- Email: [email protected]
electrolyte interface
179
J. Elecrrounul. Chem., 228 (1987) 179-l% Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ON THE ROLE OF LOCAL MODES IN THE OPTICS METAL/ELECTROLYTE INTERFACE *
A.M. BRODSKY
OF THE
and MI. URBAKH
A.N. Frumkin Institute of Electrochemistry,
Academy of Sciences of the U.S.S.R,
Moscow (U.S.S.R)
(Received 21st January 1987)
ABSTRACT The influence of surface roughness on the electromagnetic response of the metal/electrolyte interface is considered. The intensities of local surface electromagnetic modes are expressed in terms of their formation cross-sections and life times. The effect of the interactions of local modes with delocalized surface plasmons and with one another on the spectrum of electromagnetic excitation of the system and on the behaviour of the field near the interface is examined. The possibilities of using the results obtained for the description of experiments (surface enhanced Raman spectroscopy, photoemission, surface plasmon spectroscopy, electroreflection and electroscattering) at the metal/electrolyte interface are discussed.
(I) INTRODUCTION
The influence of surface roughness (inhomogeneity) on the electromagnetic response of the metal/electrolyte interface has been studied in numerous papers. The discovery of the enhancement of Raman scattering (RS) on molecules adsorbed on metal surfaces (surface enhanced RS) gave a strong impetus to investigation of the role of surface roughness in different optical phenomena. It was convincingly proved [l] that a substantial part of RS enhancement is connected with an increase in the intensity of the electromagnetic field near a rough surface. Earlier, a similar enhancement was detected when photoemission at the silver/electrolyte interface was investigated in the pioneer work of Gerischer and co-workers [2]. The effects associated with field enhancement on rough electrode surfaces were also observed in studies on non-linear optical phenomena [3]. Similar effects were revealed at the metal/vacuum interface [4]. It should be stressed that optical experiments at the metal/electrolyte interface continue to be the most effective primarily because they
* In honour of Professor H. Gerischer on the occasion of his retirement as Director of the Fritz-Haber Institute.
0022-0728/87/$03.50
0 1987 Elsevier Sequoia S.A.
180
allow potential drop modulation techniques to be employed. The data obtained by these techniques allow adequate separation of bulk and surface effects. On the other hand, the information provided by these data is extremely important for solution of the problems of the double-layer theory as well as other vital problems of electrochemistry. The effects under consideration cannot be explained on the basis of the concept, discussed in the literature [1,5], that the main role is played by surface plasmons (SP) propagating along the interface. If the formation of SP is taken into account, the field enhancement ratio on a rough surface as compared with a smooth surface [5] is much smaller than the experimentally observed values. At present, it may be considered to be established experimentally [l] that on rough metal surfaces for which the characteristic sizes of their inhomogeneities in the direction normal to the surface are of the same order as those in the direction along the surface in addition to SPs propagating along the interface, there can exist resonance localized electromagnetic modes which are also excited by an electromagnetic field and can cause a sharp increase of scattered electromagnetic radiation. It should be emphasized that according to experimental data, the decisive role in the field enhancement effect on the metal surface is played by microscopic roughnesses (mean size 10 nm) [2]. This effect disappears when roughnesses are below 1.0 nm and over several hundreds of nm in size [6]. The main theoretical results on local modes were obtained using numerical calculations for simple geometrical forms of inhomogeneities (spheres, ellipsoids, etc.). With this approach, it is impossible to establish the general features of the phenomenon under consideration which should remain valid for more realistic structures. The object of the present study was to obtain analytical expressions for different optical characteristics of rough electrodes in a long-wave approximation, in many respects similar to the zero radius potential method used in the quantummechanical theory of scattering [7]. With such an approach, all the properties of the metal and the inhomogeneities enter into the observed quantities only in terms of the coefficients of light reflection from a smooth surface and the energy characteristics of local modes on individual centres modelling the roughness. In Section (II) of this study the intensities of local surface modes are expressed in terms of their formation cross-sections and life times. It has been ascertained how the intensities of local modes enter into the expressions for the observed effects. In the Section (III), a model calculation is performed of the formation cross-sections of local modes on one centre, confirming the resonance form of these cross-sections. Section (IV) treats the effects of the interaction of local centres. Sections (III) and (IV) also examine the effect of the interactions of local modes with SPs and with one another on the spectrum of electromagnetic excitation of the system being considered and on the behaviour of the field near the surface. In many aspects, a similar approach was used earlier in considering the influence of atoms and molecules near the metal surface on light scattering [8,9]. In Sections (III) and (IV), a microscopically rough surface is represented as a combination of bumps resting on a flat substrate. The influence of these bumps is
181
modelled by introducing into the surface region, point dipole centres in whose vicinity local electromagnetic modes can be excited. This approximation can be justified quantitatively in the case where the characteristic sizes of the inhomogeneities (bumps) are less than other parameters of the problem of a dimension of length such as the light wavelength, the distance between bumps and the distance from the bump centre to the surface. In the opposite case, it is necessary to take into account the influence of multipole local modes, and, accordingly, the treatment used may claim to be no better than a qualitative description of the phenomenon. Comparison of the analytical dependences obtained with the results of numerical calculations (for those systems for which they were available) carried out taking into account modes with higher angular momenta confirms the validity of the assumption that even in this case the simple model considered gives a correct picture of the field distribution near the metal surface. In Section (V) the possibilities of using the results obtained for the description of experiments at the metal/electrolyte interface are discussed. (II) INTRODUCTION
OF LOCAL ELECTROMAGNETIC
MODES
ON ROUGH
SURFACES
The electromagnetic fields 8(x, w; /3) figuring in the problem may be considered to belong to one of three types: 1 - bulk fields (/3 = v) subdivided into s- and p-polarized waves (below we will consider only the most important p-wave contribution); 2 - surface waves (/I = sp) and 3 - singled-out resonance states localized in the vicinity of inhomogeneities with a small radius of curvature (/3 = 1; in the literature they are sometimes called local plasma modes). The interaction of the bulk and surface waves with one another will be taken into account in terms of the perturbation theory on the interaction with roughnesses, as was done in ref. 5. Since in the systems of most interest to us, surface and localized modes are long-lived, we will ignore the interference between fields of different type. To explain the assumptions made, we will consider a general expression for the scattering phase of an electromagnetic wave on a local centre for the case where the wavelength, A = 27rc/w, is much larger than the characteristic sires of the centre R,. For the largest phase S, corresponding to electric type dipole oscillations, it is possible, ignoring deviations from spherical symmetry, to write a general formula
[71:
q*(
g3 cot
6, =
a(w)R,3 +
Here, a and b are coefficients centre of spherical shape 44
= tk(4
b(4
= 0.9[(2~,
+ 2%M&4 - 44M44
c,b(o)( j)*R,’+o((g4RO) depending on the frequency w. In particular,
for a
- 41 - 41
where cc(w) and E,, are the permittivities of the metal of which the sphere is made and of the electrolyte. Equation (1) is similar to the “effective radius” formula used
182
in the quantum-mechanical theory of scattering [7] for an angular momentum equal to unity. From eqn. (1) disregarding inelastic processes, we obtain the following expression for the resonance scattering cross-section u on the centre and the reciprocal of the life time T: u(w) =12x
iI-2
2 e,’
c
(1 ;
(w - o())* + :r’
l? = 2( @~RO)‘/~
Iy__
= ( 12b(w,)
4%) + %I TR”
0
Substituting into eqn. (2), for estimation, the parameters corresponding to a sphere, we obtain a resonance life time of the order of I’-’ = 3~i5/2 1dc /dw I. During this time, the unscattered wave will move (I/4)((w/c)%a distance c/I B h, much larger than the wavelength, so that the interference between running and local modes may in fact be neglected even without allowing for other inelastic processes. This conclusion is also valid for the scattering of surface waves since their group velocity is of the order of c. These local modes can be formed both directly under the action of light incident upon a metal and as a result of the interaction of surface waves with inhomogeneities of a small radius of curvature. The last-mentioned formation mechanism of local modes is, evidently, of greatest effectiveness, at least at a small inhomogeneity concentration. The surface waves, in turn, can be excited by bulk waves on slightly rough surface sites in the region between inhomogeneities of a small radius of curvature. According to the above scheme, the relation between the number of local modes and the number of photons in the incident flux can be found from the following kinetic equations:
Here, IV, is the number of photons (in unit volume) in the incident flux; ySP( w) and N,(w) are the numbers of surface plasmons and local modes on umt surface, respectively; pi is the surface concentration of centres; and us+,= u,,(w) is the velocity of surface plasmons; in the case of a sharp metal/electrolyte interface, it is equal to C Usp =
%I
belY2kn 6,
beI
+
+ 4
+
h,,
%,)3’2 ah/au
where em(u) is the permittivity of the metal. In eqns. (3), t,,, denotes the transformation coefficient of the formation of surface plasmons from bulk waves
183
and ai,,r(~) is the cross-section for the formation of local modes from surface plasmons * attaining a maximum at 0 = w,; r, ,s(~, B = u, sp, 1) stands for the reciprocal life times of states j3 defined by their decay into states (Y, and the coefficient IYa,s accounts for the dissipative decay channel of j? fields. It is shown in ref. 5 that
k; =
;c,*sin8
(4)
where 8 is the angle of incidence of light, rp(ki) is the amplitude of the reflection coefficient of p-polarized light from a flat metal surface and (E2)*j2 and a are the root-mean-square height and the correlation length of smooth (or large radius of curvature) roughnesses. From eqns. (3) we obtain the following expression for a stationary number of local modes with frequency w on unit surface: PP,,,W N,(w)
t,,,(+N,
(5)
= rsp,l(o)
+
rd,l(u>
~la~,sP(ohp
+
ro,sP(w)
+
rd,sp(w>
The decay of local modes into radiation can be treated in a similar manner. Here, to describe RS due to an inelastic transition in the vicinity of a local centre, it is necessary to add to system (3) an equation for the number of local and surface modes with the scattered light frequency wf: dNttuf)
dN,,bf)
=
WfiNl(Q)
-Nl(Wf)(rsp,,(Wf)
+
rd,l(wf)) (6)
= N,(w,)I,,,,(+)
-
N,(wf)[p,o,,~p(w,)v,p
+
ru,s,b’f)
+
‘d,spbf)]
dt Here, wr, is the probability of an inelastic transition wherein a local mode with frequency w changes to a local mode with frequency wf. In fact, in writing expression (6), the transitions between the initial and final states were assumed to be a stochastic process; the final fields excited at different instants of time are considered to be incoherent. In principle, a situation in which the final fields are coherent is possible. In such a case, additional enhancement of
l
It is worth noting that the cross-section
o,.~~ has the dimension
of length.
184
surface RS may arise, similar to that observed in the case of stimulated RS [lo] in the bulk *. From eqns. (3) and (6) the following expression ensues for the steady-state intensity of surface waves of frequency or: rsp,lbf>
Wfi N,pbf)
=
Cp,l(~f)
+ rd*lbf)
Pl%p
bfbS,bf)
+ ~,bf) t,,“wc
m,,b)
X
+ rd.,bf)
q&4
+ rd,,w
~~~,~,bb,,(4
+ ru&4
+ rd.&4
N0
(7)
Using the approach described above, by means of formulae (5) and (7) we obtain the following expressions for the photoemission current J in the case of a rough surface (the surface regions near the inhomogeneities of small radius of curvature being assumed to make the main contribution to J) and for the radiation intensity of dY/dB formed during RS and scattered into a solid angle da = sin 0, de, dq,:
J= IMI*N,(w)
(8) Here, M is the matrix element of the optical transition and the function P,( r?,, cp,; cp) describes the angular distribution of the radiation arising du$ng decay of the surface wave propagating along the interface in the direction k = (0, cos cp, sin cp}. An expression for Psp was obtained in ref. 5 and is of the form P,,(ef,
‘Pr;
l/2
=
‘4
‘PI
I %I-
Enl I
2
X
sin
ef cos
vf +
u4 c3
-[a
43a21E,l+Cml
-2
2
exp -?- 42(~~-iks,)2]lijl+~~(~~)) ]
(-~~,/c~)l’~(l
- rp(
klf)) cos
e,J2+ 23[i ;r r:T
I; $
X~2a2exp[-~~*:)2+*~~~]~~~~a2~~~~~~{[~~~~l-~~~x,.)‘n m
xcos2ef+++~p(k~)~2 sin28fr”,s:yr
x
sin 2efcoscpf &+p
+ +
-~]~]1’21m
+ rp( k;)
r,(ki)
I2 sin e, cos
2q,
+ rd,sp + akSP
rSPSSP
X
ru.sp+ rd,SP +
It is interesting surface.
l
rSPVSP
d,sp
(9) aSP,SP
that in this case the role of mirrors
can be played
by one-dimensional
walls on the
185
and k, = k,(w) is the wave vector of a surface plasmon. Diffuse light scattering arising from the presence of inhomogeneities of small radius of curvature may be treated in a similar manner. Here, the angular distribution of scattered light, dR,JdB, is written as follows:
dR, dQ
/
N3w) =[
+r,,,w +cl,sPw]2
Plul,,(+,,(4
X
rsp,1w rsp,l(w) +
r,,+)
00)
td+N,
(III) MODEL CALCULATION OF CROSS-SECTIONS MODES ON AN ISOLATED CENTRE
FOR THE
GENERATION
OF LOCAL
In considering the problem of an isolated local centre localized near the metal surface at the point x0 = (x,0, 0,O) (the x1 axis is directed along the normal to the surface of the metal which occupies the halfspace xi < 0), we will proceed from the following algebraic equation for the electromagnetic field:
(11) Here 4:” and 0,: denote the electric field and the Green function of the electromagnetic field calculated for a semi-infinite metal without local centres and summation over identical subscripts i, j = 1, 2, 3 is to be carried out. The explicit form of the functions &,‘(x, w) and D,q(x, x’, w) is given, for example, in ref. 11. The function x(w) in eqn. (11) characterizing the properties of a point local centre not interacting with the field is related in a simple manner to the Green function G(x, x’, w) of dipole oscillations of the particular dipole centre located at the point x0 G(x,
x’, w)=P(x-x0)x(w)
83(x’-x0)
Equation (11) is equivalent to the Maxwell equations for a semi-infinite metal with a point centre near the surface. The Green function D,q(x, x0, w) contained in eqn. (11) diverges at x + x0. Because of this, rearrangement of eqn. (11) becomes imperative, similar to that carried out in deriving the multiple scattering equations in a quantum-mechanical many-centre problem [12]. For this purpose, we introduce the scattering operators
186
on the centre localized near the metal surface, i, and in the homogeneous electrolyte, t^“. The equations for i and i” in symbolic form are written as follows: i(P)
= C?(P)
+ $+0)60i(X0)
=6(x”)
+ $d(XO)B’iO(XO)
(12) i”(P)
Here, 3’ is the Green function for the Maxwell equations describing the behaviour of an electromagnetic field in a homogeneous electrolyte with permittivity z,_. In coordinate representation, the operators b)‘, t^and i” contained in eqn. (12) are of the form D,;(x,
a,,+---
X’, u)=-& i
c2 a a E,rW2 ax, ax,
i
exp ~:c~,/~]x--x’] ( (x-x’]
1 (13) \
I
t;,(x,
x’; x0) =cY3(x- x0) 63(x’ - X”)fJW)
t,;
x0) =63(x-
X0) S3(x’ - x” ) &,a( cd)
where a(~) is the polarizability of a local centre. For example, in the case of a centre modelling a spherical particle, CY(W)is written as (14) For further treatment it is essential that the local centre polarizability CX(W)has a pole at the frequency w, = 0; - il?, lying in the visible light, or near ultraviolet, region. In the vicinity of this frequency, a(~) can be written as
a(u) =
--.2L w - w,
(15)
Note that the transition from operator 6 to operator i” corresponds to taking account of the self-action effects at the point centre. Using eqn. (12) the equation for the determination of i( x o ) is written as
qx”)=P(xo)
+
$i”(x”)(bo-B’)i(x”)
(16)
The field gz(x, CO)is expressed in terms of ? in the following manner: 8,(x,
w)=C$O(x,
o)+
$D,;(x,
x0;
w)t,,(x”)E,o(xo,
w)
Taking account of the explicit form [ll] of the Green function obtain from eqn. (16) an expression for the scattering matrix:
f,,(W) = (qd)),,
=[
a-yc.@+$(P - 5f)(x”,
=s,,[a-‘(w)-~,(~)]-’
x0;
(17) D,; (x, x’; a), we
q.1 (18)
187
The analytical properties of the amplitudes r,(k,,) and r,(k,,) of the reflection coefficients of s- and p-polarized waves from a flat metal/electrolyte interface contained in eqns. (19) are discussed at length in ref. 13. When the inequality I
l
q+
E,
1+
16a( XT )’ em + eel [ 3 ( -
Qc,)3’2 2
-ii
Eln -
Z2(o) = Z,(w) =
(20)
2 %l
fG2&b) -
f(k,)3(- z)‘“-& m
where Ae2(w) = Jdx,
dx;{ e(x,,
x,l, 4
-6(x,
- x;)[%@(-x,)
+ %1~(xr)l> (21)
AC,‘(W) = Jdx,
dx;{ e-*(x1,
x;, 4
-6(x,
- x;)[%#(-x*)
+4x,>]
-‘}
The first term in Z, corresponds to the expression for classical image forces, the second accounts for the influence of the microscopic surface structure, and the third describes the possibility of decay of local resonances into delocalized surface plasmons. Note that the quantities AC, AC;’ - 1 characterizing the influence of surface properties on the interaction between centres also define the dependence of the light reflection coefficients from the metal on the surface layer structure [13]. Expressions similar to eqns. (18)-(20) were obtained in ref. 8, in which a point
188
dipole model was used in considering the optical properties of adsorbed molecules. In this case, the polarizability CX(W)can be written as follows: C+)=$
6
2 J
wJ
-
cd2 -
iy,w
(22)
where fi is the force of oscillators and oJ are the frequencies of optical transitions in the molecule. In the problem under consideration, the polarizability a(~) has a pole at the frequency wr = 0; - iI’,, which is the eigen (resonance) frequency of local plasma oscillations (LPOs) on an isolated dipole centre. The eigenfrequencies of resonance modes which arise from the interaction of LPOs of a dipole centre and SPs propagating along a flat interface are determined from the condition of the matrix elements t,j(w) becoming infinite and are roots of the equations K’(W)
-Z,(w) =o (23) The positions of the poles of the function t,,(w) (i = 1 in eqn. 23) correspond to the frequencies of normal (to the surface) oscillations whereas the positions of the poles of t,,(w) (i = 2 in eqn. 23) correspond to the frequencies of tangential (in the plane of the surface) oscillations. It follows from eqns. (20) and (23) that as a result of the above-mentioned interaction, degeneration of the LPO mode corresponding to the centre under consideration is removed: it splits into tangential and normal oscillations and new local modes appear that split from the SP band. The frequencies of the local modes formed group near two eigenfrequencies of the initial resonances: the LPO frequency w, and the limiting frequency of SPs w,$ determined as the solution of the equation em(w) + E,, = 0. In the case of polarizability of form (14) or (15), four resonance modes appear in the vicinity of the centre, two of which (tangential) can be excited both by s- and p-polarized light and the other two (normal) only by p-polarized light. The positions of the eigenfrequencies of local resonances (oscillations) relative to wr and w,; depend on the microscopic structure of the interference. In the sharp interface model (AC, ACT’ = 0), the frequencies wil of two oscillations (one tangential and one normal) resulting from LPO splitting lie below w, and the frequencies w{~ of the other two oscillations lie above w$,, the frequencies of tangential oscillations lying within the range bounded by the frequencies of normal oscillations. In most cases, the frequency ranges (0; , wi) and wi;, ~1) are less than the widths of the corresponding modes, so that only two broadened resonances should be observed, one below w, and the other above w,Op. It is to be inferred from eqns. (15), (18) and (20) that in the vicinity of each of the eigenfrequencies wil the cross-section for the formation of local modes can be written as
(24)
189
where, for example, at w, = o;
(25)
Using the above formulae, it is possible to estimate the field enhancement ratio near a rough surface. Note that formulae (24) and (25) are written assuming the distances between the frequencies of different resonances to be greater than their widths. In the opposite case, the formulae become more complex (see ref. 7) but the order of the enhancement ratio remains unchanged. (IV) OPTICS ELECTRODE
OF A SYSTEM
OF LOCAL
CENTRES
NEAR
THE
SURFACE
OF A METAL
Let us now consider the interaction of light with a system of local centres located at points x(n) = (xp, xi”)) that are distributed in a random way along the electrode surface. The total scattering operator 2” on a system of local centres can be represented as
In this case, exact solution of eqns. (16) and (26) is impossible and it is necessary to resort to additional approximations similar to those used in the disordered media theory [14]. In the present study, we will consider a compositional disorder model in which local centres are assumed to be located at the sites of a lattice on the surface, each site being occupied with the probability 8,. In order to determine the surface excitation spectrum and to calculate the light reflection coefficients, a so-called average T-matrix approximation is used in which a disordered system is replaced by an effective medium in the form of the same lattice completely occupied with equivalent local centres. The scattering matrices on individual sites of such a lattice (T(w)) are written as follows
(7(0)) = f&qw)
(27)
where 7 is the scattering matrix on one centre located near the metal surface with the form (18). Under the above assumptions, from the exact formula (26) of the multiple scattering theory, the following expression is obtained for the surface-averaged ? matrix of the system under consideration:
(f) =~(i(x’“‘)) n
+
$ c n.m#n
(i(n(“)))B”(i(x(m))) + *. *
190
The approximation used consists of the decoupling of the surface-averaged products of the matrices i(xCn))i(xCm)) * * . ‘ and represents a version of the mean field theory. This approximation becomes exact in the vicinity of two limits: for low coverages and for a completely occupied lattice. Just as in the case of quantummechanical multiple scattering problem [?I, let us express the f matrix satisfying eqn. (26) in terms of auxiliary operators T,,, f=
c T,,
(29)
m,n The operators fm,, obey the equation
In the coordinate representation,
fm,, are of the form
Ll = S3( X - X(m)) fm, 63(X’ - x(n))
(31)
The surface-averaged auxiliary operators (fmn) obey the equation (32) On account of the translation invariance (Fm,,) = fm_,,, eqn. (32) can be readily solved by expansion into a Fourier series: c = N-’ C exp[ -iq( dm) - dn))] t_, (33) m,n where N is the number of units cells in the lattice of local centres under consideration and the wave vector q is determined in the first Brillouin zone for this lattice. Substituting eqn. (33) into eqn. (32), we find
To calculate the light rejection coefficients and the law of dispersion of SPs, it suffices to estimate the q matrix at small values of q (q ( do) - x2’ 1 -=K1). If this circumstance is taken into account, the sum of Green functions l)(x”), xc”)), for example for a square lattice with period R contained in eqn. (34), assumes the form CD,;(~@), nzo
9,
r(.))exp(iq(r(‘)-x(“)))-~~((if+A!)
(35)
where U,‘=4$
c D!(d”‘, n#O
xc”)) = 9.03~,‘R-~
6,
s, = 1
a,=&=
-f
191
AU, = -4aZ,(w)
+
Xexp
(2?r)‘(m2
+ n’) - $,R2
=4%(4 -Eel
(21T)2 E -Eel
exp[-4a&n2+nz)‘/2]
-~--&(m2+n2)“2 R3
l/2 11
AU, = AU,
=
exp
-4aZ,(w)
+ f(
l/2 1 l/2 11
g3c m,n
x(~(~)‘[r~(~(m,,~2)l~2)-rp(~(m2+n2)1/2)] + ( m2+ ?z’) g( m2 + r~~)l’~ f el
I
-47qw) xexp
'P
- +( y’1 _4T?$(m2+
[
)i
(
c (m’ + n2)1’2 CA m,n
1mel
n2)1/2 s
(36)
In calculating the values Aq, the expressions for the Green functions D,T(x, x’) obtained in ref. 11 were used and transition was accomplished from summation over direct lattice sites to summation over reciprocal lattice sites. By this transformation it is possible to speed up the convergence of the series determining the form of AU,. It follows from eqns. (34) and (35) that with increasing degree of occupancy of the lattice 8,, the probability of decay of resonance states arising on a rough surface in the delocalized SPs decreases in proportion to (1 - t9,). The probability of decay of the resonance states in radiation increases in proportion to 8,. It should be emphasized that in expression (34) the terms accounting for the interaction between centres are contained in the denominator of the T matrix, i.e. they represent corrections not directly to the T matrix or the Green function but to the polarization operator or, in other words, to the self energy operator. An advantage of this form of wgting lies in that the expressions obtained remain valid in the vicinity of the poles T,(o), where the ordinary expansion into a series of the perturbation theory does not hold. Using the approach under consideration, account can be taken of the fact that the resonance frequencies for different centres on the surface may differ from one
192
another (non-uniform broadening of the spectra of states). For this purpose, we will use expression (15) for a(w) and assumed that the centres with different resonance frequencies wr are distributed over the lattice in a random manner with the distribution function (37) Just as in the case-of electron levels in disordered systems [14], let us average expression (34) for T, with the weight (37). As a result, we have
It is evident from eqn. (38) that taking account of the non-uniform broadening of the spectrum of local states leads to smoothening of the peaks in the T matrix and, accordingly, in the quantities discussed below. Formula (34) implies that the coefficients of specular reflection of light from the lattice of local centres considered above are formally of the same type as the corresponding reflection coefficients from an anisotropic layer on the metal surface. The normal C: and tangential ef, components of permittivity of this layer are determined by the equalities
0El I
(4 - %d)d= (iq22/sI )-I - c,‘)d= c;‘( &&/s,
where S, is the unit cell area for ,the lattice of centres under consideration and d is the effective thickness of the layer. The imaginary components of E:, and E: have maxima at frequencies Q,T and Q: , respectively, which are the resonance frequencies of the matrix q. Frequencies &Ii and Q; lie in the vicinity of the resonance frequency w, of LPOs on an isolated centre and frequencies QI; and sll are in the vicinity of the limiting frequency of SPs on a smooth surface. An approximate averaging method similar to that described above was also used in ref. 15 which dealt with the reflection of light from a partially occupied lattice of local centres. No account, however, was taken in ref. 15 of the presence of a metal substrate, interaction with which, as shown above, results in a qualitative change of the characteristics of local resonances. Expressions of the forms (34) and (39) can be used to describe light reflection from the lattice of atoms and a molecule on the metal surface. A similar approach to the solution of this problem was developed in ref. 9. The calculation method described there consists in generalization of the effective medium approximations [16] widely used in considering the optical properties of three-dimensional non-homogeneous media for the case of a disordered system of local centres (or molecules) located near the surface. The results presented here show that direct application of the above approximations to surface optics problems, as was done in numerous studies (see, for example ref. 17), may lead to grave errors.
193
By means of the proposed method it is also possible to study the influence of surface inhomogeneities on the dispersion relation of SPs, w,,(k,,). The form of the dependence w,(k,,) is determined by the position of the poles of matrix T considered as a function of w and k,,. With the use of eqns. (18) and (34), the equation for the determination of w,,(k,,) and Rk, -C 1 is written as follows: -1
1
Here, $,(k,,) is the surface plasmon dispersion relation on a smooth surface. Comparison of eqns. (39) and (40) shows that when local centres appear on the surface the dispersion relation of SPs changes most drastically (as compared with a smooth surface) in the vicinity of two frequencies, Qi and 0; , corresponding to tangential and normal plasma oscillations. At these frequencies, extrema should appear on the o,,(k,,) dependence, whose position and height depend on the concentration of local centres 8,. (V) CONCLUSION
Let us discuss the experimentally verifiable consequences of the theoretical treatment presented above. Let us consider first of all the increase in the electric field intensity near the surface in the presence of local modes. According to eqns. (5) and (24) this increase of g(x, w) at the point x near the centre located at the point x0 = (x,0, 0, 0) can be estimated as follows: g(x,
0) = IB(x,
WI I 2/e=
m4
I cpl(4
t SP,U c
2 ~lal,spusp
+
rU.SP
k,‘r2 +
rd.sp
(
rsp, +
I */N”Pl 2
-3
SP.1 rd
1)3
Ix
-‘:O
I6
(41)
where g,, is the field amplitude in the incident wave and cp,(x) = [cp,/l x x o I 3](( dd/c>cy ) -3’2 is the normalized solution of the Maxwell equation near the centre *. At values of = f’+~l,sp%p
+
ru,,
+
10
rsp,l/rd.l
e
1
rd,sp
( rsp,l/rd,l)2R;6( :C:f2)-3k&,2:
= lo3
l The quantity cp,/Td,, has been estimated from the condition of matching with the results of classical calculations for an isolated centre in the framework of macroscopic electrodynamics.
Fig. 1. Frequency dependence of the enhancement ratio of the electric field intensity g(s, w) at point x = (0.1 run, 0, 0), estimated for the cases of s- and p-polarized light incident upon a silver surface at an angle B = 45 O. The local centre is located at the point with coordinates x o = (5 nm, 0,O) and models a spherical particle from silver of radius 5 nm.
obtained using the perturbation theory on the interaction of surface and bulk waves on a slightly rough surface [S] and the model of a spherical centre with radius R,= 5 nm considered in Section (III), the ratio g(x, w) for silver at hw = 3 eV proves to be of the order of 104. The model calculation carried out in Section (III) shows that the values of g(x, w) in the vicinity of resonance frequencies 011.~ are several orders higher than those near the frequencies wl;Il . The value of g( X, w) as a function of the coordinate x r&aches a maximum on the axis of symmetry of the system in the space between the local centre and the surface. In this region, g( X, W) may exceed significantly (by more than one order) the maximal resonance enhancement, which takes place if the objects are isolated. Figure 1 shows the frequency dependence of the enhancement ratio of the electric field g( X, w) at the point with coordinates x = (D, 0, 0) located between the centre and the surface. In performing the calculations, the polarizabihty a(~) was taken in the form of eqn. (14), the centre and substrate being considered to consist of silver and no account being taken of the presence of smooth rot&messes on the surface. It should be emphasized that the main features of the frequency and coordinate dependences of the enhancement ratio g( x, o) found with the use of the model just considered are confirmed by the results of numerical calculations performed for a sphere [18] near the metal surface taking into account a large number of modes with an angular momentum L > 1. A sharp increase (approximately 105-lo6 fold) of the photoemission current in the vicinity of local centres on the Ag surface revealed in ref. 19 can be explained by means of eqns. (8) and (41) provided that ( I’pi’fId2)*R,“(( 0/c)@-~1c;,‘r u,,/c) = 104. This value of (r,,,/r,,)*R,“((w/c)~~~ )- k,*(u,,/c) is larger than that obtained in the spherical centre model. It is possible that when considering centres of a different kind, for example, of the type of cracks (especially when account is
195
taken of the influence of various possible polaritons in the adsorption layer), other values of g(x, w) can be obtained explaining the observed phenomena. The effects leading to enhancement of RS and diffuse scattering of light in the vicinity of local centres can be estimated in a similar manner. A significant fact (which can be verified experimentally) should be mentioned here, that in the approach considered above the angular dependences (on 8 and 8,, vr) of the radiation arising during RS and diffuse scattering should be identical. It follows from eqn. (3) that in the presence of local centres, the light reflection coefficients from the metal surface should have minima in the vicinity of the eigenfrequencies of local resonances Q,f and 0:. This conclusion agrees with the results of experiments [20] on the reflection of light from very rough metal surfaces. According to eqn. (40), at the frequencies Q,il, extrema should also appear on the dispersion curves w,(k,,) for surface plasmons. Such behaviour of the dispersion curves was observed in ref. 21 for a strongly loosened Ag surface in 0.1 M K,SO, solution. At the metal/solution interface, the eigenfrequencies of local resonances can depend on the potential drop E in the double layer and/or on the adsorption on the electrode. This dependence can be, for example, due to a change in the electron density inside local centres (inhomogeneities) resulting from a change of the potential drop or charge transfer during adsorption. In this case, using the approach being discussed, it is possible to obtain from eqn. (39) the following expression for the electroreflection signals on a rough surface, e.g. in p-polarized light,
<,,)( Ed SinZe - E, c0s2e)] -’
(42)
It is evident from eqn. (42) that loosening of the electrode surface should give rise to extrema on the frequency dependence of electroreflection signals lying in the vicinity of the resonance frequencies of local modes. Such behaviour of the electroreflection signals was observed, e.g. in ref. 22. Similar behaviour of electroscattering spectra (derivatives of the diffuse scattering intensity with respect to the potential) was revealed in ref. 23. This effect can also be explained on the basis of the treatment presented above (see formula 10) taking into account the circumstance that the frequencies of local resonances depend on the potential drop. REFERENCES 1 R.K. Chang and T.E. Furtak (Eds.), Surface Enhanced 1982.
Raman Scattering, Plenum Press, New York,
196 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
23
J.K. Sass, R.K. Sen, E. Meyer and H. Gerischer, Surf. Sci., 44 (1975) 515. C.K. Chen, A.R.B. de Castro and Y.P. Shen, Phys. Rev. Lett., 46 (1981) 515. M. Moskovits, Rev. Mod. Phys., 57 (1985) 783. A.M. Brodsky and MI. Urbakh, Surf. Sci., 167 (1986) 231. V.A. Kosobukin, Poverkhnost’, 12 (1982) 5. R.G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill New York, 1980. P.J. FiebeIman, Phys. Rev. Sect. B, 22 (1980) 3654. A. Bachi, B. Barrera and R. Fuchs, Phys. Rev. Sect. B, 25 (1982) 7086. C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1979. A.M. Brodsky and MI. Urbakh, Surf. Sci., 115 (1982) 417. F.I. Dahdchik and G.K. Ivanov, Teor. Eksp. Khim., 8 (1972) 9. A.M. Brodsky and M.I. Urbakh, Prog. Surf. Sci., 15 (1984) 121. H. Ehrenreich and L. Schwartz in F. Seitz, D. TumbuB and H. Ehrenreich (Eds.), Solid State Physics, Vol. 31, Academic Press, New York, 1976, p. 5. B.N.J. Persson and A. Liebsch, Phys. Rev. Sect. B, 28 (1983) 4247. V. Twersky, J. Math. Phys., 19 (1978) 215. V. Brusic, M.A. Genshaw and J.G’M. Bockris, Surf. Sci., 29 (1972) 653. P.K. Aravind and H. Metiu, Surf. Sci., 124 (1983) 506. A.M. Foontikov, S.K. SigaIaev and V.E. Kazarinov, Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Khim., 17 (1986) 47; J. Electroanal. Chem., 228 (1987) 197. T. Lopez-Rios, Y. Borensztein and G. Vuye, Phys. Rev. Sect. B, 30 (1984) 659. M. Abraham and A. Tadjeddine, Surf. Sci., 173 (1986) 65. V.V. Marinyuk, R.M. Lazorenko-Manevich and Ya.M. Kolotyrkin in Ya.M. Kolotyrkin (Ed.), Advances in Physical Chemistry. Current Development in Electrochemistry and Corrosion, Mir, Moscow, 1982, p. 148. A.M. Foontikov, S.K. SigaIaev and V.E. Kazarinov in Electrodynamics and Quantum Phenomena at Interfaces, Telavi (U.S.S.R.), 1984, Izd. Academii Nauk Gruz. SSR, Tbilisi, 1984, Ext. Abstr., p. 276.
J. Elecrrounul. Chem., 228 (1987) 179-l% Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ON THE ROLE OF LOCAL MODES IN THE OPTICS METAL/ELECTROLYTE INTERFACE *
A.M. BRODSKY
OF THE
and MI. URBAKH
A.N. Frumkin Institute of Electrochemistry,
Academy of Sciences of the U.S.S.R,
Moscow (U.S.S.R)
(Received 21st January 1987)
ABSTRACT The influence of surface roughness on the electromagnetic response of the metal/electrolyte interface is considered. The intensities of local surface electromagnetic modes are expressed in terms of their formation cross-sections and life times. The effect of the interactions of local modes with delocalized surface plasmons and with one another on the spectrum of electromagnetic excitation of the system and on the behaviour of the field near the interface is examined. The possibilities of using the results obtained for the description of experiments (surface enhanced Raman spectroscopy, photoemission, surface plasmon spectroscopy, electroreflection and electroscattering) at the metal/electrolyte interface are discussed.
(I) INTRODUCTION
The influence of surface roughness (inhomogeneity) on the electromagnetic response of the metal/electrolyte interface has been studied in numerous papers. The discovery of the enhancement of Raman scattering (RS) on molecules adsorbed on metal surfaces (surface enhanced RS) gave a strong impetus to investigation of the role of surface roughness in different optical phenomena. It was convincingly proved [l] that a substantial part of RS enhancement is connected with an increase in the intensity of the electromagnetic field near a rough surface. Earlier, a similar enhancement was detected when photoemission at the silver/electrolyte interface was investigated in the pioneer work of Gerischer and co-workers [2]. The effects associated with field enhancement on rough electrode surfaces were also observed in studies on non-linear optical phenomena [3]. Similar effects were revealed at the metal/vacuum interface [4]. It should be stressed that optical experiments at the metal/electrolyte interface continue to be the most effective primarily because they
* In honour of Professor H. Gerischer on the occasion of his retirement as Director of the Fritz-Haber Institute.
0022-0728/87/$03.50
0 1987 Elsevier Sequoia S.A.
180
allow potential drop modulation techniques to be employed. The data obtained by these techniques allow adequate separation of bulk and surface effects. On the other hand, the information provided by these data is extremely important for solution of the problems of the double-layer theory as well as other vital problems of electrochemistry. The effects under consideration cannot be explained on the basis of the concept, discussed in the literature [1,5], that the main role is played by surface plasmons (SP) propagating along the interface. If the formation of SP is taken into account, the field enhancement ratio on a rough surface as compared with a smooth surface [5] is much smaller than the experimentally observed values. At present, it may be considered to be established experimentally [l] that on rough metal surfaces for which the characteristic sizes of their inhomogeneities in the direction normal to the surface are of the same order as those in the direction along the surface in addition to SPs propagating along the interface, there can exist resonance localized electromagnetic modes which are also excited by an electromagnetic field and can cause a sharp increase of scattered electromagnetic radiation. It should be emphasized that according to experimental data, the decisive role in the field enhancement effect on the metal surface is played by microscopic roughnesses (mean size 10 nm) [2]. This effect disappears when roughnesses are below 1.0 nm and over several hundreds of nm in size [6]. The main theoretical results on local modes were obtained using numerical calculations for simple geometrical forms of inhomogeneities (spheres, ellipsoids, etc.). With this approach, it is impossible to establish the general features of the phenomenon under consideration which should remain valid for more realistic structures. The object of the present study was to obtain analytical expressions for different optical characteristics of rough electrodes in a long-wave approximation, in many respects similar to the zero radius potential method used in the quantummechanical theory of scattering [7]. With such an approach, all the properties of the metal and the inhomogeneities enter into the observed quantities only in terms of the coefficients of light reflection from a smooth surface and the energy characteristics of local modes on individual centres modelling the roughness. In Section (II) of this study the intensities of local surface modes are expressed in terms of their formation cross-sections and life times. It has been ascertained how the intensities of local modes enter into the expressions for the observed effects. In the Section (III), a model calculation is performed of the formation cross-sections of local modes on one centre, confirming the resonance form of these cross-sections. Section (IV) treats the effects of the interaction of local centres. Sections (III) and (IV) also examine the effect of the interactions of local modes with SPs and with one another on the spectrum of electromagnetic excitation of the system being considered and on the behaviour of the field near the surface. In many aspects, a similar approach was used earlier in considering the influence of atoms and molecules near the metal surface on light scattering [8,9]. In Sections (III) and (IV), a microscopically rough surface is represented as a combination of bumps resting on a flat substrate. The influence of these bumps is
181
modelled by introducing into the surface region, point dipole centres in whose vicinity local electromagnetic modes can be excited. This approximation can be justified quantitatively in the case where the characteristic sizes of the inhomogeneities (bumps) are less than other parameters of the problem of a dimension of length such as the light wavelength, the distance between bumps and the distance from the bump centre to the surface. In the opposite case, it is necessary to take into account the influence of multipole local modes, and, accordingly, the treatment used may claim to be no better than a qualitative description of the phenomenon. Comparison of the analytical dependences obtained with the results of numerical calculations (for those systems for which they were available) carried out taking into account modes with higher angular momenta confirms the validity of the assumption that even in this case the simple model considered gives a correct picture of the field distribution near the metal surface. In Section (V) the possibilities of using the results obtained for the description of experiments at the metal/electrolyte interface are discussed. (II) INTRODUCTION
OF LOCAL ELECTROMAGNETIC
MODES
ON ROUGH
SURFACES
The electromagnetic fields 8(x, w; /3) figuring in the problem may be considered to belong to one of three types: 1 - bulk fields (/3 = v) subdivided into s- and p-polarized waves (below we will consider only the most important p-wave contribution); 2 - surface waves (/I = sp) and 3 - singled-out resonance states localized in the vicinity of inhomogeneities with a small radius of curvature (/3 = 1; in the literature they are sometimes called local plasma modes). The interaction of the bulk and surface waves with one another will be taken into account in terms of the perturbation theory on the interaction with roughnesses, as was done in ref. 5. Since in the systems of most interest to us, surface and localized modes are long-lived, we will ignore the interference between fields of different type. To explain the assumptions made, we will consider a general expression for the scattering phase of an electromagnetic wave on a local centre for the case where the wavelength, A = 27rc/w, is much larger than the characteristic sires of the centre R,. For the largest phase S, corresponding to electric type dipole oscillations, it is possible, ignoring deviations from spherical symmetry, to write a general formula
[71:
q*(
g3 cot
6, =
a(w)R,3 +
Here, a and b are coefficients centre of spherical shape 44
= tk(4
b(4
= 0.9[(2~,
+ 2%M&4 - 44M44
c,b(o)( j)*R,’+o((g4RO) depending on the frequency w. In particular,
for a
- 41 - 41
where cc(w) and E,, are the permittivities of the metal of which the sphere is made and of the electrolyte. Equation (1) is similar to the “effective radius” formula used
182
in the quantum-mechanical theory of scattering [7] for an angular momentum equal to unity. From eqn. (1) disregarding inelastic processes, we obtain the following expression for the resonance scattering cross-section u on the centre and the reciprocal of the life time T: u(w) =12x
iI-2
2 e,’
c
(1 ;
(w - o())* + :r’
l? = 2( @~RO)‘/~
Iy__
= ( 12b(w,)
4%) + %I TR”
0
Substituting into eqn. (2), for estimation, the parameters corresponding to a sphere, we obtain a resonance life time of the order of I’-’ = 3~i5/2 1dc /dw I. During this time, the unscattered wave will move (I/4)((w/c)%a distance c/I B h, much larger than the wavelength, so that the interference between running and local modes may in fact be neglected even without allowing for other inelastic processes. This conclusion is also valid for the scattering of surface waves since their group velocity is of the order of c. These local modes can be formed both directly under the action of light incident upon a metal and as a result of the interaction of surface waves with inhomogeneities of a small radius of curvature. The last-mentioned formation mechanism of local modes is, evidently, of greatest effectiveness, at least at a small inhomogeneity concentration. The surface waves, in turn, can be excited by bulk waves on slightly rough surface sites in the region between inhomogeneities of a small radius of curvature. According to the above scheme, the relation between the number of local modes and the number of photons in the incident flux can be found from the following kinetic equations:
Here, IV, is the number of photons (in unit volume) in the incident flux; ySP( w) and N,(w) are the numbers of surface plasmons and local modes on umt surface, respectively; pi is the surface concentration of centres; and us+,= u,,(w) is the velocity of surface plasmons; in the case of a sharp metal/electrolyte interface, it is equal to C Usp =
%I
belY2kn 6,
beI
+
+ 4
+
h,,
%,)3’2 ah/au
where em(u) is the permittivity of the metal. In eqns. (3), t,,, denotes the transformation coefficient of the formation of surface plasmons from bulk waves
183
and ai,,r(~) is the cross-section for the formation of local modes from surface plasmons * attaining a maximum at 0 = w,; r, ,s(~, B = u, sp, 1) stands for the reciprocal life times of states j3 defined by their decay into states (Y, and the coefficient IYa,s accounts for the dissipative decay channel of j? fields. It is shown in ref. 5 that
k; =
;c,*sin8
(4)
where 8 is the angle of incidence of light, rp(ki) is the amplitude of the reflection coefficient of p-polarized light from a flat metal surface and (E2)*j2 and a are the root-mean-square height and the correlation length of smooth (or large radius of curvature) roughnesses. From eqns. (3) we obtain the following expression for a stationary number of local modes with frequency w on unit surface: PP,,,W N,(w)
t,,,(+N,
(5)
= rsp,l(o)
+
rd,l(u>
~la~,sP(ohp
+
ro,sP(w)
+
rd,sp(w>
The decay of local modes into radiation can be treated in a similar manner. Here, to describe RS due to an inelastic transition in the vicinity of a local centre, it is necessary to add to system (3) an equation for the number of local and surface modes with the scattered light frequency wf: dNttuf)
dN,,bf)
=
WfiNl(Q)
-Nl(Wf)(rsp,,(Wf)
+
rd,l(wf)) (6)
= N,(w,)I,,,,(+)
-
N,(wf)[p,o,,~p(w,)v,p
+
ru,s,b’f)
+
‘d,spbf)]
dt Here, wr, is the probability of an inelastic transition wherein a local mode with frequency w changes to a local mode with frequency wf. In fact, in writing expression (6), the transitions between the initial and final states were assumed to be a stochastic process; the final fields excited at different instants of time are considered to be incoherent. In principle, a situation in which the final fields are coherent is possible. In such a case, additional enhancement of
l
It is worth noting that the cross-section
o,.~~ has the dimension
of length.
184
surface RS may arise, similar to that observed in the case of stimulated RS [lo] in the bulk *. From eqns. (3) and (6) the following expression ensues for the steady-state intensity of surface waves of frequency or: rsp,lbf>
Wfi N,pbf)
=
Cp,l(~f)
+ rd*lbf)
Pl%p
bfbS,bf)
+ ~,bf) t,,“wc
m,,b)
X
+ rd.,bf)
q&4
+ rd,,w
~~~,~,bb,,(4
+ ru&4
+ rd.&4
N0
(7)
Using the approach described above, by means of formulae (5) and (7) we obtain the following expressions for the photoemission current J in the case of a rough surface (the surface regions near the inhomogeneities of small radius of curvature being assumed to make the main contribution to J) and for the radiation intensity of dY/dB formed during RS and scattered into a solid angle da = sin 0, de, dq,:
J= IMI*N,(w)
(8) Here, M is the matrix element of the optical transition and the function P,( r?,, cp,; cp) describes the angular distribution of the radiation arising du$ng decay of the surface wave propagating along the interface in the direction k = (0, cos cp, sin cp}. An expression for Psp was obtained in ref. 5 and is of the form P,,(ef,
‘Pr;
l/2
=
‘4
‘PI
I %I-
Enl I
2
X
sin
ef cos
vf +
u4 c3
-[a
43a21E,l+Cml
-2
2
exp -?- 42(~~-iks,)2]lijl+~~(~~)) ]
(-~~,/c~)l’~(l
- rp(
klf)) cos
e,J2+ 23[i ;r r:T
I; $
X~2a2exp[-~~*:)2+*~~~]~~~~a2~~~~~~{[~~~~l-~~~x,.)‘n m
xcos2ef+++~p(k~)~2 sin28fr”,s:yr
x
sin 2efcoscpf &+p
+ +
-~]~]1’21m
+ rp( k;)
r,(ki)
I2 sin e, cos
2q,
+ rd,sp + akSP
rSPSSP
X
ru.sp+ rd,SP +
It is interesting surface.
l
rSPVSP
d,sp
(9) aSP,SP
that in this case the role of mirrors
can be played
by one-dimensional
walls on the
185
and k, = k,(w) is the wave vector of a surface plasmon. Diffuse light scattering arising from the presence of inhomogeneities of small radius of curvature may be treated in a similar manner. Here, the angular distribution of scattered light, dR,JdB, is written as follows:
dR, dQ
/
N3w) =[
+r,,,w +cl,sPw]2
Plul,,(+,,(4
X
rsp,1w rsp,l(w) +
r,,+)
00)
td+N,
(III) MODEL CALCULATION OF CROSS-SECTIONS MODES ON AN ISOLATED CENTRE
FOR THE
GENERATION
OF LOCAL
In considering the problem of an isolated local centre localized near the metal surface at the point x0 = (x,0, 0,O) (the x1 axis is directed along the normal to the surface of the metal which occupies the halfspace xi < 0), we will proceed from the following algebraic equation for the electromagnetic field:
(11) Here 4:” and 0,: denote the electric field and the Green function of the electromagnetic field calculated for a semi-infinite metal without local centres and summation over identical subscripts i, j = 1, 2, 3 is to be carried out. The explicit form of the functions &,‘(x, w) and D,q(x, x’, w) is given, for example, in ref. 11. The function x(w) in eqn. (11) characterizing the properties of a point local centre not interacting with the field is related in a simple manner to the Green function G(x, x’, w) of dipole oscillations of the particular dipole centre located at the point x0 G(x,
x’, w)=P(x-x0)x(w)
83(x’-x0)
Equation (11) is equivalent to the Maxwell equations for a semi-infinite metal with a point centre near the surface. The Green function D,q(x, x0, w) contained in eqn. (11) diverges at x + x0. Because of this, rearrangement of eqn. (11) becomes imperative, similar to that carried out in deriving the multiple scattering equations in a quantum-mechanical many-centre problem [12]. For this purpose, we introduce the scattering operators
186
on the centre localized near the metal surface, i, and in the homogeneous electrolyte, t^“. The equations for i and i” in symbolic form are written as follows: i(P)
= C?(P)
+ $+0)60i(X0)
=6(x”)
+ $d(XO)B’iO(XO)
(12) i”(P)
Here, 3’ is the Green function for the Maxwell equations describing the behaviour of an electromagnetic field in a homogeneous electrolyte with permittivity z,_. In coordinate representation, the operators b)‘, t^and i” contained in eqn. (12) are of the form D,;(x,
a,,+---
X’, u)=-& i
c2 a a E,rW2 ax, ax,
i
exp ~:c~,/~]x--x’] ( (x-x’]
1 (13) \
I
t;,(x,
x’; x0) =cY3(x- x0) 63(x’ - X”)fJW)
t,;
x0) =63(x-
X0) S3(x’ - x” ) &,a( cd)
where a(~) is the polarizability of a local centre. For example, in the case of a centre modelling a spherical particle, CY(W)is written as (14) For further treatment it is essential that the local centre polarizability CX(W)has a pole at the frequency w, = 0; - il?, lying in the visible light, or near ultraviolet, region. In the vicinity of this frequency, a(~) can be written as
a(u) =
--.2L w - w,
(15)
Note that the transition from operator 6 to operator i” corresponds to taking account of the self-action effects at the point centre. Using eqn. (12) the equation for the determination of i( x o ) is written as
qx”)=P(xo)
+
$i”(x”)(bo-B’)i(x”)
(16)
The field gz(x, CO)is expressed in terms of ? in the following manner: 8,(x,
w)=C$O(x,
o)+
$D,;(x,
x0;
w)t,,(x”)E,o(xo,
w)
Taking account of the explicit form [ll] of the Green function obtain from eqn. (16) an expression for the scattering matrix:
f,,(W) = (qd)),,
=[
a-yc.@+$(P - 5f)(x”,
=s,,[a-‘(w)-~,(~)]-’
x0;
(17) D,; (x, x’; a), we
q.1 (18)
187
The analytical properties of the amplitudes r,(k,,) and r,(k,,) of the reflection coefficients of s- and p-polarized waves from a flat metal/electrolyte interface contained in eqns. (19) are discussed at length in ref. 13. When the inequality I
l
q+
E,
1+
16a( XT )’ em + eel [ 3 ( -
Qc,)3’2 2
-ii
Eln -
Z2(o) = Z,(w) =
(20)
2 %l
fG2&b) -
f(k,)3(- z)‘“-& m
where Ae2(w) = Jdx,
dx;{ e(x,,
x,l, 4
-6(x,
- x;)[%@(-x,)
+ %1~(xr)l> (21)
AC,‘(W) = Jdx,
dx;{ e-*(x1,
x;, 4
-6(x,
- x;)[%#(-x*)
+4x,>]
-‘}
The first term in Z, corresponds to the expression for classical image forces, the second accounts for the influence of the microscopic surface structure, and the third describes the possibility of decay of local resonances into delocalized surface plasmons. Note that the quantities AC, AC;’ - 1 characterizing the influence of surface properties on the interaction between centres also define the dependence of the light reflection coefficients from the metal on the surface layer structure [13]. Expressions similar to eqns. (18)-(20) were obtained in ref. 8, in which a point
188
dipole model was used in considering the optical properties of adsorbed molecules. In this case, the polarizability CX(W)can be written as follows: C+)=$
6
2 J
wJ
-
cd2 -
iy,w
(22)
where fi is the force of oscillators and oJ are the frequencies of optical transitions in the molecule. In the problem under consideration, the polarizability a(~) has a pole at the frequency wr = 0; - iI’,, which is the eigen (resonance) frequency of local plasma oscillations (LPOs) on an isolated dipole centre. The eigenfrequencies of resonance modes which arise from the interaction of LPOs of a dipole centre and SPs propagating along a flat interface are determined from the condition of the matrix elements t,j(w) becoming infinite and are roots of the equations K’(W)
-Z,(w) =o (23) The positions of the poles of the function t,,(w) (i = 1 in eqn. 23) correspond to the frequencies of normal (to the surface) oscillations whereas the positions of the poles of t,,(w) (i = 2 in eqn. 23) correspond to the frequencies of tangential (in the plane of the surface) oscillations. It follows from eqns. (20) and (23) that as a result of the above-mentioned interaction, degeneration of the LPO mode corresponding to the centre under consideration is removed: it splits into tangential and normal oscillations and new local modes appear that split from the SP band. The frequencies of the local modes formed group near two eigenfrequencies of the initial resonances: the LPO frequency w, and the limiting frequency of SPs w,$ determined as the solution of the equation em(w) + E,, = 0. In the case of polarizability of form (14) or (15), four resonance modes appear in the vicinity of the centre, two of which (tangential) can be excited both by s- and p-polarized light and the other two (normal) only by p-polarized light. The positions of the eigenfrequencies of local resonances (oscillations) relative to wr and w,; depend on the microscopic structure of the interference. In the sharp interface model (AC, ACT’ = 0), the frequencies wil of two oscillations (one tangential and one normal) resulting from LPO splitting lie below w, and the frequencies w{~ of the other two oscillations lie above w$,, the frequencies of tangential oscillations lying within the range bounded by the frequencies of normal oscillations. In most cases, the frequency ranges (0; , wi) and wi;, ~1) are less than the widths of the corresponding modes, so that only two broadened resonances should be observed, one below w, and the other above w,Op. It is to be inferred from eqns. (15), (18) and (20) that in the vicinity of each of the eigenfrequencies wil the cross-section for the formation of local modes can be written as
(24)
189
where, for example, at w, = o;
(25)
Using the above formulae, it is possible to estimate the field enhancement ratio near a rough surface. Note that formulae (24) and (25) are written assuming the distances between the frequencies of different resonances to be greater than their widths. In the opposite case, the formulae become more complex (see ref. 7) but the order of the enhancement ratio remains unchanged. (IV) OPTICS ELECTRODE
OF A SYSTEM
OF LOCAL
CENTRES
NEAR
THE
SURFACE
OF A METAL
Let us now consider the interaction of light with a system of local centres located at points x(n) = (xp, xi”)) that are distributed in a random way along the electrode surface. The total scattering operator 2” on a system of local centres can be represented as
In this case, exact solution of eqns. (16) and (26) is impossible and it is necessary to resort to additional approximations similar to those used in the disordered media theory [14]. In the present study, we will consider a compositional disorder model in which local centres are assumed to be located at the sites of a lattice on the surface, each site being occupied with the probability 8,. In order to determine the surface excitation spectrum and to calculate the light reflection coefficients, a so-called average T-matrix approximation is used in which a disordered system is replaced by an effective medium in the form of the same lattice completely occupied with equivalent local centres. The scattering matrices on individual sites of such a lattice (T(w)) are written as follows
(7(0)) = f&qw)
(27)
where 7 is the scattering matrix on one centre located near the metal surface with the form (18). Under the above assumptions, from the exact formula (26) of the multiple scattering theory, the following expression is obtained for the surface-averaged ? matrix of the system under consideration:
(f) =~(i(x’“‘)) n
+
$ c n.m#n
(i(n(“)))B”(i(x(m))) + *. *
190
The approximation used consists of the decoupling of the surface-averaged products of the matrices i(xCn))i(xCm)) * * . ‘ and represents a version of the mean field theory. This approximation becomes exact in the vicinity of two limits: for low coverages and for a completely occupied lattice. Just as in the case of quantummechanical multiple scattering problem [?I, let us express the f matrix satisfying eqn. (26) in terms of auxiliary operators T,,, f=
c T,,
(29)
m,n The operators fm,, obey the equation
In the coordinate representation,
fm,, are of the form
Ll = S3( X - X(m)) fm, 63(X’ - x(n))
(31)
The surface-averaged auxiliary operators (fmn) obey the equation (32) On account of the translation invariance (Fm,,) = fm_,,, eqn. (32) can be readily solved by expansion into a Fourier series: c = N-’ C exp[ -iq( dm) - dn))] t_, (33) m,n where N is the number of units cells in the lattice of local centres under consideration and the wave vector q is determined in the first Brillouin zone for this lattice. Substituting eqn. (33) into eqn. (32), we find
To calculate the light rejection coefficients and the law of dispersion of SPs, it suffices to estimate the q matrix at small values of q (q ( do) - x2’ 1 -=K1). If this circumstance is taken into account, the sum of Green functions l)(x”), xc”)), for example for a square lattice with period R contained in eqn. (34), assumes the form CD,;(~@), nzo
9,
r(.))exp(iq(r(‘)-x(“)))-~~((if+A!)
(35)
where U,‘=4$
c D!(d”‘, n#O
xc”)) = 9.03~,‘R-~
6,
s, = 1
a,=&=
-f
191
AU, = -4aZ,(w)
+
Xexp
(2?r)‘(m2
+ n’) - $,R2
=4%(4 -Eel
(21T)2 E -Eel
exp[-4a&n2+nz)‘/2]
-~--&(m2+n2)“2 R3
l/2 11
AU, = AU,
=
exp
-4aZ,(w)
+ f(
l/2 1 l/2 11
g3c m,n
x(~(~)‘[r~(~(m,,~2)l~2)-rp(~(m2+n2)1/2)] + ( m2+ ?z’) g( m2 + r~~)l’~ f el
I
-47qw) xexp
'P
- +( y’1 _4T?$(m2+
[
)i
(
c (m’ + n2)1’2 CA m,n
1mel
n2)1/2 s
(36)
In calculating the values Aq, the expressions for the Green functions D,T(x, x’) obtained in ref. 11 were used and transition was accomplished from summation over direct lattice sites to summation over reciprocal lattice sites. By this transformation it is possible to speed up the convergence of the series determining the form of AU,. It follows from eqns. (34) and (35) that with increasing degree of occupancy of the lattice 8,, the probability of decay of resonance states arising on a rough surface in the delocalized SPs decreases in proportion to (1 - t9,). The probability of decay of the resonance states in radiation increases in proportion to 8,. It should be emphasized that in expression (34) the terms accounting for the interaction between centres are contained in the denominator of the T matrix, i.e. they represent corrections not directly to the T matrix or the Green function but to the polarization operator or, in other words, to the self energy operator. An advantage of this form of wgting lies in that the expressions obtained remain valid in the vicinity of the poles T,(o), where the ordinary expansion into a series of the perturbation theory does not hold. Using the approach under consideration, account can be taken of the fact that the resonance frequencies for different centres on the surface may differ from one
192
another (non-uniform broadening of the spectra of states). For this purpose, we will use expression (15) for a(w) and assumed that the centres with different resonance frequencies wr are distributed over the lattice in a random manner with the distribution function (37) Just as in the case-of electron levels in disordered systems [14], let us average expression (34) for T, with the weight (37). As a result, we have
It is evident from eqn. (38) that taking account of the non-uniform broadening of the spectrum of local states leads to smoothening of the peaks in the T matrix and, accordingly, in the quantities discussed below. Formula (34) implies that the coefficients of specular reflection of light from the lattice of local centres considered above are formally of the same type as the corresponding reflection coefficients from an anisotropic layer on the metal surface. The normal C: and tangential ef, components of permittivity of this layer are determined by the equalities
0El I
(4 - %d)d= (iq22/sI )-I - c,‘)d= c;‘( &&/s,
where S, is the unit cell area for ,the lattice of centres under consideration and d is the effective thickness of the layer. The imaginary components of E:, and E: have maxima at frequencies Q,T and Q: , respectively, which are the resonance frequencies of the matrix q. Frequencies &Ii and Q; lie in the vicinity of the resonance frequency w, of LPOs on an isolated centre and frequencies QI; and sll are in the vicinity of the limiting frequency of SPs on a smooth surface. An approximate averaging method similar to that described above was also used in ref. 15 which dealt with the reflection of light from a partially occupied lattice of local centres. No account, however, was taken in ref. 15 of the presence of a metal substrate, interaction with which, as shown above, results in a qualitative change of the characteristics of local resonances. Expressions of the forms (34) and (39) can be used to describe light reflection from the lattice of atoms and a molecule on the metal surface. A similar approach to the solution of this problem was developed in ref. 9. The calculation method described there consists in generalization of the effective medium approximations [16] widely used in considering the optical properties of three-dimensional non-homogeneous media for the case of a disordered system of local centres (or molecules) located near the surface. The results presented here show that direct application of the above approximations to surface optics problems, as was done in numerous studies (see, for example ref. 17), may lead to grave errors.
193
By means of the proposed method it is also possible to study the influence of surface inhomogeneities on the dispersion relation of SPs, w,,(k,,). The form of the dependence w,(k,,) is determined by the position of the poles of matrix T considered as a function of w and k,,. With the use of eqns. (18) and (34), the equation for the determination of w,,(k,,) and Rk, -C 1 is written as follows: -1
1
Here, $,(k,,) is the surface plasmon dispersion relation on a smooth surface. Comparison of eqns. (39) and (40) shows that when local centres appear on the surface the dispersion relation of SPs changes most drastically (as compared with a smooth surface) in the vicinity of two frequencies, Qi and 0; , corresponding to tangential and normal plasma oscillations. At these frequencies, extrema should appear on the o,,(k,,) dependence, whose position and height depend on the concentration of local centres 8,. (V) CONCLUSION
Let us discuss the experimentally verifiable consequences of the theoretical treatment presented above. Let us consider first of all the increase in the electric field intensity near the surface in the presence of local modes. According to eqns. (5) and (24) this increase of g(x, w) at the point x near the centre located at the point x0 = (x,0, 0, 0) can be estimated as follows: g(x,
0) = IB(x,
WI I 2/e=
m4
I cpl(4
t SP,U c
2 ~lal,spusp
+
rU.SP
k,‘r2 +
rd.sp
(
rsp, +
I */N”Pl 2
-3
SP.1 rd
1)3
Ix
-‘:O
I6
(41)
where g,, is the field amplitude in the incident wave and cp,(x) = [cp,/l x x o I 3](( dd/c>cy ) -3’2 is the normalized solution of the Maxwell equation near the centre *. At values of = f’+~l,sp%p
+
ru,,
+
10
rsp,l/rd.l
e
1
rd,sp
( rsp,l/rd,l)2R;6( :C:f2)-3k&,2:
= lo3
l The quantity cp,/Td,, has been estimated from the condition of matching with the results of classical calculations for an isolated centre in the framework of macroscopic electrodynamics.
Fig. 1. Frequency dependence of the enhancement ratio of the electric field intensity g(s, w) at point x = (0.1 run, 0, 0), estimated for the cases of s- and p-polarized light incident upon a silver surface at an angle B = 45 O. The local centre is located at the point with coordinates x o = (5 nm, 0,O) and models a spherical particle from silver of radius 5 nm.
obtained using the perturbation theory on the interaction of surface and bulk waves on a slightly rough surface [S] and the model of a spherical centre with radius R,= 5 nm considered in Section (III), the ratio g(x, w) for silver at hw = 3 eV proves to be of the order of 104. The model calculation carried out in Section (III) shows that the values of g(x, w) in the vicinity of resonance frequencies 011.~ are several orders higher than those near the frequencies wl;Il . The value of g( X, w) as a function of the coordinate x r&aches a maximum on the axis of symmetry of the system in the space between the local centre and the surface. In this region, g( X, W) may exceed significantly (by more than one order) the maximal resonance enhancement, which takes place if the objects are isolated. Figure 1 shows the frequency dependence of the enhancement ratio of the electric field g( X, w) at the point with coordinates x = (D, 0, 0) located between the centre and the surface. In performing the calculations, the polarizabihty a(~) was taken in the form of eqn. (14), the centre and substrate being considered to consist of silver and no account being taken of the presence of smooth rot&messes on the surface. It should be emphasized that the main features of the frequency and coordinate dependences of the enhancement ratio g( x, o) found with the use of the model just considered are confirmed by the results of numerical calculations performed for a sphere [18] near the metal surface taking into account a large number of modes with an angular momentum L > 1. A sharp increase (approximately 105-lo6 fold) of the photoemission current in the vicinity of local centres on the Ag surface revealed in ref. 19 can be explained by means of eqns. (8) and (41) provided that ( I’pi’fId2)*R,“(( 0/c)@-~1c;,‘r u,,/c) = 104. This value of (r,,,/r,,)*R,“((w/c)~~~ )- k,*(u,,/c) is larger than that obtained in the spherical centre model. It is possible that when considering centres of a different kind, for example, of the type of cracks (especially when account is
195
taken of the influence of various possible polaritons in the adsorption layer), other values of g(x, w) can be obtained explaining the observed phenomena. The effects leading to enhancement of RS and diffuse scattering of light in the vicinity of local centres can be estimated in a similar manner. A significant fact (which can be verified experimentally) should be mentioned here, that in the approach considered above the angular dependences (on 8 and 8,, vr) of the radiation arising during RS and diffuse scattering should be identical. It follows from eqn. (3) that in the presence of local centres, the light reflection coefficients from the metal surface should have minima in the vicinity of the eigenfrequencies of local resonances Q,f and 0:. This conclusion agrees with the results of experiments [20] on the reflection of light from very rough metal surfaces. According to eqn. (40), at the frequencies Q,il, extrema should also appear on the dispersion curves w,(k,,) for surface plasmons. Such behaviour of the dispersion curves was observed in ref. 21 for a strongly loosened Ag surface in 0.1 M K,SO, solution. At the metal/solution interface, the eigenfrequencies of local resonances can depend on the potential drop E in the double layer and/or on the adsorption on the electrode. This dependence can be, for example, due to a change in the electron density inside local centres (inhomogeneities) resulting from a change of the potential drop or charge transfer during adsorption. In this case, using the approach being discussed, it is possible to obtain from eqn. (39) the following expression for the electroreflection signals on a rough surface, e.g. in p-polarized light,
<,,)( Ed SinZe - E, c0s2e)] -’
(42)
It is evident from eqn. (42) that loosening of the electrode surface should give rise to extrema on the frequency dependence of electroreflection signals lying in the vicinity of the resonance frequencies of local modes. Such behaviour of the electroreflection signals was observed, e.g. in ref. 22. Similar behaviour of electroscattering spectra (derivatives of the diffuse scattering intensity with respect to the potential) was revealed in ref. 23. This effect can also be explained on the basis of the treatment presented above (see formula 10) taking into account the circumstance that the frequencies of local resonances depend on the potential drop. REFERENCES 1 R.K. Chang and T.E. Furtak (Eds.), Surface Enhanced 1982.
Raman Scattering, Plenum Press, New York,
196 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
23
J.K. Sass, R.K. Sen, E. Meyer and H. Gerischer, Surf. Sci., 44 (1975) 515. C.K. Chen, A.R.B. de Castro and Y.P. Shen, Phys. Rev. Lett., 46 (1981) 515. M. Moskovits, Rev. Mod. Phys., 57 (1985) 783. A.M. Brodsky and MI. Urbakh, Surf. Sci., 167 (1986) 231. V.A. Kosobukin, Poverkhnost’, 12 (1982) 5. R.G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill New York, 1980. P.J. FiebeIman, Phys. Rev. Sect. B, 22 (1980) 3654. A. Bachi, B. Barrera and R. Fuchs, Phys. Rev. Sect. B, 25 (1982) 7086. C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1979. A.M. Brodsky and MI. Urbakh, Surf. Sci., 115 (1982) 417. F.I. Dahdchik and G.K. Ivanov, Teor. Eksp. Khim., 8 (1972) 9. A.M. Brodsky and M.I. Urbakh, Prog. Surf. Sci., 15 (1984) 121. H. Ehrenreich and L. Schwartz in F. Seitz, D. TumbuB and H. Ehrenreich (Eds.), Solid State Physics, Vol. 31, Academic Press, New York, 1976, p. 5. B.N.J. Persson and A. Liebsch, Phys. Rev. Sect. B, 28 (1983) 4247. V. Twersky, J. Math. Phys., 19 (1978) 215. V. Brusic, M.A. Genshaw and J.G’M. Bockris, Surf. Sci., 29 (1972) 653. P.K. Aravind and H. Metiu, Surf. Sci., 124 (1983) 506. A.M. Foontikov, S.K. SigaIaev and V.E. Kazarinov, Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Khim., 17 (1986) 47; J. Electroanal. Chem., 228 (1987) 197. T. Lopez-Rios, Y. Borensztein and G. Vuye, Phys. Rev. Sect. B, 30 (1984) 659. M. Abraham and A. Tadjeddine, Surf. Sci., 173 (1986) 65. V.V. Marinyuk, R.M. Lazorenko-Manevich and Ya.M. Kolotyrkin in Ya.M. Kolotyrkin (Ed.), Advances in Physical Chemistry. Current Development in Electrochemistry and Corrosion, Mir, Moscow, 1982, p. 148. A.M. Foontikov, S.K. SigaIaev and V.E. Kazarinov in Electrodynamics and Quantum Phenomena at Interfaces, Telavi (U.S.S.R.), 1984, Izd. Academii Nauk Gruz. SSR, Tbilisi, 1984, Ext. Abstr., p. 276.