Non-perturbative approach to the electrodynamics of rough metal surfaces

Surface Science 167 (1986) 231-260 North-Holland. Amsterdam

231

NON-PERTURBATIVE APPROACH TO THE ELECTRODYNAMICS ROUGH METAL SURFACES

OF

A.M. B R O D S K Y a n d M.I. U R B A K H A.N. Frumkin Institute of Electrochemistry. Academy of Sciences of the USSR, Moscow, USSR

Received 11 May 1985; accepted for publication 9 September 1985

Surface plasmon (SP) effects on the optical properties of rough metal surfaces are considered. Angular and frequency dependences of radiation, arising from SP decay and diffuse light scattering, as well as the photoemission current from a rough metal surface have been calculated. The above effects have been shown to be strongly enhanced when SP attenuation lengths stipulated by SP decay into vacuum and absorption in the metal, are much larger than the attenuation length stipulated by transformations into other SP states. Possibilities of comparing theoretical calculations with experimental data are considered.

1. Introduction In recent years optical p r o p e r t i e s of rough metal surfaces have a r o u s e d great interest. Intensive d e v e l o p m e n t of these studies was p r o m o t e d by the discovery of some new p h e n o m e n a , related to the e n h a n c e m e n t of an e l e c t r o m a g n e t i c field near the rough surface, such as surface e n h a n c e d R a m a n scattering (SERS) a n d second h a r m o n i c generation e n h a n c e m e n t [1,2]. A similar enh a n c e m e n t was discovered in investigating luminescence of molecules on the surface and in m e t a l - s e m i c o n d u c t o r - m e t a l tunnel j u n c t i o n s [3], as well as in p h o t o e m i s s i o n e x p e r i m e n t s [4,5]. A significant i m p r o v e m e n t of e x p e r i m e n t a l techniques also allowed us to t h o r o u g h l y study the characteristics of optical surface excitations on rough surfaces: their dispersion relation, decay, a n d the role in diffuse light scattering. A p r i n c i p a l result was o b t a i n e d in ref. [6]: it was shown that in the presence of roughness the surface p l a s m o n (SP) s p e c t r u m on silver undergoes not a mere shift but a two-fold splitting. E x p e r i m e n t a l a n d theoretical investigations allowed us to c o n c l u d e that in the range of frequencies at which rather long-lived surface excitations can exist (as is the case with silver), the d e s c r i p t i o n of the interface optical p r o p e r t i e s with the use of the first n o n - v a n i s h i n g term of p e r t u r b a t i o n theory in the interaction with roughnesses, leads to qualitatively inaccurate results. Such theories (often used in the recent literature [7 9]) d o not p e r m i t a c o m p r e h e n sive e x p l a n a t i o n of the p h e n o m e n a observed. In particular, in c o n s i d e r i n g the 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 ~ Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

232

A.M. Brodsl
finite number of perturbation theory terms, the form of the angular distribution of surface plasmon radiation, remains unexplained [10]. In considering the perturbation theory series there arises another problem. Corrections to the polarization operator corresponding to the n th term of the perturbation theory series, turn out to be proportional to [~(~0)+ 1] ", where ~(~) is the metal dielectric permittivity [11,12]. At c(0~) ~ - 1 this value tends to infinity. So, we observe a case of special interest from the point of view of the field theory which can experimentally be realized, displaying a specific strong interaction between the electromagnetic field and roughness. This necessitates summing up a certain infinite subset of terms. The present paper develops a technique of describing light scattering on rough metal surfaces, overstepping the boundaries of the perturbation theory in the light-roughness interaction. In calculating the field strength near the surface and in the waves reflected from the metal, a mathematical technique based on the Green function representation for the electromagnetic field in the form of functional integrals is used. The problem of statistical averaging over roughnesses is solved by means of introducing functional integrals over the so-called superfields, simultaneously comprising commutating and anticommutating variables, similarly to the analogous problem of two-dimensional conductivity in a random medium [13]. The corresponding calculations are given in section 4, which may be neglected by those who are not interested in calculational details. The SP dispersion relation, angular distributions for SP radiation and diffuse light scattering, as well as field enhancement near the surface have been calculated using the above technique. Many papers describing the roughness effect on metal optical properties contain inaccuracies related to the fact that combinations of fields undergoing drastic changes on the surface are used in the calculations [8,91. In this paper the abovementioned difficulties are overcome by means of transformed equations containing only combinations of field components and their derivatives slowly varying at the surface. Possibilities of comparing the results obtained with the experimental data are discussed in section 6.

2. Initial expressions The equation of the metal surface located in the left semispace x r < () is given by the function ,~(Xll)

_r~:~(xll).

x,=(_',-~.x~).

(1)

Assume that this function obeys the Gaussian distribution, and for averages

A.M. Brodsky, M. 1. Urbakh / Electrodynamics of rough metal surfaces

233

over this distribution, denoted by angular brackets, the following equations hold (~)=0,

(~(kll) ~ ( k ~ ) ) = ( 2 ~ r ) 2 8 ( k l l + k i l ) ~ g ( k l l ),

= fd2x,,

e x p ( - i k l l . Xu) ~(Xll ),

(2a) (2b)

g(kll ) = ~ra 2 e x p ( - a2k1~/4),

(2c)

where kll = (k2, k3) is a two-dimensional vector, (~2)1/2 is the root-mean-square value of the roughness and a is the correlation length. The electric field E(x, t) is real and satisfies the Gauss relation div

c(x, co) E(x,

co)= 0.

(3)

In (3) the dielectric function of the system investigated at frequency co is denoted c(x, co). For simplicity let us confine ourselves to considering the local dielectric function c(x, co) having the form ,(x,

c o ) = , ( x , - ~(xH), w ) =

1,

xj - ~(xll ) > 6,

,(co),

x,-

< -8,

(4)

corresponding to the smooth change on atomic distances 8 from metal dielectric permittivity to outer medium (vacuum) dielectric permittivity. In this case, far from the interval of volume plasmon frequencies, space dispersion may be neglected [11,14]. In order to reduce intermediate calculations, assume for the time being that absorption is absent in the system under consideration, i.e. Im ~(x, co) = 0. Final expressions for the observed values will be given for any arbitrary value of Im ~(x, co). In the following, to carry out calculations, take into account only the electromagnetic field components corresponding to p-polarization, as they constitute the main contribution to the field intensity near the surface and, thus to all the observed values discussed in this paper. Additional account of field components corresponding to s-polarization presents no extra difficulties. Under the assumptions made, the vector field E(x, co) can be expressed via the scalar function a(x~, kll, co) as follows

E(x,

ktl, co)= '

[ dzkl , l(x,,

k l l - k ~,co)

j (2~r) 2

e'p(k,,kll)=(1/kll)[k,ki-Si,(k

2 +kl})],

ep(k,, k~) a(x,,kll, co),

(5a)

i=1,2,3,

(5b)

where

, '(x,, k,, co)=fdbq,

co) exp(-iknn.XN),

k, = - i ;)/i)x,.

(6)

,4. M. Brod*k.v, M. 1. Urbakh / Electrodvnantics ~I rouglt metal ~mlaces

234

Let us introduce, as wa~s suggested in ref. [15], a c o m p l e t e set of eigenfunctions a°(.,cl, to, /"11' & ) o b e y i n g the following o n e - d i m e n s i o n a l equation d 1(_,,,to) ~ d a O ( x i , to, ~-,,, &) - / 2 <> I (,-~, to) u,/( ~,,to. a,c~) dx~<, -

- (c~/< .e) u"(x,,

to.

,~,,a,).

iT)

having a structure similar to that of the equation for the p - p o l a r i z e d field in the a b s e n c e of roughnesses. In (7) the p a r a m e t e r O 2 having the m e a n i n g of an eigenvalue is introduced. The function ~(x, to) being d e t e r m i n e d by eq. (4) a t se-= 0 is d e n o t e d ( o ( x l , 0o). It is convenient to choose solutions a°(.vl, o0. k i, &) as real a n d n o r m a l i z e d in the following way

/dx,

<,':{v,. to. * . ~ ) <,,~. "' v ,. to . X-~,. ,~') =,S(a0 ~ -(.C/)-~).

(Sa)

f

<," (_¥,,

(Sb)

dxI

f dx,

to" ~'=I . C , )

l/l~tp(.¥ 1 to. /~,, /.~) ,

" " a~p(-*l

<,;'( _,-,

'

co. /"ii" C,,')=
, tO, 1(I1' t~.~') : ().

(So)

Here indices sp and r denote solutions c o r r e s p o n d i n g to the discrete and c o n t i n u o u s spectra, respectively. At the same time we assume that the ine q u a l i t y * e ( t o ) < 0 is valid and thus the c o n t i n u o u s s p e c t r u m exisis at values of £-" located within the region nt, consisting of two intervals &e > / , ec: and c a - < / < i . c - / ~ ( t o ). F o r simplicity let us assume that there is only one branch of the discrete spectrum,

a,~ = a4,(<~,/,-

).

t<~l

lri particular, m the case of a sharp interface.

cQ=*,7< -~ r(to),

r ( ~ ) = [~(~)+ l]/~tto).

I10)

F r o m the form of the a s y m p t o t i c s o l u t i o n s tg'<]p(X 1, to, /, . ~2') at .t 1 • +± ~ il follows that the inequality e ( ~ ) < 0 is the c o n d i t i o n for the existence of the discrete s p e c t r u m in (7), that is the frequency to must bc lower than the \'olunle p l a s m o n frequency %,. Let us e x p a n d the function a ( x , to) from (Sa) in the solutions of eq. I7)

<,(.,<. to)=

y'

fdU, l,,,(u,,) ~,',I(.,,.

to. U,,)exp(ikH-.r,,).

(11)

¢t = ~,p,r

* Since SP~, can exist o11]}' ;.it ~1 ~') < O. f u r t h e r o n v.'c ~,hal] bc c o n c c r n c c l o n ] \ ,~ith thi> Ircc.lucnc } region.

A.M. Brodsky, M.I. Urbakh / Electrodynamics of rough metal surfaces

235

Here, the following notation is introduced ~sp = {kll},

(12a)

~2r = {kll, ~ } ,

d2kll , fd~=f (2qr)2

fdnr__= " d2kll ,

j (T~)~Ld,;?.

Substituting (5) and (11) into the Maxwell equations for the field obtain a system of equations to determine the b.(~2~)

(12b) E(x, ~o) we

fl = sp,r -2

(13)

~2.

Here

f~,B = c2fdx' [ (kll "kll ),3{ (x 1, kll, - k lq) ~11k,,~112~,3 (%'(X')d-~la°(xl'°:'$2"))

x

eol(x

-kltkll

d o )-g-~xla,(x~,,~,~2,))

&-'(fx_2~:_kij--k~) aO( x,, (kll-k~)

{o, as) a~( x, {o, £~")1"

(14,

In (14) the quantities 8{(xl, kll ) and 3{-1(xl, ktl ) which differ from zero only at atomic distances (for the coordinate xl) from the surface, are introduced and have the form 8{(x,, kll)={(Xl, kll)-(27r)28(kll) {o(Xl) -- - f ( k l l )

deo(X,)/dx ,,

8{ l ( x l , k l l ) = { - l ( x l ,

= -~(kll )

(15a)

kll ) -(2~)2~j(kll) {oI(Xl)

d % ' ( x l ) / d x ,.

(15b)

The conservation of terms of only the first order in ~(kll ) in (15) is motivated by the fact that the addition of the terms of second order in f(ktl ) to 3{ and 3{ 1 can be reduced (in a first non-vanishing approximation over the interaction with roughnesses) to a change in the real components of D2 and 62p in eq, (13). Additional effects appearing during the process are reduced to an insignificant shift of the SP spectrum and to a slight change in the specular reflection coefficient. Due to eq. (7) the functions a~,(xl, 0 0 ,o, ~2,,) and {o1(Xl) da,~(xl, ~o, ~2,,)/dx 1 slowly (exactly at distances of the order of the wave length of light) change,

236

,4. M. Brodsky. M. 1. Urbakh / Electrodvnamics of rou,gh metal surface.s

with changing x~. Let us expand these functions in terms of x~ around .v1 = 0 under the integral in (14), where they enter in the form of a product with the functions &(Xl, kll ) and 8e l(x I, k ) . Conserving in (14) only the main terms of the expansion in the parameter 8 ~ / c we obtain the following expressions which we shall need in the future

j't~p,sp(kll, k ~ ) = 2 c 2 [ - e ( ~ 0 ) ] l':(k,ikl)m'e[-(k,,'k~)+kH/,~],

(16a) ,14

km,)( + l ) - ( ~ e / c e ) e ]

~

x

,

-,l<,e(-~

t, 4

~

''2

,)2_co-

-I,7,{

(l
('-

(

~

-

~t

"

,

&2 > c2(/,~)2,

(lOb) l

.[,.,,(kH, C0, kH, E ) = T r

1 k,7((+l)-

('~

,

/,~)

~,,k,. a-,~- 7~

e

('

(/,ii)-" -

co-'>c~/,-,~, (~')~>c~(s,-i) :.

2

2

~

(16c)

Here all(x ,, 0o, ~2,,) and % I(x,) du(l(.\l, ~, ~,,)/dx, are substituted while conserving the accuracy under discussion for the corresponding functions obtained when solving the sharp interface problem. In using arguments resulting in the expressions (16), there arise no problems connected with the expansion of the functions &(x~, klm), & l(_v~, kll) into a series at the singular points or with the appearance under the integrals of products of singular functions. For example, the authors of refs. [8,9] were faced with such difficulties. The formal solution of eq. (13) can be presented in the form

b,,(&,)=l',',)(&,) -

E /J'.y

f d~2/,d*2~[V,,i,(&,,a/,) E(l
- sp,r

×.lj,.<(arj. ~7~) b4'(a.,)].

(17)

Here b~;,(/%)=(2~r)28(kl) klt') b21, and b;)(kH, &)= (2vr)2(S(kii klt') 8(& 2 0o2) bl} stand for the coefficients of expansion of a(x, 0o) occurring in (11) for

A.M. Brodsky, M.L Urbakh /Electrodynamics of rough metal surfaces

237

~ 2 = 0; and the Green function components for eq. (13) are denoted D.a. Equations for D.a have the form

D.~(t2, t2~)= D ° ( ~ . )

fd<[ D.~(~., <) ~(k~- k;~')

(~..~- Y'~ Y

×/~(~, ~) D2(~)],

(18)

where DO=

1 , ~o2 - 602p-'~in

DO

1 ~o2_ &2_ i71 '

~ ~ +0

(19)

are the Green functions for the case of a smooth metal surface. Fields bsp(kll) and br(kll, &) can be written by means of eqs. (17) and (18) as follows

3r"Dsp,r (k[I , k,~, ¢0)[ D?(k]~, 60)]-1 bO '

(20a)

= 0 0 )] - 1 b2p ~rik,,, ~)Dr,~(k,,, ~, k,~) [ Os~(k,,

o +Dr.r(kll, &, kll,

60)[oo(k,~, ~)]-1 ~o

(20b)

Eqs. (20) are similar to the corresponding equations of scattering theory [16]. They are justified in the absence of eigensolutions to eq. (13) localized along the surface. The possibility of existence and properties of localized solutions, which are not considered in this paper, are at present under discussion [17]. At sufficiently small ~7 they do not appear for sure.

3. Expressions for experimentally measured quantifies The calculations of such observed quantities as the surface plasmons dispersion relation, intensity of the electromagnetic field near the surface as well as angular distributions for the SP radiation and light diffuse scattering are of great interest for rough metal surface optics. Since in the range of parameters corresponding to SP generation, the intensity of the surface waves near the interface is much higher than the intensity of the volume r-waves, in the following, the r-wave interaction and their interaction with the sp-waves will be taken into account in the first non-vanishing order of perturbation theory. The chief attention will be attached to the description of the interaction between sp-waves.

A.M. Broddg,, M.I. Urhakh /Eleetrodvnamies of rough metal suUhees

238

Using (4), (11) and (20), the mean value of the electromagnetic field intensity near the surface, at x~ = ,~(xil)+ .,c~pcan be presented in the form

J,,(x;')_--
:l{o,(x; ,) ~;(k,, k,?) .r{}(xl,{'

,f

i{,i'(x?)

~o, kl? .

~) t,~'l ~

{,;{a., k,, ,,','~(x;'. ~. k,,) t,l']-

(21a)

k,,, t,l,'. ,o )) o a." ~)<'[~ t{0'( x,),, e'p(k,, ,,,,).~ ,4'(xl, ~, ,,.

X (/)~p.r(k,i,

kll ), {o)D,~;.r(

d2k~

d2kll

" ~, a-,,) I~ + l/(2~.)2 {2~)2 I{,, 'I-<'t e'pla-,, k,,t ,,~p(~,,"

× F ~(Ikl, k,?[)]/~.r(k', k",. ~) h:' r['~, (21b) where S is the surface area tending to infinity. An expression for the angular distribution of the diffusely scattered light is obtained in a similar manner. The average over time and surface energy flux in the direction q/q where q-9 1/-~ {(~2/c2 - q4[) ~" qii } is equal to Pdiff {q} =

{.2 q _ r d2kl,

d2k,i f[ep(k"k,l)Iqep ( k l t t

k,i)]]

(2~V (2~)-

q,,)f dD 2 d(D') 2 a;}( x,,

× (2vr)23(kllO/

×artxl,

t

t

~, kll,

~')(&~(k I, ,5, kl?, ~o

× o~;(k,~, c/, kt,', ~)) [D:'(k;,'. ~)] c2

d2kll

co, k N, &

)[qep(k I q,,)]]

d2kll [ e p ( k , ,

- 87,~ ~ f (2=)~ (2rr) 2

b {} 2 t

qll

'

X f dD 2 d(D') 2 a ° ( x , , w, qll" &) a {'' r [ x l ', w, qll" D') l

× D,°(qll, &)( Df'(qil, D'))* [hpl: ~~- g(lqll- kill)

(22a)

A.M. Brodsky, M. 1. Urbakh / Electrodynamics of rough metal surfaces

X { frr(qll'

. -, kll, 0 g~' k~l' ~) fr,(qlt' 0~,

239

0~)(2~r)48(kN_kl0 )

X 6 ( k ~ - kl?)+ fr.sp(qll, ~, kll) /r,sp(q,,, 6~', kll) 1

x
kfi) Ds; sp g(Ik - k, l) (22b)

Here k I = - i 0 / a x ] and k~ = - i a / a x ~ . Only the most important terms corresponding to the so-called ladder approximation [18] are conserved in the expression for (Drr D*) when eq. (22a) is written down. It should be noted that for the calculation of backscattering in the narrow range of angles at [kl? + qllllsp < 1 (where lsp is the SP attenuation length) it is necessary to take into account additional terms corresponding to so-called loop diagrams of perturbation theory for {Drr D*) [19]. These diagrams can be graphically represented as follows k~

q II

k0

q II

Here dashed lines correspond to the Green functions D ° and D °* , solid lines to (D.~p,sp) and (D~,sp), and wavy lines to the interactions with roughnesses. For qtt = - k ~ the contribution of loop diagrams is equal to that of ladder diagrams. Deviation from the backward direction leads to a fast decrease of loop diagram contributions at Iqtl+ k~ [lsp > 1 due to the small additional factor ( [qtk+ktt0 I/sp) - 2 . Accordingly the evaluation of the contribution of loop diagrams to Pdi, (q) is essential only in a narrow range of angles, for example for ~"da2= 10 6 ~4, 2,B.C//0) = 5 X 10 3 A and c --- 5 at Iqll + k~ I < 1 0 - 4 w / c . This phenomenon is known in the wave theory in disordered media [19]. Until now there are no measurements of diffuse light scattering intensity with such small deviations from the backward direction. Other terms dropped in (22b) for all scattering angles contain additional small factors which can be evaluated as follows: (ksp/sp) 2 where ksp is the wave vector of SP. The first term in eq. (22b), describing the direct transition of incident light into diffusively scattered light, was calculated earlier in ref. [8], in which the interaction between the sp- and r-type electromagnetic waves and roughnesses was taken into account only in the first non-vanishing perturbation theory order. The second term in (22a) describes the generation of surface waves, transitions between them, and finally their radiation. Below we show that in

A.M. Brodsky. M. 1. Urbakh /Electroc!vnamics of rough metal ~'lgrf£lccA

240

the case of well reflecting metals the second term not considered in the previous papers may exceed the first one. And, finally, the expression for the angular distribution of SP radiation, is written as follows c2 q 8~w q

P'P(q)

f

d2kll d2kll (2~r) + (2~r) 2 [ep(kl" k )[qep(ki" k[)]]

X(2w)23(kll-qll) f,d&2 d(&')2a~.(xl, oo, kll, do) a ,u. ( x ,,, ~ . k ~ , & 0

')

. k u, , ~-,, kii',/, ~\[DOik × {D~.~p( k u, &, kll~) D~.~p( spt "llu ) ['~'PIeli>'/ 'i ~;c

q c

+ 8~~ q J ~ l

'a

tept' " ' q u

o × f,,d& 2 d(&') 2 a~(x,. ~o, q~,, go) a~}(.Vl, ~o, qu" &') D, o (q!l, C~')

× ( D~'(qll, &))* F-g(Iq,,-kll[) /r.~p(qll" &, ku) f,*w(qH, &'" kll)~1 X{D~p,p(k,,,ki,') . . . D~p.~p(k,,,k,'l')} . [D°(k,t,')]p,

' b~;, .

23)

4. Functional integration Presenting the Green functions Dsp.sp(kll, kll ) and O,*p,~p(kI , kll) contained in (21)-(23) in the form of functional integral ratios [20], J,;(x? ), Pui.t and P,p can be expressed via the following integral d2kli dZk~ lr J = - S J (2~r): {2rr) : h ' ( k u ) h2(kll)(Dw'~P(k'l' k6) D~*v~P ku" kl'l}} D~Db~p Db~ exp - f

=

d~P~(p) (2~) +

~*(p)~ x(p

× f d-kll d kll- h,(kl,) h2(k;I) b~)(kll)(b~p,(kl,)), .

(2~) ~ (2~) +

b~pl)(k~))*tt2,gk Osp ,,--j ' )

A.M. Brodsky, M.1. Urbakh / Electrodynamics of rough metal surfaces

241

The concrete form of the functions h~ and h 2 in (24) results from the comparison of (24) with eqs. (21)-(23); the functional S(bsp, b~, ~) is an action whose variation gives eq. (13) 2

[

_

d2kkL

S([)sp, ~)r,~)=in=lZfJ~(bsp

(.)

*

(kl,))

× [( o~z - ~s2p(kll, ~o))(-1) " + ' - i7/] b~;)(k,,)

+f

f

d2kll

,md 2

X [( (.02 --

*

~)2)(__ 1)"+' -- art] b[")(kll, &)

+ ( - 1 ) " ~,1~y'-sp,rAS~'B(b~)' b~")' ~)}'

[

1 b = t b(~) b(2)}, Q=

(b(")(kll))* =

fd2x,,

~/~ +0,

(25a)

(b(')(xll))* exp(-ik,,xll ),

fD~ exp -j (-~)2~(k,,)~*(k,,) g(k,)

.

(25b)

Indices 1 and 2 in (25b) correspond to the Green functions Osp,s p and D~,sp, respectively. Functional integration over Db in (24) is determined according to ref. [20]. At the same time, (24) represents perturbation theory contracted series (which is easy to check). Let us now rewrite (24) in the form of a unique functional integral over the superfields '/'; it contains commutating b and anticommutating (Grassmann) variables 9( as its components. Using notation similar to that in ref. [13], we introduce the eight-component supervectors ' P ( ' ) = ~ v (") ' ~p = (~,p)Tr,

u ( ' ) = v~- 1 X(") n = 1, 2.

'

v(')=--v~-

b (")

,

(26a) (26b)

242

.4. M. Br
In (26b) tile symbol Tr denotes transposition and 6' is a matrix, having the form 0

~'

= .0

] ,

(27a)

<=(0-,) c=(0, , 1

0

"

-

1

(27b)

0

The mathematical properties of objects, containing supervectors and different operation over them, are thoroughly described in ref. [13]. Integral (24) is written by means of the q" superfields as follows

,f

J = 4-s

x Q

'f "

d2k l dekll

(27)-

,

,~(i,)

/,,(kL,) h:(k,:) 7~,,~(k )<21 + 2 : ) % ( k )

(2¢~): (2~):

. . . . .

× >e(k;) (2,- i'~) %(<)exp[-S( %, ~',, ~)] ;,/

<2s)

where (Z',)3.:

=

(A')7~

=

S(,/%. q',. ~ ) = i

(Ze)~;=-( .

E

+i

.

~'" )7.~ = 1 ,

~ 2

(~",d,)=0. t/

i.i~-21.

(29a)

fd&,'I",,(&,)(::-co~-i'li)'l:,(S,,) fdSX d~%~'t,,,(&,)~(-, -k')

E a,[~= sp.r

x./i#~( ~2~, ~2# ) 'P#( ~2# ).

29b)

The advantage of eq. (28), containing integrals over superfields, consists in that it allows us to carry out the averaging over ,~ in the general form before a direct calculation. It is convenient to calculate the J value as a derivative over the parameters c~,, /{~ (i = 1, 2) from the generation functional Z(~,, ,8,) J =

T gate, + ~ .

i a~,

a&

z(,~,./~,)1,< ,/~. ,>.

where

• - S ( q ' , e ' q'r, ~)[I Q - ' ]J

(2~y

~:x(p)

A.M. Brodsky, M.L Urbakh / Electrodynamicsof rough metal surfaces -

d2*ll

243

__

S( ~-tsp, ~/tr, ~)= S( x/-Psp, ifr, ~)+ ij~-~ { %(,,,) 2, %(,,) × [

+

-i~sp(kl,) X2 ifsp(kN) [/~,hl(*,,) q- ]~2h2(*U)] )Upon functional integration over the fields ~ and ifr in (30) we arrive at an effective action of the form soff= s 0 ( % ) + s(if~ o, h,, h2)

-1'-l~'~f d2*ll, d2'12) d2'13' d2*~4' ((2,~)2~(*I1' "4-'12)+ *13)-~"-*I 4,) (2~r) 2 (2~') 2 ( 2 7 ) 2 (2~r) 2

)'(g([*ll)'q-*12)l) (~so(*l 1)) X//sp(*12)))(~so(*13)) ~/'¢sp(*14))) X fsp,sp(--*ll) , *I 2,) fsp.sp(--k6 3>, *14))),

(31)

dependent only on ifsw Here

So('i%) = if

" 032-

d:*l[ --

6Os2(kll)-iFr(k'') A -Ar(kl,)] x/-tsp(*ll), (32a)

.-

d2kll_

--i~2 [ j~lh' (kll) "~-~2h2( kil)] ) ifsp(*l,), d2*ll

=

(32b)

K

~ - ~ - ? g ( I k l , - *NI) fs2p.,(kl, , *1~, o~).

(32c)

The quantity Fr(ktt ) characterizes the SP attenuation, conditioned by their decay into the r-waves, going away from the metal. The SP spectral shift Ar(kll ) appearing as a result of the interaction of the sp- and r-fields, is reduced to an insignificant change in the function &2p(ktt ), and henceforth it will be neglected. A t ~2/a2 < T 2 the last term in the expression for'Sef r turns out to be (3) and k N(4) values close to the ksp value which is essential only at k II(1), k21(2) I , k211 the root of equation o~ - &sp(ktt, co) = 0. Thus, if the inequalities

-ff'2/a2 < y 2,

kspa < ]

(33)

244

A.M. Brodslqv. M. 1. Urbakh / Electrodvnamics of rough metal suffiwe,*

are satisfied, the calculated results do not depend on the detailed way in which

g(Ikril) decreases at large k ll. All the above considered, we can considerably reduce our calculation when satisfying (33), if we substitute ~i J jj.

Direct calculations show that the account of the additional terms of the order

(a2kll'kf') '' in the expression for g( ]kll~'+ klr2~l)leads, in case (33)is satisfied, to minor corrections of the relative order (aksp)2" in the final expressions. From (31) it follows that the determination of the generating functional is reduced to the solution of the non-linear problem for '/%(kii). We shall find an approximate solution by means of the Hubbard Stratonovich transformation [21], allowing us to linearize the action through the introduction of an additional field. In this case three types of linearization, corresponding to different variants of dividing the four

'>)

included in Setr into two pairs, are possible. In the case of approximate calculations it is convenient [13] to choose one of the two equivalent divisions when the four in &rr from (32) are divided into pairs ~P~p(k~l") ~;p(k,', 3') and ~.(1) klt~p(k~ 2)) ~tt'~p(kill41), or, into pairs -g'~p("il '~, x/tsp(kl!4)) a n d x[t~p(klll ) ) ~/1,p(k,, ) ' ~2) respectively. According to refs. [13,22] if the Hubbard-Stratonovich transformation (in the process of which the commutating and anticommutating fields happen to be bound) is applied in such a way, it becomes relatively easy to single out the most important collective mode contribution. In accordance with the aforesaid we obtain Z=

f

D'/',p D(~Dfi~DRexp

+2

E i = 2,3

I

~(q) i~,(-q)+

-(~a:)

]

SSp

O(q) (~(-q)

"

^

i~i',,(q) R,,(-q)]

E

J

~..l -- 2,3

-s(,I%, o, P, r ~ ) - s ( ~ e, h~, h:)}, S(g',p,. O, P , , q ) = S o ( g ' ~ p ) + 2 v l ' e c e ( - ~ }

×

O(kl, )

(35a) 1,2f(2~r) ed2kll (2v):d~Up,p(k )

klllkll+ql + 2 Y'~ /~,(q)(kll)i Ikii+ql i-2,3

+ £ R,l(q ) k, (kii + q ) e } ~ , p ( - k l l - q ) t. i=2,3 172

× (klllk,~ + q] )

[ 2[ exP,t- l a [ki~ +(k,, q+) - ' ] J . ,

(35b)

245

A.M. Brodsky, M.L Urbakh / Electrodynamics of rough metal surfaces

Each matrix Q, ~ and Rij, included in (35), consists of four (4 × 4) supermatrices, for example Q=

Q2~

Q22 , a 2

(6H)+

i6121

i~n , .[ (hlZ) +

d :=i (62') + ib12] ' (~21=--11i(~12)+' (~22 = (

a22 i(622) +

(36a) 621 i(bl2)+

(36b)

'

i622) ib22 ,

(36c)

where ( ~lrnn ) + ~lmn

01 ~lmnTr 01 ,

( ( ~ m n ) + = 0 2 ~mnTr

( ~)mn ) +~ ~)nm ~ ~'~2 ~mn T r 0Tr 2 ,

cTr

The matrix elements a ' " and b'n represent commutating variables, and 6 m" anticommutating ones. The structure of the supermatrices, introduced here, is similar to the one described in ref. [13]. The symbol SSp describes a supertrace over matrix variables, which is determined by the equation SSp Q = SSp {~11 q_ SSp (~22 = Sp(a,1 + •22) _ iSp(bH + 522).

(37)

A comparison of eq. (24) with (21)-(23) shows that the functions hi(k,) and h2(ku), included in the expression for S(g'sp, h~, h2), can be presented in the form hn( kll) = 2rr1/2c2( - . '"'~ - a/21," [['4(n)b2 "11/"0 "11 + 2 E

i=23

d~')kl)ki +

Y'. d~;)kik/ ] ,

i,j=2,3

n = 1, 2,

(38)

where d o, d i and dij are coefficients independent of the wave vector kl). Taking (38) into account, let us make a linear substitution in (35) 0 ( q ) -* 0 ( q ) - (2~r)2(~ (q){i~, [aado(', + azdo(2)] + 22 [flad(o ', +/~2d0(2)] }, (39a) ~(q)---, ~ ( q ) - ( 2 ~ r ) 2 8 ( q ) { i 2 ~ [ a l d }

1} + azd} 2)] + 22[B~d} ~) + B2dI2)] },

(39b) i~ ij (q ) ---+i~ i /(q ) _ ( 2 7r )2 ~(q ) { i~l [[°tl a ij'(1)_F 0~2d~2)] _1_~2 [131ai - -(1) _F t,.,2.i j l - .4(2)]1}. 4

(39c) As a result of the substitution of (39) the contribution S(q%, h a, h2) to the

A.M. Brodvk v M.I. Urbakh / Electrodvnamics of rough metal surfaces

246

functional (35) is canceled. After the derivatives over a 1, a2, /91 and /92 are calculated, we obtain the following expression for the integral J

i,j

+

2,3

,~

E

7, .tIlb¢,2,'[ e x p [ - g ( ( ~ . /b /~)]

i,1,I,m= 2,3

- 2(~ a2)

'[ d['l'd['2)+ i=~'2.3d)"d:2)+

d~l'd'2']

,. ,,=Y"2.3 ,, ,, 1.

(40)

The functional F(O, /~, /~) in (40) has the form

F(O. P.

e) ' Spf

O(-,,)+

a(2rr)'-" [

+ ~

&,(q) &/(-q)] J

i,/=2,3

-lnfsq%

S P,(,,) P,(-',)

i=2.,

e x p [ - S(~,p, Q, /5 R)].

(41)

Since pre-exponential factors under the integrals in (39) contain variables only at q = 0, it suffices to obtain the explicit form of the functional F of the fields (~(q), /~,(q) and /~',/(q) at values close to zero. Similarly to refs. [13,22] let us present the functional F in the form of an expansion over the deviations 80, 8/~, and 8R,~ from the minimum Q = (~0 /~, = / 5 0 and R,~ = sfq~, ~'hich can be found via equating the variation from (41) over the variables O, t~, and Re/ to zero. The analysis of the equations 8F/6Q(q)=O. 6 F / 6 ~ ( q ) = O and 6F/6Pl,/(q) = O, carried out by perturbation theory, shows that the minimum (41) is attained in the case of the matrices (~0 /~0 and fi/7 being constant in the coordinate space. With regard to the aforesaid we can obtain from (35) and (41) the following equations to determine 0 °, /~0 /~0

0" = - it~_a2c2 ,/2(_~) 1/2f ~d2k" b... (k,,) /,,~ exp( - ¼a2/,,~), (2¢r)2 ~p~"

(42a)

n

/~,{}= _i~2 42c2wl/2 ( _ e )

"~ , e x p ( - 44 1 2k H 2 ) ,~2 [ - d~kll £),p.~p(k ) klTk j

fil~,~,- --i~a2c2wl/2(--e)

(2rr)2

,/2 [ d2k

j (2rr)2

(428)

II

~ /"Jsp,sp[

'kll ) k,,k,k/exp(

aa-kT) __

1

~

~

(42c) ,

A.M. Brodsky, M.L Urbakh /Electrodynarnics of rough meml surfaces

247

Dsp,sp is the G r e e n function, corresponding to the action S ( q % , O 0 p0,

where

t~°), b~p,~p(kll ) = {o~2 - G~2p(khh)- if'r (kll) A - 2mA/2c2i(--l[)-l/2kl I

x(o0

k l2l + 2

E

~A0 k l l k 1+

•=2,3

E

ROz,,,k,k m ) e x p ( - ¼ a 2 k l ~

))

1

/,m=2,3

(43) The right-hand integrals of eqs. (42) can be presented in the form of the sum of the residues at the poles the position of which is determined by the condition ^ _1 Det D~p,~p = 0, and the integrals over the semi-axis kll = t exp[i(¼7r - 8)] where t changes within the range (0, ~ ) and 0 < 8 < 7r/2. When the inequalities (33) are satisfied the integrals over the semi-axis are small as c o m p a r e d to the residues at the poles, and can be neglected. As a result we obtain the following expressions * for the matrices oo, ~0, and Ri° l~ 0

q./1/2 -

_

_

3

~2 a 2 3

C2(__E)I/2y ksp

15l°33-- 15l°22= 1 Do,

A,

Rii° = O,

(44a)

I~liO/=O,

i --#j.

(44b)

F r o m eqs. (42) and (44) it follows that the coefficient before the matrix A in the expression for the G r e e n function Dsp,~p is equal to F r + F~p where __

E,p(kH)=,~2im["

s

d2k ' I1 l

gt

]f,p..,p(kii, kfi)l 2 <2r(k )2-

(45)

describes SP attenuation caused by transitions between different SP states. It is worth mentioning that the inequality F~p >> F r holds in the frequency range where the SPs are generated most effectively ( I c(~0) I --- 1). F r o m the structure of the matrices O0, /~io, and Ri° it follows that the value of F in the m i n i m u m equals zero. The expansion procedure around the

* The system of non-linear equations (42) at sufficiently large ~ has several solutions. Following ref. [13], we have conserved only the one that in the ~ 0 limit continuously changes into the perturbation theory results. The question concerning the validity of the other solutions calls for additional discussion.

A.M. Brodsko', M.1. Urbakh /Electrodvnamics of rough metal surfaces

248

minimum is clear from the results given in the appendix. Restricting ourselves to small deviations from the minimum at F~p >> F,. we obtain 1+o

E

•

E

× SSp{8012(q) + ½[ ^12

q2

~

"p

X { 3 0 2 ' ( - q ) + 2 [ ~! , [,£1~21 , 3 3 ( - q ) + 3 R 2 2 ^21 (-q)]} + SSp{ ½[80'2 (q) - 6Ft'~(q) - 8 / ~ (q)] X [SQ2'(-q) - 6R33(-q)"21 ^21 6R22(-q)] + ¢~[ 3/~,~2(q)

-

^12

8R33(-q)

-8R~

^2, __(-q)]

Y'. l=2.3 ^12 +~[3R32(q)_3~t123(q)][_ 1

3R32 (^21 _q)_3~t2, t_q)]23,

32(--q) + 3R23( - q ) ] (46) Substituting (46) into (40), after calculating the Gaussian integrals over 30, [

3/~, and

3fi?,:, we arrive at

the following expression for the integral J

o

1

-

1

J - (4vc)'y :=-2 X £2~dqo dqg' h,(k,pk) h2(k,pk) exp[i/(q~ - 99')],

(47)

where ~:=(cosqp, sincg),

k'=(cosqo',sinq~'),

~ 0 = 23,

~+l

=

2,

K

~ ~. = ~ I-

Taking account of the terms of order (3C)) 4, (3/~,) 4 and (6fit~:) 4 in the F expansion gives minor corrections * of relative order

to the J value. Note that the reduction of the component i[p(k,p) in the denominator of J is carried out by the following exact relationship (the sum law) d2ki'l d2k[l' Cp(kll)= f(2~r)~ (2~r)~

U(k,l kl)Im(D,p.,p( ,

'

kll, k:!'

)}+0( F,. ) ,~

.

(48)

• The analysis of the effect of these high order terms in 6Q, 6P, and 8/~t~ on F c a n be carried out by, the renormalization group equations, similarly to considering surface conductivity [13,22].

A.M. Brodsky, M.1. Urbakh / Electrodynamics of rough metal surfaces

249

Here U(k n, k~) denotes the kernel of the Bethe-Salpeter type equation [11,12,18] for . The validity of (48) becomes evident when the sequence of irreduceable diagrams for F~p(kll) and U(k n, k ~) is compared as is done when considering conductivity in random media [23]. Note that the expression (47) for the integral J can also be obtained as a result of direct summation of the ladder sequence of the so-called skeleton diagram.

5. Surface plasmon dispersion relation The SP dispersion relation w ~sp(kll) which can directly be observed, for example, with the help of the ATR technique is determined by the position of the poles of the Green function =


D~p(k,,, ~,) = 2 _ c2y(w)k,~ _ ~:(k,,) -iFr(k,, ) -zar(k,,) '

(49)

where ~(kll ) satisfies the equation " d2kll ,r •(kll) = 4c 4 ~ ( _ ¢ ) - l j _ ~ k l l k l l l k l l k

~ _ (kllk~)12g(ikll_ k~l)

X {o~2-c27(w)k'~-Z(kil)-iFr(k~)-ar(k~)

} '

(50)

Eqs. (49) and (50) are obtained as a result of summation of the main terms of the perburbation theory series for Z(kn) which can be presented graphically in the form of diagrams containing no crossings. This summation is carried out similarly to the case of electron motion in a medium with mixtures, described in ref. [18]. By satisfying the inequality kspa < 1 the expressions (49) and (50) can also be obtained by the above method of functional integration, and they follow from eqs. (43) and (44). Within the non-relativistic limit (c ~ ~ ) eqs. (49) and (50) are transformed into equations obtained earlier in ref. [12] via the electrostatics equations. In ref. [12] these equations were solved numerically by the iteration technique, which led, in our opinion, to a number of inaccuracies. Below we present the results of the approximate analytical solution of eq. (50). Yet, we shall not take into account the components ]rr(kt0 and A~(kn) in (49) and (50) which characterize the interaction of the sp- and r-fields, since according to the results obtained in refs. [9,11] the contribution of this interaction in the frequency range we are concerned with, where I~(w) I : 1, is small.

25{}

A.M. Brods'ky. M. 1. Urbakh / Electrovlvnarnics of rough metal vurj~tces

Let us first consider the asymptotics of the large wave vectors klla >> 1. In this case, carrying out the substitution kll = k i l - kll under the integral in (50) and maintaining the leading terms in the parameter kta. we obtain ~ (~,,)=~(-{) 0

, 4F/-

I,-,~ ,/- - c2v/,-,{ - z o (/~,,)

×f~k

d2kll - 4

~ 2~2 ~ exp(-la ),

~

I (1-cos-{p)

"2 ( 2 ~ } ' -

51)

where kl~ = ~ll( c°s ~' sin qp). From (51) the following expression is obtained for 2'o(kll)

-',,(~,,)

=

~(02-~-,:yke)+{'(,0 ~ ,-~k,~)-~+6{ u ( F / . , : ) ~ : / . - ) l~_

-

~

•

.

( 5 2 )

The choice of the sign before the root in (52) is determined by the condition Z ' 0 ( k l i ) ~ 0 at ~ 2 / a 2 ~ 0 and lm "~'o>~0. The next term of the expansion ~(kll ) in the parameter (kua) 1 has the form

.,f12{ 1 F a

~k1{[12c2{ ' 7 - 5 k , , ' d * ' o ( k , , ) / d k , i - d - Y o ( k : , ) / d k , { ] , 1

×.( [ / - - ~.-~k,.- >:,,(k,,)]: - 6{ u ( F / . .

.

.

.

2)k,T/. ~ ,~

(53)

and as follows from a comparison of (52) and (53) at k!la >> 1 it can be neglected. Under the condition that

>> 2¢6-(-{)

{54}

the expression (52) for the polarized operator 2(kll) assumes the form ~ ( '/ell ) ~ - 6 {

lc4(F/a2 )( k:~/a 2 )( 022 --

c2ykl~) 1

(55)

This result was previously obtained in ref. [24] upon conservation of only the lowest-order term of the perturbation theory for .S(kil). Substituting (55) into expression (49) for the Green function Dsp(ktl), the authors of ref. [24] concluded that the density of states psp(kil, 02)= ( l / v ) Im Dsp(kll, 0o),

(56)

as the function 02 has, at fixed kii, two &type singularities, and correspondingly the SP spectrum splits. In fact, the solution found in ref. [24] lies beyond the validity region (54) of perturbation theory. Substitution of the whole expression (52) for ~ ( k ii) into (49) shows that the SP density of states has only one maximum (at fixed kll ) at the point 02"-= c2yki~ and, thus, if the condition

A.M. Brodsky,M.L Urbakh/ Electrodynamicsof roughmetalsurfaces

251

ktla >> 1 is satisfied, the SP spectrum does not split. It should be stressed that the expression (52) for the polarization operator has no singularities at ~02= c2ykl~. Let us now study the SP dispersion relation within the limit of small wave vectors kua << 1. To attain this we divide the right-hand part of eq. (50) into two terms, one of which represents the integral over the region k~a ~ 1 and the second the integral over the region k~a > 1. It is easy to show that the first of the above integrals is proportinal to (klla) 3, and the second to (klla) 5, and, thus, in the klla << 1 limit can be negelcted. So at kua << 1 eq. (50) can be rewritten in the form

Z(kll) -- --3,-lC4 ~

a2k~fol/adk ~k'" 2-c27(k )2z(k )

(5'7)

An analysis of eq. (57) shows that at y2(w)<<~/a2

and

(~o2/c2)a 2<<(~)a/2/a,

(58)

the polarization operator becomes purely imaginary

.~( kll ) = i(3)1/2(- c ) - ' / 2 c2 ( ~ )

1/2k~"

(59)

Correspondingly, in this frequency range the SP spectrum widens and assumes the form (d'Y(t'°) ) -ol I (Ws%)2 Wsp(kll)=¢Os% + ~ ,~,p c2kl~

i(3)1/2(~)1/2k11],

(60)

0 is the SP limiting frequency (at kll ~ ~ ) on the smooth surface. where %p Within the most interesting range of parameters, at ~o2a2 < c2y and klla < 1, the function S(kH) satisfies the equation ~(kH)=

_C2kl3ay t __2y'-~ 1 '

l t 2 q- 3 ~~7 1~2 t -~-3 ~~ 71 t4q- 23 ~~ ~t2 From (61) it follows that

~ 1

at

(61a) ln(1-t)=O.

(61b)

~2(a2"/2)-1 < 0.2

c2k~a

Z(kll) = - a2 2/ 1 - 3 ( f f 2 / a 2 ) ( l / y 2 )

(62)

At (ff2/a2)/'y2 << ] this expression is transformed into the expression obtained in ref. [24] within the framework of perturbation theory. However, as in the

A.M. Brodsky,M.1. Urbakh/ Electrodynamicsof roughmetalsurfaces

252

5

4

2 t I

O,q6

0.98

tOO

~.02

t.04

~/w- °,p

Fig. 1. SP density of states on the rough surface for the following parameter values: (~-) ":/a = 0.l, ktla=0.5, ~(~)=1 Oap/~2. case of ktta >> 1 the conclusion concerning the splitting of the SP spectrum at klla << 1, made in ref. [24], is not valid since the solution found in ref. [24] lies beyond the validity region of perturbation theory. Finally, at ( ~ 2 / a 2 ) / y 2 = 1 / 3 the solution of eqs. (61) can be written in the form Z(kil)=-c2kt~a7{0.63+[0.21(0.95-3~

a 2 y1-2 ) ] '/2}

(63)

Substituting (63) into (49) it is easy to show that in this case the SP density of states Psp(kll, w) has two m a x i m a (see fig. 1). Thus, if the conditions (~2/a2)/y2 = 1 / 3 and ktta < 1 are satisfied, the SP spectrum in fact undergoes a two-fold splitting.

6. Possibilities of comparison with experiment The results obtained in the previous sections allows us to write out expressions for different experimentally measurable quantities. Thus, for example, from (22) and (47), taking into account the functions a~p(Xl, ~, kH), ar(xl,O ao, ktl, go) in their explicit form, it is possible to obtain the following expression for

A.M. Brodsky, M.I. Urbakh / Electrodynamics of rough metal surfaces

253

the partial intensity of the diffuse light scattering Pdiff(0f, q~f) into the solid angle d~2f = sin Of d0f depf

°°4~"2a2exp[_¼a2(qll_klo)2]

I'--1] 2 ediff = 1~5~c~7 0 C4

X [1 +rp(qll)][1 +rp(k~)]

COS OCOS Of COS qOf

- , - 1 1 1 - rp(qll)] [1 - rp(kl~)] sin 0 sin Or 2

[,_ 112 4

2

+ 24-4cos 0 c 4 ~ a2{[ I'l-'[1 -rp(q'l)l

sin20f

-ro(k,°,) i

+½11 +r~(q,,) n2

+½11 + r~(k~) 12cos20]

sin~O

(G~') -' (,:;')-' + (,:p')-'

+ 21,l ' Im rp(qll) Im rp(k~) sin 20 sin 20f cos q0f

×

(](r)] -1 q_(/(m)) -sp ! \-sp

X C0S20 COS20f COS

1

+ 73(~,sp / ( s p ) ~!

2qor

-1

+ ~[1 + r~(q,) 121 + rp(k~?) 12

(l:;,)-' + (t~;",)-I +_2 ~.,., ~ , 3k'sp

×exp{-aa1 2 [(kll0 ) 2 +ql~ +

2k2p]

]

]

}Io(~a kllksp ) Io(½a2qllksp) ~ 2 o

(64)

Here k,0 = (~/c)(0, sin 0, 0), qll = (~0/c)(0, sin Of cos epr, sin Of sin fpf) are the projections on the surface plane of the wave vectors of the incident light waves and of the light scattered on the metal, rp(kH) is the amplitude of the reflection coefficient of p-polairzed light from the smooth metal surface calculated by the Fresnel formula, the angle Or is lying in the plane of incidence and is counted from the normal to the surface, and -sp / (r), /(sp) and /-sp(m) are SP attenuation -sp lengths caused by radiation into vacuum, transition to other SP states and

254

A.M. Brod~ky, M.I. Urbakh / Electrodynamics of rough metal suU~tces

absorption in the metal *, respectively. If (33) is satisfied these can be written in the form [9,11]

(C')

1=

x - ~ 1 ~ 1 - ~ 1 + 3[1[]-(1--[C + ii 1/2 arctg (+11 1/2 , I~+11 (l~;,")

1=

~Tr

c=y(~)

Re k~p(~o)]

a 2-

-

(65a)

(65b)

c5

1/ 2 \-sp

C

~1-

(65C)

The expression for radiation intensity P~p(Of, %,) being the result of decay of the SP, moving along the surface with the wave vector k~p and amplitude 6'p, is written similarly 1¢-- 112 ¢°4~! exp[--~a2(q,i Psp(0r , qq) - 4372~ + 11 ~ a2

×

smo, cos

,.+t-

k~p) 2] Id~p 12

l

cos0,

]e - 1[= 4 + 4-~21~ +-1] c5 ~ a2l~5:P I-~ e x p [ - ~a=( qti- + k~,)] I

2

II

I

+ I l l + r , ' ( q " )l:sin20f/ll' s

- ~le[

12

(r)

'-~P i

~_

'

( (m)

i

Im rp( qIl) sin 20,. cos %.

* In writing(64) and the expressions for other observablesgiven below, it was taken into account that the dielectricpermittivityof the metal has a finite imaginarycomponent.

A.M. Brodsky, M.I. Urbakh / Electrodynamics of rough metal surfaces

255

(/[spt] -1 -sp ! -1

t t,sp,

1

+ 16]1 + rp(qkb)]2 sin 0f cos 209r

(/(sp) ]-1 X

-~p I

{

(66)

(/~p]) 1 (/(m,]-l-jr\ sp ] -[-32"(](sP)) \-sp 1J

In (66) the angle lying in the plane of incidence and counted from the SP spreading direction is denoted Or . The first terms in (64) and (66) describe direct transition of the incident light and of SPs into scattered light. The second terms in (64) and (66) take into account the possibility of incident light transition into the SP, transitions between different SP states, respectively, and then SP radiation. In the case when only the first term of the perturbation theory series is conserved, these terms do not appear. Thus, until present, expressions representing just the first terms in (64) and (66) have been used in the literature [7,8,10] for calculating Pdiff and Psp- However, when .~'/ (/~p'), ' + (/~p'>) ' (this inequality is satisfied for readily reflecting metals in the frequency range where ]c(~0)l < 7, and it holds, for instance, for silver at 2.5 < he0 < 3.5 eV) the second components in (64) and (66) exceed the first components. Furthermore, the frequency and angular dependences of the first and second terms in (64) and (66) are different, which allows us to discriminate contributions from different processes to the scattered light intensity. In particular, from (64) it follows that only the inclusion of the effects connected with SP generation and rescattering may lead to a sharp decrease in the scattered light intensity when approaching the SP limiting frequency from below (to be more exact, frequencies when kspa > 1), observed in ref. [25]. The first term in eq. (66) for the intensity of SP radiation as a function of Of at fixed q~r, has only one maximum for Or varying in the range 0 ° 180 °. The second term in (66) has two maxima, symmetric with respect to the surface normal. This situation is caused by the fact that if the condition (]lsp)-~- 1 > ( i-~p, m~ 1 + (1(,,,)~ ,-up , 1 is satisfied, the different directions of the SP wave vector equally exist on the surface, p,p(O r, q0r) curves with two maxima were observed in ref. [10] when investigating SP radiation on silver and received no natural explanation. It is also necessary to pay attention to another difference of the present results and those of ref. [8], usually used when processing the experimental data on light reflection from rough metal surfaces. The calculations carried out

A.M. Brodslg,, M.I. Urbakh /Electrodvnamics of rough metal surfaces

256

in ref. [8], while conserving only the first terms of the perturbation theory series, lead to the conclusion that even at hn e(0~)---, 0 are the SP absorbed in the metal surface layer and at ]e(,o) ] = 1 this effect makes the main contribution to the change in the specular reflection coefficient caused by the presence of roughnesses. The results of this paper show that, as might be expected, SP cannots be absorbed in the metal for lm e(o0)--+0, however, taking into account the transition between different SP states leads to an appropriate change in the specular reflection coefficient. F r o m (47) one can also obtain an expression for the intensity of the field near the surface ,/,, determined from (21). It has the form

J,,(x',')=l<,(~d,')l

2tDl(O, k,',')l z+ ~]~1 le,,(-'c~')l

t -sp

2

(ivl) -up

[IE (o, + 21, '1 Dl'(O, ki,~)=[1-rp(kll')] cosOdp,

k")

, k~pa <

E,(0, kl,')=-[1

!

1 -4- ( / ~P <'m/ !

1,

(67)

+rp(kl,')] sin0dp.

Here Eli(0, k~i~) and De(0, kll~) are the tangential field c o m p o n e n t and normal induction c o m p o n e n t at the surface for the case of a smooth metal surface [11]. This expression is similar to those previously obtained in ref. [26]. F r o m (67) it follows that in the presence of roughnesses in the frequency range where (/(sp)~ i > t(](r) 1 the intensity of the field near the surface can be -sp * -sp *t- 1 + ([~,,,I) -sp considerably enhanced (in the case of silver at h~0 = 3 eV by about 10 times) as c o m p a r e d to the intensity of the field near the smooth surface. It is worth noting that taking into account the higher-order terms ( # 0 ) 4, (#/5)4 (#fii') 4.... in (41) corresponds to corrections of the relative order

) '

l~r~ )

' +

_,,, io.,I ,

'1}

to the electric field intensity near the rough surface. This conclusion could bc established in the same way as the analogous result in the theory of electronic conductivity in disordered media [13,23]. Here the quantities %,'~m/-'/~.[(/~.~) i + (/~,m _~p ) 1]c and /,~p correspond respectively to the mean free time, frequency and Fermi m o m e n t u m introduced in the theory of conductivity. For silver the value of

(

I

'/[(Z:;')

'+ (

) '1!

is relatively small .((/(~m~p . . . .I/l(lmp . ) I + .l/~''')~p) ]]< 102 at h ~ < 3 higher-order corrections to J arc insignificant in contrast to the made in ref. [28]. The enhancement of the field leads to a n u m b e r of experimentally effects, for example, to an increase in the current of pho/oemission

eV) and statement observed from the

A.M. Brodsky, M.I. Urbakh / Electrodynamics of rough metal surfaces

257

>-~

8 6

4 2

3.3

3.4

3.5

~.6

w

~t0[ev)

Fig. 2. Comparison of experimental and theoretical dependence of 6 Yp - 1 on frequency. The solid curve is based on the experimental data from ref. [4], the dashed curve represents the values calculated from expressions (67) and (68) for t = l / [ c ( o ~ ) [ . In the calculations the optical constants of silver given in ref. [27] were used.

rough surface. Taking into account the fact that the photoemission current is proportional to the field intensity near the surface [11], we obtain the following expression for the experimentally measured ratio 6Yp of photoemission quantum yields on rough and smooth surfaces in p-polarized light, Jlll-k J22 + J33

8Yp= ]D,(0,

k~)]2t+lEil(O,k~)l 2"

(68)

The factor t introduced into (68) takes into account a possible difference m photoemission matrix elements, corresponding to transitions occurring under the influence of the normal and tangential field components. Fig. 2 presents a comparison of the results calculated from expressions (67) and (68) with the experimental data obtained in ref. [4] for photoemission from silver in a 0.5 M K 2 S O 4 solution. From fig. 2 it is clear that for the parameter ~ a 2 = 4 x 107A 4 it is possible to reach quantitative agreement between theory and experiment. Note also another important result of eqs. (67) and (68). These formulae show that with an increase in the mean square size (~2)1/2 of the roughness, the photoemission current must at first increase and then, at values ~ > ~ for I which the inequality (/~p)) 1 > ( / s(m) p) is satisfied, but (~2)1/2 is still smaller than a, must become constant. Such dependence of photoemission current on (~2)1/2 was experimentally observed in ref. [5]. The use of the above expressions

258

,4. M. Brodsl,lv, M. 1. Urbakh / Electroqvnamics' o[ rough metal s'ui:fiwe~

in the analysis of the field enhancement effect on the SERS shows that it is necessary to use additional mechanisms to explain the experimentally observed increase of the Raman scattering signal. Finally, let us discuss the results of comparing the SP dispersion relation calculated in section 5 with the experimental data in ref. [6]. In ref. [6] when measuring the silver electroreflectance it was shown that on a rough surface the SP spectrum undergoes a two-fold splitting of the order of 0.1 0.2 eV. It follows from the results obtained that two-fold splitting of the SP spectrum can occur at klla ~< 1 in the frequency range i ~ - ¢°~P

a

Id ¢ / d co1°:'41'

According to eqs. (49) and (61) to make the splitting value at klla ~ 0.5 equal to the experimentally observed value, the parameter (~2/~12)12 m u s t he equal to 0.08 0.1.

Appendix The expansion of the functional F in the deviations 8 0 . 8P, and 8 R , , from the minimum can be written as follows F=

E ( ,,:2

1),,+, (2c2)"~ ..... (-~)"'2n

f d2k d2q UPp(kll ) ' J ( 2 w ) e (2vr) 2

x[ ~O(q) t~'lku+ql+ 2 ::.~E8P,(q) /',lk, +ql +

~

~SR,,(q} k , ( k l , + q ) , / g ' p ( - k

]

~.l =:2,3

Xexp

~a2(/,-;{+ ( kll + q) ~

:) 'sspf

)>

(2~7) d2q e [~8(~(q) a 0 ( - - q

~ aR,:(q) ,SR,:(-q)]

q- 2 ,~, aP,(q) 8 P , ( - q ) + 1=2.3

q) (k,i]k + q [ ) ' :

i,/

2,3

A.1) l

Here ( , { . . . ) ) denotes a functional averaging with the measure cxp[ S( '/',p. (~, fi'. R)] where 0 = 0 °. /~, = /~," and R,, = R,;. "" Averages of thc products of the ,Pp fiehts in (A.1) are calculated by' means of the Vick theorem with respect to

A.M. Brodsky, M.1. Urbakh / Electrodynamics of rough metal surfaces

259

connected diagrams only. In calculating the Gaussian integrals over the fields q',p a n d X/tsp o n e s h o u l d b e a r in m i n d the e q u a l i t i e s

-i<<%(kL, ) ®

=

+

- - i < < ~ s p ( k l l ) ® ~sTr(kb])>> = ½(2'n')23(kll + k~) Dsp,sp(kll) 4, -i((Gr(kll)

® g'sp(kll)>> = ½(2w)23(kbt + k ~ ) ÷ Dsp,~p(kll ),

+.... =¢~rnn(02 0 ), 0 02

m, n = 1,2.

(A.2)

H e r e ~/'~p® ~/'~p d e n o t e s the_ (8 × 8) m a t r i x r e p r e s e n t i n g the b i l i n e a r c o m b i n a t i o n of the fields q'~p a n d g'~p.

Reference~ [1] R.K. Chang and T.E. Purtak, Eds., Surface Enhanced Raman Scattering (Plenum, New York, 1982). [2] C.K. Chen, A.R.B. de Castro and Y.R. Shen, Phys. Rev. Letters 46 (1981) 145. [3] A.M. Glass, A. Wokaun, Y.P. Heritage, J.R. Bergman, P.F. Liao and D.H. Olson, Phys. Rev. B24 (1981) 4906. [4] J.K. Sass, R.K. Sen, E. Meyer and H. Gerischer, Surface Sci. 44 (1975) 515. [5] J.R. Kirtley, T.N. Theis and J.C. Tsang, Phys. Rev. B24 (1981) 5650. [6] R. KiStz, H.J. Lewerenz and E. Kretchmann, Phys. Letters 70A (1979) 452. [7] A.M. Brodsky and M.I. Urbakh, Usp. Fiz. Nauk 138 (1982) 413 (in Russian); Progr. Surface Sci. 15 (1984) 121. [8] A.A. Maradudin and D.L. Mills, Phys. Rev. Bll (1975) 1392. [9] D.L. Mills, Phys. Rev. B12 (1975) 4036. [10] A.J. Braundmeier, Jr. and D.G. Hall, Phys. Rev. B27 (1983) 624. [11] A.M. Brodsky and M.I. Urbakh, Surface Sci. 115 (1982) 417. [12] G.A. Farrias and A.A. Maradudin, Phys. Rev. B28 (1983) 5675. [13] K.B. Efetov, Advan. Phys. 32 (1983) 53. [14] A.M. Brodsky and M.I. Urbakh, Surface Sci. 94 (1980) 369. [15] Yu.S. Barash and V.L. Ginzburg, Usp. Fiz. Nauk 116 (1975) 5 (in Russian). [16] R.G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966). [17] A.G. Mal'shukov and Sh.A. Shekhmametiev, Fiz. Tverd. Tela 25 (1983) 2623 (in Russian); R. Ruppin, Solid State Commun. 39 (1981) 908. [18] A.A. Abrikosov, L.P. Gorkov and 1.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Pergamon, Oxford, 1963). [19] Yu.N. Barabanenkov, Izv. Vysshikh Uchebn. Zavedeniy Radiofiz. 16 (1973) 88 (in Russian). [20] P. Ramond, Field Theory. A Modern Primer (Benjamin/Cummings, London, 1981). [21] J. Hubbard, Phys. Rev. Letters 3 (1959) 77. [22] V.E. Kravtsov and I.V. Lerner, Zh. Eksperim. i Teor. Fiz. 86 (1984) 1332 (in Russian). [23] D. Vollhardt and P. WOlfle, Phys. Rev. B22 (1980) 4666. [24] T.S. Rahman and A.A. Maradudin, Phys. Rev. B21 (1980) 2137. [25] A.M. Foontikov, S.K. Sigalaev and V.E. Kazarinov, Extended Abstracts Intern. Conf. on Electrodynamics and Quantum Phenomena at Interfaces, Telavi, USSR, 1984, p. 276.

260

A.M. Brodsl~v. M. 1. Urbakh / Electrodvnamics of rough metal surfaces

[26] A.M. Brodsky and M.I. Urbakh, Electrokhimiya 17 (1981) 364 (in Russian); A.G. Mal'shukov, Solid State Commun. 38 (1981) 907; K. Arya, R. Zeyher and A.A. Maradudin, Solid State Commun. 42 (1982) 461. [27] P.B. Johnson and R.W. Christy, Phys. Rev. B6 (1972) 4370. [28] K. Arya and J. Birman, Phys. Rev. Letters 54 (1985) 1559.