a l and s c a l a r f i e l d s and o n l y a r e s i d u a l c o n t r i b u t i o n i s r e s p o n s i b l e f o r t h e Coulomb p o t e n t i a l . E x p l i c i t c a l c u l a t i o n s r e v e a l t h a t t h e l e a d i n g r e s u l t s a r e dominated by t h o s e o f 2 t h s d i s p e r s i o n l e s s case w i t h s p a t i a l d i s p e r s i o n i n t r o d u c i n g

terms of t h e o r d e r ( 6 / c ) . The Coulomb p o t e n t i a l can t h e r e f o r e be w r i t t e n a s

where A = c/w and 11 = 2w Z / c E 2Z/A The dimensionless f u n c t i o n s Fa and FB corresp8nd r e s g e c t i v e l y t o P t h e contrig;tions a r i s i n g from t h e space modes ( a ) and t h e s u r f a c e modes (6)

.

E x p l i c i t l y w e have

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PHYSIQUE

The f u n c t i o n s Fa, FB and F a r e d i s p l a y e d g r a p h i c a l l y i n Figs2 l ( a ) and l ( b ) and compared with t h e f u n c t i o n F = - 1/p ( ~ o r r e s ~ o n d i ~ l g t o V = -e / 4 Z f .

Fig. l . ( a ) : V a r i a t i o n with = 2w Z/c of t h e f u n c t i o n s Fa and FB and comparison with t h e f u n c t i o n F = - 1/p. ~ i g ! 1. ( b ) : V a r i a t i o n w i t h p of t h e f u n c t i o n F = Fa and comzarison with Fo. I n t h e l i m i t p -t 0 we have Fa -t

+

F / 2 , F6 + -+:'and F ⁺

-

F0/2. I n t h e l i m i t y -t we have Fa+ F ~ , F + 0 agd

F " F o . B

The above r e s u l t s f o r t h e Coulomb p o t e n t i a l a c t i n g on a charged p a r t i c l e o u t s i d e t h e j e l l i u m can be compared with t h o s e i n t h e l i t e r a t u r e . E a r l i e r d e r i v a - t i o n s were made by Lucas /5/ u s i n g a v e r s i o n of quantum f i e l d t h e o r y i n t h e d i s p e r s i o n l e s s c a s e ( 6 = 0 ) and t h e q u a s i s t a t i c approximation ( c + m ) . The

q u a n t i s a t i o n smerged i n terms of s u r f a c e modes only ( t h e s u r f a c e plasmon model) and l e d t o V = -e /4Z. The d e r i v a t i o n with c f m and 6 = 0 l a t e r r e p o r t e d by Tomas and S u n j i c /6/ cqnfirmed R i t c h i e ' s c o n j e c t u r e / 7 / t h a t t h e s u r f a c e plasmon model should y i e l d V n, Z f o r l a r g e Z . T h i s c o n t r a d i c t s t h e r e s u l t V = -e /4Z of t h e quasi- 2 s t a t i c model which was a l s o d e r i v e d by o t h e r workers / 8 / . The c o n t r o v e r s y was r e c e n t l y r e v i v e d by Ekardt /9/ who provided p a r t i a l answers a s r e g a r d s t h e dependence f o r l a r g e Z. Other r e c e n t c a l c u l a t i o n s , which i n c l u d e d s p a t i a l d i s p e r s i o n

(6 # 0 ) b u t employed t h e q u a s i s t a t i c approximation ( c + m ) a t t r i b u t e d c o n s i d e r a b l e importance t o t h e e f f e c t s of s p a t i a l d i s p e r s i o n and emphasised t h e r o l e of t h e bulk modes i n determining t h e Coulomb p o t e n t i a l o u t s i d e t h e d i s p e r s i v e j e l l i u m

/lo/.

The main c o n c l u s i o n s of o u r i n v e s t i g a t i o n s a r e t h e following. The popular b e l i e f t h a t t h e i n t e r a c t i o n of an e x t e r n a l charge with a m e t a l t a k e s p l a c e p r e - dominantly v i a t h e s u r f a c e modes i s i n v a l i d . T h i s can be seen by t h e i n s p e c t i o n of t h e fun t i o n s i n F i g s . l ( a ) and l ( b ) . The q u a s i s t a t i c approximation p + 0 y i e l d s

5

V6 = +e /4Z i . e . t h e c o r r e c t magnitude b u t t h e o p p o s i t e s i g n . This i s an example of t h e m a n i f e s t a t i o n of t h e Cou1omb"signature" mentioned a t t h e o u t s e t . The l a r g e Z l i m i t i s dominated by t h e c o n t r i b u Ions from t h e space modes ( a ) g i v i n g r i s e t o

t

^'

t h e expected c l a s s i c a l r e s u l t V = -e /4Z. We a l s o found t h a t s p a t i a l d i s p e r s i o n i s

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minimum a t = 2 i . e . a t Z 'L X

.

The p o s s i b i l i t y o f a "rounding o f f " of t h e p o t e n t i a l a t s h o r t d i s t a n c e s Ras d i s c u s s e d by M i l l s ( s e e r e f . /11/ and r e f e r e n c e s t h e r e i n ) . The magnitude of t h e minimum i n Fig. l ( b ) i n d i c a t e s t h a t t h e p o t e n t i a l can o n l y g i v e r i s e t o s h a l l o w image-induced s u r f a c e s t a t e s , b e i n g two o r d e r s of magnitude s m a l l e r t h a n t h o s e a r i s i n g from o t h e r p o t e n t i a l s /12/. The form of t h e p o t e n t i a l w i l l a l s o have i m p l i c a t i o n s f o r t h e s u r f a c e phenomena i n which t h e c l a s s i c a l image p o t e n t i a l i s c u r r e n t l y b e l i e v e d t o p l a y an important r o l e .

Acknowledgements I am g r a t e f u l t o M r s . Mary Loudon f o r t h e French t r a n s l a t i o n of t h e a b s t r a c t and t o t h e SERC and t h e U n i v e r s i t y o f Essex f o r f i n a n c i a l s u p p o r t . REFERENCES

/1/ See, f o r example, GUPTA, S.N., Quantum Electrodynamics (Gordon & Breach)1977 /2/ BABIKER, M . , Proc. R. Soc. Lond.

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(1982) 485

/3/ AGRANOVICH, V.M. and GINSBURG, V.L., S p a t i a l D i s p e r s i o n i n C r y s t a l O p t i c s and t h e Theory of E x c i t o n s (Wiley) 1966

/4/ BABIKER, M . , P h y s i c a e ( 1 9 8 3 ) 339

/5/ LUCAS, A . A . , i n C o l l e c t i v e E x c i t a t i o n s i n S o l i d s (plenum) 1982; Phys. Rev.

B4 (1971) 2939

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/6/ TOMAS, M.S. and SUNJIC, M . , Phys. Rev.

g

(1975) 5363 /7/ RITCHIE, R . H . , Phys. L e t t .

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/8/ ECONOMOU, E . N . and N G A I , K.L., Adv. Chem. Phys. X X V I I (1974) 256; MAHAN, G.D.

i n Nobel 24 C o l l e c t i v e P r o p e r t i e s o f P h y s i c a l Systems (Academic) 1973;

BARTON, G . , J.Phys. (1977) 601

/9/ EKARDT, W . , S o l i d S t a t e Commun. 40 (1981) 273

/lo/

^BARTON, ^{G . ,} Rep. Prog. Phys. $2- (1973) 963; EGUILUZ, A . , Phys. Rev.

(181) 1542

/11/ MILLS, D.L., PhyS. Rev. (1972) 763

/12/ BARTON, G . , J . Phys.

2

(1982) 4727 and r e f e r e n c e s t h e r e i n