Electromagnetic Interactions. A dissertation presented to. the faculty of. the College of Arts and Sciences of Ohio University. In partial fulfillment

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1 Optical Activity of Chiral Nanomaterials: Effects of Short Range and Long Range Electromagnetic Interactions A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Zhiyuan Fan May Zhiyuan Fan. All Rights Reserved.

2 2 This dissertation titled Optical Activity of Chiral Nanomaterials: Effects from Short Range to Long Range Electromagnetic Interactions by ZHIYUAN FAN has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by Alexander O. Govorov Professor of Physics and Astronomy Robert Frank Dean, College of Arts and Sciences

3 3 ABSTRACT FAN, ZHIYUAN, Ph.D., May 2014, Physics and Astronomy Optical Activity of Chiral Nanomaterials: Effects from Short Range to Long Range Electromagnetic Interactions Director of Dissertation: Alexander O. Govorov In this dissertation, chiral nanomaterials with new plasmonic properties have been investigated. Electromagnetic interactions between well-defined building blocks in nanomaterials are modeled using classical and quantum mechanical theories. We predict several new mechanisms of plasmonic circular dichroism (CD) signals in chiral nanomaterials. The predicted CD mechanisms include plasmon-plasmon interactions of nanoparticle assemblies, plasmon-exciton interactions of molecule-nanoparticle conjugates, multipole plasmon mixing in chiral metal nanocrystals and electrodynamic effect of long range plasmon-exciton interactions. It is efficient and accurate to simulate light-matter interactions with analytic solutions. However, only a limited number of geometries can be solved analytically. Many numerical tools based on finite element methods, discrete dipole approximation or finite-difference time-domain methods are available currently. These methods are capable of simulating nanostructures with arbitrary shapes. Numerical simulations using such software have shown agreements with analytical results of our models. Hence, this study may offer a new approach to design of complex nanostructures for sensing of chiral molecules. This dissertation also reviews several experimental papers that have demonstrated successful fabrications of chiral nanostructures and nano-assemblies with new plasmonic CD signals. Our theories

4 strongly motivated the field and have been used in many experimental studies for interpretation and understanding of observations. 4

5 5 DEDICATION To my family and friends. To those who helped me along the way.

6 6 ACKNOWLEDGMENTS There are a number of people without whom the completion of this dissertation will never happen. First of all, I would like to express the deepest appreciation to my adviser, Professor Alexander Govorov, who has demonstrated the highest level of dedication to scientific research and who is a model of meticulous and rigorous scholarship. Most importantly, Professor Govorov has mentored me with his experience, wisdom and patience, teaching me both the fundamental physical theories and their applications to practical problems. It has been a great experience for me to work and study in this group. I would also like to thank my committee members, Professor Eric Stinaff, Professor Horacio Castillo and Professor Wojciech Jadwisienczak, for their valuable time devoted into the review of my dissertation. Their advice has helped me greatly in the improvement of my work and presentation. In addition, I would like to thank several experimental teams led by Professor Liedl at LMU, Professor Markovich at Tel Aviv University, Professor Willner at Hebrew University, Professor Halas at Rice University, Professor Hendry at University of Exeter, Professor Kadodwala at University of Glasgow and Professor Gun ko at University of Dublin for the excellent experimental studies of their team members and successful collaboration with our team at Ohio University. Particularly I would like to thank Professor Tim Liedl once more. It has been such a great honor for me to be invited to work and study at LMU with one of the best experimental team in the field.

7 7 And also I would like to thank our department staff, Wayne, Ennice, Tracy, Meg, Candy, Julie, Donna, Chris, Don and many more, who have cared and supported us since the first day of our orientation. You have made the department a great place to work and a warm place to call home. Finally, my gratitude goes to my friends here, Dening, Haoshuang and Peng Zhao, my collegues in the department Hui, Larousse, Ramana, Mahmoud, Heath, Arbin etc. and those who are thousands of miles away, Pedro, Mikołaj, Eva, Robert, Yongzheng Xing and Tao Zhang. Thanks for your accompany during the days we work and study together.

8 8 TABLE OF CONTENTS Page Abstract... 3 Dedication... 5 Acknowledgments... 6 List of Tables List of Figures Chapter 1: Introduction Chapter 2: Light-matter interaction in nanomaterials comprising of metal nanoparticles, semiconductors and biomaterials Introduction Excitons: Optical properties of semiconductor nanocrystals and dye molecules Plasmons: Optical properties of noble metal nanoparticles Mie solution: a rigorous model of light scattering by metal spheres Point dipole model of scattering by metal spheres Coulomb interactions Plasmon-plasmon interaction: Plasmon-exciton interaction Chirality and circular dichroism Chapter 3: Interactions between metal nanoparticles and chiral molecules Introduction Theory of circular dichroism in molecule-mnp conjugates Molecular resonance overlapping the plasmonic resonance Molecular resonance far from the plasmonic resonance Chapter 4: Plasmonic circular dichroism in chiral nanoparticle assemblies Introduction D chiral nanoparticle assemblies Defects and robustness of CD in nanocrystal assemblies Chapter 5: Circular dichroism of chiral plasmonic tetramers: multipole effects Introduction Mathematical formalism of dipole interactions... 55

9 5.2.1 General equations of interacting dipoles Discrete dipole approximation (DDA) Point dipole approximation (PDA) Plasmonic circular dichroism of chiral nanostructures Numerical simulations of plasmonic circular dichroism Geometry of nanostructures Helix and elliptic helix Pyramid Equilateral Tetrahedron Multipole effects Conclusion Chapter 6: Electrodynamic effects in chiral molecular-plasmonic structures Introduction Model: Maxwell s equations incorporating a chiral parameter Circular dichroism in the quasistatic limit Electrodynamic effects in large structures Chapter 7: Plasmonic nanocrystals with chiral shapes Introduction Formalism: multipole expansion of perturbed Poisson equation Origins of plasmonic CD: mixing of multipoles Chapter 8: Experimental observation of Plasmonic circular dichroism Chapter 9: Discussions and perspectives Introduction Anisotropy of chiral nanomaterials Optical chirality and circular dichroism enhancement Conclusions References Appendix A: A Taylor expansion of plasmonic circular dichroism of metal nanostructures Appendix B: Directional circular dichroism of nanoparticle assemblies Appendix C: Numerical simulation of circular dichroism for chiral nanomaterials using discrete dipole approximation and finite element methods

10 10 LIST OF TABLES Page Table 1: Coordinates and radii of NPs in a chiral assembly with =

11 11 LIST OF FIGURES Page Figure 1: Chiral objects...14 Figure 2: Electromagnetic mechanisms of plasmonic circular dichroism...17 Figure 3: Schematics of light scattering by a single metal sphere...26 Figure 4: Calculated extinction and scattering spectra for a gold sphere...27 Figure 5: Schematics of light scattering by a two-particle complex...32 Figure 6: A classical picture of trajectories of a helical motion and a circular motion..35 Figure 7: Experimental absorption and CD spectra of -helices...38 Figure 8: Schematics of MNP-molecule interaction...40 Figure 9: Resonant interaction between an AgNP and molecules...44 Figure 10: Off-resonance interaction between an AuNP and molecules...46 Figure 11: A 2-dimensional spiral and its mirror images...48 Figure 12: Plasmonic circular dichroism spectra of chiral AuNP complexes...50 Figure 13: Effect of disorder on the CD spectra of a AuNP helix...53 Figure 14: Models and extinction spectra of compact AuNP complexes...58 Figure 15: Comparison of CD spectra of AuNP helices and elliptic helices...62 Figure 16: Comparison of CD spectra of pyramidal AuNP tetramers...63 Figure 17: CD spectra of compact equilateral tetrahedral AuNP complexes...64 Figure 18: Models of nanoparticles and core-shell nanostructures...71 Figure 19: Calculated CD spectra for small core-shell nanostructures...74 Figure 20: Calculated CD spectra for large core-shell nanostructures...75

12 12 Figure 21: Schematics of chiral objects...80 Figure 22: Modal analysis of plasmon resonances of a chiral nanocrystal...82 Figure 23: Results for the CD response for various types of chiral Au nanocrystals...84 Figure 24: The plasmonic CD band of AgNPs assembled on DNA...87 Figure 25: Optical characterization of AuNPs functionalized with E5-peptide...89 Figure 26: Models and TEM image of AuNP helices...91 Figure 27: Theoretical and experimental CD spectra of AuNP helices...92 Figure 28: Optical rotatory dispersion of AuNP helices...94 Figure 29: Schematics and CD of two extreme cases of AgNP- molecule conjugates..95 Figure 30: Lithographically made chiral metal structures...97 Figure 31: CD spectra for complexes comprising a metal NP and a molecular shell..100 Figure A1: Calculated CD properties of a 9-NP right-handed helix Figure A2: Comparing extinction and CD spectra of a 4-NP helix Figure A3: Comparing extinction and CD spectra of a small core@shell structure Figure A4: Comparing extinction and CD spectra of a large core@shell structure...137

13 13 CHAPTER 1: INTRODUCTION With modern nanotechnology, scientists can create new materials structured at atomic and nanoscale levels. Physical and chemical properties of nanoscale systems are distinct from bulk materials due to size confinements. It has attracted lots of current interest, because they have shown excellent potential in new applications. Nanoparticles (NPs) are an important class of nanomaterials. Generally speaking, they are between 1 nm and 100 nm in dimension. Optical properties of NPs are sensitive to their geometry. Mie resonances [1] of spherical dielectric NPs is one good example that demonstrates tunable optical properties with respect to the radii of spheres. In metal nanoparticles (MNPs), oscillation of free electrons may lead to a particular type of optical resonance called localized surface plasmon (LSP) resonance. Surface plasmons are driven by electrons or visible/infrared light in metal nanostructures. They are confined to an interface of a metallic structure. As a result, electromagnetic fields are strongly enhanced near the interface [2]. This field enhancement was found responsible for interesting optical phenomena such as enhanced fluorescent emission [3, 4], enhanced [5, 6] Raman scattering and second order harmonic generation [7-9]. Other than these, the high fields of plasmons also showed the capability to enhance the efficiency of lightemitting diodes [10, 11] and solar cells [12, 13], and also potentials in medical therapy [14-16], biomedical imaging [17], lithography [18, 19], sensitive chemo- and bio-sensing [20, 21], data storage [22-24], plasmonic waveguides [25-28] and photocatalysis [29-31]. NPs become compatible with other functioned nanomaterials through molecular linkers. Controllable coupling and interaction between nanomaterial building blocks may

14 result in interesting optical properties that are distinct from each individual component. Recently, a new type of nanomaterials with chirality has been created using metal or 14 semiconductor nanoparticles that are assembled with biomolecules. New circular dichroism (CD) signals are exhibited from these chiral nanomaterials. Figure 1. Chiral objects a) Our hands and a pair of enantiomers of a generic amino acid [32] b) a metric hex cap screw [33], c) A section of DNA [34], d) Chirality in snails [35]. According to definition, chiral objects are not superimposable onto their mirror images. Some examples of chiral objects are demonstrated in Figure 1. Usually, chirality offers a smooth, efficient and strong coupling between objects of the same chirality, such as a tightening between a bolt and a nut or a handshake using the right hand. In nature, chirality also played an important role in the origin of life. For example, as essential building blocks of life, almost all natural amino acid monomers have L-configuration, while sugars are D-isomers. These naturally occurring chiral biomolecules are able to facilitate the selectivity in enzyme catalysis [36] and efficiently pack into biomaterials of different functionalities [37]. In chemistry, a pair of chiral molecules are called enantiomers if they are mirror images of each other. But, chemical properties may be different for different enantiomers

15 15 due to stereoselectivity. For example, in the human bodies, biochemical reactions sometimes are only allowed between a particular type of enantiomers in a racemic drug and targeted biomolecules [38]. Therefore, in the interest of drug safety and production cost, efforts have been made into legislation and manufacturing of homochiral drugs, since stereoisomers of other types may be less active, inactive or even responsible for adverse effects in a treatment. A chiral molecule interacts differently with right-handed circularly polarized (RCP) light and left-handed circularly polarized (LCP) light. In particular, the optical attenuation is different. Such difference in spectra is called circular dichroism (CD), and is generally less than 10-3 of the attenuation. Although CD effect is relatively weak, it is very important and characteristic, since a flipped CD signal indicates an opposite handedness in molecular structures. In chemistry, CD spectroscopy is particularly useful in differentiation of enantiomers and conformational isomers. It s like telling twin brothers apart based on their fingerprints. Alternatively, optical rotatory dispersion (ORD) measures the optical rotation of a material with respect to wavelength. ORD are related to CD just as dispersion is to absorption through the Kramers-Kronig (K-K) relations of an absorptive material. [39, 40] Homochirality is a common property of naturally occurring biomolecules. It has created excitement in the community when plasmonic CD was first demonstrated in the lab on a ligand-protected-metal nanocluster system [41, 42], indicating that an artificial chirality has been achieved. It was then suggested [43, 44] that CD from a nanocluster system can be induced by 1) intrinsically chiral metal core; 2) a chiral environment; or 3)

16 an intermediate mechanism called footprint effect. It was also proposed that optical 16 chirality could be a result of interplay of these mechanisms. [45] In 2006, a study demonstrated CD in the plasmon band of 10-nm silver nanoparticles that were grown on DNA templates. [46] The CD mechanism of a NP system may be different from a nanocluster system. [44, 46] Experimentally, such nanoparticle systems feature well-defined structures, well-controlled preparation and reusability. In this respect, they are good candidates for applications such as sensing of biomolecules. [47-50] The interest in plasmonic CD is now growing rapidly. Nanoparticle-based nanostructures have been created using a variety of techniques and plasmonic CD has also been observed. [51-67] It is now possible to fabricate chiral materials on a nanoscale. In the meantime, it is interesting for us to study the underlying mechanisms of plasmonic CD of chiral nanomaterials. This dissertation is focused on interpreting the electromagnetic origins of circular dichroism of chiral NP assemblies, NP-molecule conjugates, NPs in a chiral molecular matrix and NPs with chiral surfaces (Fig. 2). [68-71] The electromagnetic interactions include plasmon-exciton interaction, plasmon-plasmon interaction, electrodynamic effect and multipole plasmon mixing. Generally speaking, they are physically different from a vicinity effect [72], an orbital hybridization effect of surface states [73] or an effect from the chiral electronic states of a distorted nanocluster [74].

17 17 Figure 2. a) Chiral optical signal (CD) can be induced onto the plasmons by chiral molecules surrounding the metal components through the exciton-plasmon interaction [56, 71]. Adapted with permission from Ref. [56]. Copyright 2011, American Chemical Society. b) Plasmonic CD can be created in the visible band through the long-range electrodynamic interaction [68] ; a CD signal in this case is significant in the structure with a thick chiral molecular shell, c) a CD signal can be generated in a chiral metal NP assembly by the dipolar plasmon-plasmon interaction [59, 70, 75], Taken from Ref. [59], Adapted with permission from supplementary information (SI) of Ref. [59]. Copyright 2012, Rights Managed by Nature Publishing Group. d) Our new study shows that spheres with a chiral surface perturbation are able to generate a strong CD signal [69]. e) a unit cell of chiral nanostructure array is covered by a chiral biomolecular medium for sensing. The current chapter has introduced the basics of nanoparticles, chirality and circular dichroism. Chapter 2 begins with general mathematical formalisms of light matter interactions. Then plasmonic CD of chiral materials can be derived as differential absorption of LCP and RCP. A plasmonic CD can be generally described by contributions from three origins using equation (1.1): CD CD CD CD (1.1) total Abs, molecule Abs, NC Sca,

18 where the sub-indices indicate contributions from absorption of a molecular medium, absorption of nanocrystals, and scattering by a complex structure consisting of both molecular materials and MNPs. In chapter 3, CD Abs, molecule and CD Abs, NC may contribute to the total CD spectra through a plasmon-exciton interaction. Simulations show that chiral molecule-mnp conjugates may exhibit a plasmonic CD caused by a chirality transfer from molecules to MNPs. The signals are sensitive to the orientation of molecular dipoles. A model of off-resonance interaction can successfully address the phenomena of plasmon CD in several experiments. [56, 58, 61] In chapter 4, CD Abs, NC can also be a result of plasmon-plasmon interaction between MNPs on a chiral assembly. It will be shown theoretically that a helical chain of gold NPs is able to generate significant plasmonic CD in the visible band. [59, 70, 75] The interactions between plasmons are treated 18 using point dipole interactions [26, 76]. In chapter 5, the effect of multipolar electromagnetic wave scattering between MNPs on plasmonic CD will be discussed. Simulations using DDSCAT [77] show that the plasmonic CD is very sensitive to the inter particle spacing. It may be useful in the developments of compact nanoparticle assemblies for sensitive biomolecular sensing applications. Chapter 6 is focused on the NP-molecule complexes with size ranging from tens to hundreds of nanometers. In this case, scattering is able to compete with or even exceed absorption in metals. CDSca becomes significant at electrodynamic resonances. Maxwell s equations are solved with constitutive relations of chiral media, which has included a chiral parameter [1, 78, 79]. In chapter 7, we also find that plasmonic CD can be induced by a single metal NP with a chiral shape [69]. Such a shape can be created by either top-down lithography or bottom-

19 19 up wet chemistry methods. For asymmetrically shaped NPs, the multipole electromagnetic fields are created and mixed due to surface distortion. We will see that multipole mixing is the key process of CD generation. In chapter 8, several representative experiments will be reviewed, in which chiral nanomaterials were created with new CD bands as was designed. In the end, the research of plasmonic circular dichroism of chiral nanomaterials will be summarized in Chapter 9. The electromagnetic CD mechanisms have been developed theoretically and a couple of experiments have already succeeded in fabrication of chiral nanomaterials of different designs that demonstrate corresponding CD signals. In biosensing applications, far field CD signals are usually recorded for analysis. A couple of designs can be made to improve the sensitivity of the CD signals, including introducing anisotropy [68, 80], increasing near field optical chirality [81-83] and using an electrodynamic effect [60]. However, experimental realization still remains challenging when incorporation of various materials and realization of interesting geometries are desired in chiral nanomaterials.

20 20 CHAPTER 2: LIGHT-MATTER INTERACTION IN NANOMATERIALS COMPRISING OF METAL NANOPARTICLES, SEMICONDUCTORS AND BIOMATERIALS 2.1 Introduction The study of light-matter interaction concerns the principles and phenomena of light absorption and emission. Light does not carry much useful information in itself. However, its interaction with a matter can tell us about the nature of the matter, such as information of its geometry, structure and composition. The understanding of light-matter interaction has led to lots of applications such as lasers, sensors and optical communication devices. This chapter will be focused on fundamental models of interaction between plasmons and excitons. First of all, some dilute systems of small objects are treated quantum mechanically, which include atoms, molecules and quantum dots that have discrete levels due to a size confinement. In section 2.1, a two-level model is used for an introduction of excitons. Rate equations of density matrix formalism analytically describe the dynamics of density matrix elements, through which optical properties of a system such as absorption and emission can be expressed. A piece of continuous dielectric material is considered as a collection of atomic dipoles in classical electromagnetism. In the presence of an external field, polarization, susceptibility and dielectric constant are conveniently defined. [84, 85] While in practice, the dielectric function of a material may be already available from experimental measurements. [86] Then light scattering problems can be handled by solving Maxwell s equations with dielectric functions

21 21 completely specified for each component material. Noble metals are described by a complex dielectric function with a negative real part, due to which a plasmon can be excited on a metallic interface. [87, 88] Coulomb interactions take place between charges. Excitons and plasmons are polarized by external fields so that they also induce Coulomb forces through radiated fields of a charge oscillation. Such interaction usually can be identified from mode splittings in a spectrum, circular dichroism signals or Fano resonances [89, 90]. In section 2.2 and 2.3, models of plasmon-plasmon and plasmon-exciton interactions will be introduced. Circular dichroism is exhibited in most natural biomolecules. A molecular CD signal arises from the handedness of excitations that interacts differently with circularly polarized light. For example, a group of chromophores are packed into a helical polymer through hydrogen bonds. The Coulomb interaction between these monomers is able to induce one parallel mode and two perpendicular modes that are optically active [91]. This model of circular dichroism will be reviewed in section Excitons: Optical properties of semiconductor nanocrystals and dye molecules In semiconductor nanocrystals and dye molecules, an exciton may be excited by a photon when their energies are matched. In semiconductors, an exciton is a neutral quasiparticle consisting of a pair of electron and hole in the conduction and valence bands respectively, which attract each other through Coulomb forces. In a molecule, an exciton is created when an electron leaves one orbital for another after absorbing a quantum of

22 22 energy. In a different terminology, the electron is found in the lowest unoccupied molecular orbital and a hole is found in the highest occupied molecular orbital. They are also bound to each other. An exciton behaves like a hydrogen atom, but with a lower binding energy and a larger spatial dimension. In this section, the optical response of an exciton to an external field is modeled semiclassically. Excitons are treated as two-level quantum systems and optical fields are treated as classical electromagnetic waves [92]. The Hamiltonian of an interaction between an exciton and an electromagnetic field is described by: H H H, (2.1) 0 ' where H 0 is the unperturbed Hamiltonian operator for a dye molecule or a semiconductor NP and H ' is the interaction between the light beam and the exciton dipole moment. Usually the wavelength of the optical field is much longer than the size of an atom, a molecule or a nanoparticle. Then the interaction is described under point dipole approximation: H ' E, (2.2) E E t E t e e. 2 it it where is an electric dipole operator, and cos The density matrix formalism is needed when a large number of quantum objects are involved in an ensemble. Given density matrix, the expectation value of the lightinduced dipole moment is: tr tr (2.3)

23 The density matrix elements nm is going to be solved from the equation of motion of : 23 d i H, nm, dt nm nm kl kl kl (2.4) Relaxation mechanisms, which lead to a broadening of absorption lines, are introduced in equation (2.4), when we study realistic experimental systems. The terms nm, klkl are called collision terms that describe different channels of energy relaxation. Rotating wave approximation is applied when the difference between the frequency of an external field and the frequency of an exciton resonance is small. The density matrix elements are written as: it te t t,,. (2.5) If we drop the double frequency terms, we may arrive at the following equations: d i dt t i t i t E t t , 2 T2, d dt t t 2i i t * i t E t 21 t e E t 21 t e ie * (2.6) where T 2 is called a relaxation time and is called the time constant that describes relaxation of population between levels 2 and 1. In a steady state, all time derivatives of t should be 0. We can solve for the variables on the right hand side of the equations (2.6): ij

24 24 E T Re 2, t t E 0 T2 1 4 T2 2 E T 2 Im E 0 T2 1 4 T T T2 4 T2 (2.7) As is defined in Eq. (2.3), the dipole moment is T 0 2 E T2 2 0 cost sint 2 2 E 0 T2 1 4 T2 2. (2.8) Then polarizability can be found as and heat dissipation is defined as * * , (2.9) E E E 2 Q molecule (2.10) This shows a dipole transition in a molecule is driven by an oscillating electric field. A resonance is achieved when the frequency of an external field matches the internal frequency of a dipole transition, i.e. the denominator is minimized in Eq. (2.8). Also by looking at equations (2.5), (2.7) and (2.10), we find a maximized heat dissipation is

25 associated with a maximized 22 to more heat generation at a dissipation rate (Eq. 2.10). 25. Naturally, a higher probability in an excited state leads 2.3 Plasmons: Optical properties of noble metal nanoparticles Noble metals have been used in monetary exchange, jewelries and investments for more than hundreds of years. They are well known for stable chemical properties in the atmosphere, which brings gold shining glossy surfaces and long-term existence. In modern science and technology, however, gold nanoparticles (AuNPs) and silver nanoparticles (AgNPs) can be used as biosensors and nanoheaters, etc., not only because they can be reactive with other chemical compounds and biomaterials, but also they demonstrate localized plasmonic resonances in visible optical bands. A plasmon can be described as intra-band transitions of electrons which live near the Fermi level of incompletely filled bands, or in filled bands that overlap with empty bands. Typically in the visible or infrared bands, dissipation peaks take place at plasmon resonances. In the classical regime, the resonance property of a metal nanostructure can be investigated with several options, such as Mie solution, multipole expansion and point or discrete dipole approximation etc.. Usually for a compactly assembled system or large MNPs, a multipole expansion or Mie solution is needed. A point dipole model is very efficient and convenient if we study nanoparticle systems that meet the criterion of point dipole approximation. With a discrete dipole approximation, we can conduct numerical simulations for nanostructures of any shape.

26 26 Generally speaking, the electric field of surface plasmons features an extremely short spatial dimension and extremely high intensity near the interface, compared with incident fields. It allows the surface plasmons to capture the information of tiny variations nearby. Hence plasmon resonances are sensitive to geometries, surrounding materials and physical environments of a metal nanostructure Mie solution: a rigorous model of light scattering by metal spheres Figure 3. Schematics of light scattering by a single metal sphere sitting at the origin. The wave is traveling in z direction and electric field is oscillating in x direction. From Mie solution, the scattering problem has been solved rigorously for one single dielectric sphere. [1] As is shown in Fig. 3, the electromagnetic wave is propagating in z direction and its electric field is x- polarized. The sphere is sitting on the

27 origin, with a radius a r and a dielectric function m 27. Its surrounding medium has a dielectric constant 0. Mie solution [1] shows that the scattering cross section is given by: 2 sca 2 n n k n C n a b, (2.11) and the extinction cross section is: 2 ext 2 n n k n1 2 1 Re C n a b, (2.12) where a, b are frequency-dependent coefficients solved from boundary condition at r a. n n These equations can be used to predict the color of scattered light and transmitted light from the NPs floating in a solution. The simulated extinction and scattering spectra are shown in Fig. 4, which can qualitatively explain the colors observed in solutions of colloidal gold nanoparticles. [93] [94] Figure 4. Calculated a) extinction and b) scattering coefficients for gold particles with different size. [95] The inset of b) shows scattering coefficients for small gold particles with radii 2.5nm and 5nm.

28 It is understood that gold nanoparticles will show atomic or semiconductor characters when their radius is below 1.5 nm, and bulk material dielectric function m is otherwise valid when radius of AuNP is above 2.5 nm. [43] Extinction and scattering are plotted in Fig. 4 a) and b) respectively, using a normalized coefficient defined in Ref.[95]. The transmitted light observed in diluted solutions in Ref. [94] shows that its color is tuned by the size of AuNPs. Qualitatively, this tunability of size is exhibited in extinction spectra shown in Fig. 4 a). For smaller particles in relatively diluted solution, the color is dominated by absorption, compared with the scattering coefficients in Fig.4 b). From extinction spectra, we see that it exhibits scattering and absorption below 550 nm, which indicates the color of transmitted light through a dilute solution of small AuNPs is likely to be red, regardless that the scattered light may have lots of components in blue or green. As radius of particles is increased to 100 nm, the scattering will contribute about 80% in the extinction near resonance. The scattering is much stronger than a transmission. The solution becomes cloudy. And the color of scattered light of 200nm nanoparticles has more red and infrared components than 100nm nanoparticles. [94] Note that a 200nm NP has a radius of 100nm Point dipole model of scattering by metal spheres From electrostatics, the polarizability of a dielectric sphere is prescribed as [85] : m 0 m 2 0 r 3 a, (2.13)

29 where, m 0 are dielectric constants of a metal and a matrix respectively, and r a is the radius of the sphere. Given the electric field in x direction (Fig. 3) the electric dipole moment is given by: 29 p E 0. (2.14) When kra 1, the electric field of incoming wave can be treated as a static field. The electric field will induce a dipole moment on the nanosphere and drive an oscillating current. The heat dissipation of the charge oscillation can be calculated from work by the electric field:, Q t j t E t dv (2.15) where Et is a real total electric field inside the particle, and jt is the induced current. Then heat dissipation averaged over a period T is: Q Q t dt Q t t T. (2.16) Alternative forms of these equations of heat dissipation and extinction are here [96] : Q 1 * i abs, i 0 Im * 2 i p pi, 1 Q Im E p 2 * ext, i 0 ext _ i i, Q Q Q. sca, i ext, i abs, i (2.17)

30 These equations work not only for a single nanoparticle, but also for interacting nanoparticles on an assembly. The sub index i represent the i-th particle of an assembly. p i 's are electric dipole moments, E ext _ i 30 's are incident fields at the i-th particle. The total extinction and absorption inside that system will require a summation over all particles. In a phenomenological interpretation, the absorption in the i-th particle equals the work done on the charges inside the sphere by the total electric field. While the extinction contributed by the i-th particle equals the work done on the oscillating charges only by the incident beam. When only one particle is in the electric field, it can be shown that: 2 * Im 1 1 Qext 0 Im Eext Eext 0 E 0. (2.18) 2 2 The extinction is proportional to the imaginary part of its polarizability. The plasmon resonance of a spherical MNP is achieved when the denominator in decreases (Eq. 2.13). 2.4 Coulomb interactions Plasmon-plasmon interaction: Generally, it requires sophisticated models [69, 97, 98] for large NPs and compact NP assemblies. Contribution from either scattering or multipoles becomes important in the total extinction spectra. Point dipole approximation works very well on nanoparticles that have dimension much smaller than wavelength. It can also be applied to interacting NPs on an assembly when they satisfy the criterion of point dipole approximation [76]. In a metal nanoparticle assembly, plasmons interact with each other through radiated electric

31 31 fields. Degeneracy of the single NP modes will split. In a NP dimer, for example, two transverse modes and one longitudinal mode will become active. Polarizability of a nanoparticle connects a dipole response of a nanoparticle with an external field. In literature [26, 27], the dipole polarizability equation (2.13) is replaced by: , (2.19) 3 3 c or 3 3 c i a 3 1. (2.20) 2 These equations have partially corrected the electric dipole strength due to a radiation loss. Comparing to Mie solution, if we expand the Mie coefficients of electric and magnetic dipoles a1, b 1 and an electric quadrupole a [99] 2, we will find that the total extinction cross section from an electric dipole oscillation is accurate to the order of kr 1 a with partial correction of kr 3 a and kr 4 a the order of 2 kr a. Since circular dichroism signals are on [70, 71, 100], we see that point dipole approximation will be very useful and efficient for the study of NP assemblies with size of NPs and assemblies much smaller than the wavelength of incident light.

32 32 Figure 5. a) Schematic of light scattering by a two-particle complex. The wave is traveling in z direction and electric field is oscillating in x direction, or the wave travels in x direction with electric field z polarized. However in quasistatic regime, the information of direction of the wavevector is not reflected in extinction. b) Absorption spectra of this complex when external field is parallel to z-axis (longitudinal) and perpendicular to z-axis (transverse). In a simple case of a two-metal-particle system in Fig.5 a), the particles are placed along the z axis. Dipoles are induced by both incident field and scattered field of each other. To solve this problem, we need equations of dipole moments p E E 1 1 ext _1 21 p E E 2 2 ext _ 2 12 (2.21) where,,

33 33 i r 3ˆ r p rˆ p p rˆ p rˆ Emn 1 c r c r c r p r c p, a mn m mn b m 2 i 0 r 0 mn mn mn m mn m m mn m mn c e mn mn (2.22) with c c a b 2 irmn 3 ir mn c 1, 5 w 2 3 e rmn c c rmn 2 irmn 1 ir mn c 1. 3 w 2 e rmn c c rmn Dipole moments can be found from this self-consistent equation. And total absorption and extinction can be calculated following equations (2.23): Q abs 1 2 p Im pi, * i 0 * i i 1 Q Im E p. (2.23) * ext 0 ext _ i i 2 i The calculated absorption spectra are shown in Fig. 5 b). The total absorption spectra are different for a longitudinal mode and for a transverse mode. The intensity is significantly different. If we observe carefully, we will see that the peak of a longitudinal mode is also slightly shifted to the red. It has also been shown that the charge oscillations behave differently in these modes [101] Plasmon-exciton interaction As is introduced in section 2.2, the exciton-photon interaction can be solved from density matrix formalism. It was also mentioned that excitons may have transition moments (Eqs.2.3 and 2.8) that generate a dipolar electric field. So do the plasmons in

34 34 metal spheres. In a coupled system of excitons and plasmons, the dipoles can interact with each other through the radiated dipolar electric fields. The Hamiltonian needs to be revised to include these interactions. [21] Then, solving this interaction is similar to the treatment of an exciton-photon interaction. Write perturbed Hamiltonian for excitons, and include both incident field and induced fields from electric dipoles around. For plasmons, the induced field will be from both metallic spheres and excitons in the system. Then mathematically this becomes a self-consistent equation system. Further derivation will be presented specifically for molecule-mnp interactions in chapter Chirality and circular dichroism A chiral object has no mirror symmetry or central symmetry. This means we cannot overlap its mirror image onto itself by either rotations or translations. Examples of chiral objects exist with all sizes, ranging from a spiral galaxy to biomolecules like DNA double helices. In large biomolecules, chromophores usually have electronic transitions in the UV band. [39] However, most of the chromophores are not chiral by themselves. The chiral arrangement of those chromophores is needed to induce optical activity and exhibit CD signals. [91]

35 35 Figure 6. a) A two level transition from ground state to first excited state. b) A helical trajectory of motion, whose time-averaged angular momentum and linear momentum is parallel to the helical axis. c) A circular trajectory of motion, whose angular momentum and linear momentum are perpendicular to each other. In a general CD theory, the rotational strength of a transition 0 is defined as R0 Im 0 m 0, where 0 is the ground state and is the excited state. In an experimental definition, R0 d. The quantity is the difference in absorption of LCP and RCP light. These quantities, a CD measurement and a rotational strength, are directly related. If we consider the equations below: e r 0 p 0, 0 e m 0 r p 2mc the rotational strength is the imaginary part of the dot product of two dipole transitions. The electric dipole moment is analogous to a linear momentum and the magnetic moment 0,

36 36 is analogous to an angular momentum. In a classical picture of a trajectory of motion, we show two types of motion in Fig. 6, which are respectively a circular motion and a helical motion of particles. When angular momentum and linear momentum are not zero and are not perpendicular to each other, the corresponding rotational strength is non-zero, therefore CD should be exhibited. This geometrical interpretation of the rotational strength is clear in the picture. However, the motion of a real electron follows quantum mechanics and an electronic cloud of a molecule is far more complicated than a helical trajectory of electrons. It has been better explained in the Refs. [91, 102], looking into the quantum mechanical origin of rotational strength and CD signals. Classically, the proteins can be modeled as a set of chromophores, each of which is treated as an oscillating dipole. [103] The dipole moment is written as a product of the polarizability and an external field, which is the same as in the point dipole approximation (Eq. 2.14). The polarizability is a tensor with complex elements. The imaginary part is given by the extinction. Then, it determines the real part from K-K relation. Dipole interaction is solved by a set of self-consistent linear equations. The vector of angular momentum of chromophoric electrons can be treated as magnetic dipole transition moment. And linear moment is proportional to the dipole moments. But due to the difficulty in estimation of polarizability and inaccuracy in predicting the spectra, the explanation of CD mechanism needed quantum mechanics. [91] Moffitt developed a modified quantum mechanical model. The helical biomolecule was divided into N identical sub groups, each with a ground state and an excited state (Fig. 6 a). If the interaction between groups is not counted, there will be a ground state and N degenerate

37 37 excited states for the molecule. The degeneracy will be broken in the perturbation when interactions are included. The Hamiltonian was written as the following, * H H m 0l V ql qm 0ldql V ql qm, (2.24) m lm m, lm where q m is the coordinates of all chromophoric electrons in a residue, H m is the Hamiltonian of a single residue, containing the kinetic energy of chromophoric electrons, mutual interaction within the chromophore and interaction with the average field from all other electrons on the helix. The mean field interaction is removed from Coulomb interaction operator is defined as: l m H m before the V q q between residues is added. The Coulomb interaction l m lm 3 l lm m lm 5 e q q r e q r q r V ql qm. (2.25) r Following a selection rule observation by considering the overall geometry of a helical polymer, the only active modes are found to be a parallel mode and a doubly degenerate perpendicular mode, the energies of which are lm 0 Vlm 0 EN 0 0 E M E N M 0 Vlm ml N 0 2, ml 2 l m M 2 cos. 0 (2.26) The rotational strength are correspondingly, R p l p l 0 N v v t v 0., RN M RM p rl r p tl t p t p r (2.27)

38 38 0 where l l m m m, p p and m 0 is radius of the helix. m m The theory was successfully verified on the -helices. Fig. 7, taken from Ref.[104], shows that the extinction spectra and the CD spectra can be reproduced by three different modes. Within this brief introduction, it showed that the coherent interaction between helically arranged chromophores actually induces the optical activity. It was found that only a parallel mode and a doubly degenerate perpendicular mode emerged due to the overall geometry that was considered. This mechanism has also enlightened the fabrication of many chiral nanomaterials that demonstrate strong visible band CD signals. Figure 7. Experimental data of -helices. a) the absorption spectra and b) the CD spectra can decomposed into three bands. Taken from Ref. [104]. Equation Chapter (Next) Section 1

39 39 CHAPTER 3: INTERACTIONS BETWEEN METAL NANOPARTICLES AND CHIRAL MOLECULES 3.1 Introduction Several experimental papers reported new CD lines from nanostructures assembled from chiral molecules and achiral nanocrystals [42, 46, 52, 105, 106]. Lots of work has been focused onto the decryption of the underlying mechanisms. This chapter will mainly focus on the a plasmon-exciton Coulomb interaction [71, 106, 107] and reveal that a plasmonic CD in the visible band can be induced by chiral molecules. 3.2 Theory of circular dichroism in molecule-mnp conjugates In this section, equations for heat dissipation and CD of a plasmon-exciton interaction will be derived. Particularly, the heat dissipation and CD can be evaluated formally as contributions from a chiral molecule and a nanocrystal.

40 40 Figure 8: a) Schematics of MNP-dye molecule interaction. b) wave vector k is represented in the coordinate system associated with the MNP-dye hybrid. c) transition diagram of Coulomb interaction and relaxation processes in this hybrid. Reprinted with permission from Ref.[71]. Copyright 2010, American Chemical Society. In Fig. 8, it is shown that the system consists of two components. The metal nanoparticle is treated as a sphere with dielectric functions taken from Ref.[108] or Ref. [86]. The chiral molecule is quantum mechanically represented by a pair of electric and magnetic dipoles. The physical significance of these dipoles has been explained in section 2.4. Starting from Eq. (2.1), we can include the interaction between a dipole from the nanocrystal and the dipoles from the chiral molecule through dipolar electric fields.

41 41 The Hamiltonian H ' in light matter interaction Eqs. (2.2) and (2.4) now has all the information of the electric field induced by the incident beam, the plasmon and the dipole image of the molecule. Using a rotating wave approximation (2.5), we are able to solve for and. [71, 107] Correspondingly, 2e E p B m m 22 0 i G E p B m 1 e 21 i m i G , 2, (3.1) where p is electric dipole moment operator, m is magnetic dipole moment. is the vector potential created by the dipole moment of the molecule near the MNP, p12 1 p 2 and finally e G p12 p21. (3.2) m Now we have explained the physical meaning of these terms. After solving this self-consistent equation system numerically, we can obtain absorption using these equations: Q molecule , (3.3) and in in* QNP Im m dve E. 2 (3.4) Additionally, averaging process is needed, because colloidal MNP-molecule complexes are floating in a solution with random orientations. Then the CD signal is defined as:

42 42 CD Q Q (3.5), where the signs indicate that incident beam is either LCP or RCP. This average is performed as if light is coming from all directions. It is equivalent to the situation when hybrid nanostructures are oriented randomly in a solution. As is seen, total absorption and CD have two terms formally, contributed by a molecule (Eq. 3.3) and a metal nanoparticle (Eq. 3.4). QQ Q metal molecule CD CD CD metal. molecule, (3.6) (3.7) To be more specified, the CD signal of the molecules and the NPs within a dipolar limit are expressed as: 8ck CD Im ˆ molecule E P m 3 i G , (3.8) ck 12 xm21x 12 ym21 y 212 zm21z CDNP anp E0 Im NP Im NP R i G, (3.9) where the magnetic and electric dipole moments of a molecule are defined as e 2 r 1 12 e m21 2r p 1 2mc. (3.10) In the coordinate shown in Fig. 8, ˆP tensor is defined as:

43 R ˆ P 1. 3 R R (3.11) m 0 3 where a. Although formally separated, the effect from a plasmon and an NP 2 m 0 exciton is still coupled in Eqs However, using these equations, we can already predict that an enhancement of CD signal in the molecular band can be achieved through a ˆP tensor and an additional plasmonic CD signal in the visible band can be induced. These equations are ready for the following investigation of resonant and non-resonant plasmon-exciton interactions. 3.3 Molecular resonance overlapping the plasmonic resonance In the resonant regime, we consider a complex consisting of an AgNP and a chiral molecule. In the simulated extinction and CD spectra of the molecule, it shows Fano signature when the molecular band matches the plasmonic band. (Fig. 9) This indicates a strong interaction between a molecule and an AgNP. However, it was difficult to tell the existence of the molecule from a total extinction spectrum, as is shown in the inset of Fig.9 a), because the extinction of an AgNP is overwhelmingly large. Fortunately, CD spectroscopy can help us resolve the existence of the molecule near the AgNP. In Fig. 9 b), the CD spectra were demonstrated for two different configurations when the electric dipole of the molecule is perpendicular or parallel to the AgNP surface. The shape of the CD spectra is dramatically different, indicating strong configuration dependence of the

44 44 CD signals. We also found that the strength of the CD signal was amplified by about 3 times. Note that this moderate enhancement is achieved within a dipole limit. It was also reported that a strong enhancement up to 100 times was discovered. [109] Figure 9. When exciton, plasmon bands overlap, a) extinction spectra of AgNP is plotted. The inset shows contribution to total extinction from the dye molecule when it is alone (1) and when AgNP is present and // R 12 (2). b) CD spectra are plotted for different configurations, (1) shows CD of the molecule without AgNP. Reprinted with permission from Ref.[71]. Copyright 2010, American Chemical Society. 3.4 Molecular resonance far from the plasmonic resonance Most biomolecules have absorption bands in the UV range, which is far from a plasmonic resonance of a spherical gold NP. It is also important to investigate a plasmonic CD in the off resonance regime. As real examples, -helix and a molecule mimicking calixarene have been studied. In these numerical results, we can see extra peaks at plasmonic resonances in the

45 45 CD spectra of MNP-molecule complexes. (Fig. 10) This is due to the chiral field induced by the molecule, and it creates differential absorption inside MNPs. If we look at the equations (3.8) and (3.9) of CD, we find that if the difference between the plasmon band and the molecular band is large, 0, then CD 12 NP dominates the CD spectra at the plasmon frequencies. This term is a differential absorption inside MNPs which was defined by Eq. (2.15). In the study of calixarene-aunp complex, the orientation of the molecular electric dipole moment determines the sign of plasmonic CD. This shows that CD signal is sensitive to the conformational change in a MNP-molecule hybrid. The sensitivity of CD spectra to the geometrical configurations is a useful property for sensing applications to investigate the composition and configuration of nanomaterials.

46 46 Figure 10. Off resonance enhancement of CD, a) Schematics of alpha helix-agnp complex b) Theoretical fit of extinction of alpha helix. c) Calculated CD spectra of helix- AgNP complex, which shows an induced plasmonic CD band. d) Schematics of dye- AuNP hybrid, with different orientation of molecular dipole moment with respect to AuNP, and corresponding CD spectra. e) extinction spectra of dye molecule and AuNP. Reprinted with permission from Ref.[71]. Copyright 2010, American Chemical Society

47 47 CHAPTER 4: PLASMONIC CIRCULAR DICHROISM IN CHIRAL NANOPARTICLE ASSEMBLIES 4.1 Introduction In a couple of experiments, chiral NP assemblies have been successfully created [53-55, 57, 59, 106, ], some of which demonstrated the anticipated CD signals. Here we show another origin of plasmonic CD, resulting from dipolar plasmon-plasmon interaction between metal nanoparticles in a chiral configuration. [70, 75] A chiral NP assembly with dimension around 40 nm consisting of small MNPs with radius around 5 nm will demonstrate relatively strong CD signals, according to our calculations. Importantly, this has also been demonstrated experimentally [57, 59]. For densely packed complexes, our colleagues have observed that the signal in the visible optical range can be enhanced to the level competing with large biomolecules D chiral nanoparticle assemblies In Fig. 11, a 2D chiral object cannot overlap onto its mirror image by any operation in the plane unless it is lifted off. It is understood that an ideal 2D chiral object will not demonstrate CD under normal incidence if they are suspended in a uniform medium. [113] We confirmed this conclusion in several 2D chiral MNP complexes, as the extinction CD is 0 under normal incidence. However, on 2D nanoparticle assemblies, it is possible to break the symmetry and lead to non-zero directional CD by introducing an asymmetric environment or inclined incident light beam. [114]

48 48 Figure 11. A 2D chiral object is not able to be matched with its mirror image in the same plane after any rotational and translational operation. When it is lifted off from the plane, then it will be moved onto its mirror image easily. Note that strictly speaking, the dimension of the NPs shown in the figure is not negligible; hence in 3D space there is a mirror plane cutting through the centers of the NPs. 3D chiral complexes are "real" chiral complexes that will demonstrate non zero CD signal in colloidal systems. We investigated several geometries of complexes, including helices, pyramids and tetramers with symmetric or asymmetric frames and with identical or different particles. All CD signals show positive and negative bands. They are strongly dependent on the size of particles. In this point dipole regime, the radius dependence is found to be 12 r a. CD signal also shows sensitive dependence on overall

49 49 geometry. In Fig. 12 b), it shows that similar structures may exhibit totally inverted signals. The helical structure is found most efficient in creating CD signals of all geometries in Fig. 12. However, in this dipolar regime, significant CD signal is not found in an equilateral tetrahedron, which has a symmetric frame but different particles on its vertices. An analytic proof can be found in the SI of Ref.[100] and also in Appendix A. Similar observation is obtained from a pyramid with a symmetric frame. Conclusion is that a symmetric frame of a chiral MNP complex will lead to a weak CD response. An extension of the dipolar plasmonic CD theory was made for another type of metal nanocrystals, nanorods, in Ref. [115]. Since nanorods have stronger optical dipoles, nanorod assemblies may produces strong plasmonic CD signals.

50 50 Figure 12. a) Helical complexes of AuNP with identical sizes. CD is calculated for such complexes with different numbers of NPs. b) Tetramers: two structures with overall similar parameters can show inverted CD signals. c) In pyramidal structures, our study shows that the CD signal is extremely sensitive to the size of particles and complexes with a symmetric frame will show a very low CD signal in the quasi-static regime. Adapted with permission from Ref.[70]. Copyright 2010, American Chemical Society. This analysis predicted a mechanism of generating CD signal in plasmonic bands through dipole-dipole interactions between MNPs. This is different from all mechanisms

51 suggested before, such as effect of induced chiral surface electronic structures by chiral adsorption, or Coulomb plasmon-exciton interactions in the previous chapter. Now technologies are available to assemble MNPs into chiral complexes using molecular materials such as DNA scaffolds, DNA bundles or Poly(fluorene-alt-benzothiadiazole) [44]. [57, 59, 66, 111] AuNP assemblies were created using DNA strands in a few experiments. AuNP helices were successfully fabricated and the anticipated CD signal was observed by our colleagues recently. This experiment [59] will be introduced in chapter Defects and robustness of CD in nanocrystal assemblies In order to show a plasmonic CD resulting from plasmon-plasmon interactions, a carefully designed and well-defined helical gold nanoparticle assembly was created in the experiments. [59] Helical chains of MNPs, mimicking twisted natural molecules, demonstrate the best efficiency in generating plasmonic CD among all frame geometries that we have investigated. However we noticed in our study [70] that the sign of the CD signal may flip when the number of MNPs increases, as was shown in Fig. 13 a). This can possibly create challenges in measuring a CD signal from an ensemble of helical MNP assemblies, as it can be that a CD signal is averaged to zero due to this kind of randomness in a fabrication. In a following study [75], we showed the dependence of isotropic CD strength on various kinds of parametric uncertainties, where we allow the geometrical parameters of a helix to vary, including the pitch of a helix, the total number of MNPs, the size and positions of MNPs, and defects due to missing MNPs. It was shown that a moderate level of randomness in these parameters does not diminish or

52 52 change the shape of the CD signals. In Fig. 13, two cases are presented, which demonstrate the consequences of randomized particle sizes and randomized particle positions. The influence of such disorder is weak. In nature, partially-disordered molecular complexes such as random-coil proteins still demonstrate significant CD signals. From this point of view, it is not surprising that a helical chain of MNPs have a stable CD response. After these careful verifications, we are confident that a plasmonic CD from helical MNP chains with carefully selected geometric parameters should be strong.

53 53 Figure 13. Effects of disorder on the CD spectra of nine-particle helices. a) Calculated averaged CD spectra for the ideal helix and for three non-ideal helices. For the non-ideal helices, the CD spectra were averaged over a set of trials. Overall, ~80 trials with generating random positions of particles were made. Insert: The model to scale. b) Calculated CD spectra for randomized positions of helically arranged NPs as a function of the trail number. Inserts show how the NP position fluctuations were introduced. Adapted with permission from SI of Ref. [59]. Copyright 2012, Rights Managed by Nature Publishing Group.

54 Introduction CHAPTER 5: CIRCULAR DICHROISM OF CHIRAL PLASMONIC TETRAMERS: MULTIPOLE EFFECTS In this chapter, Ref. [100] and its supplementary information are reused with permission. In the previous chapter, chiral metal nanoparticle (NP) assemblies exhibit plasmonic circular dichroism in the visible spectral region. It was found that CD signals can be induced by dipolar interactions between nanoparticles of a chiral assembly. In this chapter, we show that plasmonic CD signals can be enhanced by multipole effects in tightly packed nanoparticle assemblies. We have used discrete dipole approximation (DDA) to simulate CD signals of nanoparticle tetramers of different geometries, which includes an equilateral tetrahedron, a helical structure and an asymmetric pyramid. In the point dipole model, we have shown that a stronger NP-NP interaction may lead to a larger CD response described by a power law ( CD ~ R, where R is a characteristic particle-particle distance) [59, 70], whereas from the DDA simulations, we see that the NP- NP interaction in the multipolar regime gives a significantly red-shifted CD spectrum as [70, 75] well as an extra enhancement of the CD signal compared to the point-dipole model. Our results show that the strength of CD signals of tightly packed assemblies follows a power law as 9.7 1/ R for the helices and as where R is an interparticle distance / R for the tetrahedral 4-NP complexes,

55 Mathematical formalism of dipole interactions General equations of interacting dipoles A system of N particles is described by a set of dipole moments p 1,..., equation for a dipole of the i-th NP due to an external field and interactions is i i N. The pi i Eext, i E ji, (5.1) ji where E is the external field, ext, i i is the isotropic polarizability of the i-th NP, and the field due to the neighboring NPs [84] is E ji i r r p r p p r p r 1 c r c r 2 0 3ˆ ˆ ˆ ˆ ji ji j ji j j ji j ji c e ji ji i 0 r ji. (5.2) The NP polarizability can be defined in the usual way: 3 a, where Au( ) is the Au dielectric function and 0 ( ) / ( 2 ) i NP, i Au 0 Au 0 is the dielectric constant of matrix. The external field varies at different sites by a phase factor ik r e i. The set of linear Eqs. (5.1) can be solved self-consistently for all 3N dipole moments. Then total absorption and extinction can be calculated using the following equations: Q abs 1 2 p Im pi, * i 0 * i i (5.3) 1 Q Im E p. * ext 0 ext _ i i 2 i (5.4)

56 56 [77, 116, 117] Discrete dipole approximation (DDA) In nature, all materials are composed of various kinds of atoms. When the wavelength of light is much longer than the atomic spacing, the dielectric properties of a substance can be described using Maxwell s equations written for a continuous medium. On the other hand, the dielectric properties of a substance with a cubic lattice can be related to the polarizability of individual atoms. The DDA technique was inspired by this argument. To approximate a continuous medium, a target (i.e. a nanocrystal) can be replaced by 3N polarizable dipoles defined on a cubic lattice assuming that the wavelength is much longer than the lattice period. The oscillating dipole moments 1,..., p are excited by an incident monochromatic field. The polarizability i and the i i N lattice of dipoles can be defined following the method given in Ref.[116]. Then the dipoles become iteratively computed from Eq. (5.1). Finally the absorption and extinction cross sections can be calculated from 1,..., p. i i N Point dipole approximation (PDA) When nanoparticles are located far from each other, the electrodynamics problem can be simplified introducing a set of point dipoles to represent each individual NP of an assembly. [26, 76] The dipolar approach is valid when the spacing between NPs is large enough and the size of a single NP is small enough so that the multi-pole fields become [25, 76] weak. Mathematically, the system of equations can be similar to the DDA method, although the PDA approach is applied to a different physical system.

57 Plasmonic circular dichroism of chiral nanostructures Circular dichroism is defined as the difference in absorption or extinction of LCP and RCP light. The directional CD for incident light associated with a particular wavevector k is : CD Q Q, (5.5) k where Q refers to extinction. In colloidal systems, the nanostructures are randomly oriented in a solution and averaging over orientations is needed as usual, CD Q Q. (5.6) k Instead of a numerical average of all directions, we use the following equation for small objects [69] : CD x CD x CD y CD y CD z CD z CD. (5.7) 6 The proof of this equation is given in Ref. [69] and also available in the Appendix A. In fact one can show that CD CD (see the Appendix A) and therefore the averaging k k can be done over three perpendicular directions: CD x CD y CD z CD. (5.8) 3 Since we are using a method of local dielectric function, we need to define the dielectric constants of the components. For gold NPs, we take the data for ( ) from Ref. [108]. For the matrix material, we assume water with Au

58 Numerical simulations of plasmonic circular dichroism Geometry of nanostructures Figure 14. a) Models of chiral nanoparticle assemblies. b) Extinction spectra computed from DDA for closely packed chiral nanoparticle assemblies. The coordinates and sizes of NPs can be found in table 1 for a packing parameter. The packing parameter reflects the NP-NP separations, assume larger NP-NP separations according to the equation R ij. When we increase a size of a complex, we simply Rij with 1. Reprinted with permission from Ref.[100]. Copyright 2010, American Chemical Society. Systematically, we have studied a series of nano-assemblies using DDSCAT. They include a helical structure, an elliptic helical structure, a pyramidal structure, and an equilateral tetrahedral structure, respectively. The sketches of these nano-assemblies are shown in Fig. 14 (a). The size is chosen to be 5 nm in radii for NPs in a helix, a pyramid, and an elliptic helix. In an equilateral tetrahedral complex, the asymmetry comes from

59 NPs of different size; the sizes of NPs are taken as 3nm, 4nm, 5nm and 6nm, respectively. To begin with the following investigation, we first studied assemblies in which NPs are close to each other and choose the minimum surface to surface gap within the assembly to be only 1nm, as is shown in Fig. 14 (a). We define the size parameter =1 in the structures shown in Fig. 14 (a). Then we expand the structure by multiplying the coordinates of each NP with a parameter 1. In other words, the coordinates of NPs in 0 0 our complexes will be r ( ) r, where r is the -coordinate of the i-np for i, i, 1. Table 1 shows the case of =1. The coordinates and radii of NPs are given in the following form: [x,y,z, radius]. i, 59 Table 1: Coordinates and radii of NPs in a chiral assembly with =1.

60 60 In Fig. 14 (b)., we show the computed extinction spectra for the structures with. For small AuNPs, the total extinction is roughly proportional to the total volume of AuNPs. With the geometric parameters chosen, we can predict that the total extinction of those nano-assemblies is generally of the same magnitude, although some variations of extinction appear in the spectra due to the interaction between NPs Helix and elliptic helix Helical nanoparticle assemblies create strong CD signals. [70] This is because a helical chain of nanoparticles possesses a chiral collective excitation that comes directly from the geometry of the frame. The anisotropy factor g / CD abs is the largest of all helical geometries that we have studied. However, one can optimize the CD strength by varying geometrical parameters of a helix. [75] For example, we first fixed the radius of a helical assembly made of four NPs. Then, the pitch was chosen to get the maximal CD. Using the size parameter, we can compress or expand the geometry. The elliptic helices are shown in Fig. 14 (a). This structure is similar to the complex designed in Ref. [118]. If we place the first NP in the center of coordinates, the second NP will be displaced in the x-direction by a distance d. Then, we start with the position of the second NP and make a displacement in the y-direction by the same distance. This will be a position for the third NP. The position for the fourth NP is obtained by a displacement from the third one in the z-direction. The comparison of CD computed using DDA and PDA is presented in Fig. 15. From the geometry in Fig. 14 (a), we can easily tell that the helix is right handed and the

61 61 elliptic helix is left handed. The CD responses are of the same order, but the shapes are roughly flipped due to the opposite handedness of the assemblies. At and 2, simulations using DDA and PDA give similar results, which confirms that the dipole model is valid for diluted chiral nanostructures. When, the CD strength computed from DDA is significantly different from the PDA model. It indicates that the multipole effects are becoming important. Simultaneously, the CD bands of tightly-packed complexes are red shifted, which can be attributed to a stronger interaction between plasmonic NPs.

62 62 Figure 15. a) - c) Comparison of CD spectra computed using DDA and PDA for a helix and an elliptic helix; the packing parameter, and. d) The normalized CD strength as a function of for the helix. The plot is in the logarithmic scale. Reprinted with permission from Ref.[100]. Copyright 2010, American Chemical Society Pyramid For pyramidal tetramers, we take asymmetric frames. Such asymmetry is created by introducing arms of different length in the complex, as shown in Fig. 14 (a). The coordinates of Au NPs for can be found in table 1. The CD spectra of pyramidal

63 tetramers are shown in Fig. 16 for different size parameters. Comparison between DDA and point dipole method is also presented. 63 Figure 16. a) - c) Comparison of CD spectra computed using DDA and quasistatic dipole methods for pyramidal tetramers with packing parameter and. d) The normalized CD strength as a function of. The plot is in the logarithmic scale. Reprinted with permission from Ref.[100]. Copyright 2010, American Chemical Society.

64 Equilateral Tetrahedron Figure 17. a) CD spectra computed using DDA for the tetrahedral structures with various packing parameters. b) The normalized CD strength as a function of. The plot is in the logarithmic scale. Reprinted with permission from Ref.[100]. Copyright 2010, American Chemical Society.

65 65 An equilateral tetrahedral complex has a highly symmetric frame. The asymmetry comes from NPs of different radii. The dipolar interactions between plasmons in a tetrahedral 4-NP complex are not able to create CD responses even though the complex is chiral [70]. This has been confirmed numerically by using both DDA and PDA for equilateral tetrahedral complexes with relatively large separations between NPs. This interesting property of a chiral equilateral tetrahedral complex can be also demonstrated analytically in the SI of Ref. [69]. Interesting phenomena were observed when the AuNPs are pushed closer to each other when the multipole fields become stronger. A moderate CD signal with g~10-5 was observed when the surface-to-surface gap is decreased to 1nm. The results of simulations for several packing parameters are shown in Fig. 17 (a). These CD responses are extra sensitive to the inter-particle separation in comparison to the other chiral nanostructures studied here Multipole effects As we have previously shown [70] for the dipolar regime, for small structures composed of nanoparticles with the same size, where a is the radius and R is the dimension of the nanostructure. Intuitively, the strength of CD are directly related to the strength of interaction and dipole moments; the stronger the interaction is, the stronger the CD signal will be. The exponent of 12 shows the power of enhancing the CD signal by simply increasing the size of NPs. A previous experiment was able to confirm this relation by increasing the sizes of NPs in a well-defined DNA-assembled nanostructure. [59] However, it was unclear whether a point dipole model can accurately

66 describe the relationship between the size and the magnitude of the CD signal especially when the NPs are nearly touching. Noticeably, both experiments and dipolar simulations have showed enormous CD enhancement with the increase of NP size. Here, DDA simulations allow us to look beyond the dipole interaction between NPs. The simulated dipolar CD for closely-packed AuNP helical assemblies show enhancement in CD as well as a larger red shift, which is qualitatively consistent with the experiment. In Fig. 15 (d), 16 (d), 17 (b), we plot the CD strength as a function of the packing parameter in a logarithmic scale. In these figures, we also show the linear fits. A CD strength is chosen as the peak-dip difference of a spectrum. The slopes obtained in the PDA model are: for a helix and for a pyramid. Then from DDA, we obtained for a helix and for a pyramid. These faster decay rates can qualitatively be attributed to the multipole effects; the multipole fields induced by NPs decay faster than the dipolar fields. From our observation, we found that the dipole effect is still a critical factor in the CD of assemblies with asymmetric frames. However in the equilateral tetrahedral complexes, we see that the decay of CD is much faster; the slope obtained from the linear fit is about -18. Importantly, the dipolar interaction will give no contribution to a CD response in the equilateral tetrahedral structures, and, therefore, the corresponding decrease of CD( ) in Fig. 17 (b) is much steeper compared to the other structures. This suggests that the plasmonic CD response in nanoparticle tetramers with symmetric frames can be strong only if NPs are very densely packed. 66

67 Conclusion In conclusion, we have compared the simulated spectra from the discrete dipole approximation and the point dipole approach. Both methods are consistent in the dipolar interaction regime. However the PDA method is much faster for computing. In the multipolar regime, when the NPs are tightly packed, the DDA simulations are able to reveal extra effects, such as a larger red shift of the CD band is observed compared to dipolar CD band. In the dipolar regime, chiral collective excitations of plasmonic NPs assembled on an asymmetric chiral frame are the cause of CD response. Due to the multipole interactions, an asymmetry in sizes of NPs starts to play an equally important role. Hence the equilateral tetrahedral structures also become good candidates for an observation of the plasmonic CD effect. Concerning the fabrication, these equilateral tetrahedral 4-NP structures should be densely packed in order to show an essential CD response. Overall our investigation demonstrates that the plasmonic CD effect is very sensitive to geometry of a chiral assembly. Plasmonic CD can be used as a sensitive tool to characterize a morphology and size-dispersion of three-dimensional nanostructures in a solution.

68 CHAPTER 6: ELECTRODYNAMIC EFFECTS IN CHIRAL MOLECULAR- PLASMONIC STRUCTURES Introduction In NP-molecular complexes with sizes ranging from tens to hundreds of nanometers, scattering is able to compete with or even exceed the rate of absorption in nanostructures. becomes significant at electrodynamic resonances. To include an chiral molecular medium into consideration, the constitutive relations are [78] : D E ib, (6.1) B H i E. (6.2) where is introduced as the chiral parameter. Solving the coupled Maxwell s equations with constitutive relations of a chiral medium, 1 B E, c t 1 D H, c t D 0, B 0. (6.3) we first obtain: E i 1 E i. 2 H c i H (6.4) We will see that the degeneracy in propagation constant between right-handed and lefthanded polarizations is broken. The eigenvalues of the matrix in Eq. (6.4) is found as

69 69 2 k. 2 k c (6.5) Therefore, these light beams propagate with slightly different phase difference in chiral media. The difference between and results from the chiral parameter of a chiral medium. Generally speaking, it can shift electromagnetic resonances and influence the strength of absorption and scattering once the size of nanostructures become comparable to wavelength. This process is an interplay strongly dependent on the geometrical parameters such as the size of the structure and the thickness of shells, and the dielectric functions of each component material. Due to differential scattering and absorption, CD signals is expected to emerge from large nanostructures that incorporate chiral media. In this chapter, a Mie solution incorporating a chiral parameter will be used to solve the light scattering problem of multilayer core-shell structures [1, 79]. 6.2 Model: Maxwell s equations incorporating a chiral parameter The light scattering problem of core-shell multilayer structures can be solved exactly using the Mie theory. When a chiral medium is involved, its chiral property will break the symmetry of the system and lead to different propagation constants for the RCP and LCP electromagnetic waves. It is hence convenient to expand the electromagnetic fields using circular polarized vector harmonics as basis. The coupled Maxwell s equations (6.4) can be diagonalized into the following equations: P 0 P 2 Q c 0 Q 2 (6.6)

70 70 where and are RCP and LCP spherical waves which are linear combinations of vector harmonics and [1]. The vector harmonics and are defined as the following: m m m dpn cos M emn sin m Pn cos zn( ) e cos m zn( ) e, sin d m m m dpn cos M omn cos m Pn cos zn ( ) e sin m zn( ) e, sin d m z ( ) m dp cos n n 1 d Nemn n( n 1) cos m Pn cos er cos m zn( ) e d d m Pn cos 1 d msin m zn ( ) e, sin d m z ( ) m dp cos n n 1 d Nomn n( n 1) sin m Pn cos er sin m zn( ) e d d m Pn cos 1 d m cos m zn ( ) e. sin d (6.7) Explicitly, the new basis are expressed as o e mn P M k r N k r (3)/(1) (3)/(1) o/ e, mn /, Q M k r N k r (3)/(1) (3)/(1) o/ e, mn o/ e, mn (6.8) Then, total electric or magnetic fields can be expanded using and. They are called odd/even RCP/LCP vector harmonics.,.

71 71 Figure 18. Models of nanoparticles and core shell nanostructures used in the calculations. Reprinted with permission from Ref.[68]. Copyright 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. In Fig. 18, several typical examples of the core-shell multilayer structures are demonstrated. In each layer, the dielectric functions and chiral parameters are specified. Eigen modes are conveniently found using equations (6.7) and (6.8). The total electric and magnetic fields are given as:

72 (3)/(1) (3)/(1) (3)/(1) (3)/(1) (3)/(1) (3)/(1) E A M k N k B M k N k H i o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1n n1... i i n1, (3)/(1) (3)/(1) (3)/(1) (3)/(1) (3)/(1) (3)/(1) A M k N k B M k N k o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1n (6.9) where sub-index i represents the i-th layer, and (3)/(1) indicates that the spherical Bessel 72., function in equation (6.7) is the spherical Hankel function of the first type, or the spherical Bessel function,. On the boundary of each layer, the transverse component of both electric and magnetic fields are continuous. The coefficients and of those modes can be found by solving the matrix equations given in Ref. [119]. Using a set of convenient notations for scattering coefficients, the scattered field is rewritten in the following form: (3) (3) (3) (3) E a M k N k b M k N k Sca o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1n n1... H Sca n1,... i 2 0 (3) (3) (3) (3) a M k N k b M k N k o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1 n o/ e,1n The coefficients of the scattered field contain the information of extinction, absorption and scattering of a nanostructure. [1, 68, 84] Applying the optical theorem, extinction and scattering cross sections can be found in terms of scattering coefficients: C a 2,1,1, 2 Re o n e n ext 2 n 1 k n En / 2 ia, (6.10).,

73 73 C sca a a b b n n n n n ,1,1,1,1 2 Re 2 1 o n e n o n e n n k E / 2 E / 2 E / 2 E / 2, (6.11) n 2n 1 En i E0. n n 1 where 6.3 Circular dichroism in the quasistatic limit Alternatively, the absorption is defined for the whole system using these convenient symbolic equations for small nanostructures in particular: * abs eff eff V0 Q 2 Re j E dv ; ci jeff E E B c c. (6.12) * * abs or [120] (6.13) Q 2 Im P E M B dv, V 0 where and. These two equations are essentially equivalent. Using equations (6.1)-(6.3), equation (6.13) can be written as E E Qabs dv B E E B dv V0 2 V0 2 * * * Im c Re. (6.14) In formula (6.14), the integration is taken over the whole system. It can be formally distributed into two terms if we integrate it respectively inside the molecular medium and the metal nanocrystal. As usual, the CD is defined using Eq. (3.5). The averaging over incident angle can be removed due to the symmetry of a spherical object.

74 74 Figure 19. a) Calculated CD spectra for a chiral nanoshell metal NP structure; we also show the signal from the molecular nanoshell in the absence of a metal NP. b) The inverted case: A metal nanoshell chiral particle structure; for comparison, we again show the CD signal from the molecules alone. Insets: Nanostructure models. Reprinted with permission from Ref. [68]. Copyright 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. When the nanostructure is optically small, the consistency between a semiclassical dipolar theory and this Mie solution was shown in Ref. [68]. Due to randomly orientated chiral molecules, the isotropic chiral parameter of the shell leads to zero total effect on interaction with plasmons. In chapter 9, more discussion will show that it gives a vanishing plasmon CD signal when the layer of the chiral medium is thin. Analytically, the Mie solution shows that the plasmonic CD will be:

75 CD Im k Re f, k Re f, (6.15) metal metal c c c metal c c metal The first term is due to a near field interaction, and the second term is a retardation effect, which is small for small complexes. Since the presence of chiral molecule does not 75 change much the dielectric constant of the chiral shell, the plasmonic CD vanishes when the size of the nanostructure is small. 6.4 Electrodynamic effects in large structures Figure 20. a) and b) Calculated CD spectra for large core-shell structures. c) The plasmon peak CD as a function of the shell thickness, b NS a NS. Inset shows the plasmon peak CD as a function of the volume of the chiral shell. General inset: Model of the chiral plasmonic core shell structure. Reprinted with permission from Ref.[68]. Copyright 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

76 76 When the structure is large, for example, the full Mie solution has to be used to simulate the extinction, scattering and CD spectra. Several examples of large core-shell nanostructures are given in Fig. 20. From the extinction spectra, the resonances mostly come from scattering at the plasmon band around 600nm. This is an electrodynamic effect originating from resonances in the Mie coefficients. Total extinction is computed using Eq. (6.10). Since the nanostructure is large, we are not able to predict the resonances from an expansion of spherical Bessel functions with arguments or. However, we know that these resonances in the extinction spectra are results of interplay between the dielectric functions of component materials and geometric parameters of the whole structure. One may refer to the example of a single sphere in Ref. [1], where the coefficients are solved analytically. In this particular case displayed in Fig.20, the resonance is contributed by an a coefficient, which corresponds to a TM mode or a dynamic electric dipole according to Ref. [1]. For convenience and consistency, we still name the corresponding CD signal as a plasmonic CD, which is observed at the band of the resonance. In comparison, when the core is dielectric, the interference becomes more interesting. Such case was considered recently in Ref. [80]. For example, in silicon particles, it supports magnetic resonances (coming from the b coefficients) due to its large dielectric constant ( ). The electromagnetic waves are able to oscillate inside the nanostructures, creating interferences that maximize scattered electromagnetic fields. As a consequence, a few resonances will be found in the extinction spectrum. The corresponding CD bands may

77 come from differential absorption or differential scattering. They may be associated with either a TM mode or a TE mode and CD spectra may exhibit interesting shapes. 77

78 78 CHAPTER 7: PLASMONIC NANOCRYSTALS WITH CHIRAL SHAPES 7.1 Introduction Chiral metal nanocrystals constitute a new type of nanomaterials. A chiral nanocrystal lacks mirror or central symmetry. It was shown already that chiral metal structures can greatly improve the detection sensitivity for chiral molecules. [47] Here, we present a study of chiral metal nanoparticles, using a multipole expansion of electromagnetic fields with a perturbation on the shape of NPs to solve the light scattering problem. The calculations will show a new plasmonic CD mechanism through mixing multipole plasmons. 7.2 Formalism: multipole expansion of perturbed Poisson equation The formalism to describe chiral metal nanoparticles follows a quasistatic approximation. The solution of the induced electric fields by a chiral NP can be found by solving the Poisson equation: On the boundary, r f r. (7.1) ind ind, ind n ( E E ) n Au ( E E (7.2) ), where is a normal vector on the nanocrystal s surface, is the incident field, is the induced electric field related to, which can be written as an expansion of spherical harmonics,

79 79 r A Y (, ), ( r R ) in ind l, m l, m NC l0 anc lml l l1 a B Y (, ), ( r R ) out NC ind l, m l, m NC l0 r lml (7.3) Therefore, we end up solving these linear equations: l0, lml A a B b l, m l', m'; l, m l, m l', m'; l, m 0, A a B b F l ', m' l ' m', lm l, m l' m', lm ext, l ' m' l0, lml l1, m, (7.4) where the matrix elements of a, b, a, b, and F ext are explicitly defined in the SI of Ref.[69]. Once the field is explicitly expressed through the coefficients A and lm, B, lm, the heat absorption that we are seeking is defined as: Im NC 2 NC E E dv (7.5) * where E is the field inside NC. In a small nanocrystal, the circular dichroism signal can be expanded as a power series of, using a long wave approximation [69] : 2 4 CD ak bk..., (7.6) where and are coefficients related to the dielectric function and the geometric parameters of a nanocrystal. Leading by, the CD is much smaller than the absorption, since absorption is of the power of. On the other hand, CD is still much stronger than a scattering of.

80 80 Figure 21. Schematics of chiral objects generated by various equations r R,. Adapted with permission from Ref. [69]. Copyright 2012, American Chemical Society. NC The chiral shape of a nanocrystal is defined using equations. It will be more straightforward to just list a few examples at the beginning of this chapter. The plots of these objects are shown in Fig.21. The functions that were used to plot these objects are given below: Twisters: Cos( ) / RNP(, ) R0 R e ; ( ) / / ( 2 ) / / e 1 e 1 e 1 e 1 Twister : R 1.5 nm, R 7 nm, 0.05, 0.2; 0 Anti twister : R 0.8 nm, R 7 nm, 0.05, 0.2; 0

81 81 Asymmetric pyramid: F( x, y, z) x y z x y z x y z Step 1 Step 1 Step z Step 1 ; 1 1 Step x ; x e 1 Implicit function : F( x r,,, y r,,, z r,, ) 0.1 0; 1.0 In the case, a numerical solution for F xr yr zr (,,,,,,,, ) 0.1 provides us with the function r R (, ) that has a character of a chiral pyramid. NC Tetrahedral structure: R (, ) R R( r,, ), NP 0 R( r,, ) R '( x r,,, y r,,, z r,, ); R '( x, y, z) x 1 y 1 z 1 x 1 y 1 z 1 r 3 r 3 r 3 r 3 r 3 r A e A e x 1 y 1 z 1 x 1 y 1 z 1 r 3 r 3 r 3 r 3 r 3 r A e A e ; 0.5, a 0 2 nm; A 0.35 a, A 0.75 a, A 1.15 a, A 1.55 a This function is obtained starting first from a sphere ( RNP(, ) R0 ) and then introducing a strong tetrahedral distortion of the surface R'(, ).

82 Origins of plasmonic CD: mixing of multipoles Figure 22. Modal analysis of plasmon resonances of a chiral nanocrystal using the Drude model. a) The spectrum of absorption for a small plasmon broadening. Inset: Model of the chiral nanocrystal and its comparison with an ideal sphere. b) The lowest triplet modes ( ) for various plasmon broadenings. c) CD spectra of the triplet modes, plotted again for different broadenings. Adapted with permission from Ref.[69]. Copyright 2012, American Chemical Society.

83 83 It is known that defects on spherical metal nanoparticle surface could create both splitting and mixing of plasmonic modes. If a chiral distortion is created, it generates corresponding chiral plasmonic modes by mixing plasmonic modes and therefore demonstrates chiroptical response. In order to learn more about the physical process underlying the interactions of chiral nanocrystals and incident light, we show the following modal analysis using an artificial dielectric constant from the Drude model. The Drude model parameters are selected to provide a very narrow absorption peak so that we can see the splitting of multipole plasmons clearly. The Drude model dielectric function of a metal is metal 2 p 1 i p, (7.7) in which we choose 4eV. Fig. 22 shows the results for a twister with a tiny surface p variation ( =0.05) and a small plasmon broadening ( ). The -mode will split into 2 +1 lines, as is shown in Fig. 22. The th multipole is centered at p l l By gradually increasing the broadening parameter, both the absorption lines and CD lines are broadened and then merged. (Fig. 22 b and c) These spectra are focused on the dipolar plasmon band ( =1). We see clearly how a bisignate CD signal is formed. It should be noted that in our computation, a CD signal will arise in this band only when we include more harmonics than just =1. In other words the mixing between multipoles is critical for the formation of chiral plasmons and CD signals of chiral nanocrystals. In

84 84 helical gold NP assemblies, we have learnt that dipolar interactions between plasmons could dominate the CD response. The current mechanism of multipole mixing is qualitatively different from the dipolar interaction between metal nanoparticles in a chiral arrangement, although the CD signals from helical twisters (Fig.23) also possess a bisignate feature. Figure 23. Results for the CD response for various types of chiral Au nanocrystals. (a) Right-handed twister and antitwister with R = 1.5 and 0.8 nm. (b) Asymmetric pyramid with = 1. (c) Chiral tetrahedral structure with. In all cases, Chemical Society.. Adapted with permission from Ref. [69]. Copyright 2012, American Using a dielectric function taken from Ref. [108], we have tested several functions that generate chiral surface distortion. The function describes a perturbation in geometry created on a sphere. Perturbation can be modulated by a parameter, seen in section 7.2 or the SI of Ref.[69]. In our study, we are interested in

85 85 helical twisters, pyramid-like NPs and equilateral tetrahedron-like NPs on which four bumps of different sizes are created at the vertices to mimic an equilateral tetrahedral framework. Coincidently, the helical structure again demonstrated the best potential in plasmonic CD generation, though we have discussed in the last paragraph that the mechanism is essentially different from helical AuNP chains. This suggested an interesting connection between chiral plasmon excitations and chiral geometries, which can be explained using a Taylor expansion of CD signals in a discrete dipole interaction formalism. The relation between the signs of the CD band and the handedness of a geometry can be also qualitatively explained. In a brief summary of this chapter, plasmonic circular dichroism can also be induced on metal nanoparticles with chiral surface distortion through mixing of multipoles. Helical twisters/antitwisters show the strongest CD signals. Generally, it is common to see individual chiral nanoparticles. It might be still technologically challenging to create chiral nanoparticles that are uniform in shape and size in a macroscopic ensemble in a solution. Only a couple of chiral metal nanocrystals have been made lithographically [121] or through wet chemical methods [61, 122].

86 86 CHAPTER 8: EXPERIMENTAL OBSERVATION OF PLASMONIC CIRCULAR DICHROISM Chiral plasmonic nanomaterials so far involved spherical NPs, nanorods, nanofibers, chiral mesoporous films, chiral inorganic molecules, biomolecules, etc.. [51-61, 65, 66] In this chapter, a few examples of experimental works on chiral nanoscale systems will be presented. In the field of plasmonic CD research of nanoparticles, a pioneer study was carried out on chiral silver nanoparticle assemblies by Shemer et al. [46]. In these experiments, silver ions were reduced from the solution. DNA served as a template for the growth of silver nanoparticles. The resulting silver NP complex is shown in the TEM image in Fig. 24 b). Interestingly, a new CD band was created in the plasmon band as shown in Fig.24 a), while the CD signal of DNA at 260 nm was inversed. In a control experiment, it was shown that plasmonic CD could not be created by merely mixing DNA molecules with silver nanoparticles. It was concluded that DNA had directed the asymmetric growth of silver nanoparticles [46]. Now we know that the asymmetry can be in the overall geometry of the assembly or stem from surface distortions on the NPs. It was noted that the plasmonic CD can also come from other mechanisms such as orbital hybridization or plasmon-exciton interaction.

87 87 Figure 24. The plasmonic CD band of Ag NPs assembled on DNA (a) and the geometry of the system including a TEM image of the Ag NP chain (b). Reprinted with permission from Ref.[46]. Copyright 2006, American Chemical Society. In previous chapters, it was proposed that both plasmon-plasmon and plasmonexciton interactions could induce CD signals at plasmon bands. In order to see the physical pictures more clearly, alternative designs of the nanostructures and protocols of fabrication were realized in the following experiments. First of all, on colloidal AuNPs coated with chiral peptides [56], a moderately strong plasmonic CD signal can be identified from the spectra of E5-AuNP in Fig.25 and of FlgA3-AuNP that can be found in Ref. [56]. This induced CD signal was attributed to the plasmon-exciton interaction (Fig.25 b). An -helical peptide (E5) has a stronger dipole moment than an unstructured peptide (FlgA3). The -helical peptide s interaction with the AuNP was expected to be stronger than an unstructured peptide. However, it is interesting that in this experiment [56], the CD signals measured from both peptide-aunp

88 88 conjugates are comparable. It was postulated that the adsorption of unstructured peptides to AuNPs is in a multidentate fashion, which would lead to more interaction with the gold nanoparticle, while an -helical E5 coil bonds with an AuNP through a single cysteine residue at one end of the peptide, which created a longer separation between the molecular dipole and the Au NP, leading to less chirality transfer.

89 89 Figure 25. Optical characterization of AuNPs functionalized with the E5 peptide ( ) with (a) UV-vis spectra of E5 and E5-gold nanoparticles. (Inset) Illustration of gold nanoparticle surface covalent linked to the E5 peptide via the thiol linkage (red circle). (b) The calculated CD spectra for a dipole of chiral molecule (black curve for ) and for a dipole-nanoparticle complex with two separations (R = 5.3 and 6 nm); the absorption wavelength of the dipole dipole = 200 nm. Inset shows the model used. (c) CD spectra and (d) CD spectra of UV region of E5 peptide gold nanoparticles with or without E5 peptide. Adapted with permission from Ref. [56]. Copyright 2012, American Chemical Society.

90 90 CD induced by a plasmon-exciton interaction was also identified in another experiments [58], where Gerard et al. demonstrated that the plasmonic circular dichroism can be strongly enhanced by AuNP aggregates. In a solution of gold nanoparticles attached with oligonucleotides, plasmonic CD was observed only when the gold nanoparticles were partially aggregated [58]. It suggests that the plasmonic hot spots may be the underlying physical reason for the observation of strong CD signals at the plasmon band in Ref. [58]. In a recent experiment [123], hotspot effect on the CD enhancement was also demonstrated from molecule-agnp conjugates. Calculations by Zhang et al. have shown that, in NP aggregates, the electromagnetic enhancement in the hot spot of MNP dimer can be very large for small inter-np separation, which can lead to giant CD response. For instance, the field enhancement in a hot spot can reach up to 4 10 for an Au-NP dimer with a NP radius and a separation. Therefore, by employing a NP dimer, one can expect the appearance of giant CD signals at the plasmonic wavelength with enhancement factors to be approximately 2 10 [109].

91 91 Figure 26. Gold particle helices templated with DNA origami. a) 24 parallel DNA double strands form the scaffold for nine gold particles that are arranged either in a lefthanded or a right-handed helix. b) Transmission electron micrograph of the DNA origami constructs carrying the gold particles. Scale bar: 50 nm. Adapted with permission from Ref. [59]. Copyright 2012, Rights Managed by Nature Publishing Group. In around two decades, the field of DNA nanotechnology [124] has been developing into a robust technology platform for the assembly of nanostructures. It can organize objects like nanoparticles, fluorophores and biomolecules, with nanometer precision. Our colleagues have successfully demonstrated its capability to construct helical AuNP assemblies with perfectly controlled handedness. Briefly, the protocol is presented as the following. [59, 125, 126] Single stranded DNA is folded into a rigid bundle of 24 tightlyconnected double helices, adopting the shape of a cylinder with a length of 100 nm and a diameter of 16 nm. On the surface of the DNA bundle, 9 handles were created by synthesizing 15 extra bases to the ends of the three nearest staple strands on the surface,

92 92 forming either a left-handed or a right-handed spiral staircase. AuNPs of 10 nm in diameter were covered with thiol-modified DNA sequences that were complementary to the handle sequence. After the DNA bundles and dressed AuNPs were mixed, the handles will be occupied by AuNPs, resulting in the formation of helical chains of AuNPs (Fig. 26 a). Importantly, attachment yields were greater than 97 % (Fig. 26 b). Figure 27. Calculated and experimental CD signals of DNA-origami-templated gold particle helices (left-handed helix: red lines; right-handed helix: blue lines). a) Calculated CD signal of nine gold particles (10 nm diameter) arranged in a helix with a diameter of 34 nm and a helical pitch of 57 nm. b) Experimental CD for solutions containing the 10 nm gold particle helices presented in figure 25. c) The signal strength increases dramatically with increasing particle diameter (here: 16 nm). Scale bars: 20 nm. Adapted with permission from Ref. [59]. Copyright 2012, Rights Managed by Nature Publishing Group. The self-assembled AuNP helices exhibited exactly the optical activity predicted by theory. In Fig.27 a), the calculated CD signals for an AuNP helix are shown. It

93 93 reproduced the bisignate shape dip-peak for a right-handed helix or peak-dip for a lefthanded helix at plasmon frequency of AuNPs (525 nm) in the experiments (Fig.27 b). To demonstrate the dependence of the CD strength on the NP size, our colleagues tried depositing additional gold from solution onto the pre-assembled particles. A dramatic enhancement of the CD signal round 500-fold and a shift of the CD band to a longer wavelength were observed when the diameter of AuNPs was increased from 10 nm to 16 nm (Fig.27 c). These features can be easily reproduced in a simulation using discrete dipole approximation [77] or finite element methods [127]. In further experiments, silver was deposited on the gold particles and a shift of the plasmon resonance to shorter wavelengths was observed. The collective plasmon-plasmon interactions were so strong that a frequency-dependent optical rotatory dispersion was detected when droplets of the samples were placed between two crossed linear polarizers. (Fig.28)

94 94 Figure 28. Optical rotatory dispersion of self-assembled gold nanohelices. a) Droplets of left- handed and right-handed nano helices as well as a droplet of isotropically disversed gold nanoparticle are placed between two linear polarizers. b) Transmission images are taken for clockwise (top) and counter-clockwise (bottom) rotation of the analyser by one degree out of the orthogonal configuration. The droplets of right-handed and left-handed helices appear red in the top and bottom images, respectively, as a result of optical rotatory dispersion. Reprinted with permission from Ref. [59]. Copyright 2012, Rights Managed by Nature Publishing Group.

95 95 Figure 29.Top: Schematic of the two possible extreme cases of chiral molecule coated Ag nanoparticles: Left: Monolayer-coated particles for Ag:molecule ratio 4:1; Right: molecular stack-coated particles for Ag:molecule 32:1. (a) Calculation of the CD spectrum for a dipolar particle-stack interaction at different particle-molecule average separation distances. (b) The experimental CD spectrum for a Ag:molecule 4:1 sample, showing the broad negative plasmonic CD around 400 nm. (c) A calculation of the plasmonic CD for a chiraly distorted particle taking the first 20 and 25 spherical harmonics into account. (d) The experimental induced plasmonic CD spectrum for the Ag:molecule 32:1 sample. Adapted with permission from Ref.[61]. Copyright 2012, American Chemical Society.

96 96 Another approach to create chiral plasmonic resonances is to use a large chiral molecule as a template for growth of plasmonic nanocrystals. In Ref. [61], silver nanoparticles were prepared in aqueous solutions of chiral supramolecular structures made from chiral molecular building blocks. By adjusting the metal-atom:molecule ratio, it was possible to obtain either a molecular monolayer coating on the nanoparticles (the top right scheme of Fig.29) or chiral stacks of molecules coating the nanoparticles (the top left scheme of Fig.29). In the latter case, the chiral stack-plasmon coupling induced a CD signal at the surface plasmon resonance absorption band of the silver nanoparticles (Fig.29 b). The sensitivity of this induction effect to temperature, which affected the stacking of the molecules (i.e. the level of chirality of molecules), indicated that this CD induction was governed by a Coulomb electromagnetic interaction, which could be modeled as a dipolar coupling (Fig.29 a) between plasmons and excitons. In the case of the monolayer coated silver particles, the observed significant induced plasmonic CD (Fig.29 d) could only be interpreted and modeled by the formation of metal nanoparticles with a slight chiral shape distortion (Fig.29 c).

97 97 Figure 30. a) Plasmonic Au crosses covered with a layer of chiral molecules exhibit a plasmon peak in the CD spectrum [60]. Adapted with permission from Ref.[60]. Copyright 2012, American Chemical Society. b) Lithographical plasmonic oligomer structures exhibiting strong directional CD [118]. Adapted with permission from Ref. [118]. Copyright 2012, American Chemical Society. In addition to wet chemical methods, we should also mention lithographicallymade plasmonic metamaterial structures with chirality. 3D chiral nanostructure can be fabricated using a non-chiral quasi-2d metamaterials (plasmonic crosses) and a layer of chiral molecules (Fig.30 a). The resulting CD spectrum of this structure contained a

98 plasmon peak due to long-range electromagnetic interactions between the plasmonic and molecular components [60]. Another possibility to create a chiral nanostructure is to build 98 a purely plasmonic system with a 3D chiral architecture. [118, 121] Lithographically-made two-layer structure with plasmonic oligomers exhibiting strong directional CD, one of the impressive examples, is demonstrated in Fig. 30 b).

99 99 CHAPTER 9: DISCUSSIONS AND PERSPECTIVES 9.1 Introduction Sensing of biomolecules is an attractive field, which is particularly useful in the development of medical therapies and drugs. Stereoisomers may have dramatically different properties. CD spectroscopy is widely used in biochemistry to distinguish them. The current research of plasmonic CD offers another option of sensing biomolecules using visible light as light source for CD measurements. In previous chapters, it has introduced several mechanisms of transferring a CD signal from ultraviolet band of chiral molecules to the plamson band in the visible spectral region. It was also suggested that anisotropy and local field enhancement are effective approaches to improve sensitivity of plasmonic CD measurements. This chapter will be focused on a discussion comparing these options. We will show that anisotropy can effectively improve sensitivity of CD spectroscopy in both molecular and plasmonic bands. Anisotropy can arise from orientated molecular adsorption or orientated nanostructures, leading to a higher asymmetry in nanostructures that will be detected by circularly polarized light beams. Instead of a geometrical consideration, CD response can also be enhanced by increasing molecular dipole response to enhanced local fields or hotspots. [107] Optical chirality has been proposed as another approach [81, 128], considering that the interaction between optical chirality and molecular chirality may directly contribute to molecular CD signals. We will show from a different perspective that optical chirality is a convenient parameter in small interacting nanoparticle-molecule complexes.

100 Anisotropy of chiral nanomaterials Figure 31. Oriented and non-oriented molecular shells. Calculated CD spectra for complexes comprising a metal NP and a molecular shell. The molecules in a shell can be non-oriented (a) or oriented (b and c). Insets: Models used in the calculations. Nanoshell thickness. Adapted with permission from Ref.[68]. Copyright 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. In small structures, we noticed that isotropy in chiral medium will greatly diminish the CD signal in the plasmon band. This can be verified using equation (3.9). If a molecule is randomly orientated in the cladding of a spherical MNP, the CD signal at the plasmon band will diminish, because is averaged to 0. Meanwhile, no enhancement will be observed in the molecular band. This theory shows consistency with chiral Mie solutions that give a nearly negligible

101 101 plasmonic CD response [58] for a spherical MNP coated with a thin layer of optically active medium, as is shown in Fig. 31 a). But our studies [58, 61] show that nanostructures with orientated molecular adsorption are good candidates for plasmonic CD measurements in the visible bands (Fig.9, Fig.10 and Fig.31), while the enhancement in the molecular band depends on a specific geometrical configuration. In Fig.31 b), we see that the molecular CD strength will be stronger if the electric dipole of molecules is perpendicular to the NP surface. It has been just discussed that CD signal vanishes when molecules are indeed randomly orientated in a thin shell coated on a spherical MNP. It was suggested [80] that in order to achieve significant enhancement in molecular CD, these randomly oriented molecules should be adsorbed at the bottom and the top of a spherical MNP, forming a orientated molecule-mnp conjugate, while light beams propagate along the diameter connecting the poles of the sphere. Experimental realization will require techniques to orientate the hybrid structures, such as depositing chiral materials on top of a planar NP array prepared by lithography [47] or evaporation [49]. Essentially with this type of anisotropy, CD signals are averaged over orientation of molecular dipoles instead of directions of incident beams (Eq. 3.5) that is needed for colloidal systems. Note that in experiments, the adsorption of molecules onto MNPs may be partially oriented or in a multidentate fashion due to molecular linkers.

102 Optical chirality and circular dichroism enhancement The quantity of optical chirality was previously adopted [81] to estimate the enhancement of CD in biomolecular sensing. Without an optically active medium, it was defined as [81, 83] : 1 C E 2 E B 2 B E 2 B 0 0 * Im. 0 c (9.1) where the electric field was represented by 1 2 it * it E Ee E e The parameter C is convenient in many cases and was employed in several recent papers [47, 80, 83]. However, some of the limitations were also pointed out in a recent publication [129] that the estimation of C parameter may have a bounded enhancement effect. In this section, we would like to describe the physical origin and usage of this quantity from a different perspective. In Ref. [120], the absorption in a medium is proposed as: * * Qabs 2 Im P E M B dv, (9.2) V0 where and. Using constitutive relations of a chiral medium (6.1) and (6.2), equation (9.2) can be written as a sum of two terms [68] :. Q Q Q abs abs, abs, * E E * * Im dv c Re 0 2 B E E B 0 2 dv V V * * Im Im Im 2 E E dv 0 c E B dv V V0 (9.3)

103 103 The physical meaning of this equation is that absorption occurs due to both properties of a system - the dielectric property ( Qabs, ) and the chiral property ( Qabs, ). More specifically, the first term of equation (9.3) comes from a complex dielectric function of a nanocrystal or a molecular medium and the second term results from the socalled optical chirality that interacts with a chiral medium described by. We find the second term is linear to the optical chirality parameter, if we follow the derivation of Eq. (9.1) using the Maxwell s equations. and RCP: The CD signal is the difference in equation (9.3) for the incident beams of LCP CD CD CD. (9.4) abs abs, abs, Using (9.3), the second term which is a CD contribution from the chiral properties of a molecular medium can be expressed as: * * CDabs, Im c Im E B dv Im Im 0 c E B dv V V0 V0 Im c C r C r dv (9.5) where the signs indicate that the incident beam is LCP or RCP. For examples of metal/molecule core-shell structures, analytic expressions as well as numerical results of both CD abs, and CDabs, can be found in Ref. [68]. We should note that the electromagnetic fields and in Eq. (9.3) are coupled with the optical response of a chiral material, since the total and fields are selfconsistently solved from Maxwell s equations with all boundary conditions. The

104 information carried by the electromagnetic fields is from not only the metal nanocrystal but also the chiral medium. Hence, we agree with the author of Ref.[129] that the optical 104 chirality in equation (9.5) should formally include an optical response from chiral media. If there is no dramatic change in the spectra of and interference is weak between plasmonic resonance and molecular resonance, the optical chirality may be decoupled from the optical properties of the chiral medium. Then, it is very convenient to estimate a possible enhancement of molecular CD by studying a free-standing [81, 128, 129] nanocrystal. On the contrary, if the system is strongly interacting, such as the case in section 3.3 when a plasmon and an exciton are in resonance, the total CD is also contributed by a CD abs, that is from a nanocrystal. We have seen that the shape of the total CD signal becomes complicated because of interference (Fig. 9 b). In a classical treatment, such light-matter interaction can be solved from coupled Maxwell s equations with constitutive relations (6.1) and (6.2). In addition to analytical Mie solutions of coreshell structures [68], numerical simulations of complex structures are available using finite element methods. (Appendix C) By observing Q abs, in equation (9.3) the optical chirality parameter has no direct impact on the CD signals of nanocrystals transferred from chiral molecules. For optically large hybrid structures, the scattering CD signal that we have shown in chapter 6 is not described by an optical chirality parameter, either. CD signals of chiral nanomaterials have been described by a general equation (1.1), CD CD CD CD (1.1) total Abs, molecule Abs, NC Sca.

105 105 CDAbs, NC and CD Sca may become significant comparing to CD, Abs molecule. All three contributions in a general case may be of similar magnitude. Theoretical calculations supporting this picture were performed in Ref. [68] for several model nanostructures. 9.4 Conclusions We have suggested four different mechanisms of plasmonic circular dichroism. Several experiments were able to demonstrate these mechanisms with successful fabrication of nanomaterials using well-defined building blocks. The creation of optical chirality in the visible band is interesting and attractive. Many natural molecules usually demonstrate CD in the UV band. But their interaction with plasmons creates strong plasmonic circular dichroism that enables measurements using visible light. In this respect, sensing of biomolecules may become more convenient and inexpensive. On the other hand, plasmonic circular dichroism can also be a very good method to identify materials other than biomolecules, such as heavy element Hg 2+ [48]. Using Hg 2+- mediated aggregation of Au nanorods, Zhu et al. have enabled an easy but useful application of heavy element sensing with a high sensitivity. Efforts are also being made in finding metamaterials with negative refractive index [ ], with attempted application to build novel optical devices. In the future, induced plasmonic circular dichroism may have many more applications. In experiments on plasmonic structures, we saw enhanced circular dichroism in the molecular band [136, 137]. However, it seems that the amplification of molecular CD requires a little more efforts in the design and fabrication, for example, to involve

106 106 anisotropy into engineered structures. It was shown theoretically that the adsorption of orientated molecules onto a spherical NP can lead to enhanced molecular circular dichroism [71, 107], while an orientated hybrid nanostructure with randomized molecular adsorption on the poles when light comes in vertically constitutes another possibility to improve CD signals [80]. Also a metal NP dimer structure is a good candidate to enhance molecular CD signals [107], because the orientated molecule will feel a higher field enhancement factor. At the same time, a hotspot effect in such a dimer structure can more effectively enhance the molecular CD when the NPs approaching each other closely [109]. However, a hotspot effect from random NP aggregates generally needs to be treated carefully, since complicated features will appear in the CD spectra. One more approach to generate anisotropy can be introducing 3D chiral nanocrystals or assemblies. The exploration of the optical chirality factor has provided us with a general suggestion that near-field enhancement and twisting should be combined in sensing of biomolecules [83]. Although chiral nanomaterials have demonstrated some interesting optical phenomena and potentials in future applications, the fabrication of artificial chiral nanostructures is still in its infancy and remains challenging. In particular, more efforts are needed to achieve the yield and homochirality of nanocrystals in an ensemble with a large number. Nowadays, the technologies available to construct chiral nanomaterials may be divided into two categories. In the first category, the nanostructures are assembled on an atomic level. This will include metal/semiconductor nanoclusters either with an intrinsic chiral structure or a surface chiral molecular adsorption. While in the second category, in which this dissertation is mostly interested, the fabrication

107 107 incorporates nanomaterials that interact through electromagnetic interactions. The building blocks are well defined and the techniques of assembling the nanostructures are generic and easy to adapt to future applications. Several informative reviews are available in Refs. [37, 138], which have introduced the self-assembly technologies to assemble nanoparticle structures through mechanisms of functionalized building blocks, driving external fields or direction by templates. In addition, focused ion beam lithography or electron beam lithography are available to build larger structures for optical chirality studies, such as chiral or achiral metal particles in a 2D array. In conclusion, the field of manufacturing chiral nanostructures with NPs and molecules is very active nowadays and has attracted attention from researchers of biology, chemistry and physics. Chiral nanoparticle systems have demonstrated interesting optical properties and these days, we are in the exciting situation to see new achievements being frequently reported in theory, novel designs, experimental realizations and applications. A better understanding will surely help us to find better design strategies and to manipulate the optical properties further, while the ongoing advancement of nanotechnology is another important factor that drives the creation of new and interesting chiral nanostructures.

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121 121 APPENDIX A: A TAYLOR EXPANSION OF PLASMONIC CIRCULAR DICHROISM OF METAL NANOSTRUCTURES 1. Directional circular dichroism The interacting dipole equations (5.1) and (5.2) can be found in chapter 5. According to the definition, circular dichroism is defined in Eqs. (A.1)-( A.2). In Eq. (A.1), the sub-index defines the direction of incident beam. Eq. (A.1) is therefore called directional circular dichroism. In colloidal systems, nanocrystal and nanoparticle assemblies are randomly orientated while the light beam comes into the system with a fixed direction. As was explained in the main text of this dissertation, we apply the averaging over the directions of incident beams, since they are mathematically equivalent. CD Q Q, (A.1) k CD Q Q k. (A.2) In the Gaussian-quadrature method, the accuracy of an integral will be strongly dependent on the partition of the interval of integration. This will require at least tens of directional CD to be evaluated for integration. With a simplified equation below, it is able to reduce the number of directional CD needed for an isotropic CD (A.2) by at least one order. [69] CD x CD x CD y CD y CD z CD z CD. (A.3) 6 To prove this, we need this symbolic equation rewritten from Eq. (5.1): P E MP. (A.4)

122 is the polarizability matrix. M describes the interactions between dipoles. The electric field vector in the equation above has the form as the following, 122 ik r 1 e E1, x ik r 1 e E1, y ik r 1 E1, z E e. (A.5)... ik r... n e It describes the profile of incident field at all n sites of dipoles. The phase difference between different dipoles is a key factor for CD generation. The solution of dipoles can be symbolically expressed as 1 P I M E TE. (A.6) It can be easily shown that T is a symmetric matrix. Instead of using two indices for rows and columns, I will describe T using 4 indices. In the matrix M, i and j refer to the site of dipoles; and refer to the x-,y- or z- components. These indices divide the matrices into blocks of matrices. Correspondingly, the indices are applied on T, P and E. Hence, there is From Eq. (5.4) and Eq.(A.6), the extinction is T T ij, ji,. (A.7) 1 1 * * Q Im E P Im E f T f E. (A.8) ext 0 0 i i ij, j j 2 2 ij, Note that the extinction and absorption are similar in the small structures due to the fact that scattering is proportional to k 4. Directional CD associated with a particular k, is the difference of extinction for two different circular polarizations:

123 123 1 CD Im T E f f E E f f E 2 1 Im 2 1 Im 2 1 Im 2 * * * * 0 ij, i i j j i i j j ij, T f f E E E E * * * 0 ij, i j i j i j ij, j i * * 0 ij, i j i j ij, ik r r T e E E E E j i * * ik r r T e E E E E 0 ij, ij,, (A.9) where, k e e cos sin sin sin cos, cos cos sin cos sin, sin cos 0, 1 E cos cos i sin 2 sin cos i cos sin, 1 E cos i sin 2 sin cos i cos sin (A.10) We apply Taylor expansion of the phase difference between the i-th dipole and the j-the dipole: ik r 2 3 jri 1 i e 1 ikrji, krji, krji,... 2! 3!. (A.11) 1 i 1 ikrji, krji, k rji, kkk rji, rji, rji,... 2! 3! Using (A.2), the isotropic CD can be written as:

124 124 1 ik r r CD Im T e E E E E k 2 j i * * 0 ij, ij, 1 Im T 1ik r E E E E 2 Im * * ji 0 ij,, ij, T r Im k E E * 0 ij, ji, k ij, 0 k0 Im Tij,12 rji,3 Tij,13 rji,2 Tij,21 rji,3 Tij,23 rji,1,31,2,32,1 6 T. ij rji Tij rji ij k k (A.12) In the equation above, it has used the expansion of the phase factor up to the first order of k in (A.11). The integration is non-zero only when,, in k E E is a permutation * of coordinates x,y,z. One can easily prove that these 6 terms are the same direction CD as Eq. (A.9) for small objects. When the object becomes larger, the next orders in Eq. (A.12) should be included and a reasonable number of directional CD needs to be evaluated in order to obtain isotropic CD. On the other hand, it is also found that the directional CD is the same if circularly polarized light beams travel in opposite directions. Without loss of generality, the direction of the incident beam will be fixed along +z or z directions. The target can be oriented at a particular direction since all information of geometry will be described through T matrix. Then the vector E representing LCP and RCP travelling in +z and z directions can be expressed conveniently only with x- and y- components: 1 0, 1 0, E i E i i, k // z i, k // z E 1 i 0, E 1 i 0. i, k // z i, k // z (A.13) Note the i in the sub-indices refers to the dipoles. The variables f i is the phase factor at * ik z jzi site i. It is convenient to use this identity f f e, which is a complex number only i depends on the difference in vertical axis between two dipole sites. Formally, the j

125 125 extinction can be treated as 3 terms of summation, if we use the symmetric property of T and also swap i and j when it is needed: 1 Q Im E f T f E E f T f E E f T f E E f T f E * * * * * * * * ext 0 i1 i ij,11 j j1 i2 i ij,22 j j2 i1 i ij,12 j j 2 j 2 j ji,21 i i1 2 ij 1 * * * * * * 0 Im Ei1 fi Tij,11 f je j1 Ei 2 fi Tij,22 f je j 2 2Tij,12 Re Ei1 fi f je j 2. 2 ij (A.14) The numbers 1 and 2 following i and j in the subindices refer to x- or y- component. The first two terms will give 0 CD response. Because if we plug in the polarizations defined previously in Eq. (A.13), it will be polarization independent. But the third term will be strongly dependent on the polarization as well as the phase factors. Therefore, the corresponding CD is * ikz * 0Re 1 2 Im,12 0 Re j z i ik z j z CD E i i E j e Tij Ei 1 E j2 e Im T ij,12. (A.15) ij When a beam of light propagates in the opposite direction, we not only need to flip, but also the polarization vectors need to be modified. It is because when we observe the rotation of e.g. LCP, the vector of electric field rotates clock-wisely if we look at it against the direction of propagation, and it rotates counter-clock-wisely if we look at it along the direction of propagation. Hence, using the polarization vectors (A.14) in the lab frame again, we can easily prove that CD will be the same when light beams travels in the opposite directions. ij ik z jzi CD z 20 Im e ImT ij,12, (A.16) ij ik z j zi ik z j zi CD z 2 0 Im e ImTij, Im e ImT. (A.17) ij,12 ij ij

126 Dipolar CD of equilateral tetrahedral tetramers Here we will prove that CD of an equilateral tetrahedral 4-NP complex is zero in the dipole theory. This statement has been also confirmed numerically. The interaction between two dipoles can be described by a Green s function G G r, r that is only dependent on rij ri rj. In the following proof, it is ij i j convenient to break it down into two terms: G A G B I A r r B I, (A.18) (1) ij ij ij ij ij ij ij ij where, A ij and B ij are coefficients depending on the distance r ij between two particles and also explicitly dependent on the wavelength of the incident light. Meanwhile, T in Eq. (A.6) can be expressed as: 3 a Tij ii igij j igil 1 l G 1 l1 j j igil 1 l G 1 l1l 2 l G 2 l2 j j O a l1 l1, l r 2 ij (A.19) 12. Note that i here is an isotropic polarizability of NPs. From (A.12) and (A.17), the isotropic CD signal of the first order in k 2 is Im. (A.20) 0 CD k0 Tij, rji, 3 ij, is the Levi-Civita symbol. Therefore the total CD signal (A.20) can be formally written as a summation of CD n, where n>1, CD k Im... G G... G r n. (A.21) 0 0 i j l1 l2 ln 2 il1 l1l 2 ln 2 j ji, 3 ij, l1, l2,... ln2

127 127 First of all, one may easily find CD 0. This is because the NPs are treated as 1 isolated spherical NPs. Then the interaction between two NPs is considered: CD Im G r Im G r 2 i ij j ji, i ij j ji, ij, ij, Im G r Im G r 0. i ij j ji, i ij j ji, ij, ij, (A.22) In Eq. (A.22), we have used the symmetric property of G ij. In the next order, CD Im 3 i j l G 1 il G 1 l1 j rji,, ij, l1. (A.23) We need to use Eq. (A.18) to expand the product of two matrices in the bracket. G G A A G G A B G A B G B B I. (A.24) (1) (1) (1) (1) il1 l1 j il1 l1 j il1 l1 j il1 l1 j il1 l1 j il1 l1 j l1 j il1 The last three terms does not contribute to CD like CD and 1 CD Im 3 i j l1 il1 l1 j il1 l1 j ji ijl1 2. Therefore, CD r r r r r. (A.25) One may find CD 0, because 3 r r r for any set of i, l1, j. Note that a il1, l1j, ji, directional CD of this order, represented by one of the three terms in the dot product of r r r, may differ from 0. Intuitively, for example, that is when a vector is not il1 l1j ji perpendicular or parallel to the plane spanned by the NPs with indices i, l 1 and j. CD 4 will distinguish NP assemblies with symmetric frames from those with asymmetric frames. This can be observed from a numerical simulation using Eq. (A.19), but here we only focus the CD of an equilateral tetrahedral tetramer. From its geometry, we find that all Aij s and Bij s are equal. Hence,

128 CD 4 i j l1 l2 2 ij, l12 l il 1 l1l 2 l1l 2 l2 j il 1 l2 j ji 3 A r r r r r r r Im A B ril r 1 l1l r 2 il r 1 l1l r 2 ji rl 1l r 2 l2 j rji. In the summation, we may flip. ril rl l rl l rl j And then, On the other hand, (A.26) is invariant due to the symmetry. ril rl j rji ril rl j rji rji rl i ril rl j rji rl i rij rl l rji ril rl j rl l rji rij rl l 0. r r r r r r r r r r r r il l l ji l l l j ji il l l ji l j l l ji (A.27) We may conclude that CD 0 for equilateral tetrahedral tetramers. The derivation of 4 CD will be slightly more difficult. However one may use similar arguments as we n prove CD. It can be shown exactly that CD 0 for equilateral tetrahedral 1~4 0 n tetramers in dipole limit both analytically and numerically. In summary, this result is directly derived from the symmetry of an equilateral tetrahedral tetramer, since all edges are equal and all angles between adjacent edges are equal in the structure. Starting with Eq. (A.21) where CD~k 2, this proof is valid for small equilateral tetrahedral 4-NP complex with NPs of different sizes. However, one may conveniently apply numerical simulations of Eq.(A.19) for contributions from higher orders.

129 APPENDIX B: DIRECTIONAL CIRCULAR DICHROISM OF NANOPARTICLE ASSEMBLIES 129 Figure A1. Calculated CD properties of the right-handed helix composed of 9 nanoparticles. a) Decomposition of the total CD (black spectrum) into parallel z-cd (red spectrum) and perpendicular xy-cd (blue spectrum) components. b) and c) Calculated directional CD spectra at two wavelengths which correspond to the extremes of the total CD in graph a). The similar simulation results are available for left handed helices in SI of Ref.[59]. Adapted with permission from Ref. [59]. Copyright 2012, Rights Managed by Nature Publishing Group. To explain the bisignate CD signal in the plasmon band, our study [59] showed that it originates from the splitting between a transverse(xy-) mode and a longitudinal(z-)

130 mode. The collective excitation results from the plasmon-plasmon interaction in our model. When circularly polarized light enters vertically, the transverse modes in a standing helix are excited, while longitudinal modes will be excited in a lying helix. The strength of an excitation will be slightly stronger if its handedness matches the handedness of an incident light beam. Hence it generates more heat dissipation. In Fig. A1, we show that a total CD signal can be mathematically separated. The wavelength of a longitudinal(z-) mode is slightly longer than a transverse(xy-) mode. At 522nm, we observe negative maxima at the poles of the sphere in Fig. A1 c) for a right-handed helix. At the poles, it indicates that incident beams propagate along the axis of a helix. Therefore the xy-mode of a right handed AuNP helix is slightly stronger under RCP illumination, contributing a negative CD signal to the total CD spectrum. At 552 nm, although there is no direct analogy between handedness of the excitation and handedness of the helix, we observed that the CD signal flips between a right handed helix and a left handed helix in the simulations reported in the SI of Ref. [59]. This indicates a direct correspondence between the CD signal and the handedness of a helix. Recently, the splitting of CD signals has been observed experimentally. [126] 130 Using the formalisms developed in Appendix A, we can analytically explain the CD dependence on the geometry of an assembly. First we assume that the interaction is in the point dipole regime, so that we can reduce the summation of each order (Eq. A.21), restricting to a nearest neighbor interaction, since the G functions are sensitive to a distance between two dipoles. In a helix, for example, the leading contribution of isotropic CD comes from CD. In the summation, the largest contribution comes from 4

131 131 the terms whose indices on the helical chain are l, l 1, l 2, l 3, considering that swapping any two of them may lead to significant drop of magnitude due to the G functions. In addition, the geometry of l, l 1, l 2, l 3 will be exactly the same as l ', l ' 1, l ' 2, l ' 3. Therefore the isotropic CD signals can be accumulated as contributions from many identical 4-NP groups. In comparison, in a pyramidal structure, the contribution from NPs with indices l, l 1, l 2, l 3 may be comparable with l, l 2, l 1, l 3, the sign of which might be flipped. Therefore isotropic CD signal of a pyramidal structure can be diminished compared to a helical structure. This section is focused on a directional CD of a helical chain of AuNP. Instead of CD 4, we need to look into CD and explain the flip of CD signals for two different 3 modes. As is argued in the previous paragraph, we focus on the contributions to directional CD from nearest neighbor interactions in a point dipole regime. The summation of CD comes mostly from groups of NPs with indices l, l 1, l 2 3 on the chain, i.e. CD A A r r r r r 3. Im il1 l1 ji jl1 il1 l1 j il1 l1 j ji ijl1 l, l1, l2 The variables were defined in Appendix A already. Since all particles are identical and A, A only depend on the distance between two neighboring particles, the most il1 l1j significant factor in 3 CD is the term r r r, where il1 l1j ji

132 x y z ji ji ji r r r x y z il1 l1 j ji il1 il1 il1 x y z l1 j l1 j l1 j 132 ji il 1 l1 j il1 l1 j ji il1 l1 j il1 l1 j ji il1 l1 j il1 l1 j xil y 1 il z 1 l1 j xil y 1 l1 j zil x 1 l1 j yil z 1 l1 j xl 1 j yl 1 j zil x 1 l1 j yil z 1 il x 1 il y 1 il z 1 l1 j xl 1 j yl 1 j zil x 1 il y 1 l1 j zl 1 j xil y 1 l 1 j zil x 1 il y 1 l1 j zl 1 j xl 1 j yil z 1 il x 1 l1 j yil z 1 l1 j xil yl j zil xl j yil zl j xl j yil zil xil yl j zl j xil yl j zil xil yl j zl j xl j yil zil xl j yil zl j x y z z y y z x x z z x y y x The xy-mode observed in the experiments corresponds to the CD k// z, which is the second 4 terms in the parentheses. Because the 3-NP groups on the chain may have different orientations about the axis of a helix, the signal of z-mode is averaged between CD k// and CD k// y, therefore the z-mode is contributed by the first 4 term in the sum. Comparing these terms, we just showed that the directional CD of a helix has opposite signs for z- mode and the xy-mode. Although directional CD signal may flip when direction of an incident beam is rotated 90 degrees in some cases [100], this analysis should be restricted to helical structures with NPs interacting in the point dipole regime, due to the fact that we have assumed that an order change of dipoles inside a nearest neighbor 3-NP group may lead to a dramatically reduced contribution to directional CD signals. x

133 133 APPENDIX C: NUMERICAL SIMULATION OF CIRCULAR DICHROISM FOR CHIRAL NANOMATERIALS USING DISCRETE DIPOLE APPROXIMATION AND FINITE ELEMENT METHODS A few kinds of software are available as open source or commercially available software that can be used to simulate circular dichroism for chiral nanomaterials in a classical picture. Our experience with DDSCAT and COMSOL Multiphysics [127] show that they are generally very demanding for memory and computation time. However, they offer flexibility for design of various geometries. In this section, I will show some comparison of CD simulations using analytical methods and numerical methods. DDSCAT can be easily applied to chiral nanocrystals and chiral nanoparticle assemblies. An extended formalism is also available for generalized discrete dipole approximation [139], which can be adapted to a chiral medium described by Eqs. (6.1) and (6.2). In a commercially available finite element methods (FEM) software, these constitutive relations can be conveniently implemented. It also supports anisotropic dielectric functions [140] and chiral parameters for chiral molecular media. 1. Chiral nanoparticle assemblies First of all, a set of extinction spectra and CD spectra of a 4NP-helix are compared, which have been simulated using point dipole approximation, discrete dipole approximation and finite element methods [127]. The geometry of the 4NP-helix is exactly the same as that with a size parameter in Ref.[100]. The convergence test for the DDA simulation was carefully checked in the SI of Ref.[100]. In the FEM simulation,

134 134 the meshing was chosen to be moderately fine in the software. The following plots (Fig.A2) show a very good agreement using these softwares. Most importantly, they can also be used for nanocrystals with complicated shapes. [122] Figure A2. a) Extinction spectra and b) circular dichroism spectra of a 4NP helix simulated using PDA, DDA and FEM [127], the geometric parameters of the complex are the same as for size parameters in chapter Core shell structure consisting molecular media and metal nanoparticles. Constitutive relations of medium consisting of chiral molecules are defined in a more general form that couples both electric and magnetic fields into polarization and magnetization fields. Analytic methods were extensively used in the simulation of optical properties of chiral nanomaterials. It is effective and accurate. However it is also limited to only a few number of geometries. Using COMSOL Multiphysics, the constitutive relations can be modified for chiral molecular medium. Here, one simple example will

135 show a comparison of CD simulations for a coupled system. Circular dichroism signals obtained from FEM simulations show good agreements with analytic results. 135 Figure A3. a) The model of a small core shell structure. b) CD strength at wavelength of 350nm with respect to a strength parameter (Spar). c) Extinction and absorption spectra simulated using Mie theory and COMSOL Multiphysics for a strength parameter of 40. d) CD spectra simulated using Mie theory and COMSOL Multiphysics for a strength parameter of 40. In the models, the molecular media is modeled using an isotropic dielectric function and a chiral parameter. The formula can be found as Eq. (S3) in the SI of Ref.[68]. The first structure in Fig.A3 has a molecular core and a metal shell. The radii of

136 136 the core and the shell are respectively 10nm and 20nm. The material parameters are specified in the model. Simulation showed a very good consistency between the two methods (Fig.A3 c and d). Also additional simulations show that the CD signal at the peak and dip are linear to a strength parameter, which we used to amplify the chiral parameter. This agrees with our theory in Ref. [68] that the CD signal is proportional to a chiral parameter in quasi static limit. Due to limited memory and computational errors, very small is not favored in the FEM simulations for CD signals. The strength parameter, however, helps us get around this numerical difficulty. We found that the intensity of CD strength is actually proportional to the strength parameter (Spar) from a Mie solution (Fig. A3 b). The second model consists of a metal core with a radius of 80nm and a molecular shell with a radius of 120nm. In this model, the interaction between molecular medium and the metal core is in the electrodynamic regime. The results again show good consistency between two simulations (Fig.A4).

137 137 Figure A4. a) The model of a large core shell structure. b) CD strength at wavelength of 425nm with respect to a strength parameter (Spar). c) Extinction and absorption spectra simulated using Mie theory and COMSOL Multiphysics for a strength parameter of 40. d) CD spectra simulated using Mie theory and COMSOL Multiphysics for a strength parameter of 40. In sensing application, one of the goals is to identify a type of substance of a very small quantity. Using a strength parameter here, it allows us to see an effect of interaction between molecules and metal nanostructures. Numerical observations of CD can be extrapolated to a low limit of molecular density. This formalism will be capable of facilitating design and simulations of complex nanostructures, possibly with a long range electrodynamic interaction, for future biosensing applications.

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