Quantum-size effects in SERS from noble-metal nanoparticles

Microelectronics Journal 36 (2005) 559–563 www.elsevier.com/locate/mejo

Quantum-size effects in SERS from noble-metal nanoparticles Vitaliy N. Pustovit, Tigran V. Shahbazyan* Department of Physics, Jackson State University, Jackson, MS 39217, USA Available online 18 April 2005

Abstract We study the role of a strong electron confinement on the surface-enhanced Raman scattering from molecules adsorbed on small noblemetal nanoparticles. We describe a novel enhancement mechanism, which originates from the different effect that confining potential has on s-band and d-band electrons. We demonstrate that the interplay between finite-size and screening effects in the nanoparticle surface layer leads to an enhancement of the surface plasmon local field acting on a molecule located in a close proximity to the metal surface. Our calculations, based on time-dependent local density approximation, show that the additional enhancement of the Raman signal is especially strong for small nanometer-sized nanoparticles. q 2005 Elsevier Ltd. All rights reserved. Keywords: Nanoparticle; Surface-enhanced Raman scattering; Surface plasmon

1. Introduction An recent interest in single-molecule surface-enhanced Raman scattering (SERS) stems from the discovery of enormously high (up to 1015) enhancement of Raman spectra from certain (e.g. Rhodamine 6G) molecules fixed at the nanoparticle surfaces in gold and silver colloids [1–3]. Major SERS mechanisms include electromagnetic (EM) enhancement by surface plasmon (SP) local field near the metal surface [4–8] and chemical enhancement due to dynamical charge transfer between a nanoparticle and a molecule [9–12]. Although the origin of this phenomenon has not completely been elucidated so far, the EM enhancement was demonstrated to play the dominant role, especially in dimer systems when the molecule is located in the gap between two closely spaced nanoparticles [10–13]. An accurate determination of the SERS signal intensity for molecules located in a close proximity to the nanoparticle surface is a non-trivial issue. The classical approach, used in EM enhancement calculations [4–8], is adequate when nanoparticle–molecule or interparticle distances are not very small. For small distances, the quantummechanical effects in the electron density distribution can no longer be neglected. These effects are especially * Corresponding author. E-mail address: [email protected] (T.V. Shahbazyan).

0026-2692/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2005.02.069

important in noble-metal particles where the SP local field is strongly affected by highly-polarizable (core) d-electrons. In the bulk part of the nanoparticle, the (conduction) s-electrons are strongly screened by the localized d-electrons. However, near the nanoparticle boundary, the two electron species have different density profiles. Namely, delocalized s-electrons spill over the classical boundary [14], thus increasing the effective nanoparticle radius, while d-electron density profile mostly retains its classical shape. The incomplete embedding of the conduction electrons in the core electron background [15–18] leads to a reduced screening of the s-electron Coulomb potential in the nanoparticle surface layer. The latter has recently been observed as an enhancement of the electron–electron scattering rate in silver nanoparticles [19,20]. Here we study the role of electron confinement on SERS from molecule adsorbed on the surface of small Ag particles. To this end, we develop a microscopic theory for SERS in noble-metal particle, based on the quantum extension of two-region model [15–17,21], which describes the role of the surface-layer phenomenologically while treating conduction electrons quantum-mechanically within time-dependent density functional theory. We find that the reduction of screening near the surface leads to an additional enhancement of the Raman signal from a molecule located in a close proximity to the nanoparticle. In particular, we address the dependence of SERS on nanoparticle size and show that the interplay of finite-size and screening effects is especially strong for small nanometer-sized particles.

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we obtain

2. Quantum two-region model We consider SERS from a molecule adsorbed on the surface of Ag spherical particle with radius R. For R/l, the frequency-dependent potential is determined from Poisson equation Fðu; rÞ Z f0 ðrÞ C e

2

ð

dNðu; r 0 Þ d r ; jr K r 0 j 3 0

(1)

where f0(r)ZKeEi$r is potential of the incident light with electric field amplitude EiZEiz along the z-axis, and dN(u,r) is the induced density (hereafter we suppress frequency dependence). There are four contributions to dN(r) originating from valence s-electrons, dNs(r), core d-electrons, dNd(r), dielectric medium, dNm(r), and the molecule, dN0(r). The density profile of delocalized s-electrons is not fully inbedded in the background of localized d-electrons but extends over that of localized ˚ [15–17]. Due to localized nature d-electrons by Dw1–3 A of the d-electron wave-functions, we can adopt a phenomenological description by assuming a uniform bulk-like d-electron ground-state density nd in the region confined by Rd!R with a sharp step-like edge. Then the induced charge density, edNd(r)ZKV$Pd(r), is expressed via electric polarization vector vanishing outside of the region r!Rd Pd ðrÞ Z

ed K 1 qðRd K rÞEðrÞ 4p

ZK

ed K 1 qðRd K rÞVFðrÞ; 4pe

(2)

where ed(u) is the core dielectric function which can be taken from experiment. Similarly, dielectric medium contribution, which is nonzero for rOR, is given by edNm ðrÞ Z KVPm ðrÞ; ePm ðrÞ Z K

em K 1 qðr K RÞVFðrÞ; 4p

(3)

where em is medium dielectric constant. We represent the molecule by a point dipole with dipole moment p0Za0E, where a0 is the molecule polarizability tensor, so that edN0 ðrÞ Z KVP0 ðrÞ; eP0 ðrÞ Z dðr K r0 Þp0 Z Kdðr K r0 Þa0 VFðr0 Þ;

(4)

where r0 is the vector pointing at the molecule location (we chose origin at the sphere center). Using Eqs. (2)–(4), the potential F(r) in Eq. (1) can be expressed in terms of only induced s-electron density, dNs. Substituting the above expressions into dNZdNsCdNdC dNmCdN0 in the rhs of Eq. (1) and integrating by parts,

eðrÞFðrÞ Z f0 ðrÞ C e C

ed K 1 4p

C

em K 1 4p

K V0

2

ð

ð ð

d3 r 0

dNs ðr 0 Þ jr K r 0 j

d3 r 0 V0

1 V 0 qðRd K rÞFðr 0 Þ jr K r 0 j

d3 r 0 V 0

1 V 0 qðr K RÞFðr 0 Þ jr K r 0 j

1 a V Fðr0 Þ; jr K r0 j 0 0

ð5Þ

where e(r)Zed, 1, and em in the intervals r!Rd, Rd!r!R, and rOR, respectively. Since the source term has the form f(r0)Zf(r0) cos qZKeEir cos q, we expand F and dNs in terms of spherical harmonics and, keeping only the dipole term (LZ1), obtain ð 2 eðrÞFðrÞ Z f0 ðrÞ C e2 dr 0 r 0 Bðr; r 0 ÞdNs ðr 0 Þ

K

ed K 1 2 Rd vRd Bðr; Rd ÞFðRd Þ 4p

C

em K 1 2 R vR Bðr; RÞFðRÞ 4p

K V0 ½Bðr; r0 Þcos q0 a0 V0 ½Fðr0 Þcos q0 ;

(6)

where q0 is the angle between molecule position and incident light direction (z-axis), and   4p r 0 r 0 0 0 Bðr$r Þ Z qðr K r Þ C 02 qðr K rÞ (7) 3 r2 r is the dipole term of the radial component of the Coulomb potential. The second terms in rhs of Eq. (6) is the s-electrons contribution to total induced potential, while the third and fourth terms originate from the scattering due to change of dielectric function at rZRd and rZR, respectively. The last term represents the potential of the molecular dipole. The latter depends on the molecule orientation with respect to the nanoparticle surface. In the following we assume averaging over random orientations and replace the polarizability tensor by isotropic a0. The values of F at the boundaries and at the molecule position can be found by setting rZRd, R, r0 in Eq. (6). In doing so, the total potential is expressed in terms of only s-electron induced density dNs. Within TDLDA formalism, the latter can be related back to the potential via Ð Q dNs ðrÞ Z d3 r 0 s ðr; r 0 Þ½Fðr 0 Þ C Vx0 ½nðr 0 ÞdNs ðr 0 Þ; (8) Q where s (r, r 0 ) is the polarization operator for noniteracting s-electrons, Vx0 ½nðr 0 Þ is the (functional) derivative of the exchange-correlation potential (in the endependent-particle approximation) and n(r) is electron ground-state density. The latter is calculated in a standard way using Kohn–Sham

V.N. Pustovit, T.V. Shahbazyan / Microelectronics Journal 36 (2005) 559–563

equations for jelium model, and then is used as imput in the evaluation of Ps and Vx0 . Eqs. (6) and (8) determine the selfconsistent potential in the presence of molecule, nanoparticle, and dielectric medium.

where a Z Rd =R;

ld Z

e K1 lm Z m ; 2em C 1 3. Calculation of Raman signal


K ed K 1 bðr=Rd ÞfðRd Þ eðrÞfðrÞ Z fðrÞ 3 em K 1 bðr=RÞfðRÞ; 3 Ð

0 02

0

(14) 3

h Z 1 K 2a ld lm

(9)

0


Z f0 ðrÞ C e dr r Bðr; r Þdns ðr Þ; fðrÞ

(10)

and bðr=RÞ Z ð3=4pÞR2 vR Bðr; RÞ is given by bðxÞ Z xK2 qðx K 1Þ K 2xqð1 K xÞ:

f Z 40 C d40 C d4s ;

(15)

where 40 Z f0 =eðrÞ Z KeEi r=eðrÞ

d40 ðrÞ Z

1 ½Kbðr=Rd Þf0 ðRd Þld ð1 K 2lm Þ=h eðrÞ C bðr=RÞf0 ðRÞlm ð1 K a3 ld Þ=h;

(16)

and Ð 2 d4s ðrÞ Z dr 0 r 0 Aðr; r 0 Þdns ðr 0 Þ;

(17)

with Aðr; r 0 Þ Z

e2 ½Bðr; r 0 Þ K bðr=Rd Þ½BðRd ; r 0 Þ eðrÞ

K 2alm BðR; r 0 Þld =h

where 2

ed K 1 ; ed C 2

It is convenient to separate out dns-independent contribution by writing

In the conventional SERS picture, the enhancement of Raman signal from the molecule comes from two sources: far-field of the radiating dipole of the molecule in the local nanopartricle field, and the secondary scattered field of this dipole by the nanoparticle. Accordingly, we present total potential as a sum FZfCfR, where fR is the potential of the radiating dipole. Since the molecular polarizability is very small, in the following we will restrict ourselves by the lowest order in a0, i.e. f is the potential in the absence of molecule and fR determines the Raman signal in the first order in a0. Inclusion of higher orders leads to the renormalization of molecular and nanoparticle polarizabilities due to image charges; these effects are not considered here. In the same manner, the induced s-electron density can be decomposed into two contributions, dNsZdnCdnR, originating from the electric field of incident light and that of the radiating dipole. Keeping only zero-order terms in Eq. (6), we have

C

561

(11)

C bðr=RÞ½BðR; r 0 Þ K a2 ld BðRd ; r 0 Þlm =h:

ð18Þ

Note that dfs(r) as well as the total potential f(r) are continuous at rZRd, R. Turning to Eq. (8), we use decompositions FZfCfR and dNs Z dns C dnRs to obtain decoupled equations for quantities of zero and first orders in a0. Then, expanding both parts in spherical harmonics and keeping only the dipole (LZ1) terms, we obtain Ð 2 dns ðrÞ Z dr 0 r 0 Ps ðr; r 0 Þ½fðr 0 Þ C Vx0 ðr 0 Þdns ðr 0 Þ:

(19)

The boundary values of f can be obtained by matching f(r) at rZRd, R,

Using Eqs. (15) and (17) then leads to a closed equation for dns,


dÞ ð3d C 2ÞfðRd Þ C 2að3m K 1ÞfðRÞ Z 3fðR

Ð 2 dns ðrÞ Z dr 0 r 0 Ps ðr; r 0 Þ½40 ðr 0 Þ C d40 ðr 0 Þ hÐ Ð 2 2 C dr 0 r 0 Ps ðr; r 0 Þ dr 00 r 00 Aðr 0 ; r 00 Þdns ðr 00 Þ


ð3d K 1Þa2 fðRd Þ C ð23m C 1ÞfðRÞ Z 3fðRÞ

(12)

where aZRd/R. Substituting f(Rd) and f(R) back into Eq. (9), we arrive at
d Þ K 2alm fðRÞ

K bðr=Rd Þ ld ½fðR eðrÞfðrÞ Z fðrÞ h C bðr=RÞ

lm

d Þ; ½fðRÞ K a2 ld fðR h

(13)

i CVx0 ðr 0 Þdns ðr 0 Þ ;

ð20Þ

with A(r, r 0 ) given by Eq. (18). The effect of d-electrons and dielectric medium is thus encoded in the functions A(r, r 0 ) and df0(r), which reduce to B(r, r 0 ) and 0, respectively, for edZemZ1.

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V.N. Pustovit, T.V. Shahbazyan / Microelectronics Journal 36 (2005) 559–563

Turning to the first order in a0, the equation for fR takes the form

the far-field asymptotics of fR(r). From Eqs. (16)–(18), we find for r[R

e K1 R bðr=Rd ÞfR ðRd Þ eðrÞfR ðrÞ Z f
ðrÞ K d 3

d40 ðrÞ Z

C

em K 1 bðr=RÞfR ðRÞ; 3

(21)

with Ð 2 R f
ðrÞ Z fR0 ðrÞ C e2 dr 0 r 0 Bðr; r 0 ÞdnRs ðr 0 Þ;

(22)

where fR0 ðrÞ Z Ka0 V0 ½Bðr; r0 Þcos q0 V0 ½fðr0 Þcos q0 ;

(23)

is the potential of the molecular dipole in the presence of local field and dnRs ðrÞ is the induced charge of s-electrons due to molecular potential. The frequency dependence of the Raman field fR(us, r) is determined by the Stokes shift usZu-u0, where u0 is the vibrational frequency of the molecule as determined by a0 The last two terms in Eq. (21) describe potential of molecular dipole scattered from the nanoparticle boundaries at Rd and R. For simplicity, we only consider the case when the molecule is located at the z-axis (q0Z0) so that molecular dipole potential is given by 4pa0 vfðr0 ; uÞ bðr=r0 Þ vr0 3r02   r3 Z KcðuÞeEi rqðr0 K rÞ K 02 qðr K r0 Þ ; 2r

fR0 ðrÞ Z K

(24)

(25)

Marching fR at rZRd and rZRd, we obtain f

R

Z 4R0

C d4R0

C d4Rs

Z cðuÞ½4~ 0 C d40 C d4~ s ;

(26)

(27)

df0 is given by Eq. (16) with us instead of u and Ð 2 (28) d4~ s ðrÞ Z dr 0 r 0 Aðr; r 0 Þdn~ s ðr 0 Þ; where A(r, r 0 ) is given by Eq. (18), with us instead of u, and dn~ s ðrÞZ dnRs ðrÞ=cðuÞ satisfies Eq. (20) but with 4~ 0 ðrÞ instead of f0(r) (and us instead of u). In the following we consider the case when the molecule is located at the distance of several angstroms from the nanoparticle classical boundary, so the overlap between the molecular orbitals and the s-electron wave function is small. Then we have Ð 0 02 Q Ð 2 0 ~ 0 ðr 0 Þ x dr 0 r 0 Ps ðr; r 0 Þ40 ðr 0 Þ; dr r (29) s ðr; r Þ4 leading to dn~ s ðrÞ xdns ðrÞ and, correspondingly, d4~ s ðrÞ xd4s ðrÞ. The Raman signal is determined by

(30)

ad ðus Þ Z R3 ½1Kð1 Clm Þð1 Ka3 ld Þ=h; ð N 4p 3 dr 0 r 0 dns ðr 0 ÞK½1Kð1 Clm Þð1 Kld Þ=h as ðus Þ Z 3eE0 0 ðR 3 ! dr 0 r 0 dns ðr 0 ÞK½1Kð1 Clm Þð1 Ka3 ld Þ=hR3 ð0N ! dr 0 dns ðr 0 ÞC½ð1Clm Þld =h R  ðR 3 ! dr 0 ðr 0 KR3d Þdns ðr 0 Þ ; ð31Þ RD

and using Eq. (27) we obtain for the far field fR ðrÞ Z

eEi r03 c½1C2ðad Cas Þ=r03 : 2em r 2

(32)

Turning to c, we note that for small moleculenanopatricle overlap, the local potential at the molecule location can be evaluated using the far-field expressions Eq. (30). We then obtain (33)

and, substituting into Eq. (32), we finally arrive at 4pa0 eEi ½1 C2gðuÞ½1C2gðus Þ; 3em r 2

(34)

with gZ

where   eEi r03 rqðr0 K rÞ K 2 qðr K r0 Þ ; 4~ 0 ðrÞ Z K eðrÞ 2r

eEi as ; em r 2

with

fR ðrÞ ZK 8pa0 vfðr0 Þ 3eEi r03 vr0

d4s ðrÞ Z

vf0 eE ZK i ½1 C2ðad Cas Þ=r03 ; vr0 em

where cðuÞ Z

eEi ad ; em r 2

a ; a Z ad Cas : r03

(35)

The above expression generalizes the well-known classical result [4,5] to the case of noble-metal particle with different distributions of d-electron and s-electron densities. The surface-enhanced Raman field retains the same functional dependence on the nanoparticle polarizability, however the latter contains all the information about the surface layer effect. Finally, the enhancement factor is given by the ratio of Raman to incident field intensities, Aðu; us Þ Z j1 C 2gðuÞ C 2gðus Þ C 4gðuÞgðus Þj2 :

(36)

4. Numerical results For large nanoparticle with Rw100 nm, the classical EM theory provides an enhancement of the Raman signal as large as 106K107 [6]. In reality, the EM enhancement is inhibited by various factors. In noble-metal particles, the interband transition between d-electron and s-electron bands

V.N. Pustovit, T.V. Shahbazyan / Microelectronics Journal 36 (2005) 559–563

Fig. 1. Calculated nanoparticle polarizability for different surface layer thicknesses.

563

with and without surface layer. In the presence of the surface layer, the SP energy experiences a blueshift [15,16] due an effective decrease in the d-electron dielectric function in the nanoparticle. At the same time, an increase in the peak amplitude, which accompanies the blueshift, indicates a stronger local field at resonance energy acting on a molecule in a close proximity to nanoparticle surface. In Fig. 2 we show the results of our numerical calculations of SERS with and without surface layer thicknesses, D for different nanoparticle sizes. Although the overall magnitude of the enhancemet increases with D, a more important effect is its size-dependence. For finite D, the enhancement factor descreases more slowly that in the absence of the surface layer: as nanoparticle size decreases, the signals strength ratio increases. The reason is that, as the nanoparticle becomes smaller, the fraction of the surface layer increases, and so does the contribution of the unscreened local field into SERS.

Acknowledgements This work was supported by NSF under grants DMR0305557 and NUE-0407108, by NIH under grant 5 SO6 GM008047-31, and by ARL under grant DAAD19-01-20014.

References

Fig. 2. Size-dependence of enhancement factor for different surface layer thicknesses.

reduce the SP oscillator strength leading to a weakening of the local fields. For nanoparticle radius below 15 nm, finitesize effects become important. The SP resonance damping comes from the electron scattering at the surface leading to the size-dependent SP resonance width gs xvF =R. At the resonance frequency, the size-dependence of SERS is quite strong. Indeed, if molecular vibrational frequencies are smaller that the SP width, the enhancement factor decreases as AfR4 for small nanoparticles, resulting in several orders of magnitude drop in the Raman signal. Our main observation is that, in small nanoparticle, the local field enhancement due to reduced screening in the surface layer can provide an additional enhancement of the Raman signal. Although the thickness of the surface layer (0.1-0.3 nm) is small as compared to oveall nanoparticle size [15–17], such an enhancement can be condiderable for a molecule located in a close proximity to the surface. In Fig. 1 we show the calculated polarizability

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