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SURFACE PLASMONS ON A RANDOMLY ROUGH SURFACE
G. Farias, A. Maradudin
To cite this version:
G. Farias, A. Maradudin. SURFACE PLASMONS ON A RANDOMLY ROUGH SURFACE. Jour-
nal de Physique Colloques, 1983, 44 (C10), pp.C10-357-C10-361. �10.1051/jphyscol:19831072�. �jpa-
00223530�
JOURNAL DE PHYSIQUE
Colloque CIO, supplbrnent au n012, Tome 44, ddcernbre 1983 page CIO-357
SURFACE PLASMONS ON A RANDOMLY ROUGH SURFACE
G.A. F a r i a s and A . A . Maradudin
Department of Physics, University o f CaZifornia, Irvine, California 92717, U.S.A.
~ B s u m k
-
Nous a v o n s P t u d i B l a r e l a t i o n d e d i s p e r s i o n d e s plasmons d e s u r f a c e s u r une s u r f a c e r u g u e u s e a l e a t o i r e , e n a l l a n t a u - d e l S d e l ' a p p r o x i m a t i o n l a p l u s b a s s e p a r r a p p o r t Zi l a f o n c t i o n d e p r o - f i l d e s u r f a c e . L e developpement d e l a r e l a t i o n d e d i s p e r s i o n d e s plasmons d e s u r f a c e e n p u i s s a n c e s d e l a f o n c t i o n d e p r o f i l d e s u r f a c e c o n t i e n t un sous-ensemble i n f i n i d e t e r m e s q u i s o n t t o u s du meme o r d r e d e g r a n d e u r que l a c o n t r i b u t i o n d ' o r d r e l e p l u s b a s , s e u l e c o n s i d S r 6 e d a n s l e s p r 6 c B d e n t e s d 6 t e r m i n a t i o n s t h B o r i q u e s d e c e t t e r e l a t i o n . Ce sous-ensemble d e t e r m e s c o n d u i t 2 une Bqua-t i o n i n t e g r a l e non l i n e a i r e p o u r l a s e l f - 6 n e r g i e p r o p r e d e s p l a s - mons d e s u r f a c e , e n f o n c t i o n d e l a q u e l l e l a r e l a t i o n d e d i s p e r s i o n e s t e x p r i m g e . La s e p a r a t i o n d e l a c o u r b e d e d i s p e r s i o n d e s p l a s - mons d e s u r f a c e e n deux b r a n c h e s p a r l a r u g o s i t e d e s u r f a c e , p r e - d i t e t h e o r i q u e m e n t p a r un c a l c u l e n t h B o r i e d e p e r t u r b a t i o n Zi
l ' o r d r e l e p l u s b a s , e t o b s e r v e e e x p 6 r i m e n t a l e m e n t 1 e s t p r e s e r v e e d a n s les r e s u l t a t s d e s p r e s e n t s c a l c u l s . Cependant, 2 l a f o i s l a g r a n d e u r d e l a s e p a r a t i o n e t l ' a m o r t i s s e m e n t d e s plasmons d e s u r - f a c e q u i s o n t o b t e n u s i c i s o n t p l u s g r a n d s , pour l a mEme l o n g u e u r d e r u g o s i t 6 , q u e l e s mcmes q u a n t i t 6 s o b t e n u e s S p a r t i r du c a l c u l e n p e r t u r b a t i o n 2 l ' o r d r e l e p l u s b a s , pour l a p l u p a r t d e s v e c - t e u r s d ' o r d r e du plasmon d e s u r f a c e .
A b s t r a c t
-
We have s t u d i e d t h e d i s p e r s i o n r e l a t i o n f o r s u r f a c e plasmons on a randomly rough s u r f a c e , g o i n g beyond t h e l o w e s t a p p r o x i m a t i o n i n t h e s u r f a c e p r o f i l e f u n c t i o n . The e x p a n s i o n of t h e s u r f a c e plasmon d i s p e r s i o n r e l a t i o n i n powers of t h e s u r f a c e p r o f i l e f u n c t i o n c o n t a i n s an i n f i n i t e s u b s e t o f t e r m s t h a t a r e a l l o f t h e same o r d e r o f m a g n i t u d e a s t h e l o w e s t o r d e r c o n t r i b u t i o n , t h e on1 y one c o n s i d e r e d i n p r e v i o u s t h e o r e t i c a l d e t e r m i n a t i o n s o f t h i s r e l a t i o n . T h i s s u b s e t o f terms y i e l d s a n o n l i n e a r i n t e g r a l e q u a t i o n f o r t h e s u r f a c e plasmon p r o p e r s e l f - e n e r g y , i n t e r m s o f which t h e d i s p e r s i o n r e l a t i o n i s e x p r e s s e d . The s p l i t t i n g o f t h e s u r f a c e plasmon d i s p e r s i o n c u r v e i n t o two b r a n c h e s by t h e s u r f a c e r o u g h n e s s , p r e d i c t e d t h e o r e t i c a l l y by t h e l o w e s t o r d e r p e r t u r b a t i o n t h e o r y c a l c u l a t i o n , and observed e x p e r i m e n t a l l y , is p r e s e r v e d i n t h e r e s u l t s o f t h e p r e s e n t c a l c u l a t i o n . However, b o t h t h e magni- t u d e o f t h e s p l i t t i n g and t h e damping of t h e s u r f a c e plasmon o b t a i n e d h e r e a r e l a r g e r , f o r t h e same c o r r u g a t i o n s t r e n g t h , t h a n t h e same q u a n t i t i e s o b t a i n e d from t h e l o w e s t o r d e r p e r t u r b a t i o n c a l c u l a t i o n , f o r most s u r f a c e plasmon wave v e c t o r s .Recent t h e o r e t i c a l ( l r 2 ) and e ~ p e r i m e n t a l ( ~ 1 s t u d i e s h a v e shown t h a t , i n t h e p r e s e n c e of s u r f a c e r o u g h n e s s , t h e d i s p e r s i o n c u r v e f o r s u r f a c e plasmons c o n s i s t of two b r a n c h e s , i n c o n t r a s t w i t h t h e d i s p e r s i o n c u r v e f o r s u r f a c e plasmons on a f l a t s u r f a c e which c o n s i s t s of a s i n g l e b r a n c h .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831072
C10-358 JOURNAL DE PHYSIQUE
In the results obtained theoretically in Refs. 1, 2, the effects of the surface roughness were taken into account only to the lowest approximation in the surface profile function. However, in an
analysis of higher order terms in the expansion of the surface plasmon dispersion relation in powers of the surface profile function, we found that there is an infinite subset of terms that are all of the same order of magnitude as the lowest order contribution. For this reason, in this paper we study the dispersion relation of surface plasmons, going beyond this lqwest order approximation.
The system of interest consists of vacuum in the region z
> c(gn)
( = ;x +
&)
and a dielectric medium characterized by an isotropic, frequency dependent dielectric constant ~ ( a ) in the region z<
~(x,,). + The surface profile function 5(zl)
is a stationary stochastic process.-+
-
*Its Fourier transform possesses the properties (kB= xkx
+
yky)= 0 (la)
where the angular brackets denote an average over the ensemble of realizeions of the surface profile function, 62 is the mean square departure of the surface from fTatness, and g(k ) is t h x r y a c e structure factor. Higher order correlation funci!ions are evaluated on the assumption that
C(g,,)
is a Gaussianly distributed random variable.We seek the solution of Laplace's equation for the potential in each
>
of the regions z < t(;[), subject to the boundary conditions that the potential and the normal component of the electric displacement vector be continuous across the surface, and that the potential vanish as
121 + w .
The solution for the potential in the region z > C(~,,)max, that vanishes at infinity, can be written as
We use the extinction theorem form of Green's theorem(4) to eliminate the potential
ig
the dielectric medium, and obtain an integralequation for A(kl lw) which, in the small roughness limit, is given by
where y (w) = (E(w) + ~)/(E(w)
-
1).Because 6 (Z1 ) is a random function so is the solution A (k -+
1
a). Wetherefore solve for its average value, <A($, la)>. With the use of
I, '
projection operators(5) we can transform Eq. (3) into a homogeneous algebraic equation for <A($!
1
w)>.
The solvability condition for the latter equation gives the dispersion relation for surface plasmons on a rough surface. By the use of a diagrammatic analysis of the terms entering the expansion of the dispersion relation in powers of thesurface profile function the expansion can be resummed to yield the equation
where
1
, (;,Iw)
has the nature of a proper self-energy. It sat isf ies a nonlinear integral equation+ +
where
r
(qll ;pll la) is a vertex function that in the lowest approximation.. ..
is (1
-
q,, *Pi 1 .To present our results for the surface plasmon dispersion curve we calculated the surface plasmon spectral density
where G(qn
1
w) is the surface plasmon propagator given byIn writing Eq. (7) we used the fact that
1
(q,,1
w),
and hence G (q1
w),
depends on the wave vector
G,,
only through its magnitude, due to the restoration of isotropy to the dielectric/vacuum system by the averaging over the ensemble of realizations of the surface profile function. The poles of G(ql w),
according to Eq. ( 4 ) , occur at the frequencies of surface plasmons on a randomly rough surface. We used the free electron model for the dielectric constant, ~ ( w )-
2 = 1-
-
up/ (w (w+iy) ),
the lowest order approximation to the vertex function,2 2 2
and a Gaussian form for g(k,), viz. g(k,) = xa exp(- k,,a /4), where a is the so-called transverse correlation length, in solving Eq. ( 5 )
numerically by iteration. In Fig: (1) we present the dimensionless surface plasmon spectral density p (Sw) = ap (q!w), where 5 = aq n '
together with the result corresponding to the lowest order approxima- tion for
Iw)
obtained in Ref. 2. The value of 5. = a q l = 1 chosen is the one for which the maximum splitting of the surface plasmon dispersion curve occurs in Ref. 2. To obtain a splitting of 0.15-
0.2 eV in the surface plasmon dispersio curve for Ag, as has been observed experimentally by ~ a t z et al. ('), a value of 6/ab0.05 is required for 5 = 1 on the basis of the present results. The widths of the peaks in the present work are due to the attenuation of the
surface plasmon by multiple scattering from the peaks and valleysas it propagates along the rough surface, Finally, in Fig. 2, we have plotted the frequencies of the two peaks in the spectral density as functions of 5., together with the dispersion curve obtained in Ref. 2, for 6/a = 0.05. It is seen that the splittinq of the dispersion curve obtained in present calculations exceeds that-calculated in Ref. 2 for 0.3 -" 5. < 2.
JOURNAL DE PHYSIQUE
Pig.
Fig.
-
- -
- -
-
0.96 0.98 1.00 1.02 1.04
w'wsp
1: The spectral density of surface plasmons on a randomly rough surface calculated self-consistently ( ) , and in the lowest order approximat ion in the surface profile function
(-
- -
) , at = 1, for 6/a = 0.05.1.04 t I I I
0.96 I I I
0.00 1.00 2.00 3.00
€
2: The frequencies of surface plasmons on a randomly rough surface as obtained from the positions of the peaks in the spectral density p ( E , w ) ( )
,
and from the lowest order approximation in the surface profile function (-- -
f r for 6/a = 0.05.In conclusion we observe that it is necessary to include the congribu- tions beyond the lowest order in the surface profile function C ( x ) in the calculation of the surface plasmon dispersion curves, and thae the results of this work have implications for the theoretical study of all surface excitations on randomly rough surfaces.
Acknowledgement
The work of G.A.F. was supportad by cNPq-Brasil Proc. No. 200.601/81- FA. The work of A.A.M. was supported in part by NSF Grant DMR 82- 14214. He acknowledges gratefully a grant f r ~ m the Graduate Council of the University of California, Irvine, that made possible his participation in this conference.
References
1. KRETSCHMANN E., FERREL T. L., and ASHLEY J. C., Phys. Rev. Lett.
42 (1979) 1312.
2.
-
R5HMAN T. S. and MARADUDIN A. A., Phys. Rev. 821 (1980) 2137.3. KOTZ R., LEWERENZ H. J., and KRETSCHMANN E.
PFS.
Lett.E,
(1979) 452.
4. See, for example, WOLF E., in Coherence and Quantum Optics ( eds.
L. Mandel and E. Wolf (Plenum, New York, 1973), p. 339.
5. RAHMAN T. S. and MARADUDIN A. A., Phys. Rev. g z (1980) 504.
