Plasmonic Enhancement of Raman Optical Activity in Molecules near

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J. Phys. Chem. C 2010, 114, 7390–7400

Plasmonic Enhancement of Raman Optical Activity in Molecules near Metal Nanoshells: Theoretical Comparison of Circular Polarization Methods† Richard Lombardini,‡,§,| Ramiro Acevedo,‡,§,|,⊥ Naomi J. Halas,‡,§,|,# and Bruce R. Johnson*,‡,§,| Department of Chemistry, Rice Quantum Institute, Laboratory for NanoPhotonics, Applied Physics Program, and Department of Electrical and Computer Engineering, MS 60, Rice UniVersity, Houston, Texas 77005 ReceiVed: October 8, 2009; ReVised Manuscript ReceiVed: December 11, 2009

Surface-enhanced Raman optical activity (SEROA) in molecules near a spherical metal nanoshell is examined for different experimental polarization schemes. This is accomplished within extended Mie theory, including both the nanoparticle surface plasmon modes and the radiating molecular multipole fields. Analytical simplification of the scattering expansion is accomplished through judicious use of the spherical harmonic addition theorem. Special attention is paid to the dependence of backscatter circular polarization difference signals on molecular direction from the spherical nanoparticle and to the conditions that ensure the differences occur only for molecules of chiral character. Dual circular polarization strategies are determined to have special advantages in these circumstances, and the corresponding excitation profiles for a simple chiroptical model are analyzed in detail to suggest preferred excitation wavelengths. I. Introduction It was recognized during the first age of surface-enhanced Raman scattering1 (SERS) that surface plasmons, that is, collective oscillations of conduction electrons, account for much of the ability of noble metal substrates to strongly enhance molecular Raman scattering intensities. Strong local electromagnetic (EM) fields are generated by interaction of light and the surface plasmon modes of small metal particles or surface features, accounting for a major part of the molecular scattering enhancement.2 Additional nonelectromagnetic enhancement contributions are also important for adsorbed analytes1,3 and are generally molecule-specific in character. There is great interest in developing chemical and biomolecular sensors using the SERS effect,4,5 and this has brought attention to the possibility of developing surface-enhanced Raman optical activity6-19 (SEROA) as a more powerful version of ROA, a vibrationally specific spectroscopy sensitive to biomolecule chirality and secondary/tertiary protein folding structure.20 We have recently developed a SEROA formalism for molecules near spherical metal nanoparticles21 by combining elements of the classicalfields SERS model of Kerker et al.,22 and the original far-fromresonance ROA analysis of Barron and Buckingham.23 This study identified a first set of detector/polarization choices leading to chiral-molecule selectivity as obtained in unenhanced ROA, and excitation spectra across plasmon resonances were calculated for Ag and Au nanoshells using a one-electron chiroptical model. There are both disadvantages and advantages of restricting the model to unbound analyte molecules. While SEROA may be carried out in aqueous solution, the dominant interest in initial experiments so far has been application to adsorbates,9,10,13,14,16,19 so common in SERS. Indeed, there is a rich set of challenges †

Part of the “Martin Moskovits Festschrift”. * Corresponding author. Department of Chemistry. § Rice Quantum Institute. | Laboratory for NanoPhotonics. ⊥ Applied Physics Program. # Department of Electrical and Computer Engineering. ‡

in trying to understand the impacts of the combined electromagnetic and chemical effects on SEROA, as well as various bonding geometries, surface morphologies, and so on.17,18 Many of these difficulties are set aside by stipulating that the molecule is unattached and subject only to the local electromagnetic field enhancements. Theory can then be carried out much more precisely. Nevertheless, this has not happened before, and it turns out that our understanding of the effects of EM enhancement alone needs upgrading. For example, the local EM field around even a spherically symmetric plasmonic substrate generally gives rise to differential scattering contributions that can arise from either chiral or achiral molecules, while ordinary ROA is selective for chirality. Such issues will also be of concern for adsorbed molecules, of course, but are better first addressed in the simpler scenario. Metal nanoshells with dielectric cores are distinguished by the ability to tune the dipole surface plasmon resonance across the visible spectrum.24 They have been used in many SERS applications and also in a backscatter SEROA instrument built at Rice.19 They have well-known Mie coefficients, that is, expansion coefficients for vector spherical harmonic solution of the scattering of an EM plane wave from the nanoshell.25 Similarly, transition multipole fields (electric dipole, magnetic dipole, electric quadrupole) of the molecule outside the particle lead to scattered waves that can also be solved using vector spherical harmonic expansions. Together, both absorption and emission steps of the ROA process are enhanced by nanoparticle-scattered waves and lead to SEROA enhancement curves21 in a manner similar to that of Kerker et al.,22 for SERS enhancement near metal nanospheres (electric dipole only). Moreover, these models allow for specific consideration of incident and scattered polarizations and detector direction. The full scattering solutions contain multidimensional sums over nanoparticle plasmon multipole modes that become laborious at higher order. These are simplified to one-dimensional sums involving a few angular derivatives through appeal to the addition formula for spherical harmonics. These results are useful both in programming the numerical evaluation of the total fields and in the derivation of analytical formulas through

10.1021/jp909654n  2010 American Chemical Society Published on Web 01/22/2010

Plasmonic Enhancement of Raman Optical Activity

J. Phys. Chem. C, Vol. 114, No. 16, 2010 7391 scattered waves can be obtained by expansion of all fields in vector spherical harmonics and matching of coefficients at material interfaces so as to satisfy the EM boundary conditions. The full solution including the higher multipoles in the scattering was put in the form21

Erad,t ) T(E1) · d + T(M1) · m + T(E2):Q

Figure 1. Schematic of SEROA experiments considered here. Chiral molecules in the electromagnetic enhancement range of a metal nanoshell with a dielectric core are probed, modulating between circular polarization senses for incident and scattered light: red ) right-handed, blue ) left-handed.

symbolic algebra for inspection of different polarization schemes. This allows us to understand numerical results derived by Acevedo et al., for backscatter SEROA using incident circular polarization26,27 (ICP) and particularly scattered circular polarization28,29 (SCP), the earlier polarization setups used in ROA. More importantly, we have investigated dual circular polarization30-32 (DCP) arrangements, which were introduced later in ROA and which appear to have distinct advantages in the version of SEROA examined here. Specifically, averaging over molecular rotational coordinates alone guarantees chiral selectivity for DCP, whereas additional ensemble averaging was needed for ICP and SCP.21 This finding is potentially important for carrying SEROA to lower analyte concentrations. Even though the laser frequency may be far-from-resonance with molecular transitions, it is likely to be very close to resonance with surface plasmon modes. Ag and Au nanoshell excitation profiles are investigated throughout the visible spectrum for two different forms of DCP polarization experiments. Excitation of both dipole and quadrupole plasmon modes are investigated, and it is delineated how total and differential scattering depend in those cases on the polar angle of the molecule relative to the incident beam direction. The results obtained indicate potential advantages to making dipoleplasmon-region scattering measurements somewhat off-resonance and to lower energy.

where d, m, and Q are induced electric dipole, magnetic dipole, and electric quadrupole moments, respectively. They depend on the molecular hyperpolarizability tensors (vide infra) and the incident-plus-scattered field quantities Et, Bt, and 3Et at the position of the molecule in the near zone. The three T tensors are of size 3×3, 3×3, and 3×3×3, respectively (the tensor notation : is used for contraction on two indices). Their elements contain the information on the solution of the scattering problems, including complicated sums of products of vector spherical harmonics in the molecular position r′ and the direction of observation r. They also depend on Mie scattering coefficients at the Raman-scattered frequency ω (as would be calculated if the incident plane wave were at that frequency). The detailed expansions obtained using the dyadic EM Green tensor33,34 may be found in the earlier paper. Here a strong simplification is used. The nested summations involved can become computationally expensive for high accuracy, as remarked even in the SERS case by Kerker, et al.22 There are three indices, l, m, and σ. The nanoparticle multipole index l (1 for dipole, 2 for quadrupole, etc.) relates to spherical Bessel functions jl(kr), outgoing Hankel functions h(1) l (kr), and their derivatives (k is the wavevector in the external medium). Both l and the azimuthal index m are used for the associated Legendre polynomials Pml (cos θ) and their derivatives, while σ (values e or o) are related to the even and odd azimuthal trigonometric functions cos(mφ) and sin(mφ). We have been able to perform all sums over m and σ analytically. If γ is the angle between r and r′

cos γ ) rˆ · rˆ′ ) cos θ cos θ′ + sin θ sin θ′ cos[m(φ φ′)] (2) we can use the spherical harmonic addition theorem in the form35

II. SEROA Formalism The earlier investigation developed the classical-fields approach to scattering of light by a molecule near a metal nanoshell or nanosphere.21 As indicated in Figure 1, the current work additionally studies the situation with SEROA experiments in which circular polarization is used for both incident light excitation and scattered light analysis. The overall picture is the following. An incident plane wave field E of frequency ω0 scatters from the nanoparticle, producing another field Es. The molecule is thus exposed to a total local field Et ) E + Es, which induces oscillation of the molecular multipoles at the Raman-shifted frequency ω. The resulting electric dipole (E1), magnetic dipole (M1), and electric quadrupole (E2) fields E(E1) , E(M1) , E(E2), respectively, each also scatter off the nanoparticle. Thus, the total radiative field reaching the detector is (E1) + E(M1) + E(E2) Erad,t ) E(E1) ) E(E1) + Es(E1), and t t t , where Et so on. Each of the electric fields naturally has an accompanying magnetic field. The treatment is a higher-multipole extension of the wellknown E1 SERS model of Kerker et al.22 that allowed for enhancement of both absorbed and emitted fields. As there, the

(1)

l

Pl(cos γ) )

Pm(cos θ)Plm(cos θ′) ∑ (2 - δm,0) (l(l +- m)! m)! l

m)0

cos[m(φ - φ′)] (3) to express all the m and σ summations in the T tensors in terms of low-order derivatives of Pl (cos γ). For the E1 case, one ultimately ends up with a single l-sum of the form ∞

Et(E1)(r) )

2l + 1 ˆ (E1) ik3 [Rtl Pl(cos γ)] · d πεε0 l)1 4

∑

(4)

ˆ (E1) is a 3×3 tensor operator acting on Pl (cos γ). The where R tl quantity ε0 is the electrical permittivity of free space and ε is the relative permittivity. In spherical coordinates, the rows of ˆ (E1) are labeled by r, θ, and φ and the columns by r′, θ′, and R tl φ′. The tensor is defined component-by-component in Table 1, utilizing the following definitions

hl(1)(kr)ula(kr′) ∂ ∂ ηlh(kr)wlb(kr′) ∂ ∂ + l(l + 1) ∂θ ∂θ′ l(l + 1)sin θsin θ′ ∂φ ∂φ′

Lombardini et al.

ula(kr′) ) jl(kr′) + al(ω)hl(1)(kr′)

(5)

ulb(kr′) ) jl(kr′) + bl(ω)hl(1)(kr′)

(6)

wla(kr′) ) ηlj(kr′) + al(ω)ηlh(kr′)

(7)

wlb(kr′) ) ηlj(kr′) + bl(ω)ηlh(kr′)

(8)

ηlj(x) )

1d xj (x), x dx l

ηlh(x) )

1 d (1) xh (x) x dx l

(9)

The coefficients al(ω) and bl(ω) are the Raman-shifted Mie scattering coefficients, well-known for either nanospheres36-38 or nanoshells.25 Equation 4 thus provides an explicit form for T(E1) ∞

ulb(kr′) 1 ∂ kr′ sin θ ∂φ

hl(1)(kr)ula(kr′) ∂ ∂ ηlh(kr)wlb(kr′) ∂ ∂ + l(l + 1)sin θ′ ∂θ ∂φ′ l(l + 1)sin θ ∂θ′ ∂φ

T(E1) )

ηlh(kr)

l(l + 1)

hl(1)(kr) ulb(kr′) kr kr′ u (kr′) ∂ lb ηlh(kr) kr′ ∂θ

hl(1)(kr) ∂ wlb(kr′) kr ∂θ′ hl(1)(kr)ula(kr′) ∂ ∂ ηlh(kr)wlb(kr′) ∂ ∂ + l(l + 1)sin θ sin θ′ ∂φ ∂φ′ l(l + 1) ∂θ ∂θ′

hl(1)(kr) wlb(kr′) ∂ kr sin θ′ ∂φ′ hl(1)(kr)ula(kr′) ∂ ∂ ηlh(kr)wlb(kr′) ∂ ∂ + l(l + 1)sin θ ∂θ′ ∂φ l(l + 1)sin θ′ ∂θ ∂φ′

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ˆ tl(E1) Tensor Operator in Spherical Coordinates for r > r′ TABLE 1: Components of the R

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2l + 1 ˆ (E1) ik3 [Rtl Pl(cos γ)] πεε0 l)1 4

∑

(10)

that is efficient for general geometries with r > r′. For r < r′, ˆ (E1) operator can the corresponding (R, β) components of each R tl be obtained from the (β, R) components in Table 1 by interchanging the coordinates {r, θ, φ} and {r′, θ′, φ′}. For example, using the (r,φ′) element from the table, we find for r < r′ that

(E1) Rˆtlφr′ ) wlb(kr)

hl(1)(kr′) 1 ∂ kr′ sin θ ∂φ

(11)

This symmetry arises from symmetry of the dyadic Green tensor, which also leads to various EM reciprocity relations of use near small particles.39 For the M1 case, the reduced expression for the electric field is

Et(M1)(r) ) -

k3cµµ0 ∞ 2l + 1 (M1) ˆ tl Pl(cos γ)] · m [R πn l)1 4 (12)

∑

where c is the speed of light, µ0 is the free-space magnetic permeability, µ is the relative permeability (usually equal to one for these applications), and n is the index of refraction. The tensor operator components for r > r′ are listed in Table 2. Equation 12 provides a compact formula for the tensor T(M1) analogously to the E1 case shown in eq 10. For the E2 case, there are more components and more complexity. Nevertheless, from eqs 28-36 of Acevedo et al., the total E2 field can be expressed as ∞

Et(E2)(r) )

2l + 1 ik3 [∇′ · Q · Rˆtl(E1)T]TPl(cos γ) 6πεε0 l)1 4

∑ ∞

)

2l + 1 ˆ (E1) ik3 r Rtl Pl(cos γ)∇′:Q 6πεε0 l)1 4

∑

(13)

hl(1)(kr)wla(kr′) ∂ ∂ ηlh(kr)ulb(kr′) ∂ ∂ l(l + 1) sin θ′ ∂θ ∂φ′ l(l + 1) sin θ ∂θ′ ∂φ ηlh(kr)ulb(kr′) hl(1)(kr)wla(kr′) ∂ ∂ ∂ ∂ l(l + 1) sin θ sin θ′ ∂φ ∂φ′ l(l + 1) ∂θ ∂θ′ -hl(1)(kr)

ula(kr′) 1 ∂ kr′ sin θ ∂φ hl(1)(kr)

0

ula(kr′) ∂ kr′ ∂θ

hl(1)(kr) ∂ ulb(kr′) kr ∂θ′ hl(1)(kr)wla(kr′) ηlh(kr)ulb(kr′) ∂ ∂ ∂ ∂ l(l + 1) sin θ sin θ′ ∂φ ∂φ′ l(l + 1) ∂θ ∂θ′ ˆ tl(M1) Tensor Operator in Spherical Coordinates for r > r′ TABLE 2: Components of the R

-

hl(1)(kr) ulb(kr′) ∂ kr sin θ′ ∂φ′ hl(1)(kr)wla(kr′) ∂ ∂ ηlh(kr)ulb(kr′) ∂ ∂ + l(l + 1) sin θ ∂θ′ ∂φ l(l + 1) sin θ′ ∂θ ∂φ′

Plasmonic Enhancement of Raman Optical Activity

J. Phys. Chem. C, Vol. 114, No. 16, 2010 7393 where the multiple transposes on the first line serve only to correctly contract ∇′ · Q with the indices of the primed variables. In the equivalent last line, the gradient with the left arrow acts to the left to avoid use of transposes. (Symmetry of the Q matrix is also used here.) This provides a compact form for the T(E2) tensor using the E1 tensor and the gradient operator that we use in calculations. All field calculations in this paper use these reduceddimension expansions, which simplify computation sufficiently that coding can be executed in Mathematica. The magnetic field components may be obtained directly from the electric field components using Maxwell’s curl equations. The above results all apply to r > r′, though symmetry such as in eq 11 can be used to derive similar results for r < r′ if needed in other applications. III. Polarizations in SEROA A. Polarization-Specific Intensity Formulas. Intensity at the detector is evaluated using cycle-averaged components of the Poynting vector in the far zone.23 The influence of the nearzone scattering remains in the Mie coefficients multiplying radial scattering functions in kr′. The transformation to the far zone is achieved by using the asymptotic forms of the radial functions in kr.22 For large observation distance r, we have that Erad,t, Brad,t, and the propagation vector are all at right angles, and the intensity can be expressed in terms of Erad,t alone. Using the expressions for the induced molecular multipoles in terms of the total initial fields, the radiated intensity along direction λ (to the first couple orders in the multipole expansion) is proportional to21

|Erad,t,λ | 2 ≈

∑

(E1)* (E1) E*t,τTλ,σ Tλ,ν Et,ν〈Rσ,τRµ,ν〉 +

σ,τ,µ,ν

2Re{

∑

(E1)* E*t,τTλ,σ

σ,τ,µ,ν

1 (E1) T 3 λ,µ

∑ ∂r∂ ν′ Et,F〈Rσ,τAµ,ν,F〉 + F

]

∑

(E2) Tλ,µ,ν Et,F〈Rσ,τAF,µ,ν〉

∑

(E1)* (E1) ′ 〉E*t,τTλ,σ [Tλ,µ Bt,ν〈Rσ,τGµ,ν

2

F

+ 2Im{

[

}

σ,τ,µ,ν (M1) ′ 〉]} (14) Tλ,µ Et,ν〈Rσ,τGν,µ

The far-from-resonance molecular response tensors r, G′, and A are of types E1-E1, E1-M1, and E1-E2, respectively, as described in the book by Barron.40 We see that the intensity terms group into “RR”, “RG”, and “RA” terms, depending on the particular products of response tensor components. The angular brackets indicate averaging over rotational variables of the molecule, which proceeds in the same way for both ROA and the present version of SEROA. There are two principal differences between eq 14 and its ROA limit (no nanoparticle). The first is the use of the total local fields and gradients, Et, Bt, and 3′Et, that depend on the incident frequency ω0 and are evaluated at the molecular position r′. These have contributions from the scattering of the plane wave by the nanoparticle that do not all point in the same directions as their plane wave components. The second is the makeup of the T tensors, which depend both on the plasmonic nanoparticle and on details of the molecular scattering via multipole fields at frequency ω. There are other slight differences since, even far-from-resonance, there are two distinct types of RG terms and two distinct types of RA terms in eq 14, corresponding to whether the upward or downward transition

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Lombardini et al. R ) β2(R)

)

G′ ) β2(G′) ) β2(A)

Figure 2. Polarization of incident and scattered light in Figure 1 for different circular polarization ROA/SEROA experiments. Linearly- or unpolarized light is measured in scattering for ICP and is used in incidence for SCP experiments. Intensity sums and differences in each case are formed between the top and bottom scattering processes shown.

)

1 3 3 2 1 3 3 2 1 2

∑R , ∑ ∑R R ∑ G′ , ∑ ∑ R G′ ∑ ∑R A σσ

σ

στ στ

σ

-

τ

∑R ∑R σσ

σ

ττ

τ

(15)

σσ

σ

στ

σ

τ

σ

τ

στ

-

1 2

∑ R ∑ G′ σσ

σ

ττ

τ

ε στ στ

In the latter we employ the double contraction of the tensor A with the Levi-Civita antisymmetric tensor (εσµν elements are +1 for indices in cyclic order, -1 for anticyclic order, 0 otherwise)23 ε Aστ )

∑ ∑ εσµνAµντ µ

is E1.21 These terms combine more gracefully in the far-fromresonance ROA formalism.23 This limit can be verified by eliminating the nanoparticle, that is, making all Mie scattering coefficients zero, but we have so far not obtained a simplification of the RG and RA cross-terms in eq 14 when the nanoparticle is present. All of the ROA polarization schemes shown in Figure 2 have counterparts in SEROA. The original ICP arrangement26,27 modulates between left and right circular polarization input beams, recording the differences in scattered intensity. The total and differential ICP signals are IR ( IL, where the intensities are rotationally averaged. The SCP experiment uses linear or depolarized incident light and analyzes the right and left circular polarization character of the emitted light,28,29 with corresponding subscripted notation IR ( IL for the total and differential signals. The DCP experiments modulate between circular polarizations for both the incident and scattered light.30-32 There are four intensities that can be measured by modulating input and output polarizations, IRR , IRL , ILR , ILL. The DCPI arrangement measures in-phase modulation sums and differences IRR ( ILL, while DCPII measures out-of-phase modulation ILR ( IRL. Initial experiments with SEROA instruments have used both DCPI9 and SCP10,19 arrangements. Equation 14 can be modified for each of the different polarization schemes. Let us take right/left circular polarization vectors as (xˆ - i yˆ)/
2 for propagation along the positive z axis as usual. For ICP, eq 14 applies straightforwardly by using E ) E0 (xˆ - i yˆ)/
2 on incidence, calculating the scattering wave from the nanoparticle, then constructing the total local fields and gradients at the molecule, Et,ν (r′), Bt,ν (r′), (∂/∂rν) Et,p (r′). For SCP SEROA, one uses linear incident polarization for these fields in eq 14, but combines the λ ) x and y cases with complex coefficients. (For light propagating in the backscatter geometry, the R and L polarizations as measured by a common observer will switch from their definitions in the incident beam.) For DCP SEROA, these considerations must be combined. The rotational averages in eq 14 are all expressible in terms of the frame-invariant quantities41

1 2

(16)

ν

These ROA invariants28,32,41,42 in eq 15 are an extension of the far-from-resonance Placzek invariants used in ordinary Raman scattering.43 All of the molecule-specific information is contained in these scalar quantities, which may be calculated in the molecular frame if electronic structure programs are used. The quantities that these rotational averages multiply in eq 14 are considerably different than in ROA, but it is still possible to express all the intensities for each polarization scheme in terms of these reduced constants. B. Selectivity in ICP and SCP. Ordinary ROA is selective for chiral molecules due to rotational averaging. Specifically, the differential scattering intensity in SCP IR - IL (ICP provides equivalent information far-from-resonance) has no RR contributions, whatever the chirality of the molecule. In contrast, the RG and RA contributions will be nonzero if the molecule is chiral.40 It will still be true that the averaged IR - IL is 3 or more orders of magnitude smaller than the total scattering IR + IL, for which the RR contributions reinforce rather than cancel. As a consequence, if full rotational averaging is hindered, it is rather easy for RR terms in IR - IL to dominate and for chiral selectivity to be lost. An obvious example where this might occur is adsorption of the analyte on the plasmonic substrate. It was less obvious that the electromagnetic enhancement alone makes RR contributions nonvanishing,21 even with a spherical nanoparticle, full rotational averaging and no adsorption. In examining the high-symmetry SCP backscatter geometry, it was determined that certain specific combinations of polarization and molecular position outside the nanoparticle did indeed make (IR - IL)RR ) 0. With initial propagation chosen along the z axis, initial polarization along the x or y axis, and final circular polarizations chosen as (xˆ ( i yˆ)/
2, placing the molecule along the y axis gives one such configuration. Most geometries fail in this regard, but a symmetry was observed in the molecular azimuthal angle φ′ such that averaging of this variable as well would make RR difference terms vanish. Such positional averaging would arise for an ensemble of molecules with a uniform concentration around the spherical nanoparticle. Thus, even though they were kept well away from the single molecule limit, a first concrete set of circumstances was identified in which SEROA is predicted from theory to have the same advantages as ROA.

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Further insight can be gained by focusing on a single nanoparticle plasmon multipole and on just the RR intensity contributions. Let us consider the frequency range for which the l ) 1 (dipole plasmon) mode is the chief contributor to T(E1) in eq 10. The radiative fields may be calculated according to Table 1 in terms of the molecular position r′, θ′, and φ′ for backscatter observation along r f ∞, θ ) π, φ ) 0. The molecular position also enters the RR intensity terms on the right hand side of eq 14 through the total local field. If the incident plane wave is taken to propagate along positive z and to have its polarization vector in the x-y plane with values E0x and E0y, the l ) 1 local field at the molecular position can be worked out to be (in spherical components)

Et )

[

3 u1b0 sin θ′E′0x, (iu1a0 + w1b0 cos θ′)E′0x, 2 k0r′

]

(17)

The u and w functions of r′ are as defined in eqs 5-8, except that the subscript zero implies k f k0 and ω f ω0. The field quantities with primes are rotated versions of the incident field vectors

′ ) E0x cos φ′ + iE0y sin φ′, E0x ′ ) -E0x sin φ′ + iE0y cos φ′ (18) E0y If we take θ′ ) π/2, for example, with E0x and E0y both real, it is found that the rotationally averaged RR contributions to the differential intensity are

( )

3k2 2 3 [(E2 - E20y) sin 2φ′ 4πεε0 160 0x 4 2E0xE0y cos 2φ′] × -2β2(R) Im(u*1b0w1b0u1bw*1b) + k0r′kr′ 4 Im(u*1a0w1b0u1aw*1b) + [9R2 - β2(R)] 3 4 (19) Im(u*1b0w1b0u*1bw1b) + Im(u1a0w*1b0u1aw*1b) k0r′kr′

{

[

[ ]

{[

|

|

|

]}

This has factored into r′ and φ′ parts. Previously, we have investigated the cases E0x ) 0 and E0y ) 0, for which eq 19 has an overall factor of sin 2φ′. These results also hold for other values of l and of θ′. This simple angular dependence is therefore the behavior in φ′ noted empirically before.21 The new consideration is that cross-terms in cos 2φ′ will occur if both E0x * 0 and E0y * 0. Then these RR differential contributions will not vanish at φ′ ) π/2, though they will still vanish on averaging over φ′. C. Selectivity in DCP. DCP is one situation in which both E0x * 0 and E0y * 0, but it is a special case since the amplitudes are of the same magnitude and precisely 90° out of phase. The analytical behavior of the RR terms was therefore examined again in this case. For l ) 1 and θ′ ) π/2 once more, using R incident polarization, it was found that the leading intensity terms for R (upper sign) and L (lower sign) scattered polarizations are

( | || |

|

)}

This is completely independent of φ′. Furthermore, when L incident light is used, one finds the same results, that is,

(IRR)RR ) (ILL)RR,

(w1b0 + iu1a0 cos θ′)E′0y

(IR - IL)RR )

]|

27k4 2 6R2 - β2(R) u1au1a0 3 5120π2ε2ε20 u1b u1b0 2 2 u1b 2 u1b0 2 w1bw1b0 - 4 +β (R) 32 + kr′ k0r′ kr′ k0r′ u1b0 u1b 2|u1au1a0 | 2 + 2|w1bw1b0 | 2+4 u1a + u 2+ k0r′ kr′ 1a0 u1b0 u1b (20) w 2 + |u1aw1b0 - u1a0w1b | 2 4 w1b k0r′ kr′ 1b0

R (IR/L )RR )

(ILR)RR ) (IRL)RR

(21)

This is important information. The DCPI in-phase difference then reduces to

IRR - ILL ) (IRR - ILL)RG + (IRR - ILL)RA

(22)

while the DCPII strategy out-of-phase difference reduces to

ILR - IRL ) (ILR - IRL)RG + (ILR - IRL)RA

(23)

both of which are free of any contamination from RR terms. This leads directly to the question of other values of θ′. It is straightforward to continue the analytical analysis using symbolic algebra, though the increased number of terms for general θ′ is much too large to present here. The result is that the quantities (IRR)RR, (ILR)RR, (IRL)RR, (ILL)RR all remain independent of azimuth, and eq 21 is verified to hold for all polar and azimuthal molecular directions. It was furthermore verified analytically for l ) 2, the quadrupole plasmon mode. Thus, any RR contributions to differential intensities, which can be due to either chiral or achiral molecules, appear to be completely avoided in backscatter DCP SEROA. One can similarly calculate RG and RA contributions through symbolic algebra, though the intermediate calculations are also too lengthy to include. In the final analysis, we find that backscatter DCP gives the specific results

(ILL)RG ) -(IRR)RG,

(IRL)RG ) -(ILR)RG

(24)

(ILL)RA ) -(IRR)RA,

(IRL)RA ) -(ILR)RA

(25)

so that the normalized CIDs for DCPI and DCPII simplify to

∆I )

IRR - ILL IRR + ILL

)

(IRR)RG + (IRR)RA (IRR)RR

)-

(ILL)RG + (ILL)RA (ILL)RR

(26)

∆II )

ILR - IRL ILR + IRL

)

(ILR)RG + (ILR)RA (ILR)RR

)-

(IRL)RG + (IRL)RA (IRL)RR

(27) These relations hold for both ROA and our SEROA model.

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Lombardini et al.

TABLE 3: Intensity Factors for 600 nm DCP Backscattering with Model H2S2 1 nm Outside 40/50 Ag Nanoshell with θ′ ) O′ ) π/2a

a

component

|FRR|2

|FLR|2

|FRL|2

|FLL|2

RR RG RA

4.72261994e-30 -3.44471119e-36 -8.15333967e-36

8.12869194e-30 -2.24170169e-36 2.37864046e-35

8.12869194e-30 2.24170169e-36 -2.37864046e-35

4.72261994e-30 3.44471119e-36 8.15333967e-36

Results are independent of φ′. Each column sum is overall positive, though individual components need not be. Units are V2.

TABLE 4: Intensity Factors for 600 nm DCP Backscattering with Model H2S2 1 nm Outside 40/50 Ag Nanoshell with θ′ ) π/4, O′ ) π/2a

a

component

|FRR|2

|FLR|2

|FRL|2

|FLL|2

RR RG RA

2.35250995e-31 -5.47910886e-38 1.55005584e-36

5.62814512e-31 1.39189307e-36 -3.26793431e-35

5.62814512e-31 -1.39189307e-36 3.26793431e-35

2.35250995e-31 5.47910886e-38 -1.55005584e-36

Results are independent of φ′. Each column sum is overall positive, though individual components need not be. Units are V2.

While DCP scattering has some advantages in ordinary ROA,30,31,44 they appear to be considerably more dramatic here. In particular, it is not necessary to invoke additional ensemble averaging to eliminate possible interference from achiral molecules. One can consider the DCP investigation of small analyte concentrations without concern that scattering intensity differences are contaminated by achiral molecules for some directions around the spherical nanoparticle. In an indirectly related vein, the experiments by Kneipp et al.,9 already used a backscatter DCPI SEROA arrangement for adenine adsorbed on isolated silver colloidal particles. While adenine is not chiral by itself, the authors suggest that the differential scattering observed arises from formation of a chiral complex with the surface. The significance of the circular intensity differences has been the subject of some debate12,20 that will probably take some time to resolve. Certainly the current results do not apply directly to the experiment by Kneipp et al., because adsorption is not considered here. They do, however, argue that it would be interesting to have more experimental investigations of DCP SEROA. IV. DCP Excitation Curves We apply the simple chiroptical Twisted-Arc model of Trost and Hornberger,45 used earlier for SCP backscatter excitation curves.21 This approximates delocalized electron orbitals in chiral H2S2 using 1D motion along two arcs twisted from each other by the dihedral angle χ of the two S-H bonds. Simple wave functions and dynamical (excitation-frequency-dependent) molecular-response tensors are available from this model that suit our spectroscopic enquiries well, though fully ab initio methods11,17,18,46-49 continue to be improved. It should be emphasized, however, that our focus here is to assess whether SEROA is viable even in simple circumstances, not to make predictions for a specific molecule. As before, we consider 40/50 Ag and Au nanoshells with silica cores (40 nm inner radius, 50 nm outer radius). The dielectric functions for Ag and Au, widely used in SERS, are obtained by frequency-interpolating the tabulation of Johnson and Christy.50 The dielectric constant for the silica core is taken as 2.04 and that for an aqueous surrounding medium as n2, where the index of refraction n ) 1.33. The r, G′, and A tensors are actually only strictly appropriate for Rayleigh scattering. As usual for ROA calculations, the formalism for Raman scattering proceeds in the same way, except that the first derivatives of the tensors (“derived tensors”) with respect to the Raman mode are used

instead.21,23,51 The torsional coordinate χ that governs the chirality of the molecule can approximately be taken as the 420 cm-1 ν4 normal coordinate of H2S2.52 One can readily differentiate the parametric χ dependence of the tensors from the Twisted Arc model to obtain dr/dχ, dG′/dχ, and dA/dχ appropriate to Raman excitation. This is done in the calculations, though we continue to use the r, G′, and A notation for simplicity in the text. The intensity in direction (θ, φ) is proportional to |F|2, where22

F(θ,φ) ) [re-ikrErad,t(r, θ,φ)]rf∞

(28)

We can therefore label F in each polarization channel by the same subscripts and superscripts used above, for example, FRR. From eqs 26 and 27, one can express the CIDs directly in terms of these quantities. For the Ag nanoshell, 600 nm excitation, and the particular molecular position r′ ) 51 nm, θ′ ) φ′ ) π/2, each of the different cases of |F|2 is decomposed into RR, RG, and RA contributions in Table 3. Each component is calculated to convergence using lmax ) 12 for calculation of incident wave scattering and 20 for calculation of molecular multipole field scattering and is independent of the choice of φ′. Equations 21, 24, and 25 are fully verified here, as can be seen from identical RR values and identical-but-opposite RG and RA values across the columns. Table 4 shows the same result for an Au nanoshell with a different value of θ′ to demonstrate generality. If one replaces the Mie coefficients in Tables 3 and 4 with zero, the ordinary ROA results will follow this same pattern 2 2 except that the out-of-phase intensity factors |FLR|RG ) |FRL|RG R2 L2 ) |FL |RA ) |FR|RA ) 0. In this limit, the far-from-resonance backscatter DCPII arrangement ROA signal vanishes, and ICP/ SCP/DCPI ROA gain equivalence.53,54 The DCPII intensities actually only vanish exactly in this limit for Rayleigh scattering, though they are small for Raman scattering to the extent that (ω-ω0)/ω is small. With inclusion of the plasmonic effects of the nanoshell, DCPII becomes somewhat stronger and so we include comparison of both forms of DCP here. 2 2 Figure 3 shows components |FRR|RR and |FLR|RR for the Ag nanoshell as computed with the molecule positioned at three polar angles θ′ ) 0, π/2, and π. The total DCPI and DCPII scattering intensities are proportional to these two factors, respectively, as can be seen from eqs 21, 24, and 25. One can observe the nanoshell dipole mode at longer wavelengths and the quadrupole and octupole modes at shorter wavelengths. It is immediately clear that the dipole surface plasmon mode

Plasmonic Enhancement of Raman Optical Activity

Figure 3. Calculation of the 40/50 Ag nanoshell DCP RR total 2 2 scattering intensity factors |FRR|RRand |FLR|RR as functions of excitation wavelength at three different polar angles of the molecule and 1 nm outside the shell radius. The nanoshell-dipole features are most pronounced for equatorial molecules, while nanoshell-quadrupole features are most pronounced in the polar regions. Units are V2.

dominates for the molecule near the equator, while quadrupole modes dominate for molecules near the poles. The modes are well-separated energetically, and therefore the relative importance of different polar angles can change markedly with excitation energy. It is further observed that the DCPI Raman scattering is generally larger than the DCPII Raman scattering. As pointed out by the reviewer, this is consistent with the b vibrational symmetry of the torsional mode of H2S2 in C2 geometry and the depolarized character of Raman measurements for DCPI in backscatter geometries. Figure 4 shows the corresponding RG and RA contributions. Each of the frames has the same ranges to simplify comparisons. While not fully shown, the quadrupole contributions are once again strongly enhanced near the north and south poles. In contrast, the contributions in the nanoshell-dipole region are of similar magnitudes for the different values of θ′. Similar to the situation shown for SCP before, each of the RR, RG, and RA terms have their own excitation/enhancement curves. This is certainly true for the rapid variations seen in the l ) 2 and higher spectral regions to the short wavelength side, but is also evident in the l ) 1 nanoshell dipole region. In fact, it is even clear that, while DCPI still exhibits stronger intensities as a general rule, the shapes of the DCPI and DCPII excitation curves show significant differences. In particular, DCPI shows a strong tendency to change sign. That is, a particular vibrational feature in off-resonant SEROA may apparently exhibit an abrupt sign change as a function of excitation, unlike the situation for ordinary off-resonant ROA. A further conclusion that mirrors the findings from the SCP investigation is that the RA terms strongly dominate the RG terms. This is in opposition to ordinary ROA, where the latter tend to dominate49 and is a reflection (in part) of the strong variations in local fields near the plasmonic nanoparticle. Here we are able to make this point quantitatively.

J. Phys. Chem. C, Vol. 114, No. 16, 2010 7397 Figure 5 shows the normalized CIDs corresponding to eqs 26 and 27. Larger CIDs are preferable experimentally. It is seen for θ′ ) π/2 that the CIDs are strongly suppressed in the nanoshell dipole portion of the excitation profile. Based on scaling laws, Janesko and Scuseria11 give an extended discussion of the fact that SEROA with a dipolar substrate can decrease the normalized CIDs relative to ROA and that quadrupolar substates may very well be preferable. The spherical substrates used here have both types of modes, which may be selected by excitation frequency. The dipole plasmon spectral region does indeed give a small CID for equatorial molecules, as seen in the middle frame of Figure 5. The reason is that the RG + RA numerators shown in the last column of Figure 4 are of similar magnitudes for different θ′, while the RR denominator terms displayed in Figure 3 are strongly peaked equatorially. The other observation is that the DCPI CIDs for molecules at the north and south poles grow in magnitude significantly as we leave this region and head to longer wavelengths where the dipole enhancement tails off. The combined conclusion is that the dipole plasmon peak resonance is definitely not the preferred excitation energy to use. The RR contributions for the Au nanoshell are shown in Figure 6. The nanoshell quadrupole contributions are smaller and closer to the nanoshell dipole region, as can be seen (with effort) for θ′ ) 0. Those for θ′ ) π are not visible, whereas they were merely smaller for Ag in Figure 3. Once again the dipole contributions are significantly larger equatorially and DCPI signals are generally more intense than DCPII signals. Figure 7 shows that the RG terms are once again significantly smaller than the RA terms. The latter are of similar magnitude for most values of θ′, except for the larger DCPI response in the quadrupole plasmon region λ < 600 nm for θ′ ∼ 0. Comparing Figures 4 and 7, a common pattern emerges. Near the peak of the nanoshell-dipole scattering curve, the RG + RA cross-terms for DCPII are relatively small but roughly follow the shape of the RR curve (to within a sign). The DCPI crossterms are generally much larger but go through a zero-crossing near the peak of the RR dipole plasmon excitation curve. The normalized CIDs in the nanoshell-dipole region of Figure 8 also follow the same pattern as for Ag in Figure 5. They are noticeably suppressed for equatorial molecules where the RR peak is strong. For molecules at either the north or south pole, the DCPII CID is relatively flat for the dipole plasmon region, while the DCPI CID crosses through zero and continues to rise in magnitude for an extended region to longer wavelengths. The experimental importance of the normalized CIDs is that they largely remove effects due to, for example, variations in detector efficiency and laser intensity. Furthermore, signal-tonoise ratios achieved in SEROA differential scattering will play a part in determining the smallest magnitudes of CIDs worth pursuing. Nevertheless, surface plasmon enhancement has been shown here to complicate the interpretation of the CIDs compared to the benign situation for ROA. Each of the RR, RG, and RA contributions to the Raman scattering exhibit their own enhancement curves. The CIDs therefore depend on the details of the enhancement curves in both the numerator and the denominator, and optimal dipole-peak enhancement in fact leads to suboptimal CIDs. This is not the issue in the quadrupole plasmon excitation region, but there large swings of different terms and the resultant CIDs are exhibited. The undesirable consequence is that small changes in excitation frequency can easily lead to sign reversals for some vibrational features in SEROA (a problem also found for different binding geometries with adsorption by Janesko and Scuseria).18

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Lombardini et al.

Figure 4. DCP RG components |FRR|RG and |FLR|RG (col. 1), RA components |FRR|RA and |FLR|RA (col. 2) and their sums (col. 3) corresponding to the Ag nanoshell data of Figure 3 at three different polar angles of the H2S2 molecule. The magnitudes in the dipole plasmon spectral regions are relatively insensitive to polar angle, as opposed to Figure 3. Units are V2. 2

2

Figure 5. DCP CIDs ∆I and ∆II of eqs 26 and 27 corresponding to the Ag nanoshell intensity components in Figures 3 and 4. The CIDs for ordinary DCPI and DCPII ROA are shown for comparison in each frame, though they are independent of θ′.

On balance, the most promising recommendation for experimental investigations appears to be use of the DCPI configuration somewhat to the longer-wavelength side of the nanoshelldipole peak. There is presumably a balance to be achieved between decreasing overall signals and increasing CIDs in these regions. While there are no ROA or SEROA instruments reported capable of tunable excitation, it is possible to instead prepare a series of nanoshells with systematically shifted surface plasmon resonance peaks.55

2

2

Figure 6. Calculation of the 40/50 Au nanoshell DCP RR total 2 2 scattering intensity factors |FRR|RR and |FLR|RR, analogous to the Ag nanoshell calculations of Figure 3.

V. Summary and Discussion A SEROA model developed earlier for molecules moving near spherical metal nanoshells has been investigated in more detail. Vector spherical harmonic expansions of incident and molecular multipole fields, as well as their enhancing counterparts scattered from the plasmonic nanoparticle, have been simplified by dimensional reduction of the summations involved. The previous backscatter SCP SEROA analysis exhibited

Plasmonic Enhancement of Raman Optical Activity

J. Phys. Chem. C, Vol. 114, No. 16, 2010 7399

Figure 7. DCP RG, RA, and RG + RA components for the Au nanoshell, analogous to the Ag nanoshell calculations of Figure 4.

Figure 8. DCP CIDs ∆I and ∆II for the Au nanoshell intensity components in Figures 6 and 7, analogous to the Ag nanoshell calculations of Figure 5.

electric-dipole/electric-dipole (RR) contributions to the differential intensities that depended on azimuthal position of the molecule around the nanoparticle, and this has now been analyzed analytically. Symbolic algebra was used to determine that DCP versions of far-from-resonance backscatter SEROA would have vanishing RR differences regardless of molecular position, thus, eliminating the need to ensemble-average positions around the nanoparticle in order to achieve the chiral selectivity present in ordinary ROA. DCPI and DCPII backscatter SEROA intensities were then investigated numerically within the chiroptical model used

earlier for H2S2 near Ag and Au nanoshells with silica cores. Excitation profiles throughout the nanoshell dipole and quadrupole regions were calculated for the separate components, that is, RR for total scattering, RG and RA for differential scattering. The enhancement curves depend on polar angle of the molecule around the nanoshell, though not on azimuthal angle. The nanoparticle dipole plasmon RR peaks were shown to be dominated by molecules near the equator, though the RG and RA curves were not, resulting in CID ratios becoming strongly reduced for equatorial molecules and resonant dipole plasmon excitation. Of the two DCP spectroscopies, DCPI is generally the more intense except for zero-crossings near the dipole resonance. The initial suggestions we derive for chirally selective CID measurements are to use DCPII with excitation to the red side of the dipole resonance if tunable excitation is possible or to use metal nanoshells with the dipole resonance tuned to the blue side of the laser if not. It is encouraging to find theoretical evidence that there are SEROA configurations for which chiral selectivity can be guaranteed, especially without invoking the positional averaging arising in the earlier SCP analysis. Rotational averaging alone appears sufficient in DCP SEROA for our simple scenario of a spherical substrate and far-from-resonance excitation. This is a far cry from trying to explain all possible SEROA experiments. In adsorption, for example, the different possible chemical effect influences17,18 represent a particularly challenging direction of theoretical research. Nevertheless, there are already facets in nonadsorbed SEROA, where the enhancement is safely regarded as dominantly electromagnetic in origin, that should be of broader interest. It is to be expected, for example, that different excitation curves for different molecular multipole contributions will be the rule under any circumstances. There is a clear need for experiment and theory to work synergistically if SEROA is to become fully realized. Acknowledgment. Communications with L. A. Nafie, S. C. Hill, B. E. Brinson, B. G. Janesko, S. Lal, and P. Nordlander are gratefully acknowledged, and useful reviewer comments are

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acknowledged as well. This material is based upon work supported by the U.S. DoD/ARO under the Multidisciplinary University Research Initiative (MURI) Award W911NF-04-010203, by National Science Foundation Award No. CHE0518476, and by the Robert A. Welch Foundation Award Nos. C-0016 and C-1685. References and Notes (1) Moskovits, M. ReV. Mod. Phys. 1985, 57, 783. (2) Schatz, G. C.; Young, M. A.; Van Duyne, R. P. Electromagnetic Mechanism of SERS. In Surface-Enhanced Raman Scattering: Physics and Applications; Kneipp, K., Moskovits, M., Kneipp, H., Eds.; Springer: Berlin, 2006; Vol. 103; p 19. (3) Otto, A.; Mrozek, I.; Grabhorn, H.; Akemann, W. J. Phys. (Paris) 1992, 4, 1143. (4) Yonzon, C. R.; Stuart, D. A.; Zhang, X.; McFarland, A. D.; Haynes, C. L.; Van Duyne, R. P. Talanta 2005, 67, 438. (5) Lal, S.; Link, S.; Halas, N. J. Nat. Photonics 2007, 1, 641. (6) Efrima, S. J. Chem. Phys. 1985, 83, 1356. (7) Hecht, L.; Barron, L. D. Chem. Phys. Lett. 1994, 225, 525. (8) Hecht, L.; Barron, L. D. J. Mol. Struct. 1995, 348, 217. (9) Kneipp, H.; Kneipp, J.; Kneipp, K. Anal. Chem. 2006, 78, 1363. (10) Abdali, S. J. Raman Spectrosc. 2006, 37, 1341. (11) Janesko, B. G.; Scuseria, G. E. J. Chem. Phys. 2006, 125, 124704. (12) Etchegoin, P. G.; Galloway, C.; Le Ru, E. C. Phys. Chem. Chem. Phys. 2006, 8, 2624. (13) Abdali, S.; Johannessen, C.; Nygaard, J.; Nørbygaard, T. J. Phys.: Condens. Matter 2007, 19, 285205:1. (14) Johannessen, C.; Abdali, S. Spectroscopy 2007, 21, 143. (15) Bourˇ, P. J. Chem. Phys. 2007, 126, 136101. (16) Abdali, S.; Blanch, E. W. Chem. Soc. ReV. 2008, 37, 980. (17) Jensen, L. J. Phys. Chem. A 2009, 113, 4437. (18) Janesko, B. G.; Scuseria, G. E. J. Phys. Chem. C 2009, 113, 9445. (19) Brinson, B. Nonresonant Surface Enhanced Raman Optical Activity, Ph.D. thesis, Rice University, Houston, TX, 2009. (20) Barron, L. D.; Zhu, F.; Hecht, L.; Tranter, G. E.; Isaacs, N. W. J. Mol. Struct. 2007, 834-836, 7. (21) Acevedo, R.; Lombardini, R.; Halas, N. J.; Johnson, B. R. J. Phys. Chem. A 2009, 113, 13173. (22) Kerker, M.; Wang, D.-S.; Chew, H. Appl. Opt. 1980, 19, 4159. (23) Barron, L. D.; Buckingham, A. D. Mol. Phys. 1971, 20, 1111. (24) Oldenburg, S. J.; Averitt, R. D.; Westcott, S. L.; Halas, N. J. Chem. Phys. Lett. 1998, 288, 247. (25) Aden, A. L.; Kerker, M. J. Appl. Phys. 1951, 22, 1242.

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