Plasmon-Induced Circular Dichroism of a Chiral Molecule in the

Apr 1, 2011 - This CD band is positive and corresponds to the positive Cotton effect. ...... Molotsky , T.; Tamarin , T.; Moshe , A. B.; Markovich , G...
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Plasmon-Induced Circular Dichroism of a Chiral Molecule in the Vicinity of Metal Nanocrystals. Application to Various Geometries Alexander O. Govorov* Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, United States

bS Supporting Information ABSTRACT: The Coulombic interaction between a chiral molecule and a nonchiral metal nanocrystal creates new plasmonic lines in a circular dichroism (CD) spectrum. In many cases, optical lines in CD spectra of chiral molecules and plasmon absorption bands in metal nanocrystals are not in resonance. Typically, CD lines of chiral molecules are in the UV spectral range, whereas plasmon bands of metal nanocrystals are in the visible range. Nevertheless, plasmon excitations can strongly interact with dipoles of chiral molecules and become chiral. We demonstrate this effect theoretically and derive a general equation for the CD spectrum of a complex composed of metal nanocrystals and a single chiral molecule. In our theory, new plasmon peaks in the CD spectra come from an effect of interference between external and induced fields inside a hybrid complex. Our models involve single spherical nanoparticles, nanoparticle pairs, and nanoshells. The results obtained here can be used to design new optical materials and hybrid nanosystems with sensor properties.

’ INTRODUCTION Chirality is a very intriguing property of molecules. This property is traditionally employed to determine conformation states of biomolecules1 and also to control concentrations of enantiomers in a solution. Enantiomers are molecules which are mirror images of each other. Since many biomolecules and many drugs are chiral, the ability to measure chirality is of great importance. The circular dichroism (CD) spectroscopy measures chirality of molecules using the two chiral states of light. A difference between absorbences for the left- and right-handed photons provides us with the CD signal from a solution containing chiral molecules. This CD signal is nonzero only for a solution with chiral objects. A nanoscale structure may also have the chiral property. However, it is challenging to create artificial chiral nanostructures since it requires a high degree of homogeneity in an ensemble. So far, chiral nanoscale assemblies reported in the literature utilized mostly chiral molecules and nonchiral nanocrystals. In these structures, chiral molecules become adsorbed on a surface of nanocrystals. There may be various mechanisms of interaction between chiral molecular adsorbate and nanocrystals.2-12 In the case of ref 13, CD lines of chiral adsorbate were plasmonically enhanced. In other cases, a hybrid complex acquires new or strongly modified CD lines which were often interpreted as lines coming from a chiral adsorbate or from chiral surface states of nanocrystals.4-7 In an assembly comprising only metal nanocrystals, purely plasmonic CD signals can appear if an assembly has a chiral geometry.14 In this case, a new CD activity comes from Coulombic and electromagnetic interactions between plasmonic nanoparticles.14 Another recently proposed mechanism of new CD lines in nanoscale assemblies is based on the Coulombic interaction r 2011 American Chemical Society

between a molecular dipole and a metal nanocrystal.15 This Coulombic mechanism was involved recently in interpretations of the experimental data.9,10,12 In this paper, we will focus on the Coulombic mechanism of interaction and generalize the results of ref 15 to a nanocrystal complex of arbitrary geometry. Here, we study theoretically the CD effect of a chiral molecule placed in the vicinity of metal nanocrystals. We generalize the results obtained previously for a single spherical nanoparticle.15 In particular, we derive convenient equations for the CD spectrum of a chiral molecule interacting with a metal nanocrystal complex of arbitrary geometry. We note that the metal nanocrystal complexes themselves involved in this theoretical study are nonchiral. Considering a few geometries, we find that a nanoshell and a nanoparticle dimer are able to create strong plasmonic CD lines and also to enhance natural CD transitions of a molecule. Our discussion includes both resonant and offresonant Coulombic interactions between a molecular dipole and a plasmon band. The case of the resonant exciton-plasmon Coulombic coupling has relevance to the recent experiments on plasmonically enhanced photosynthetic molecules.16 Importantly, the CD lines of molecules and biomolecules are often not in resonance with the plasmon bands of commonly used metal nanocrystals. Typical spectra of metal nanocrystals (gold and silver) and a dye molecule (chromophore) are shown in Figure 1a. The commonly studied CD bands of strongly chiral molecules (proteins and DNAs) lie in the UV spectral region Received: December 21, 2010 Revised: February 6, 2011 Published: April 01, 2011 7914

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Figure 2. (a) Schematics of a complex composed of a metal nanoparticle and a dye molecule. A dye molecule has both electric and magnetic dipole moments, μ12 and m21. (b) Quantum transitions in the system. The solid vertical arrows represent light-induced transitions, and the horizontal blue arrow depicts the Coulombic coupling. The dotted vertical arrows show relaxation processes. (c) Spherical system of coordinates used in the text.

Figure 1. (a) Normalized extinction coefficients of metal nanoparticles (gold and silver) and a dye molecule with λ0 = 200 nm. (b) Enhancement factors for the external electric field at the surface of gold and silver spherical nanoparticles. (c) Circular dichroism (CD) and optical rotatory dispersion (ORD) signals of a generic molecule. This CD band is positive and corresponds to the positive Cotton effect. (d) Typical calculated CD signals of molecule-nanoparticle hybrids.

(150-250 nm), whereas the plasmon bands of gold and silver nanoparticles are located in the visible range, at about 520 and 400 nm, respectively. Then, we clearly see that the molecular exciton transitions and the plasmon bands are not in resonance. Nevertheless, the CD spectrum of a molecule-nanocrystal complex acquires new plasmon-induced lines. The origin for this effect is in a strong plasmon enhancement of electric fields (Figure 1b) that also leads to an enhancement of the Coulombic interaction between a chiral molecule and a metal nanocrystal. The plasmon-enhancement factor in Figure 1b shows peaks at the plasmonic wavelengths. The factor P in Figure 1b corresponds to the enhancement factor Pzz defined later in the text. A chiral molecule with CD resonances in the UV range has nevertheless an optical chiral response in the visible spectral interval, as shown in Figure 1c. This optical, nonabsorbing chiral response is a rotation of light polarization (an optical rotatory dispersion) which remains strong at wavelengths far from absorption peaks of a chiral molecule (Figure 1c). Therefore, it is possible to expect plasmonic lines in a CD spectrum of hybrid complex with an off-resonant interaction between a molecular exciton and a plasmon (Figure 1d). In chiral media, the CD effect and the effect of optical rotatory dispersion (ORD) come always together. They are coupled by the Kramers-Kronig transform.17 The CD effect is described

by a molar ellipticity, θ(ω). This parameter describes an ellipticity of linearly polarized light coming through a chiral medium. The ORD spectrum, R(ω), gives a rotation of polarization of incident light. In a chiral medium, the photon dispersions of circularly polarized photons become18,19 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 kL, R ¼ ( ω 3 ξc þ ε0 ω2 þ ðω 3 ξc Þ2 where ξ0 is a chiral medium parameter entering the equation for the electric displacement field D = ε0E þ iξcB. Then, the difference in the absorption coefficients for the left- and righthanded photons Δσ = 2ω 3 Im ξc/c0, when ξc , 1. Similarly, the difference in the refractive indices for the left- and right- handed photons Δn = 2 3 Re ξc. The CD and ORD spectra are proportional to Δσ and Δn/λ, respectively.17 In the vicinity of an isolated optical transition ξc ∼

1 1 þ pω - pω0 þ iΓ pω þ pω0 þ iΓ

where ω0 and λ0 are a frequency and a wavelength of an isolated molecular resonance. Then, CD and ORD spectra can be written as θðωÞ ¼ aθ ω 3 Im ξc , RðωÞ ¼ aR ω 3 Re ξc The above equations are defined for both negative and positive frequencies ω. The Kramers-Kronig transform couples θ(ω) and R(ω) Z ¥ ω2 3 θðω0 Þ 2 -1 dω0 , RðωÞ ¼ a 3 0 02 2 π 0 ω ðω - ω Þ Z ¥ ω 3 Rðω0 Þ 2 dω0 θðωÞ ¼ - a 3 02 2 π 0 ðω - ω Þ where a = aθ/aR. In the vicinity of the resonance (ω ≈ ω0), we now obtain a Lorentzian line shape for the CD spectrum and a peak-dip shape for the ORD20 7915

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-Γ , ðpω - pω0 Þ2 þ Γ2

RðωÞ ¼ aR ω

pω - pω0 ðpω - pω0 Þ2 þ Γ2

Importantly, the CD signal is mostly present in the vicinity of the transition, whereas the ORD strength exists at any wavelength. Mathematically, we can see the following from the asymptotic behaviors of the CD and ORD spectra: θ(ω)  Δω-2 and R(ω)  Δω-1 for pΔω = |pω - pω0| . Γ. The CD spectrum decays fast for |pω - pω0| . Γ, whereas the ORD strength decreases slower as the light frequency moves away from the molecular resonance. In the case of the Lorentzian band, the optical responses are power-law functions of Δω when Δω . Γ. But, absorption and CD bands of real molecules are often Gaussian: θ(ω)exp[-(pΔω)2/Γ2]. Then, the Kramers-Kronig transform from the Gaussian band will give a power-law function for the ORD spectrum outside the absorption band: θ(ω)Δω-1 for p|Δω| . Γ.20 We now see a striking difference between the CD and ORD spectra: The CD signal decays very fast (exponentially) and practically vanishes outside the molecular resonance, but the ORD signal remains strong outside the molecular absorption band. This is an important property of many molecules and biomolecules which typically have CD lines in the UV range. Simultaneously, these molecules have nonvanishing ORD signals in the visible range and, therefore, rotate light in a much wider wavelength interval. Prominent examples are proteins (λ0 ∼ 200 nm) and DNAs (λ0 ∼ 250 nm), which absorb light in the UV range but are able to rotate light also in the visible wavelength interval. Figure 1c illustrates this situation. The CD signal rapidly decreases as the wavelength moves away from the molecular resonance (Figure 1c). Simultaneously, the ORD signal remains strong. Then, it is interesting to look at calculated CD spectra of the molecule-nanocrystal complex with the off-resonant interaction between an exciton and plasmon (λ0 6¼ λplasmon). Such CD spectra (Figure 1d) show plasmon peaks originating from the Coulombic interaction between a chiral molecule and a strongly absorbing plasmon band. A physical mechanism for the appearance of the plasmonic CD peaks is interference between external and induced fields inside a hybrid complex. These plasmonic CD bands come, of course, from the light absorption inside a metal component where external and induced fields add up constructively or destructively. This interference phenomenon is similar to the exciton-plasmon Fano effect studied recently in several papers.21-24 Another study relevant to the above discussion concerns a CD effect in the presence of superchiral electromagnetic fields created in a metamaterial or in an optical resonator.25-27 In this case, chiral optical responses of molecules can be strongly enhanced, which offers interesting opportunities for an ultrasensitive detection. In purely molecular systems, an effect of transfer of chirality from one molecular element to another is also possible. One example is a complex composed of chiral DNAs and porphyrins.28 However, we should note that the characteristics of chiral interactions in purely molecular systems and in hybrid molecular-metal structures are very different. In particular, hybrid molecular-metal structures allow shifting of CD signals further to the long wavelength region via tailoring of plasmon resonances.

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1. MODEL Our system incorporates a chiral molecule and metal nanocrystal (NC) interacting via the Coulombic forces (Figure 2a). The vector connecting the centers of the components is denoted as R, the corresponding center-to-center distance is given by R, and the separation between the surface of NC and the center of ^. the molecule is Δ. The molecule is located on the z-axis, i.e. R||Z A chiral molecule should have both electric and magnetic dipole moments, μ12 and m21, and these moments should not be perpendicular to each other.1,29,30 The CD strength of an insolated molecule is CD0 ∼ Imðμ12 3 m21 Þ To describe the optical responses of metal nanocrystal and matrix, we use the classical approach involving the local dielectric function approximation. In our model, a metal NC (gold or silver) and a matrix (water) are described with bulk dielectric constants, εNC(ω) and ε0, respectively; ε0 = εwater = 1.8. A convenient method to calculate optical responses of molecule is the density-matrix formalism.31 For simplicity, we model a chiral molecule as a two-level quantum system with a ground state |1æ and an excited state |2æ (Figure 2b). The system experiences an electromagnetic field of an incident wave 

Eext ¼ Eext, ω 3 e-iωt þ Eext, ω 3 eiωt , 

Bext ¼ Bext, ω 3 e-iωt þ Bext, ω 3 eiωt 1/2

where Eω,ext = e0E0ei(ε0) k 3 r and Bext,ω = (ε0)1/2[k  Eext,ω]/k are the complex amplitudes of the electromagnetic field, e0 is the polarization vector, and ω and k are the frequency and wave vector in vacuum, respectively; ω = c0k, where c0 is the vacuum light speed; k = 2π/λ, where λ is the photon wavelength in vacuum. Employing the spherical coordinates, we now consider an incident light wave approaching our system at the angles θ and j (Figure 2c). Then, the equation of motion of the density matrix written for the quantum state of molecule takes the form p

∂Fnm ^ FÞ ^ 0 þ V^ jmæ - ðΓ ¼ iÆnj½^ F, H 3 nm ∂t

ð1Þ

where |næ are the involved quantum states with n = 1, 2; their energies are E1 and E2. The corresponding absorption energy pω0 = E2 - E1. The molecular emission wavelength in vacuum λ0 = 2πc0/ω0. The relaxation matrix in eq 1 has the following elements: (Γ̂ 3 F)12 = Γ12F12, (Γ̂ 3 F)21 = Γ21F21, (Γ̂ 3 F)22 = Γ22F22, ̂0 þ V ̂ consists ̂ F)11 = Γ22F22. The molecular Hamiltonian H and (Γ 3 ̂ of the internal energy of molecule (H0) and the light-matter interaction ^ 3 Etot - m V^ ¼ - μ ^ 3 Btot ¼ V^ ω 3 e-iωt þ V^ωþ 3 eiωt

ð2Þ

where Etot = Etot,ω 3 e-iωt þ E*tot,ω 3 eiωt is the electric field acting on the optical electric dipole of molecule in the presence of a metal NC and Btot = Btot,ω 3 e-iωt þ B*tot,ω 3 eiωt is the total dynamic magnetic field applied to the molecule. The electric and magnetic ̂ = -|e|r and m dipole operators are given by μ ^ = -(|e|)/(2mc0)^ [r  P], respectively. It is important to note that eq 1 involves the plasmon excitations of NC through the total electromagnetic field Eω, tot ¼ Eω, ind þ Eω, NC1 , Bω, tot ¼ Bω, ind þ Bω, NC1 7916

ð3Þ

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where Eω,ind is the external field inside the molecule in the presence of a NC. The field Eω,NC1 describes a “self-action” of the molecular dipole and comes from the surface changes of NC induced by the molecular dipole. The magnetic fields Bω,ind and Bω,NCI are defined in the same way as the electric fields above. We are interested in systems of small size (λ0 . R), and we first consider the fields Eω,ind = Eω,ext þ δEω,ind and Bω,ind = Bω,ext þ δBω,ind. It follows from the Mie theory that a small NC strongly perturbs the electric field, Eω,ext ∼ δEω,ind, whereas a change in the magnetic field is small: δBω,ind ∼ (R/λ0)Bω,ext , Bω,ext. These results can also be seen directly from the analysis of Maxwell’s equations. Within the Mie theory, Eω,ind = E(0) ω,ind þ n (n) E(1) ω,ind þ ..., where the term Eω,ind  (R/λ0) . For a small NC, E(0) ω,ind is the leading term. Regarding the CD effect, the influence of the field E(1) ω,ind was analyzed in ref 15, and it was shown that this term does not create CD signals for symmetric NCs. To summarize, we will use below the following approximations: (0) 0 Eω,ind ≈ E(0) ω,ind and Bω,ind ≈ Bω,ext, where Eω,ind ∼ (R/λ0) Eω,ext is the first term in the Mie-theory expansion of the electric field in terms of the parameter R/λ0. Now we look at the fields Eω,NC1 and Bω,NC1. The field Eω,NC1 can be calculated in the quasi-static 0 approximation (Eω,NC1 ≈ E(0) ω,NC1 ∼ (R/λ0) Eω,ext) since the electrodynamics correction to Eω,NC1 is small (ΔEω,NC1 ∼ R/λ0). Also, ΔEω,NC1  |k|; it means that the CD signal coming from this term will vanish after averaging over the solid angle.15 The magnetic field Bω,NC1 ∼ (R/λ0)Eω,ext is small, and, in addition, its contribution to the interaction given by eq 2 has the characteristic small parameter R = e2/pc0. The general solution of eq 1 within the rotating-wave approximation31 was obtained in ref 15

energy transfer from a molecule to metal NCs is expressed as32

F11 ¼ σ 11 , F22 ¼ σ22 , F21 ¼ σ 21 e-iωt , F12 ¼ σ 12 eiωt

where Æ...æΩ is the averaging over the solid angle. This averaging is needed since hybrid complexes are typically studied in a solution, and, therefore, they have random orientations. Qþ(-) are the rates of absorption for two incident electromagnetic waves with the following polarizations: e0,L = e0þ = (eθ þ iej)/21/2 and e0,R = e0- = (eθ - iej)/21/2, where eθ(j) are the unit vectors along the two directions perpendicular to the wave vector and the ^ . It is convenient to split the total CD signal into vector ej ^ Z two parts

ð0Þ ^ 3 ðEω, ind Þjr ¼ R - m ^ 3 ðBω, ext Þjr ¼ R V^ω, ind ¼ - μ ^ Æ2jV ω, ind j1æ σ21 ¼ , pω - pω0 þ iΓ12 - Gω 2Γ12 3 jÆ2jVω, ind j1æj2 1 σ 22 ¼ Γ22 jpω - pω0 þ iΓ12 - Gω j2 Gω ¼ μ21 3 ðrðΦω 3 μ12 Þjr ¼ R Þ

ð4Þ

(y) (z) The vector function Φ(r) = (Φ(x) ω , Φω ,Φω ) defines the field induced by the surface changes of NCs in the presence of r(Φω 3 dω) = an oscillating molecular dipole dω, Eω,NC1 = -B (y) (z) -dω,x r B Φ(x) -d r B Φ -d r B Φ . In other words, ω ω,y ω ω,z ω the scalar function jω,dipole = j0ω,dipole þ Φω 3 dω is an electric potential of a dipole d ω in the presence of metal NCs, where j0ω,dipole = dω 3 r/(ε0r3) is the potential of molecular dipole in the absence of metal objects. The corresponding total field induced by 0 B jω, the molecular dipole Etot ω,dipole = -r dipole = Eω,dipole þ Eω,NC1. The total potential of the molecular dipole can be written conveniently as jω,dipole = Φtot ω 3 dω, where r Φtot þ Φω ω ¼ ε0 r 3

2 Im½Gω0  p The optically induced dipole moment of molecule is expressed via the density matrix γFRET ¼ -

dω, exc ¼ σ 21 3 μ12 The absorption rate of the system is calculated as Q ¼ Qmolecule þ QNC

ð5Þ

where σ 22 3 Γ22 ¼ ω0 3 σ 22 3 Γ22 p Z Z ω in ¼ dV Æj 3 Eæt ¼ ImðεNC Þ Ein ω Eω 2π metal metal Qmolecule ¼ pω0 3

QNC

ð6Þ

The integral in QNC is taken over the volume of metal NCs, Æ...æt is the standard time averaging, and j is the electric current. The complex amplitude of electric current inside metal NCs jω = -iω(εNC - 1)/4π 3 Ein ω. The total electric field inside NCs ð0Þ

ð0Þ

tot tot B Ein ω ¼ Eω, ind þ Eω, dipole ¼ Eω, ind - rðΦω 3 dω Þ

Also, note that Im(εNP) > 0. The strength of circular dichroism (CD) is defined as CDmolecule - NC ¼ ÆQþ - Q- æΩ

ð7Þ

CDmolecule - NC ¼ CDmolecule þ CDNC where CDmolecule = ÆQmolecule,þ - Qmolecule,-æΩ and CDNC = ÆQNC,þ - QNC,-æΩ come from the absorption rates of dye molecule and NCs, respectively. The averaging in eq 7 can be done over all directions of incident-wave wave vector k, assuming R that a hybrid complex has a fixed orientation, Æ...æΩk = (4π)-1 ... sin [θk] dθkdjk. This averaging is equivalent to the averaging over all orientations of complex for a fixed direction of k.

2. CIRCULAR DICHROISM Using eqs 4-7 and performing the averaging over Ωk, we obtain CDmolecule

¼ E20

The function Gω gives the broadening of the dipole transition due to energy transfer to metal NCs; this energy transfer is similar to the well-known FRET effect. The rate of

8 pffiffiffiffi Γ12 ^ ðtrÞ 3 μ12 Þ 3 m21  ε0 ω 0 Im½ðP 3 jpω - pω0 þ iΓ12 - Gω j2

ð8Þ 7917

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^ (tr) is a matrix transposed from the field-enhancement where P matrix 0 1 Pxx Pxy Pxz C BP C ^ ¼B P @ yx Pyy Pyz A Pzx Pzy Pzz

are the components of the vector Φ tot where Φ tot(R) ω ω . In the following sections, we will employ eq 10 for the three particular geometries: (1) a dipole and a single spherical NC, (2) a dipole in the center of NC dimer, and (3) a dipole and a nanoshell. The other contribution to the NC CD

This matrix is defined in the following way: E(0) ω,ind(R) = ^ 3 e0), where the field E(0) E0(P ω,ind(r) should be taken at the position of the molecule, r = R. In the next step, we calculate CDNC

CDNC;dipole-dipole  ¼

CDNC ¼ CDNC;dipole-external field þ CDNC;dipole-dipole CDNC;dipole-external field ¼ Z ω ð0Þ Bj - ImðεNP Þ 3 2Re½Æ ðEω, ind, þ 3 r ω, dipole, þ 2π metal

Z JðωÞ ¼ metal

¼ metal

CDNC, dipole - dipole Z ω 2 2 Bj B ¼ ImðεNP Þ 3 Æ ðjr ω, dipole, þ j - jrjω, dipole, - j ÞdV æΩ 2π metal

where the contributions CDNC,dipole-external field and CDNC,dipole-dipole come from the interference of the dipole and external fields inside the metal. The first term in CDNC is reduced to pffiffiffiffi ω 2 4 ε0 CDNC;dipole-external field ¼ ImðεNP Þ 3 E0 3 2π 

 Z 1 B ðΦtot μ ÞÞ m ^ ðtrÞ 3 r dV ðK ^ ω 3 12 3 21 3 pω - pω0 þ iΓ12 - Gω metal

ð9Þ The position-dependent matrix K̂(r) determines the external field inside a NC 0

Kxx B ð0Þ B ^ ðrÞ 3 e0 Þ, K ^ ¼ @ Kyx Eω, ind ðrÞ ¼ E0 ðK Kzx

Kxy Kyy Kzy

1 Kxz C Kyz C A Kzz

The above eqs 8-9 are a generalization of the formulas obtained in ref 15 to a nanocrystal of arbitrary geometry. For the case of symmetric NC complexes (two orthogonal mirror planes coming through the position of a dipole), the matrix P̂ is diagonal and the components of K̂(r) have the following spatial symmetry: KRβ ∼ R 3 β, where R(β) = x, y, z. Due to the spatial symmetry, the integral in eq 9 can be simplified to Z  B ðΦtot μ ÞÞm ^ ðtrÞ 3 r ðK ω 3 12 ^ 21 3 dV VNP

Z

¼

∑ μ12R 3 m21R R¼x, y, z β¼x, y, z

VNP



KβR

B ðΦtot μ ÞÞ ðr B ðΦtot,  μ ÞÞ dV ðr ω 3 12 3 ω 3 21 3

Z

ð0Þ

Bj - Eω, ind, - 3 r ω, dipole, - ÞdV æΩ 

3 Im

9 8 ðtrÞ pffiffiffiffi 2 4 > > > > ^ < Im½ð P ε E μ Þ m  0 0 3 12 3 21 = ω 3 ImðεNP Þ 2 > 2π > > ; : jpω - pω0 þ iΓ12 - Gω j >

∂ΦtotðRÞ ω dV ∂β 3

ð10Þ

Edipole Edipole ω ω 3 dV ¼

-4π Im Gω ImðεNP Þ

B ðΦtot μ Þ ¼ -r Edipole ω ω 3 12

ð11Þ

Here Edipole is the field induced by an effective moleω cular dipole μ12. One can see that the integral J(ω) can be expressed via the function Im[Gω] that describes energy transfer from the molecular dipole to the metal. To obtain the above relation between J(ω) and Im[Gω], one should employ the energy conservation law and neglect the radiation from the system. We should comment that, in most cases, CDNC,dipole-external field . CDNC,dipole-dipole. The contribution CDNC,dipole-external field and CDNC,dipole-dipole can only be comparable for very small moleculesurface separations. The reason is that CDNC,dipole-external field ∼ μ12 3 m21, whereas CDNC,dipole-dipole ∼ (μ12 3 m21)μ122. A typical dipole moment of a molecule is a relatively small number: μ12/ |e| , d, where d is any typical length associated with the complex. More explicitly, μ12/|e| ∼ Å and d ∼ 1 ÷ 10 nm. In the next section, we give explicit analytical equations for the case of a single NC, and we can see that CDNC;dipole-dipole μ212 ∼ 3 CDNC;dipole-external field d jpω - pω0 þ iΓ12 j for Γ12 . |Im[Gω]|. The length d is defined in the next two lines. When R . aNP, d ∼ R, where aNC is the NC radius. For relatively small molecule-surface separations (such as Δ = R - aNC , aNC), the length d ∼ Δ. The latter case, however, assumes that the distance Δ cannot be very small because of the condition Γ12. |Im[Gω]| = const/Δ3 (Δ f 0). An important observation from eqs 8 and 9 is that the term CDNC,dipole-external field is proportional to (ω - ω0)-1 for large detunings |ω - ω0| . Γ12/p, whereas CDmolecule  (ω - ω0)-2 under the same conditions.15 It means that the dissipation inside NCs (i.e., the term CD NC,dipole-external field ) can give CD signals far from the molecular resonance for a molecule-NC complex with an off-resonant interaction (ω0 . ωplasmon). 7918

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Figure 3. Panels (a)-(c) are related to a complex based on an Au-NC. (a) Extinction spectra of isolated molecule and Au nanocrystal. (b, c) CD spectra of the hybrid complex with three separations. Panels (d)-(f) describe a complex with an Ag-NC. (d) Extinction spectra of isolated molecule and Ag nanocrystal. (e, f) CD spectra of the Ag-based hybrid complex.

 0 β 1 B R3 B B ^ ¼B 0 P B B B @ 0

Extinctions in the standard units (cm-1 3 M-1) can be calculated using eqs 8-11 εmolecule-NC

2π NA 3 10-4 ÆðQmolecule þ QNP ÞæΩk ¼ pffiffiffiffi 2 c0 ε0 E0 0:23

1 0  1-

0

β R3



0

ΔεCD ¼ ΔεCD, molecule þ ΔεCD, NC 2π NA 3 10-4 CDmolecule - NC ¼ pffiffiffiffi 2 c0 ε0 E0 0:23

C C C C 0 C  C C 2β A 1þ 3 R

ð12Þ Inside a spherical NC, the induced electric field Eind(0) = ω, ^ is diagonal and independent of E0γ(ω)e0 and the matrix K(r) the coordinates

where the absorption rates Q are now in the cgs units.

3. APPLICATIONS 3.1. A Single Spherical Nanoparticle and a Chiral Molecule. In this case, it is easy to calculate the enhancement matrix

using the dipole field induced by the NC and the polarizability of a metal sphere ð0Þ β εNP - ε0 E Bω, ind ¼ eB0 þ 3 ð3 3 e0z 3 ^z - eB0 Þ, β ¼ a3NP R ðεNP þ 2ε0 Þ

where a NP is the nanoparticle radius and ε NP is the di^ electric constant of metal nanoparticle. Then, the matrix P becomes

KRR ¼ γðωÞ ¼

3ε0 ð2ε0 þ εNP Þ

We arrive at the exact analytical equation for CDNC,dipole-external field that is valid for any molecule-NC distance R 8 a3 CDNC;dipole-external field ¼ ImðεNP Þ NP3 9 ε0 R

 2    pffiffiffiffi 2  ðμ12x 3 m21x þ μ12x 3 m21x - 2 3 μ12z 3 m21z Þ 3ε0  ω 3 ε 0 E0    3 Im ð2ε0 þ εNP Þ pω - pω0 þ iΓ12 - Gω

We note that this equation contains only the term R-3 because the matrix elements KRR are position-independent and 7919

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Figure 4. Distance dependences of the CD contributions for the parameters given in the graph. The photon wavelength corresponds to the Au-plasmon resonance, λ = 523 nm.

the integral includes only the first spherical harmonic of Φ(R) ω R (R) (see section R 1 in the(R)Supporting Information): -3metal∂Φω / ∂R 3 dV  metal(∂Φω /∂R) 3 Y00(θ,j) 3 dΩ  R . However, the exciton broadening Gω is an infinite sum over the terms R-n (Supporting Information). The first term for the function Gω is given here β Gω ¼ ðμ μ þ μ21y μ12y þ 4μ21z μ12z Þ þ ::: ε0 R 6 21x 12x Note that Im[Gω] > 0; this corresponds to energy transfer from the molecule. In an important limit Δ = R - aNP , aNP, the function Gω diverges33,34 Gω ¼ -

ðμ21x μ12x þ μ21y μ12y þ 2μ21z μ12z Þ 1 ðεNC - ε0 Þ 8ε0 Δ3 ðεNC þ ε0 Þ

This divergence at Δ f 0 indicates fast energy transfer from an exciton to a metal NC at small surface-to-molecule separations. Another note about CDNC,dipole-external field is that it creates Fano-like line shapes in the CD spectrum. The quantities μ12R 3 m21R are imaginary; i.e., μ12R 3 m21R = iaR, where aR are the real numbers and R = x, y, z. Then, we see that for Γ12 . |Im[Gω]| and p|ω-ω0| . |Re[Gω]   1 CDNC;dipole-externaldipole ∼ Re pω - pω0 þ iΓ12 ¼

pω - pω0 ðpω - pω0 Þ2 þ Γ212

This is a contribution with both positive and negative CD bands. The function CD NC,dipole-dipole is given by an infinite series in terms of R -n . The integral J(ω) has the following limits JðωÞ ¼

π 1 ðμ μ þ μ12y μ21y þ 2μ12z μ21z Þ Δ3 12x 21x jε0 þ εNC j2

where Δ = R - a NP , a NP . The above limit corresponds to small molecule-to-surface distances. For long moleculeNC distances, we use the dipole approximation and

Figure 5. (a) Model of a system composed of a nanocrystal dimer and a chiral molecule. The material parameters correspond to the regime of an off-resonance exciton-plasmon coupling. (b) CD spectra of the hybrid dimer complex for two cases: R1 = R2 = 7.5 nm and R1 = R2 = ¥. We also show the CD spectrum for a single NP for R = 7.5 nm.

obtain JðωÞ ¼

4π 3 3 a ðμ μ þ μ21y μ12y þ 4μ21z μ12z Þ R 6 NP jεNC þ 2ε0 j2 21x 12x

where R . a NP . Figure 3 presents numerical data obtained from the above equations. The molecular dipoles can be written as μ12 = |e| 3 r12 and (μ12 3 m21)/μ12 = -i 3 |e|r0 3 r12 3 ω0/(2c0), where r12 is the dipolar matrix element and r0 is a parameter describing an amplitude of the molecular CD effect. In Figure 3, we used the following parameters for the chiral molecule: λ0 = 300 nm, r12 = 2 Å, r0 = 0.1 Å, and Γ12 = 0.3 eV. For the metal dielectric functions, we used empirical data from ref 35 (gold) and ref 36 (silver). These parameters give typical numbers for the amplitudes of the optical responses of an isolated molecule: ε0 ∼ 3  104 cm-1 M-1 and ΔεCD ∼ 20 cm-1 M-1 at λ0 = 300 nm. We see that the CD signal at the plasmon frequency becomes very strong for small molecule-surface separations. The CD data in Figure 3 are given for three separations Δ = R - aNP = 1 nm, 3 nm, ¥, assuming that aNP = 5 nm. Other observations from Figure 3 are the following: (1) a sign of CD signal depends on an orientation of a molecular dipole with respect to the NC surface and (2) plasmon-induced CD for Ag-NCs is much stronger than that for Au-NCs. The observation (2) comes from the fact that the Ag-plasmon wavelength (∼400 nm) is located much closer to the molecular band wavelength (300 nm). It is also interesting to look at the distance dependence of the plasmonic CD. In Figure 4, we show calculated plasmonic CD signals at λ = 523 nm as a function of the molecule-NC distance R. We can see that the leading contribution to the CD signal is CDNC,dipole-external þ CDmolecule, whereas the contribution CDNC,dipole-dipole is only essential at very small moleculesurface separations. In section 2, we already discussed this behavior of CDNC,dipole-dipole. In addition, at the plasmonic wavelength, CDNC,dipole-external . CDmolecule, and, in Figure 4, the function CDNC,dipole-external þ CDmolecule ≈ CDNC,dipole-external. Mathematically, all contributions approach zero at Δ f 0 because of the plasmon-induced broadening of molecular transition Im 7920

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The Journal of Physical Chemistry C

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Figure 6. CD spectra for a hybrid complex with two Ag NCs in the regime of resonant exciton-plasmon interaction. The plot also shows the CD spectra of a single NC, for comparison. The parameters used in the calculations are the following: r12 = 2 Å 3 , r0 = 0.02 Å, and Γ12 = 0.1 eV.

Figure 7. CD spectra for a molecule-NC complex in the regime of offresonance exciton-plasmon interaction. This system mimics a hybrid photosystem-NC complex investigated experimentally in ref 16. A photosynthetic molecule (photosystem I) has optical dipoles at λ0 ∼ 680 nm. The other parameters are r12 = 1 Å, r0 = 0.01 Å, and Γ12 = 0.05 eV.

Gω∼Δ-3 f ¥. In particular, both CDNC,dipole-external and CDNC,dipole-dipole are proportional to Δ3 for very small Δ. We should comment here that the plasmon enhancement at the position Δ = 0 is maximal, but there is a stronger effect that leads to the vanishing of the calculated CD signals at the metal surface. Mathematically, the resonances at very small Δ become strongly broadened and also shifted. This behavior is given by the expression for the pole in eqs 9 and 11: pω = pω0 iΓ12 þ Gω. 3.2. A Nanoparticle Dimer and a Chiral Molecule. The geometry of the complex with two metal NCs is shown in Figure 5a. NCs have the same radius (aNP). To obtain analytical results for the dimer geometry, we employ here the dipole approximation for the treatment of the Coulombic NC-NC interaction. It helps that the dipolar fields rapidly decay in space. Then, this approximation is valid if NCs are not very close to each other (D ∼ 3aNC or D > 3aNC).37 Here, D is the center-to-center distance between two NCs (see Figure 5). Within this approach, the field-enhancement matrix and other matrices (eqs 8-10) are diagonal  cR β 1 1   þ PRR ¼ 1 þ , cR β R1 3 R2 3 1- 3 D

0

CDmolecule ¼ E20

1

B cR β 1C C B  3 C, KRR, i ¼ KRR ¼ γðωÞB1 þ  @ cR β D A 1- 3 D totðRÞ

∂Φω, insideNC1 ∂R totðRÞ

!! γðωÞ cR β D3 c2R c3R β ¼ þ þ 3 , ε0 R1 3 D6 - c2R β2 3 R2 3 R1

∂Φω, insideNC2 ∂R

!! γðωÞ cR β D3 c2R c3R β þ þ 3 ¼ ε0 R2 3 D6 - c2R β2 3 R1 3 R2

where cx = cy = -1 and cz = 2. Details of the derivation are given in the Supporting Information. Here we omit for simplicity the term CDNC,dipole-dipole which contributes only at very small molecule-surface separations. The components of the CD signal are now calculated as shown in eq 13, where the index

8pffiffiffiffi Γ12    ε0 ω 0 2 Im½Pxx ðμ12x 3 m21x Þ þ Pyy ðμ12y 3 m21y Þ þ Pzz ðμ12z 3 m21z Þ 3 jpω - pω0 þ iΓ12 - Gω, 2NC j

pffiffiffiffi 8 CDNC ¼ ImðεNP Þ a3NP ω 3 ε0 E20 3 90 2 6 6 6 6 Im6 6i ¼ 1, 2 6 4

∑

totðxÞ ∂Φω, inside NCi @ðμ12x m21x ÞK  xx 3

∂x

totðyÞ  ∂Φω, inside NCi þ ðμ12y 3 m21y ÞKyy

∂y

pω - pω0 þ iΓ12 - Gω, 2NC

7921

13

totðyÞ  ∂Φω, inside NCi A7 þ ðμ12z 3 m21z ÞKyy 7 7

∂y

7 7 7 7 5

ð13Þ

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The Journal of Physical Chemistry C i = 1, 2 is the NC number. An approximated equation for Gω,2NC is given in the Supporting Information. We mostly can neglect Gω,2NC compared to Γ12 except for very small molecule-surface separations. We now apply our results to a few particular cases. Figure 5 shows the data for the hybrid complex with an offresonant exciton-plasmon interaction (λ0 < λplasmon). We see that the dimer geometry has a strongly increased CD signal at the plasmonic wavelength compared to the case of a single NC. This increase is mostly due to the interference of the molecular dipole with two isolated plasmons. For the chosen NP-NP gap (5 nm), the NP-NP plasmonic interaction is relatively weak, and the resultant amplification of the electric fields in the gap between the NCs is not so strong. In the systems with closely located NCs (such as bow-tie antennas, gap antennas, split gates, etc), electricfield enhancements in the gap region can be much stronger,38,39 and this would be a way to achieve much stronger enhancements of CD signals. Another possibility to achieve strong plasmonic enhancements for an external field and for Coulombic interactions is to use complex geometries such as rods, prisms, tips, etc.40,41 In such structures, chiral molecules should be placed in selected regions with a strong plasmon enhancement. Importantly, eqs 8-11 are fully applied to structures with an arbitrary,

Figure 8. (a) Models of Au nanoparticle and Au nanoshell. (b) Calculated field-enhancement factors for a dipole located in the vicinity of a nanoparticle and nanoshells.

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^ ^ , Φtot complex geometry. However, the functions (P ω , K ) in eqs 8-11 should be calculated numerically. In Figure 6, we show CD data for a complex with a resonant Coulombic coupling (λ0 ≈ λplasmon). In this, resonant regime, CD lines have Fano-like shapes. These shapes come from the both terms CDmolecule and CDNC,dipole-external field. In this case, we see a significant amplification of CD lines that comes from the plasmon enhancement effect for external and internal fields. The CD amplification and the plasmon enhancement are about 10-fold. The other offresonant regime (λ0 > λplasmon) is relevant to the recently observed enhancement of intrinsic CD lines of the photosynthetic molecules (photosystem I) in the red absorption band with λ0 = 680 nm.16 In Figure 7, we make an attempt to mimic a CD signal from a chromophore placed in a gap between two Au NPs. The molecular transition and the Au plasmons are not in resonance. Nevertheless, one can see an enhancement of CD for a molecule with a perpendicular dipole. 3.3. A Nanoshell and a Chiral Molecule. Another model allowing for an exact solution incorporates a nanoshell (NS) and a dipole (Figure 8a). The nanoshell geometry has stronger plasmon resonances compared with spherical NCs (Figure 8).23,42 This stronger plasma enhancement comes from the fact that a NS has a more nonuniform geometry with an additional inner surface. In addition, these plasmon resonances are red-shifted and have narrower plasmon-enhancement peaks (Figure 8b). A magnitude of plasmon enhancement and a position of plasmon peak in a NS strongly depend on the ratio b/a (Figure 8b). As the metal nanoshell width (b - a) decreases and b/a f 1, the plasmon peak becomes further shifted to the red wavelength region and the plasmon enhancement becomes stronger. However, our approach will not be valid for very thin shells since thin shells cannot be described with a local dielectric function taken from a bulk metal. To calculate the CD spectrum for the molecule-NS complex, we should again calculate the functions entering eqs 8^ (r), and Φtot 11: Gω, PRR, K ω . Due to the spherical symmetry, there functions can be calculated as series in terms of spherical harmonics. In the Supporting Information, we outline the corresponding derivations. As expected, the chiral molecule-NS system demonstrates stronger plasmonic CD signals

Figure 9. CD spectra for the chiral molecule-NS complex in the regime of an off-resonant coupling. (a) A molecule with λ0 = 300 nm. (b) A molecular transition with λ0 = 200 nm. The geometrical parameters are a = 8, b = 10, and R = 11 nm. 7922

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The Journal of Physical Chemistry C in comparison with the molecule-NC complex (Figures 3 and 9). For the molecule with λ0 = 300 nm, the intensities of the plasmon CD peaks in the case of a NS (Figure 9a) are much stronger than those for a NP (Figures 3b and c). We also can see that the NS system allows us to create strong plasmonic CD lines in the red spectral interval.

4. CONCLUSIONS In this paper, we investigated the effect of plasmonic nanocrystals on CD spectra of chiral molecules. Using the formalism developed here, one can model CD spectra of hybrid molecule-nanocrystal complexes with arbitrary geometry. The plasmon enhancement and the interference effect change a CD spectrum of a chiral molecule in two ways. First, a CD spectrum of a hybrid complex acquires new CD lines at plasmonic wavelengths. In other words, plasmon excitations become chiral. Using this effect, one can shift the molecular chirality of many biomolecules from the UV spectral range to the visible. Second, plasmonic nanocrystals can amplify natural CD lines of molecules through the plasmon-enhancement and interference effects. To amplify the plasmonic effects in CD spectra, one should use nanostructures with a strong plasmonic enhancement, such as nanoparticle pairs, split-gate structures, nanoshells, etc. ’ ASSOCIATED CONTENT

bS

Supporting Information. Details of analytical derivations. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Phone (740)-593-9430; fax (740)-593-0433; e-mail govorov@ helios.phy.ohiou.edu.

’ ACKNOWLEDGMENT This work was supported by the Air Force Research Laboratories (Dayton, OH), NSF (Award: CBET-0933782), and Volkswagen Foundation. ’ REFERENCES (1) Circular Dichroism and the Conformational Analysis of Biomolecules; Fasman, G. D., Ed.; Plenum: New York, 1996. (2) Gautier, C.; B€urgi, T. ChemPhysChem 2009, 10, 483. (3) Santizo, I. E.; Hidalgo, F.; Perez, L. A.; Noguez, C.; Garzon, I. L. J. Phys. Chem. C 2008, 112, 17533. (4) Molotsky, T.; Tamarin, T.; Moshe, A. B.; Markovich, G.; Kotlyar, A. B. J. Phys. Chem. C 2010, 114, 15951. (5) Cathcart, N.; Mistry, P.; Makra, C.; Pietrobon, B.; Coombs, N.; Jelokhani-Niaraki, M.; Kitaev, V. Langmuir 2009, 25, 5840. (6) Ha, J.-M.; Solovyov, A.; Katz, A. Langmuir 2009, 25, 10548. George, J.; Thomas, K. G. J. Am. Chem. Soc. 2010, 132, 2502. (7) Govan, J. E.; Jan, E.; Querejeta, A.; Kotov, N. A.; Gun’ko, Y. K. Chem. Commun. 2010, 46, 6072. (8) Goldsmith, M.-R.; George, C. B.; Zuber, G.; Naaman, R.; Waldeck, D. H.; Wipf, P.; Beratan, D. N. Phys. Chem. Chem. Phys. 2006, 8, 63–67. (9) Oh, H. S.; Liu, S.; Jee, H. S.; Baev, A.; Swihart, M. T.; Prasad, P. N. J. Am. Chem. Soc. 2010, 132, 17346.

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