Chiral Nanocrystals: Plasmonic Spectra and Circular Dichroism

May 16, 2012 - In the examples described in this paragraph, CD signals can be utilized as a sensor parameter and as a measure of geometrical chirality...
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Chiral Nanocrystals: Plasmonic Spectra and Circular Dichroism Zhiyuan Fan and Alexander O. Govorov* Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, United States S Supporting Information *

ABSTRACT: The life is inherently chiral. Consequently, chirality plays a pivotal role in biochemistry and the evolution of life itself. Optical manifestation of chirality of biomolecules, so-called circular dichroism, is a remarkable but relatively weak effect appearing typically in the UV. In contrast to the biomolecules, plasmonic nanocrystals offer an interesting opportunity to create strong circular dichroism (CD) in the visible spectral range. Here we describe plasmonic properties of single chiral nanocrystals and focus on a new mechanism of optical chirality originating from a chiral shape of a nanocrystal. After careful examination, we found that this CD mechanism is induced by the mixing between different plasmon harmonics and is qualitatively different from the previously described dipolar CD effect in chiral assemblies of spherical nanoparticles. Chiral plasmonic nanocrystals studied here offer a new approach for the creation of nanomaterials with strong chiroptical responses in the visible spectral interval. KEYWORDS: Nanocrystal, plasmon, chirality, circular dichroism, optical properties he field of chiral nanocrystals and nanostructures is rapidly developing (Figure 1). The concept of chirality is really intriguing and came from biology and chemistry.1,2 Most naturally occurring molecules are chiral. Chirality of molecules plays an important role in chemical reactions providing the recognition function to the molecules; molecules with different chirality may react differently. Another example is that the majority of modern drugs are chiral because when used they should interact with chiral biomolecules in a human body. Therefore, it is important to sense chirality. Circular dichroism (CD) spectroscopy is one of the central methods to probe chiral states of molecules. In the CD spectroscopy, chirality of molecular and nanoscale objects is sensed using circularly polarized light (left-handed and right-handed). The difference in the absorptions of circularly polarized beams tells us about handedness (chirality) of molecules. The emerging chiral nanostructures3−16 (Figure 1) have new and enhanced properties and are natural candidates for applications in chiral sensing and optical materials. In that respect, plasmonic chiral nanostructures12,17−24 look especially interesting because they exhibit strong plasmon resonances and are able to enhance locally incident electromagnetic fields. Recent developments in this field of chiral plasmonic nanostructures include biosensing with plasmon-enhanced electromagnetic fields,18,25 induced plasmonic chirality using interacting biomolecules and nanocrystals,3,13−15 chiral nanocrystal assemblies with well-controlled geomentries,9,10,26 and so forth. By definition, a chiral NC has no mirror symmetry planes and therefore should be anisotropic (Figure 2). Anisotropic NCs are a new, interesting direction of nanotechnology.8,27−30 However, to date the majority of reported anisotropic NCs appeared to be achiral. It is technologically challenging to create asymmetric (chiral) nanoparticles that are uniform in shape and

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© XXXX American Chemical Society

size in a macroscopic ensemble in a solution. One recent success in this direction is a report on fabrication of chiral colloidal nanorods and hexagons8 (Figure 1b), although the nanocrystal solution reported in ref 8 appeared to be racemic (i.e., macroscopically achiral) and did not show measurable CD signals. Here we describe theoretically a mechanism of optical chirality which is characteristic for single plasmonic nanocrystals (NCs) with chiral shapes (Figure 2). A chiral distortion of NC surface creates both splitting and mixing of plasmonic modes. Simultaneously with the above effects, a NC acquires optical chirality and strong CD responses at the plasmon frequency. In particular, we found that chiral nanocrystals with twisted, helical distortions (Figures 2b,c) display the strongest CD responses. The CD mechanism identified here comes from the mixing between plasmon harmonics with different angular momenta whereas the central plasmonic CD mechanism described previously10,12,26 originated from the plasmon− plasmon interaction in a nanoparticle assembly with a chiral arrangement of nanoparticles. The CD mechanism coming from the plasmon−plasmon interaction10,12,26 resembles a dipolar CD mechanism introduced in a seminal paper31 to describe a protein α-helix. In that respect, the CD mechanism proposed here has no direct analogs in the molecular spectroscopy since it is specific to the optically active nanocrystals. Along with interesting physical properties (such as CD and optical rotation), chiral nanocrystals have also relevance to chiral sensing, plasmonic materials, and methods of assembly at Received: April 11, 2012 Revised: May 8, 2012

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Figure 1. Chiral nanocrystals available in the literature. (a) A semiconductor quantum dot conjugated with chiral molecules. Reproduced with permission from ref 7. Copyright 2007, The Royal Society of Chemistry. (b) Chiral silver nanorods. Reproduced with permission from ref 8. Copyright 2011, American Chemical Society. (c) Chiral gold-nanoparticle helices assembled with a help of DNA. Reproduced with permission from ref 9. Copyright 2009, the American Association for the Advancement of Science. (d) Theoretical model of plasmonic helix fabricated and studies in ref 10. (e) Chiral colloidal complexes assembled in a magnetic field. Reproduced with permission from ref 11. Copyright 2008, Nature Publishing Group.

E = E0 + δ E,

B = B0 + δ B

where E0 and B0 are the fields of an incident electromagnetic wave and δE and δB are the perturbations coming from a plasmonic NC. The incident fields are E0 = E0,ω·e−iωt + E*0,ω·eiωt and B0,ω = (ε0)1/2[k × E0,ω]/k, where Eω,0 = 1/2 e0E0ei(ε0) k·r is a complex amplitude, e0 is the polarization vector, and ω and k are the frequency and wavevector in vacuum, respectively; ω = c0k, where c0 is the vacuum light speed; k = 2π/λ0, where λ0 is the photon wavelength in vacuum; ε0 is the optical dielectric constant of matrix (water). Employing the spherical coordinates, we consider an incident wave approaching our system at the spherical angles θk and φk (Figure 2). A full solution for a NC of an arbitrary shape should be found from the Maxwell’s equations and can be expressed via the electromagnetic potentials

Figure 2. Chiral nanocrystals studied in this paper and the system of coordinates used to calculate CD responses.

the nanoscale. They can be used as building blocks of more complex nanoscale constructions in which interactions between nanocrystals depend on chirality. Potentially chiral nanocrystals can be crystallized, like chiral molecules, and in this way nanocrystal enantiomers can be separated. Interactions between nanocrystal enantiomers and chiral molecules may depend on chirality that can be potentially used for sensors. Such interactions can be chemical or via dynamic Coulomb forces32 under optical excitation. In the examples described in this paragraph, CD signals can be utilized as a sensor parameter and as a measure of geometrical chirality and structural homogeneity of NCs in a solution. 1. Formalism. 1.1. General Equations. We now consider a metal NC with a complex shape (Figure 2). A convenient formalism to describe the system is the approach of local dielectric function, ε(r). For the material of NCs, we assume gold. Then, we take the dielectric function of NC (εAu) from the empirical tables.33 A matrix medium is water with an optical dielectric constant ε0 = 1.8. A NC is excited by an incident monochromatic wave of optical frequency (Figure 2). Then, the total electromagnetic fields in our system can be written as

δ E = −∇δφind −

1 ∂δ A ind , c ∂t

δ B = ∇ × δ A ind

(1)

where δφind and δAind are the potentials induced by the plasmonic charges on a NC. The formalism becomes strongly simplified for NCs with small dimensions such that R ≪ λ, where R is a nanocrystal size. In particular, for small nanocrystals (R ≪ λ), the induced vector potential δAind is weak and can be neglected (Supporting Information) and, therefore, δE≈ − ∇δφind, δB ≈ 0, and also Eω ,0 ≈ e0E0(1 + i ε0 k·r)

(2)

Then, the solution of the problem can be found from the Poisson equation ∇⃗ ·ε(r )∇⃗δφind = f (r)

(3)

in which the right-hand side contains the expanded external field (eq 2) f (r) = ∇⃗ ·ε(r)Eω ,0 = ∇⃗ ·ε(r)(e0E0[1 + i ε0 k·r]) B

(4)

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CD. Simultaneously, the scattering rate Qscat ∼ k4 for small NCs and, therefore, can be neglected. It follows from our derivations above that the longwavelength approximation employed here is valid for relatively small radii R0, such that (kR0)2 ≪ 1. Indeed, the parameter (kR0)2 ∼ 0.008 for the conditions used below (R0 = 7 nm and λ ∼ 500 nm). A nonchiral object has zero CD strength and, therefore, a CD response can be taken as a figure of merit for chirality of an absorbing object. Convenient units for an extinction and CD are 1/M·cm and the corresponding conversion is performed with the following formulas

In the following calculations, we will see that the term linear in k in the expansion of Eω,0 is the key term to obtain nonzero CD signals from small NCs. The optical absorption by a NC is calculated in the usual manner Qk =

∫NC dV ⟨j·E⟩t = Im(εNC) 2ωπ ∫NC Eω·E*ω dV

(5)

where the integral is taken over the volume of metal NC; ⟨...⟩t is a standard time averaging, and j is the electric current density. The complex amplitude of electric current inside metal NCs jω = −iω(εNC − 1)/4π·Eω. The CD signal is defined as the difference between two absorptions CD = ⟨Q k, + − Q k, −⟩

εext =

where an average over the solid angle, ⟨...⟩, is needed since NCs are assumed to be in a solution and, therefore, have random orientations. The rates of absorption, Qk,+(−), should be calculated for two incident electromagnetic waves with different circular polarizations: e0⃗ ± =

ΔεCD =

2

φind|− = φind|+ ,

where the symbols ∓ denote the internal and external sides of the NC surface and n is the normal to the NC surface. Importantly, the above boundary conditions are written at the surface of NC which can be defined via an equation

r = RNC(θ , φ) The function RNC(θ,φ) now defines a NC shape. Then, by using various functions RNC(θ,φ), one can generate various chiral NCs, shown in Figure 2. Concrete functions RNC(θ,φ) generating the chiral shapes used here (Figure 2) can be found in Supporting Information. A general solution for the quasi-static problem can be conveniently expanded in terms of spherical harmonics

Then, CD measured in an experiment can be calculated as CD = ⟨CDk⟩Ωk. Computationally, it is demanding to calculate ⟨CDk⟩Ωk as an integral over θk and φk. Instead, it is sufficient to calculate the CD strength for six main directions2

in φind =

∑ l≥0

⎛ r ⎞l Al , m ⎜ ⎟ Yl , m(θ , φ), ⎝ aNC ⎠

(insideNC)

l ≥ m ≥−l

∫ CDk ·sin θk dθk dφk

out φind =

∑ l≥0

= (CDk || x ̂ + CDk ||− x ̂ + CDk || y + CDk ||− y + CDk || z + CDk ||− z )/6

εAu n ·(E0 + δ E)|− = ε0 n ·(E0 + δ E)|+ (7)

CDk = Q k, + − Q k, −

1 4π

NA × 10−4 2π CD 0.23 c0 ε0 E02

where the absorption rates Q and CD should have the cgs units. 1.2. Formalism to Compute Electromagnetic Fields. Equations 3 and 4 should be now used to calculate the induced electric field δE = −∇φind. The standard boundary conditions for the induced potential can be written as

eθ⃗ ± ieϕ⃗

where eθ⃗ (ϕ) are the orthogonal unit vectors along the directions perpendicular to the wave vector: eθ⃗ (ϕ)⊥k⃗ and the vector eϕ⃗ lies in the x−y plane. The averaging ⟨...⟩ in the above equations can be done in two ways: (1) A NC is rotated, whereas the incident angles of the electromagnetic wave (θk and φk) are kept constant; (2) Internal axes of a NC are fixed and the electromagnetic waves with polarizations ± approach a NC from all possible directions. These ways to perform the averaging are equivalent. In experiments with a solution, NCs have random orientations but the wavevector of incident light (k) has a well-defined direction. In other words, in a solution, one should perform an averaging over all possible orientations of a NC. But, here we will use the method 2 since it is more convenient for calculations. In other words, we will perform an averaging over all directions of k. It is also convenient to define a “directional” CD strength as

CD =

NA × 10−4 2π ⟨Q ⟩Ωk , 2 0.23 c0 ε0 E0

⎛ a ⎞l + 1 Bl , m⎜ NC ⎟ Yl , m(θ , φ), ⎝ r ⎠

(outsideNC)

l ≥ m ≥−l

(8)

(6)

Equation 8 defines the induced potential for two spatial regions, inside and outside of a NC; aNC is a constant and can be taken as an averaged radius of a NC. The boundary conditions 7 should be now applied to the functions 8 at a curved surface. This procedure leads to a system of coupled linear equations

The proof for eq 6 is given in Supporting Information and valid for any small system (kR ≪ 1). If a plasmonic system is small, the CD strength can be calculated using the dissipation integral (eq 5) and, then, can be expanded in terms of k:12 CD = ak 2 + bk 4 + ...

∑

where a and b are constants which depend on a NC geometry . Since kR is a small parameter, it will be sufficient for us to keep in the following only the first term: CD ≈ ak2. We also note that the absorption rate of a NC Q ∼ k and, therefore, Q ≫

Al , mal ′ , m ′ ; l , m + Bl , mbl ′ , m ′ ; l , m = 0

l ≥ 0, −l ≥ m ≥ l

∑ l ≥ 0, −l ≥ m ≥ l

C

Al , m ·al̃ ′ , m ′ , lm +

∑ l ≥ 1, m

Bl , m ·bl̃ ′ , m ′ , lm = Fext, l ′ m ′ (9)

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Figure 3. Convergence of the method. (a,b) Absorption and CD spectra as a function of wavelength calculated for various numbers of involved harmonics Lmax. (c) CD signal at a particular wavelength (λ = 550 nm) as a function of Lmax. Inset: Model of the calculated nanocrystal (right-handed twister).

Figure 4. Modal analysis of plasmon resonances of a chiral nanocrystal within the Drude model. The absorption rate averaged over all incident directions is shown as a function of wavelength for a twister with a very weak surface variation. Inset: Model of the chiral nanocrystal and its comparison with an ideal sphere.

where the matrix elements a, b, ã, b̃, and Fext are defined in the Supporting Information. In addition, the Supporting Information gives a detailed derivation of eq 9. 1.3. Convergence of the Method. The infinite system of eq 9 should be first truncated at l = Lmax and then solved numerically. Correspondingly, the number of involved harmonics for a given Lmax is nhar = (Lmax + 1)2. We first test the formalism using an analytically solvable model, a metallic

ellipsoid.34 By taking Lmax sufficiently large, we found an excellent agreement between our computational results and the exact solutions for prolate and oblate ellipsoids. The second step is an application of the formalism to a NC with a more complex shape. As the first example, we choose a twister (Figure 3). This twister has a relatively small variation of radius, on purpose. This NC is close to a sphere since the surface distortion is relatively small, δR/R0 = 0.14, where R0 and δR are D

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Figure 5. Mechanism of CD coming from the mixing between plasmonic modes within the Drude model. (a) The spectrum of absorption for a small plasmon broadening. (b) The lowest triplet modes (l = 1) for various plasmon broadenings. (c) CD spectra of the triplet modes, plotted again for different broadenings. Inset: Model of plasmonic twister and its comparison with a sphere.

plasmonic resonances, it is instructive to use first the Drude model with a small plasmonic broadening γp

the average radius and the amplitude of radius variation, respectively. In this way, we also like to test the precision in our calculations of CD signals which are expected to be relatively small. Indeed, the absorption spectrum (Figure 3a) is converging very rapidly with Lmax and close to that of a spherical particle. The CD signal, as expected, is a few orders of magnitude smaller and also converges very well, but the convergence occurs at larger numbers Lmax (∼15). Then, the corresponding number of involved harmonics needed to compute a CD signal for this NC is ∼250. In the following examples of NCs with relatively strong surface irregularities, we will be using Lmax ∼ 30 and, correspondingly, nhar ∼ 960. This will be sufficient to obtain convergence for our NCs within the numerical method employed here. 2. Plasmonic Circular Dichroism. 2.1. Modal Analysis. We start numerical simulations with the modal analysis of a chiral NC. Since gold nanoparticles have strongly broadened

εmetal = 1 −

ωp2 ω(ω + iγp)

in which we choose ωp = 4 eV. Figure 4 shows the results for a twister with a very small surface variation (δR/R0 = 0.05) and a small plasmon broadening (γp = 10−5 eV). As expected, each lmode is now split into a multiplet with 2l + 1 lines. The multiplets are centered at ωl = ωp/(1 + ((l + 1)/l)ε0)1/2. A chiral shape of a NC lifts the degeneracy of the l-modes since a chiral NC has no spatial symmetries. Figure 4 shows the absorption in the logarithmic scale. This scale allows us to distinguish many sharp peaks and also shows that the dipolar transitions (l = 1) dominate the spectrum since the NC is close to spherical. This result is an additional proof for the multipleexpansion method introduced by us above in Section 1. E

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2.2. Mechanism of CD: Mode Mixing. To reveal the mechanism of CD in our nanostructures, we will now look at the formation of CD signals in the Drude model with narrow plasmon lines. This is, of course, an artificial model, but this model will allow us to observe the formation of plasmonic CD spectra of NCs with chiral shapes. Figure 5 focuses on the dipolar transitions. With increasing γp, the plasmon resonances first broaden and then merge, as expected (Figure 5b). The corresponding CD lines behave in an interesting way. For a very small broadening, one can see individual asymmetric lines, whereas for larger γp, the lines merge and form a typical bisignate spectrum with positive and negative bands. In our calculations, we clearly observe that the CD signals appear only for chiral shapes and simultaneously with the splitting between plasmon modes in a multiplet. The splittings in the plasmonic multiplets are an effect coming from the mixing between plasmon modes owing to a chiral surface distortion. Therefore, we can conclude here that the mechanism of CD originates from the mixing between plasmonic modes in a nonspherical NC. Additional analysis shows that the CD signal appears only if we involve in our calculation more than 4 harmonics, that is, the parameter lmax should be larger than 1 (see Supporting Information). This also tells us the mixing between multiplets (i.e., between the modes with l = 1 and l = 2) is critical for the formation of CD signals. It is interesting to compare the CD mechanism described here with the well-established mechanism of CD in protein α-helices. Our mechanism comes from the interaction (mixing) of plasmonic harmonics on a single NC owing to a chiral surface distortion, whereas the CD mechanism in a molecular α-helix originates from the Coulomb interaction of point dipoles arranged into a helical geometry. 2.3. CD Spectra of Au Nanocrystals. We now consider plasmonic CD signals of gold NCs with a dielectric function taken from ref 33. Using various functions r = RNC(θ,φ), we now generate a variety of chiral shapes (Figures 2 and 6). The Supporting Information includes particular analytical expressions for twisters, antitwisters, asymmetric pyramids, and chiral tetrahedron-like structures (Figure 6a−c). All these shapes involve two parameters: R0 (average radius) and δR (amplitude of variation of radius). Additional parameter Γ involved in the functions r = RNC(θ,φ) defines a smoothness of a NC surface. The largest CD was calculated for the twisters and antitwisters (Figure 6a) since these shapes have certainly strong geometrical chirality coming from the helical band on the NC surface. The weakest optical chirality was found in the spherical NC with a tetrahedral surface distortion (Figure 6c). Interestingly, it qualitatively agrees with our previous calculations on chiral nanoparticle assemblies.12 Tetrahedrons composed of spherical nanoparticles (NPs) calculated in ref 12. exhibit nearly zero CD signals, whereas asymmetric pyramids and helices composed of interacting NPs have typically rather strong CD responses.12 Fabrication of chiral nanocrystals remains a challenge. One approach is to use chiral templates (biomolecules or nanotubes) to grow metal nanoparticles.4,6,16 In this case, NCs can acquire a chiral geometry from a template and it can be that a chiral distortion of metal surface becomes stabilized by ligands of chiral template. Remarkably, recent experiments with chiral molecular templates and metal NCs demonstrated the appearance of plasmonic bisignate CD bands4,6,16 that are very similar to the calculated spectra in Figure 6. Another fabrication method is to assemble preformed particles using biomolecules10,26,35 or porous silica films.36 Moreover, a chiral discrete bioassembly of plasmonic particles can be transformed

Figure 6. Results for the CD response for various types of chiral Au nanocrystals. (a) Right-handed twister and antitwister with δR = 1.5 and −0.8 nm. (b) Asymmetric pyramid with α = 1. (c) Chiral tetrahedral structure with a0 = 2 nm. In all cases, R0 = 7 nm.

into a continuous NC using an overgrowth in a solution.10 This process can strongly amplify CD signals since collective interactions become much stronger in tightly packed12,37 or continuous nanostructures.38 Another class of plasmonic chiral nanostructures is the oriented lithographically fabricated 3D metamaterials that typically display strong directional CD signals.22,23 In particular, a very recent paper23 reported the fabrication of 3D molecular-like oligomer structures with chiral plasmon resonances. Conclusions. We have investigated theoretically plasmonic properties of nanocrystals with chiral shapes. The peculiar property of these nanoscale objects is strong CD signals. A F

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(21) Guerrero-Martínez, A.; Auguie, B.; Alonso-Gomez, J. L.; Dzolic, Z.; Gomez-Grana, S.; Zinic, M.; Cid, M. M.; Liz-Marzan, L. M. Angew. Chem. 2011, 50, 5499. (22) Helgert, C.; Pshenay-Severin, E.; Falkner, M.; Menzel, C.; Rockstuhl, C.; Kley, E.-B.; Tnnermann, A.; Lederer, F.; Pertsch, T. Nano Lett. 2011, 11, 4400. (23) Hentschel, M.; Schäferling, M.; Weiss, T.; Liu, N.; Giessen, H. Nano Lett. 2012, 5 (12), 2542−2547. (24) Li, Z.; Zhu, Z.; Liu, W.; Zhou, Y.; Han, B.; Gao, Y.; Tang, Z. J. Am. Chem. Soc. 2012, 134, 3322. (25) Abdulrahman, N. A.; Fan, Z.; Tonooka, T.; Kelly, S. M.; Gadegaard, N.; Hendry, E.; Govorov, A. O.; Kadodwala, M. Nano Lett. 2012, 12, 977. (26) Shen, X.; Song, C.; Wang, J.; Shi, D.; Wang, Z.; Liu, N.; Ding, B. J. Am. Chem. Soc. 2012, 134, 146. (27) Zhang, J.; Langille, M. R.; Personick, M. L.; Zhang, K.; Li, S.; Mirkin, C. A. J. Am. Chem. Soc. 2010, 132, 14012. (28) Kelly, K. L.; Coronado, E.; Zhao, L.; Schatz, G. C. J. Phys. Chem. B 2003, 107 (3), 668. (29) Zhang, S.; Bao, K.; Halas, N. J.; Xu, H.; Nordlander, P. Nano Lett. 2011, 11, 1657. (30) Geitner, N. K.; Doepke, A.; Fickenscher, M. A.; Yarrison-Rice, J. M.; Heineman, W. R.; Jackson, H. E.; Smith, L. M. Nanotechnology 2011, 22, 275607. (31) Moffitt, W. J. Chem. Phys. 1956, 25, 467. (32) Govorov, A. O.; Fan, Z. Y.; Hernadez, P.; Slocik, J. M.; Naik, R. R. Nano Lett. 2010, 10, 1374. (33) Johnson, P. B.; Christy, R. W. Phys. Rev. B. 1972, 6, 4370. (34) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley-VCH: Weinheim, 2004. (35) Xiong, H.; van der Lelie, D.; Gang, O.. J. Am. Chem. Soc. 2008, 130 (8), 2442. (36) Qi, H.; Shopsowitz, K. E.; Hamad, W. Y.; MacLachlan, M. J. J. Am. Chem. Soc. 2011, 133, 3728. (37) Fan, Z.; Govorov, A. O. J. Phys. Chem. C 2011, 115, 13254. (38) Zhang, Z.-Y.; Zhao, Y.-P. J. Appl. Phys. 2008, 104, 013517.

typical CD spectrum of a chiral NC exhibits a bisignate Cotton effect2 centered at the main plasmonic resonance. The mechanism of plasmonic CD described here originates from the mixing of plasmonic harmonics induced by a chiral NC shape. Chiral nanocrystals are a novel, interesting type of nanostructures with potential applications in chiral sensing and optical devices.

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ASSOCIATED CONTENT

S Supporting Information *

Details of modeling. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the NSF (project: CBET0933782) and by the Volkswagen Foundation.

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REFERENCES

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