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The enhancement of raman and fluorescent intensity by small surface roughness. changes in dipole emission
Volume
74. number
2
THE ENHANCEMENT BY SMALL SURFACE
CHEMICAL
PHYSICS
LETTERS
1 September
1980
OF RAMAN AND FLUORESCENT INTENSITY ROUGHNESS. CHANGES UU DIPOLE EMISSION
P K. ARAVIND and Horia METIU Departmetrt of Cltemsny, Santa Barbara Caltfornra
Un~~emty of Caltfornia. 93106, USA
Recewed
III final form 28 Aprd
14 March 1980,
1980
WC propose that surface roughness mfluences the Raman and fluorescence spectra of adsorbed molecules since (1) it allous cwitauon of surface evanescent fields whch mcnzasc the induced dipole moment of the molecule. (2) it allows plasmon radlatlon modulated by the oscdlatmg nuclei, and (3) It mcreases the radration of the mduced dipole. We present numerIcal results for the thud mechanism
l_ Introduction
Recently, it has become apparent that roughness plays an rmportant role in the surface enhanced Raman scattering (SERS). In fact, there is, as yet, no experimental evidence for the existence of SERS on a flat surface (thrs means that the enhancement, If any, is smaller than the three orders of magnitude needed to observe SERS). There are many ways to roughen a surface and study its effects on SERS. The tradttional one [l] is surface anodization; sometimes [2,3] this causes the formatron of salver “spheres” on the surface and these act as “roughness”. The inherent roughness of metal films formed through vapor depositron [4] or even mechamcal roughenmg [S] of the surface, seems to be sufficrent for the appearance of SERS. Some degree of controlled roughness can be achieved by depositing metal films [6] on CaF, layers [7], or by making drffraction grattngs on the metal surface
WI.
Furthermore, both island fiis [9] and colloidal partrcles [lo] should be considered mtrinsrcalty rough systems, even d their geometric surface is perfectly smooth; due to the small size of these systems, no parallel momentum conservation 1s required, just as tn the case of extended rough surfaces. The first attempt to explain the role of roughness III SERS has been made in a pioneering study by
Moskovits [l 11_ He models the rough surface by assunung that it consists of silver islands seated ORthe top of the Bat sdver surface. The molecules are located on the rsland and form - through a mechanism that is not specified in detail - a strongly coupled system. It is then assumed that SERS consists of resonant Raman scattering from the whole (island f molecules) system; the rsland provides the resonating electronic states, which are the islands electromagnetic modes, while the molecules participate by providing vibrational nuclear motion. Moskovits has shown that his model can fit some of the obsenred (2,121 SERS excttation spectra. In passing we note that very similar results can be obtained by startmg from the model proposed by Efrima and Metiu [ 131 for resonant Raman, by replacing the semi-infirute metal with a finite one (e.g., a very small sphere, etc.). The role of the metal islands has also been considered by Burstein et al. [ 141; other mechanisms have been analysed in unpublished work of Ku-tfey et al. [15] and Gersten [16]. Kn ths letter we present some preliminary results which are part of a phenomenologicaf model rrying to descnbe the role of roughness in Raman scattering or fluorescence *_ We distinguish three
* There IS some evidence face roughness [ L7]_
that fluorescence
IS :nfIuenced
by sur-
VoIume 74, number 2
CHEMICAL PHYSICS LETTERS
possible mechanisms. (1) The ncldent field excites a plasmon and the surface ftdd thus created mcreases the total field mcident on the molecule_ Thrs will mcrease the induced molecular dtpole moment, hence produce a larger Raman or fhrorescence intensity. This is similar to the spectroscopic effects [ 1S] tnduced through ATR [ 191, but roughness rather than an ATR prism is used to excrte th2 evanescent field. (2) The radiation from the rough surface is modulated by the vrbrational matron of the nucIei, smce the effectrve nuclear charges interact with the surface fields (modtfied by roughness). Thrs IS the type of process that has also been discussed by K.ttdey et al. [ 1.~1. (3) The reduced molecular dipole moment has a component ascdlatmg at the incident frequency mums the vrbratronal fr2quency. Thrs radiates Raman photons and the properties of thus radiatron are modified by the presence of roughness. To compute these effects we use phenomenologrcal electrodynarmcs hlazwell’s equattons are solved by a method developed by hlaradudin and Mrlls f20] and modified to correspond to the situation of interest here. In the present letter we describe the results obtained for mechamsm (3).
2. Brief description of the model
W2 assume that th2 incrdent hght mduces a dtpoIe moment p(r).whrch IS modulated by the vrbratronal motion. Therefore, we can wnte
i(r, t) Gj(r, =
-1w
mode of
interest and ro(f) IS the mduced dipole when the nuclei are at equdrbrium (Q = 0) The second term m eq. (1) is responsible for F&man scattering, smce it has a component that oscdlates at the frequency 0 s w. - w,, here w. IS the frequency of the Incident hght and o, that of the nuclear vrbrattons. If we denote rl(t)= (ae/aQ)* Q(~)EJL~(w)~-‘~~, the quantrty j(t) f awl(t)/& ISa current which, according to the MaxweLl equations, radiates hght of frequency w. - w, G w. if there are N molecules, located at rcr,cY= 1.. _,N, having the dipole pI__*, the corresponding current density IS 302
w) e--rwf
g1
)I]**
(W)6(f -
f&e-iwr.
(2)
We can compute the Raman inrensny by solving the Maxw2U equattons to fiid the radiatron emitted by j(r, t) m the presence of the rough surface. Fluorescence can be treated m a similar manner; the only difference being the meaning of the magmtude of cl,u(o) m (2).
In order to treat the effect of roughness we assume that the posrtion of the metal surface IS given by z = $(x,_v)- We assume a stochastic model 12 I], for the surface. Thrs accepts the fact that given x and _V(coordinates “parallel’” to the metal) we do not know the posrtlon of z on the surface; it assumes however that we know the probabrhty P(=, x,y) that at the pomt (x,~) z has a given vaIue. We now defme a plane z = 0 such that th2 average height, at every pomt (x,~) IS zero. We can then wnte the dielectric constant of the system in the form E(X,Y, z, w) = e(-J(z; w) + Ae(x,y, z, w). Here eo(z; w) is the dielectrrc constant of a ficrrttous system wrth a flat surface located at z = 0 having the metal dielectric constant l O(w) for z > 0 and the vacuum value for z C 0. E(x,)-‘, z, o) IS rhe true drelectric constant of the system wrth the rough surface, and AE is the difference. We can now consider Ae to be a perturbation and solve Ma\weIl’s equations by ustng the Born approxrmation The result is
I= 1, + 6,6, where C?(r) is the amphtude of the normal
1 September 1980
*DO:i,
fj,j*]_
(3)
Here so is the dyadrc Green function for the Maxwell equations m the presznce of the j7at surface. Cl o is the asymptottc form of DO I is the total intensity, whrle 1, is the intensity for the case of the flat surface. The purpose of tlus schematic outline IS to point out that there are four quantities of physical mterest, as described below. (I) The quantity that starts the process 1s the current-current comlatron function [i(r; w),j(r’.
w)] * 6(f -r’)Nw*1J+(w)l?
(4)
We assume here mcoherent scattering neglecting thus ah momentum transfer processes that coherence entak
Volume 74,
number
2
CHEMICAL
PHYSICS
(II) The radiation enutted by the molecular currents is propagatedJo_tte rough surface by the pau of Green’s functrons DoDo. These two quanti~es_kve denominators winch become small (hence DoDo becomes very large) when the parallel momentum k, and the frequency w are those corresponding to the surface plasmon. (III) The radiatron emitted by [j,j] and propagated by 6u6u is scattered by the roughness. The strength of this scattering process is given (in the Born approximatron) by the correlation function (Ae(r, w) X Ae(r’, w)>_ If one thmks of surface roughness as a density “fluctuation” rn the surface region, mducing a change AE in the dielectric constant, then the appearance of IS completely sinular to the role this quantity plays rn hght scattenng by density fluctuations m fluids. The correlation function (AEAE) IS proportional to <{(x,JJ){(x’, y’)). The Founer transform of this quantity IS taken to be [2 I] (<(kn)<*(k~))=
(2a)‘6(ks
+k;,)6”a2
exp[-($akd)2]. (9
Here 6’ is the mean square amphtude of the roughness, Q 1s its correlation length and k, and k’, are parallel momenta. The Born approxrmation requires 8 to be small (=30 A) but imposes no limitatrons on CI. The correlation length depends on how the surface has been prepared. (IV) The last elemenlcin Eq_ (3) IS the pair of asymptotic Green’s functions CD, CZIu.They propagate the bght scattered by roughness to the detector. This quantity has no resonant behavior and mfluences the depol arizatx~~
ratio as well 35 the angular
and frequen-
LLTTERS
1 September 1980
.P;
a=ZoH IO
. l t
z
2 2
i
___*___&--~-
_a---
__a-----
_-
_I’
: : :
‘. ‘.
‘. P sa= 150 a ‘. ‘. ___o_____-o--__--o----O-= ‘4_ --o a=!500 a _o.____~___--o___~___o-___-~-.~ _.
01
I
20
I
1
24
I
I
26
w
I
32
,I,
-\ ‘*
36
,
40
Inev-
Fg. 1 fht = IO is the intcnuty radiated by the dipole in the is the diff21-21122 between presence of the flat surface. Irou the emission for rough and that Por a flat surface. The parameters are- the mean amplitude L = 3OA, the angle of detecnon from normal is tJ= 49, the metal is silver. The curves correspond to three different correlaaon lenggs a_ The arrow mdicates the posltion of the unretarded plasmon frequency.
correlation length a is rather small (of the order of 20-50 a). If this requirement is fulfilled the enhancement factor caused by roughening the surface could vary between 3 and 50, depending on the frequency and the value ofa. The enhancement factor is largest at frequencies_close to and below the plasmon frequency 1~~/2r’~, and becomes extremely small for frequencies above it. Sinular features are seen for Cu and Au.
4_ Comments
cy dependence m a “rruld” way.
3. Results In fig. 1 we present the dependence of the quantity j(a) G (I- lu)/lo on frequency, for a variety of correlation lengths. The dipoles are taken to be perpendicular to the tktitrous flat surface. The drstance of the dipole to that surface is 2 A. From eq. (3) we see that f(w) is the intensrty increase caused by surface roughemng, relative to the rntensity present if the surface were flat. It 1s obvious from fig. 1 that mechanism (3) yields a significant rntensrty enhancement only if the
(a) It is customary to atrnbute effects such as those described here to the following mechanism_ The induced dipole being close to and below the plasmon frespatially inhomogeneous field on the metal. Fourier transforming this field over the spatial coordinates expresses it as a superposition of a number of planar waves, with a variety of wavevectors and the same frequency w. If the drpole is very far from the surface the field is described by a single wavevector given by w/c. When the dipole is close, many higher wavevectors are needed to descnbe the field. Due to this for any given frequency there is a Fourier component in the dipole field which can excite the surface plasmon even rf the surface is flat. The excited plasrnon cannot how303
Volume
74. number
2
CHChlICAL
PHYSICS
ever radiate and the plasmon’s
role IS to deplete the induced dipole of energy. Consequently, less energy will be radiated. Roughenmg the surface allows plasmon radtation, hence an Increase of intensuy compared to the flat surface case We fiid that thts sunple prcture IS not quote adequate. The structure of I - IO, gtven by (3) and (4), IS f-IO=
exp[-(iall-,,)‘] J k,,dk,
0
X Q(k,, ~)exp[-irL(k,,,
w. d] .
(6)
Here the term exp[-(iok,,)‘] comes from the roughness correlatton functton WE&), eq (5)__@&, o) and exp[-i$(k,,, w, d)] come from the term DODO_ If we denote by Q,, = Q,,(w) the plasmon dtspersron relation, then Q(k,,. o) has a peak at k E Q,,(o) and then varies hke ki for lugh values of k,, The correlatton length controls, through the gaussran correlatton functton appeanng in eq. (6), wluch values of k,, contrtbute to I - I,. If a is extremely large then k,,,,, = 1/a 1s so small that k, ma, < Q,,(w) and the plasmon IS not exctted For mtermedtate values of a, one can have k Um;LII> Qil(o) and the plasmon contnbutes, but k,‘is not large enough to reach the values for winch d(kU.w)=k$_We find that.mvanably,msuchcases I - I, IS small. It is only when Q is rather small (2050 A) thar iill ma, becomes so larse tint p(k,, , U) a k: for kz, k 1: then the factor k; app2aring II~ the integrand contnbutss heawly to the total ermssion This contribution IS substantrally larger than that of the
plasmon peak We cannot assoctate any elementary eycitation wrth tlus addrtronal contnbution. (b) The distance of the dipoles from the ficttttous flat surface enters m I - IO only through I,!&“, W, d). Wefindthat forsmallk,,.$(k,,,o,d)= l.Whenk,, IS large enough G(k,, w, d) decays hke e\p(-k,,d). Therefore in the product e\p[-(ial;,,)‘] exp(-k,d). as long as 1 /d Z+k, B- 1/a. tile expression for I - IO 1s independent of d. Thus happens as long as d a, I - IO will depend on d but be mdependent of Q. fn such a caSe the roIe of the lengths d and Q IS reversed. The dtscussron at (a) will have to use d instead of a (c) When a 1s small the mechanism (3) discussed here contnbutes an enhancement factor of an order of 30-t
LETTERS
1 Septembrr 1980
magnitude, on top of the contrtbutton at a flat surface. no special enhancement mechamsm the Raman scattering at a flat surface is enhanced by an estimated factor of 30, as compared to the gas phase. This 1s due to muror effects of the metal and the fact that the surface molecules do not tumble (the so called “mmor enhancement” [ 131). Therefore the total enhancement from mechanism (3), as compared to the gas phase srtuatton, IS of order 300. We esttmate that mechamsm (2) may add at most a factor of 100. We base this guess on the fact that m mechanism (2) one uses roughness to exctte the plasmon evanescent field, we expect this to act as tt does in ATR expertments on flat surfaces. Calculatrons for ATR predict (for a flat surface) an enhancement factor of 200. One never sees experimentally such a large effect, probably because roughness shortens the plasmon mean free path (hence the hfetime) and therefore dumps to some extent the evanescent field [22]. The outcome 1s a factor of about 100. If we assume the same factor for mechanism (2) then the combined enhancement from both mechanisms could be at most 109. (d) We emphastze now the limitations of the pres2nt model. (d,) We have neglected the role of spattal dtspersion (non-local dtelectrtc constant) and of the contmuous change of the dtelectnc constant on the mterface [23]. The mcluston of such effects m the calculauon IS hkely to increase the number of posstble excitations of the metal by the dipole (a great number of 212ctron-hole parr e\cttations are excluded by our present use of opttcal electric constant) Kmematically, roughness would allow some of them to radiate. Whether they chose to do so or they would rather
WA
prefer to de-excite radiationlessly dynamxs and could be established
IS detennlned
by
only by detailed calculations. Therefore, we do not know whether the rnclusron of these effects would enhance or dump the dipole radtatton. (d,) We have used the Born approximation and drd not renormalize the plasmon to include the effect of roughness on it [22] _ In other words, m our calculatton roughness allows the plasmon to radrate, but the radiatmg plasmon is that of the flat surface not that of the rough one. We believe that if such a renormabzatton IS done, the overall effect of roughness will be very sbghtly dimmished (dg) It IS clear that a dtfferent kmd of roughness (e.g., metal particles lymg on top of a flat metal surface) will produce drfferent effects. This is qutte apparent rn
Cl-IEM ICAL. PHYSICS LETTERS
VoIume 74, number 2
Iight emitting tunnehng junctions where the CaF, roughening [I41 gives different spectra than roughenmg by deposition of metal “spheres” OR the surface [15j _To &stingwsh the effects of different kinds of roughness one needs detaded studies of angular disulbutIon, polarization and excitation spectrum.
Acknowledgement We are grareful to NSF (Cm-78-1 618 1 and CHE79-18773) and the Research Camm~ttee at UCSB for supporting this work We wish to thank E. Burstem, H lðer, J. Gersten, J. Tsang, J. Ku-tley, W. Weber, B. Pettinger, M. mpott, P. Felbelman, A. Otto and D. F.fdls for sendmg us their u~pubh~ed work and/or for useful conversations.
References f 11 hf. ~~elschrn~,
P J Hendra and AJ. McQuillan, Chem. Phys Letters 26 (1974) 163, D-L Jeanmaue and R P. van Duyne. 3 Eiectroanal. Chem. 84 (1977) 1. hf G. Aibrccht and J A Creighton, J. Am. Chem. SOC 99 (1977) 5214. [21 B Petwger and U Wennmg, C&m. Phys Letters 56 (1978) 253, B. Pezuo&ter, U_ We-g and D hl. Kolb. Ber- Bunsen@zs_ Phystk Chem. 82 (1978) 1326 f3]
B. Fetuqer, Letters
U.
Wenmng
and
t-t. Weael.
Chem
Ph>s
67 (1979) 192.
[I Wenning, B Pzthnger and H Wetrel, Chem Phys Lcners 70 (1980) 49, B Perttnger, A TadJeddme and D hf_ Kolb, Chem. Phys Letters 66 (1979) 544 I41 T-f-t Wood and M-V KJem, J Vacuum Sci Technot 16 (1979) 359 [5] J Bdman. G. Kovacs and A Otto, Surface Sci 92 (1980) 153. [6] J C Tsang, J R Kutley and J A. Bradley, Phys Rev. Leners 43 (1979) 772
i September LPSO
J G. Endria and WE. Spicer, Phys Rev. Letters 24 (1970) 64, Phys. Rev. B4 (1971)4144. A Gubtndo, MR PhiIpoff, D. Heitmann. J-D. SwaIen and R Santa, J. C&em. Phys , to be published. C-Y. Chen, I. Davoir and ELBurstein, to be published J A. Cre@hton, C-G. Bbtcbford and M-G. AJbrecht, J Chem. Sac. Faraday Trans. It 75 (1979) 790. hf. hfoskovits. J. Chem Phys. 69 (1978) 4159; Solid State Commun. 32 (1979) 59. J.A. Cretghton, M.C. Albrecht. R-E. Hester and J.A hfatthew, Cbem. Phys Letters 55 (1978) 55. S. Efnma and H. hfetiu, J. Cl-tern. Phys 70 (1979) 1939; Chem Phys Letters 60 (1978) 59; f. C&m. Phyr 70 (1979) 1602,2297; Surface SC&92 (1980) 417.433; fsraef J. Chem. 17 (1979) 18. E. Burstein, C Y. Chen and S. Luudquist. in. Li#tt scattering m solids, eds. LX_. Birman, H 2. Cummtus and K K. Rebane (Plenum Press, New York, 1979); C Y. Chen, E Burstein and S. Lundqu~~ Sobd Strrte Commun. 32 (1979) 61 J R. Krrtley. S S. Jha and J-C Tsang, to be published. J I Gersten, to be pubbhed A Adams, R.W. RendeB, W-P. West, H P. Broida, PK. Hansma and H. Menu, Phys. Rev. B, to be pubhsbed; L. BNS. private communication. Y-J Chen, W P. Chen and E. Burstem, Pbys. Rev. Letters 36 (1976) 1207, WI_ Chen, G. Ritchre and E. Burstem, Phys Rev. Letters 37 (1976) 993; A. BrtBante, 1. Pockrand, Iif&. Philpott and J.D. Swafeu, Chem. Phys. Letters 57 (1978) 395. A. Otto, 2 Physik 216 (1968) 398; E Kretschmann, Opt. Commun. 26 (1978) 41. .A_k Aldus and D-I, Mrlls, Phys. Rev. Bll(19751 139.2 J.hI. Etson and RN. Ritchie. Phye F&v_ B-+ (L971) 4X29. D-J_.-MIUS,Phgs. Rev. B12 (1975) 4036: A_A hlaradudin and W. Zterau, Phys Rev. 614 (1976) E Kroeger and E. Kretschmann, Phys. Stat Sol. 76b (1976) 5 1.5. [23] W.H. Weber and G-W. Ford, to be pub&bed; G Koneniewshf, T. hfaniv and H. Metiu, fIbem_ Phys, Letters 73 (1980) 212.
305
74. number
2
THE ENHANCEMENT BY SMALL SURFACE
CHEMICAL
PHYSICS
LETTERS
1 September
1980
OF RAMAN AND FLUORESCENT INTENSITY ROUGHNESS. CHANGES UU DIPOLE EMISSION
P K. ARAVIND and Horia METIU Departmetrt of Cltemsny, Santa Barbara Caltfornra
Un~~emty of Caltfornia. 93106, USA
Recewed
III final form 28 Aprd
14 March 1980,
1980
WC propose that surface roughness mfluences the Raman and fluorescence spectra of adsorbed molecules since (1) it allous cwitauon of surface evanescent fields whch mcnzasc the induced dipole moment of the molecule. (2) it allows plasmon radlatlon modulated by the oscdlatmg nuclei, and (3) It mcreases the radration of the mduced dipole. We present numerIcal results for the thud mechanism
l_ Introduction
Recently, it has become apparent that roughness plays an rmportant role in the surface enhanced Raman scattering (SERS). In fact, there is, as yet, no experimental evidence for the existence of SERS on a flat surface (thrs means that the enhancement, If any, is smaller than the three orders of magnitude needed to observe SERS). There are many ways to roughen a surface and study its effects on SERS. The tradttional one [l] is surface anodization; sometimes [2,3] this causes the formatron of salver “spheres” on the surface and these act as “roughness”. The inherent roughness of metal films formed through vapor depositron [4] or even mechamcal roughenmg [S] of the surface, seems to be sufficrent for the appearance of SERS. Some degree of controlled roughness can be achieved by depositing metal films [6] on CaF, layers [7], or by making drffraction grattngs on the metal surface
WI.
Furthermore, both island fiis [9] and colloidal partrcles [lo] should be considered mtrinsrcalty rough systems, even d their geometric surface is perfectly smooth; due to the small size of these systems, no parallel momentum conservation 1s required, just as tn the case of extended rough surfaces. The first attempt to explain the role of roughness III SERS has been made in a pioneering study by
Moskovits [l 11_ He models the rough surface by assunung that it consists of silver islands seated ORthe top of the Bat sdver surface. The molecules are located on the rsland and form - through a mechanism that is not specified in detail - a strongly coupled system. It is then assumed that SERS consists of resonant Raman scattering from the whole (island f molecules) system; the rsland provides the resonating electronic states, which are the islands electromagnetic modes, while the molecules participate by providing vibrational nuclear motion. Moskovits has shown that his model can fit some of the obsenred (2,121 SERS excttation spectra. In passing we note that very similar results can be obtained by startmg from the model proposed by Efrima and Metiu [ 131 for resonant Raman, by replacing the semi-infirute metal with a finite one (e.g., a very small sphere, etc.). The role of the metal islands has also been considered by Burstein et al. [ 141; other mechanisms have been analysed in unpublished work of Ku-tfey et al. [15] and Gersten [16]. Kn ths letter we present some preliminary results which are part of a phenomenologicaf model rrying to descnbe the role of roughness in Raman scattering or fluorescence *_ We distinguish three
* There IS some evidence face roughness [ L7]_
that fluorescence
IS :nfIuenced
by sur-
VoIume 74, number 2
CHEMICAL PHYSICS LETTERS
possible mechanisms. (1) The ncldent field excites a plasmon and the surface ftdd thus created mcreases the total field mcident on the molecule_ Thrs will mcrease the induced molecular dtpole moment, hence produce a larger Raman or fhrorescence intensity. This is similar to the spectroscopic effects [ 1S] tnduced through ATR [ 191, but roughness rather than an ATR prism is used to excrte th2 evanescent field. (2) The radiation from the rough surface is modulated by the vrbrational matron of the nucIei, smce the effectrve nuclear charges interact with the surface fields (modtfied by roughness). Thrs IS the type of process that has also been discussed by K.ttdey et al. [ 1.~1. (3) The reduced molecular dipole moment has a component ascdlatmg at the incident frequency mums the vrbratronal fr2quency. Thrs radiates Raman photons and the properties of thus radiatron are modified by the presence of roughness. To compute these effects we use phenomenologrcal electrodynarmcs hlazwell’s equattons are solved by a method developed by hlaradudin and Mrlls f20] and modified to correspond to the situation of interest here. In the present letter we describe the results obtained for mechamsm (3).
2. Brief description of the model
W2 assume that th2 incrdent hght mduces a dtpoIe moment p(r).whrch IS modulated by the vrbratronal motion. Therefore, we can wnte
i(r, t) Gj(r, =
-1w
mode of
interest and ro(f) IS the mduced dipole when the nuclei are at equdrbrium (Q = 0) The second term m eq. (1) is responsible for F&man scattering, smce it has a component that oscdlates at the frequency 0 s w. - w,, here w. IS the frequency of the Incident hght and o, that of the nuclear vrbrattons. If we denote rl(t)= (ae/aQ)* Q(~)EJL~(w)~-‘~~, the quantrty j(t) f awl(t)/& ISa current which, according to the MaxweLl equations, radiates hght of frequency w. - w, G w. if there are N molecules, located at rcr,cY= 1.. _,N, having the dipole pI__*, the corresponding current density IS 302
w) e--rwf
g1
)I]**
(W)6(f -
f&e-iwr.
(2)
We can compute the Raman inrensny by solving the Maxw2U equattons to fiid the radiatron emitted by j(r, t) m the presence of the rough surface. Fluorescence can be treated m a similar manner; the only difference being the meaning of the magmtude of cl,u(o) m (2).
In order to treat the effect of roughness we assume that the posrtion of the metal surface IS given by z = $(x,_v)- We assume a stochastic model 12 I], for the surface. Thrs accepts the fact that given x and _V(coordinates “parallel’” to the metal) we do not know the posrtlon of z on the surface; it assumes however that we know the probabrhty P(=, x,y) that at the pomt (x,~) z has a given vaIue. We now defme a plane z = 0 such that th2 average height, at every pomt (x,~) IS zero. We can then wnte the dielectric constant of the system in the form E(X,Y, z, w) = e(-J(z; w) + Ae(x,y, z, w). Here eo(z; w) is the dielectrrc constant of a ficrrttous system wrth a flat surface located at z = 0 having the metal dielectric constant l O(w) for z > 0 and the vacuum value for z C 0. E(x,)-‘, z, o) IS rhe true drelectric constant of the system wrth the rough surface, and AE is the difference. We can now consider Ae to be a perturbation and solve Ma\weIl’s equations by ustng the Born approxrmation The result is
I= 1, + 6,6, where C?(r) is the amphtude of the normal
1 September 1980
*DO:i,
(3)
Here so is the dyadrc Green function for the Maxwell equations m the presznce of the j7at surface. Cl o is the asymptottc form of DO I is the total intensity, whrle 1, is the intensity for the case of the flat surface. The purpose of tlus schematic outline IS to point out that there are four quantities of physical mterest, as described below. (I) The quantity that starts the process 1s the current-current comlatron function [i(r; w),j(r’.
w)] * 6(f -r’)Nw*1J+(w)l?
(4)
We assume here mcoherent scattering neglecting thus ah momentum transfer processes that coherence entak
Volume 74,
number
2
CHEMICAL
PHYSICS
(II) The radiation enutted by the molecular currents is propagatedJo_tte rough surface by the pau of Green’s functrons DoDo. These two quanti~es_kve denominators winch become small (hence DoDo becomes very large) when the parallel momentum k, and the frequency w are those corresponding to the surface plasmon. (III) The radiatron emitted by [j,j] and propagated by 6u6u is scattered by the roughness. The strength of this scattering process is given (in the Born approximatron) by the correlation function (Ae(r, w) X Ae(r’, w)>_ If one thmks of surface roughness as a density “fluctuation” rn the surface region, mducing a change AE in the dielectric constant, then the appearance of
(2a)‘6(ks
+k;,)6”a2
exp[-($akd)2]. (9
Here 6’ is the mean square amphtude of the roughness, Q 1s its correlation length and k, and k’, are parallel momenta. The Born approxrmation requires 8 to be small (=30 A) but imposes no limitatrons on CI. The correlation length depends on how the surface has been prepared. (IV) The last elemenlcin Eq_ (3) IS the pair of asymptotic Green’s functions CD, CZIu.They propagate the bght scattered by roughness to the detector. This quantity has no resonant behavior and mfluences the depol arizatx~~
ratio as well 35 the angular
and frequen-
LLTTERS
1 September 1980
.P;
a=ZoH IO
. l t
z
2 2
i
___*___&--~-
_a---
__a-----
_-
_I’
: : :
‘. ‘.
‘. P sa= 150 a ‘. ‘. ___o_____-o--__--o----O-= ‘4_ --o a=!500 a _o.____~___--o___~___o-___-~-.~ _.
01
I
20
I
1
24
I
I
26
w
I
32
,I,
-\ ‘*
36
,
40
Inev-
Fg. 1 fht = IO is the intcnuty radiated by the dipole in the is the diff21-21122 between presence of the flat surface. Irou the emission for rough and that Por a flat surface. The parameters are- the mean amplitude L = 3OA, the angle of detecnon from normal is tJ= 49, the metal is silver. The curves correspond to three different correlaaon lenggs a_ The arrow mdicates the posltion of the unretarded plasmon frequency.
correlation length a is rather small (of the order of 20-50 a). If this requirement is fulfilled the enhancement factor caused by roughening the surface could vary between 3 and 50, depending on the frequency and the value ofa. The enhancement factor is largest at frequencies_close to and below the plasmon frequency 1~~/2r’~, and becomes extremely small for frequencies above it. Sinular features are seen for Cu and Au.
4_ Comments
cy dependence m a “rruld” way.
3. Results In fig. 1 we present the dependence of the quantity j(a) G (I- lu)/lo on frequency, for a variety of correlation lengths. The dipoles are taken to be perpendicular to the tktitrous flat surface. The drstance of the dipole to that surface is 2 A. From eq. (3) we see that f(w) is the intensrty increase caused by surface roughemng, relative to the rntensity present if the surface were flat. It 1s obvious from fig. 1 that mechanism (3) yields a significant rntensrty enhancement only if the
(a) It is customary to atrnbute effects such as those described here to the following mechanism_ The induced dipole being close to and below the plasmon frespatially inhomogeneous field on the metal. Fourier transforming this field over the spatial coordinates expresses it as a superposition of a number of planar waves, with a variety of wavevectors and the same frequency w. If the drpole is very far from the surface the field is described by a single wavevector given by w/c. When the dipole is close, many higher wavevectors are needed to descnbe the field. Due to this for any given frequency there is a Fourier component in the dipole field which can excite the surface plasmon even rf the surface is flat. The excited plasrnon cannot how303
Volume
74. number
2
CHChlICAL
PHYSICS
ever radiate and the plasmon’s
role IS to deplete the induced dipole of energy. Consequently, less energy will be radiated. Roughenmg the surface allows plasmon radtation, hence an Increase of intensuy compared to the flat surface case We fiid that thts sunple prcture IS not quote adequate. The structure of I - IO, gtven by (3) and (4), IS f-IO=
exp[-(iall-,,)‘] J k,,dk,
0
X Q(k,, ~)exp[-irL(k,,,
w. d] .
(6)
Here the term exp[-(iok,,)‘] comes from the roughness correlatton functton WE&), eq (5)__@&, o) and exp[-i$(k,,, w, d)] come from the term DODO_ If we denote by Q,, = Q,,(w) the plasmon dtspersron relation, then Q(k,,. o) has a peak at k E Q,,(o) and then varies hke ki for lugh values of k,, The correlatton length controls, through the gaussran correlatton functton appeanng in eq. (6), wluch values of k,, contrtbute to I - I,. If a is extremely large then k,,,,, = 1/a 1s so small that k, ma, < Q,,(w) and the plasmon IS not exctted For mtermedtate values of a, one can have k Um;LII> Qil(o) and the plasmon contnbutes, but k,‘is not large enough to reach the values for winch d(kU.w)=k$_We find that.mvanably,msuchcases I - I, IS small. It is only when Q is rather small (2050 A) thar iill ma, becomes so larse tint p(k,, , U) a k: for kz, k 1: then the factor k; app2aring II~ the integrand contnbutss heawly to the total ermssion This contribution IS substantrally larger than that of the
plasmon peak We cannot assoctate any elementary eycitation wrth tlus addrtronal contnbution. (b) The distance of the dipoles from the ficttttous flat surface enters m I - IO only through I,!&“, W, d). Wefindthat forsmallk,,.$(k,,,o,d)= l.Whenk,, IS large enough G(k,, w, d) decays hke e\p(-k,,d). Therefore in the product e\p[-(ial;,,)‘] exp(-k,d). as long as 1 /d Z+k, B- 1/a. tile expression for I - IO 1s independent of d. Thus happens as long as d
LETTERS
1 Septembrr 1980
magnitude, on top of the contrtbutton at a flat surface. no special enhancement mechamsm the Raman scattering at a flat surface is enhanced by an estimated factor of 30, as compared to the gas phase. This 1s due to muror effects of the metal and the fact that the surface molecules do not tumble (the so called “mmor enhancement” [ 131). Therefore the total enhancement from mechanism (3), as compared to the gas phase srtuatton, IS of order 300. We esttmate that mechamsm (2) may add at most a factor of 100. We base this guess on the fact that m mechanism (2) one uses roughness to exctte the plasmon evanescent field, we expect this to act as tt does in ATR expertments on flat surfaces. Calculatrons for ATR predict (for a flat surface) an enhancement factor of 200. One never sees experimentally such a large effect, probably because roughness shortens the plasmon mean free path (hence the hfetime) and therefore dumps to some extent the evanescent field [22]. The outcome 1s a factor of about 100. If we assume the same factor for mechanism (2) then the combined enhancement from both mechanisms could be at most 109. (d) We emphastze now the limitations of the pres2nt model. (d,) We have neglected the role of spattal dtspersion (non-local dtelectrtc constant) and of the contmuous change of the dtelectnc constant on the mterface [23]. The mcluston of such effects m the calculauon IS hkely to increase the number of posstble excitations of the metal by the dipole (a great number of 212ctron-hole parr e\cttations are excluded by our present use of opttcal electric constant) Kmematically, roughness would allow some of them to radiate. Whether they chose to do so or they would rather
WA
prefer to de-excite radiationlessly dynamxs and could be established
IS detennlned
by
only by detailed calculations. Therefore, we do not know whether the rnclusron of these effects would enhance or dump the dipole radtatton. (d,) We have used the Born approximation and drd not renormalize the plasmon to include the effect of roughness on it [22] _ In other words, m our calculatton roughness allows the plasmon to radrate, but the radiatmg plasmon is that of the flat surface not that of the rough one. We believe that if such a renormabzatton IS done, the overall effect of roughness will be very sbghtly dimmished (dg) It IS clear that a dtfferent kmd of roughness (e.g., metal particles lymg on top of a flat metal surface) will produce drfferent effects. This is qutte apparent rn
Cl-IEM ICAL. PHYSICS LETTERS
VoIume 74, number 2
Iight emitting tunnehng junctions where the CaF, roughening [I41 gives different spectra than roughenmg by deposition of metal “spheres” OR the surface [15j _To &stingwsh the effects of different kinds of roughness one needs detaded studies of angular disulbutIon, polarization and excitation spectrum.
Acknowledgement We are grareful to NSF (Cm-78-1 618 1 and CHE79-18773) and the Research Camm~ttee at UCSB for supporting this work We wish to thank E. Burstem, H lðer, J. Gersten, J. Tsang, J. Ku-tley, W. Weber, B. Pettinger, M. mpott, P. Felbelman, A. Otto and D. F.fdls for sendmg us their u~pubh~ed work and/or for useful conversations.
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P J Hendra and AJ. McQuillan, Chem. Phys Letters 26 (1974) 163, D-L Jeanmaue and R P. van Duyne. 3 Eiectroanal. Chem. 84 (1977) 1. hf G. Aibrccht and J A Creighton, J. Am. Chem. SOC 99 (1977) 5214. [21 B Petwger and U Wennmg, C&m. Phys Letters 56 (1978) 253, B. Pezuo&ter, U_ We-g and D hl. Kolb. Ber- Bunsen@zs_ Phystk Chem. 82 (1978) 1326 f3]
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