Overwhelming Analogies between Plasmon Hybridization Theory and

Subscriber access provided by University of Florida | Smathers Libraries

C: Plasmonics, Optical Materials, and Hard Matter

Overwhelming Analogies Between Plasmon Hybridization Theory and Molecular Orbital Theory Revealed: The Story of Plasmonic Heterodimers Hadiya Mecheri Abdulla, Reshmi Thomas, and Rotti Srinivasamurthy Swathi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00503 • Publication Date (Web): 13 Mar 2018 Downloaded from http://pubs.acs.org on March 15, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Overwhelming Analogies Between Plasmon Hybridization Theory and Molecular Orbital Theory Revealed: The Story of Plasmonic Heterodimers Hadiya Mecheri Abdulla, Reshmi Thomas and Rotti Srinivasamurthy Swathi* School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Vithura, Thiruvananthapuram 695551, India

ABSTRACT: Plasmon hybridization theory (PHT), an analogue of molecular orbital theory (MOT) for plasmonic molecules, has enjoyed tremendous success over the last decade in discerning the optical features of hybrid nanostructures in terms of their constituent monomeric nanostructures. Dimers of metal nanoparticles served as prototypes in elucidating many of the key aspects of plasmon hybridization. Employing quantum two-state model, in conjunction with the quasi-static approximation and the finite-difference time-domain simulations, we demonstrate that the analogy between PHT and MOT can be further propelled by a theoretical estimation of the plasmon coupling strengths and the relative contributions of the unhybridized monomeric states toward the hybrid dimeric states in plasmonic Ag-Au nanorod heterodimers. The aspect ratio of the constituent nanorods and the gap size between the monomeric nanorods can further be used as handles to tune the relative contributions of (i) the bonding and the anti-bonding modes to the total extinction and (ii) the monomeric states toward the dimeric states, with meaningful implications for surface-enhanced

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 19

spectroscopy. The tunability in light absorption properties of heterodimers in the 400-800 nm region arising as a result of broken symmetry is also suggestive of their potential role as plasmonic rulers for measuring distances.

INTRODUCTION Metal nanostructures have gained an unprecedented importance in the last couple of decades, thanks to the development of successful synthetic methodologies for preparing nanoparticles of various shapes and sizes and the demonstration of their utility for a variety of applications ranging from electronics, energy and environment to medicine. 1-4 Interaction of metal nanostructures with incident electromagnetic radiation can cause collective conduction electron oscillations, the so-called localized surface plasmon resonances (LSPRs) possessing giant dipole moments thereby giving rise to large electric fields in vicinity. 5 Excitation of various LSPR modes in metal particles, namely, dipolar, quadrupolar and other multipolar modes can be represented by spherical harmonics, similar to the way the electron distribution in various orbitals in atoms is represented. 6 Metal nanostructures owe a lot of their charm to LSPRs as they provide a handle to the researchers for manipulating light at the nanoscale, leading to the emergence of a plethora of new spectroscopic techniques 7-9 like surface-enhanced Raman scattering, enabling ultrasensitive detection of analytes, down to the single molecule limit.10 The quest for better plasmonic properties and high-end applications has led to investigations on controlled assemblies of nanostructures. The optical features in assembled metal nanostructures have been demonstrated to be far more interesting, both from a purely fundamental point of view as well as from the point of view of their superior performance to isolated nanostructures for a variety of applications. 11-18 In an assembled nanostructure, the LSPR modes on individual

ACS Paragon Plus2Environment

Page 3 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


particles hybridize to form new hybrid bonding and anti-bonding modes, analogous to the way in which the atomic orbitals on individual atoms hybridize to form molecular orbitals. Such a description, referred to as plasmon hybridization is intuitively rather appealing and can be used to decipher the optical features in assembled nanostructures. 6 Dimers of metal nanoparticles have been employed by several researchers as prototype systems for unravelling the plasmonic features of assemblies.13-14, 19 According to the plasmon hybridization approach, the plasmonic modes in a dimer can be thought of as formed from the bonding and the antibonding combinations of the monomeric plasmonic modes. In a homodimer, depending on the state of polarization of the incident light, either the bonding mode or the anti-bonding mode is optically bright, while the other one is dark. Consequently, one often observes either a red-shifted or a blueshifted coupled plasmon mode.6, 20 Asymmetry can now be introduced into dimer systems to form heterodimers in three ways: (i) by considering two particles of same geometry, but with varying composition21-23 (ii) by considering two particles of varying geometry, but with same composition2425

and (iii) by considering two particles of varying geometry as well as composition.26-27 One of the

major differences between a homodimer and a heterodimer is that, both the hybrid modes are bright modes in the latter, while only one of them is a bright mode in the former. Experimental characterization of the relative phases of the individual LSPR modes in dimers is currently possible due to techniques like electron energy loss spectroscopy.28 There are several ways in which symmetry breaking in heterodimers can be utilized to full glory. The range of wavelengths available from a heterodimer system can be increased by manipulating the geometry and the composition of the monomeric systems. Selective excitation of hot spots can further be attempted based on the choice of the wavelength.22 In case of homodimers,20 both the monomeric states contribute equally to the hybrid states, in analogy with the equal contributions of the 1s atomic orbitals of hydrogen atoms to the



Page 4 of 19

molecular orbitals of H2. However, in heterodimers, the monomeric states make unequal contributions toward the hybrid plasmonic states, in parallel with the unequal contributions of the 1s orbital of the hydrogen atom and the 2p atomic orbital of the fluorine atom toward the molecular orbitals of HF.29 Thus, there exists a compelling analogy between molecular orbital theory (MOT) and plasmon hybridization theory (PHT). Furthermore, simple molecular orbital calculation enables quantification of the contributions of the 1s and the 2p atomic orbitals to the molecular orbitals in HF by their corresponding molecular orbital coefficients. 29 Despite several advances in the area of plasmonics, surprisingly, the contributions of the monomeric unhybridized states to the hybrid states in heterodimers have not been theoretically estimated, to date. Employing the formalisms of quasi-static (Q-S) approximation, finite-difference time-domain (FDTD) method and quantum two-state model, in this article, we report an MOT-like approach for estimating the plasmon coupling strengths and the contributions of the unhybridized monomeric states to the hybrid dimeric states in Ag-Au nanorod heterodimers. Q-S approximation is one of the powerful approaches for describing the plasmonic features of metal nanostructures and their assemblies, at almost no computational cost. 26, 30-32. The optical properties of particles whose sizes are smaller than the wavelength of the incident light can be described within the quasi-static limit, which is essentially a first order Mie theory approach. In such an approach, electronic excitation features beyond the dipolar terms are neglected. Although approximate, the Q-S approximation can provide important insights into the optical features of nanostructures. More accurate computational approaches like the FDTD method are currently widely used in the literature. 27, 33-34 The FDTD method is based on the Yee’s algorithm and is a numerical analysis technique belonging to the general class of grid-based methods for computational electrodynamics. The method attempts a solution of the Maxwell’s equation by employing finite differences as approximations to both the spatial and the temporal


Page 5 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


derivative terms. Quantum two-state models are employed by theorists in describing the hybrid states across a spectrum of topics, including molecular orbital theory, 29 spectroscopy,35 plexcitonics,36 and quantum computing.37 Indeed, such approaches for describing hybrid states are quite generic. Classical coupled oscillator model has been extremely popular in elucidating the coupling energies and the hybrid states in strongly coupled systems.38-39 Herein, quantum two-state model is employed, in conjunction with the Q-S and the FDTD methods to derive new, fundamental physical insights on plasmon hybridization. This opens up opportunities for novel interpretations on plasmonic nanoarchitectures based on PHT. Our current study of plasmon coupling in Ag-Au nanorod heterodimers reveals that they exhibit tailored optical responses, well beyond the possibilities offered by the homodimers. Nanorod heterodimers with constituent nanorods of same geometry (aspect ratio; AR), but with varying compositions (Ag and Au) are considered as a prototype system for the demonstration of our proposition of estimating the contributions of the monomeric states toward the hybrid dimeric states. It is interesting to note that, properties of nanorod heterodimers with varying geometries (ARs) but with same composition are investigated by some researchers in the last few years.24, 40 However, plasmonic features of nanorod heterodimers with varying composition of the constituent nanorods are rarely described in the literature. Thakkar and co-workers recently employed the Heisenberg-Langevin approach to demonstrate quantum beats in mixed metal heterodimers. 41 The AR of the nanorods and the gap size (g) between the nanorods in heterodimers offer handles in meaningfully deciphering the monomeric contributions toward the hybrid states. THEORETICAL METHODOLOGIES The spectral cross-sections within the Q-S approximation can be calculated from the single particle (effective) polarizabilities of the individual nanorods (nanorod dimers). Lin and co-workers recently analysed the plasmon coupling in metal nanorod homodimers as a function of the orientation of the ACS Paragon Plus5Environment


Page 6 of 19

nanorods by employing the Q-S treatment.42 We have adopted their formalism for the analysis of the optical features of Ag-Au nanorod heterodimers using the Q-S approximation (see Supporting Information). All numerical calculations employing the Q-S approximation are performed using Mathematica.43 The FDTD simulations reported herein are performed using the program FDTD Solutions (version 8.16).44 The nanorod geometries are modelled as cylinders with semispherical caps. We used Johnson and Christy bulk dielectric data for Au and Palik dielectric data for Ag. All simulations are performed with water as the dielectric medium, with a refractive index of 1.33. A total field-scattered field source of light, consisting of plane waves in the wavelength range 300-900 nm, is used as the incident beam for the simulations of nanorods with AR=1.5, 2.0 and 2.5 and plane waves in the wavelength range of 300-1200 nm are used for nanorods of AR=3.0. The incident light is polarized along the nanorod axis (|| polarization) in all the calculations. We note that, in view of the lower dipolar strengths of the transverse modes of excitation of nanorods, plasmon coupling involving the transverse LSPR modes is not pursued. The scattering and absorption cross-sections are evaluated using a set of power monitors to measure the net flow in certain locations. The extinction crosssections are evaluated as the sums of the absorption and the scattering cross-sections. Perfectly matched layer boundary conditions are applied in the simulations. Symmetric and anti-symmetric boundary conditions are used for homodimers of nanorods to reduce the simulation time. For all the FDTD simulations, we used a mesh size of 0.35 nm, after prior testing for the convergence of the numerical results.


Page 7 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


RESULTS AND DISCUSSION l

l

g w

w

l = 30, 40, 50 and 60 nm l = 30, 40, 50 and 60 nm w = 20 nm w = 20 nm

g = 3, 6, 9, 12, 15 and 18 nm

g

g

g = 3, 6, 9, 12, 15 and 18 nm

g = 3, 6, 9, 12, 15 and 18 nm

Figure 1. Schematic representation showing the nanorod monomers and dimers under investigation. Herein, we have investigated the plasmonic properties of Ag and Au nanorod monomeric as well as dimeric (homodimers and heterodimers) systems. A schematic representation of various systems under consideration, along with the geometrical parameters is given in Figure 1. Initially, the monomeric states and the hybrid states are established by computing the extinction spectra of the monomers and the heterodimers using the Q-S approximation (see Supporting Information) as well as the FDTD method. It is interesting to assess the performance of the computationally cheap Q-S method against the computationally intensive, 42 but more accurate FDTD method toward a description of plasmon coupling. Figure 2a shows a comparison of the extinction spectra of the monomeric Ag and Au nanorods, computed using the Q-S and the FDTD methods for varying AR (AR=1.5, 2.0, 2.5, and 3.0; Figure 1). The extinction maxima are found to red shift and the extinction cross-sections are found to increase with increase in AR, in agreement with previous reports. 31 This is attributed to the stronger dipolar resonances that are created in nanorods with larger ARs. In Table S1 in Supporting Information (SI), we report the extinction maxima computed for various nanorods using the Q-S and the FDTD methods and the percent deviations of the Q-S results from the FDTD results. The Q-S method is



based on the dipolar approximation and is therefore expected to work well for nanorods of lower AR. The percent deviations clearly indicate (Figure 2b)

(a)

(b) Q-S Au

Ag

2.4 1.2 0.0

Q-S

AR=1.5 AR=2.0 AR=2.5 AR=3.0

Ag

FDTD Au

FDTD

2.4

600

500

1.2 0.0

Ag FDTD Au FDTD Ag Q-S Au Q-S

700 max (nm)

Cross-section (104nm2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 19

500

600

700  (nm)

500

600

700

1.5

2.0 2.5 Aspect Ratio

3.0

Figure 2. (a) Extinction spectra of monomeric Ag and Au nanorods computed using the Q-S and the FDTD methods for varying aspect ratio and (b) the deviations of the λmax values computed using the Q-S method from those computed using the FDTD method. the largest deviations for nanorods with AR=3.0 (~5-8%) due to multipolar contributions becoming important for nanorods with large ARs. The deviations are on the higher side for the Ag nanorods compared to the Au nanorods. This can be attributed to the nature of the optical constants of Ag and Au. Besides, our comparative analysis of the optical properties computed from the Q-S approximation and the FDTD method is noteworthy for assessing the feasibility of employing simple, analytic approaches toward a description of plasmonics.


Page 9 of 19

(b)

(c)

(d) Cross-section (104 nm2)

(a)

g=3 nm

AR=1.5 AR=2.0 AR=2.5 AR=3.0

4.0

Energy

2.0

0.0

(e) Cross-section (104 nm2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.0

Ag Au Effect of aspect ratio

Ag-Au heterodimer

Effect of gap size

1.5

0.0

400

600

800  (nm)

g=3 nm g=6 nm g=9 nm g=12 nm g=15 nm g=18 nm

1000

AR=2.5

550 600 650 700 750  (nm)

Figure 3. Schematics of the variation in the energies of the plasmonic states in Ag-Au nanorod heterodimers with variation in the constituent nanorod aspect ratio and the gap size (a-c). The colour coded scale bars denote the relative contributions of the monomeric nanorod states toward the hybrid dimeric states. (d) Aspect ratio and (e) gap size dependence of the extinction spectra of Ag-Au nanorod heterodimers computed using the FDTD method. Next, we investigate the plasmon coupling in Ag-Au nanorod heterodimers as a function of the gap size and the AR of the nanorods (Figure 1). The gap size is varied so as to assess the extent of plasmon coupling in each of the dimer systems and to arrive at values of g beyond which plasmon coupling is negligible (retrieval of monomeric optical response). Such an analysis is extremely relevant for plasmon ruler applications. The plasmon coupling strengths are also expected to vary as a function of the AR as the strengths of dipolar resonances that are created on the nanorods are different for nanorods with various ARs. In addition, we have performed FDTD simulations on the nanorod heterodimers to assess the performance of the Q-S approximation in describing plasmon coupling in nanorod heterodimer systems. Figure S2 in SI shows a comparison of the extinction spectra computed using the Q-S as well as the FDTD methods for Ag-Au nanorod heterodimers for varying gap sizes and ARs. In Table S2 in SI, we report the extinction maxima computed for various



Page 10 of 19

heterodimer systems using the Q-S and the FDTD methods and the percent deviations of the Q-S results from the FDTD results. Unlike in case of homodimers (Figure S3 in SI), both the bonding as well as the anti-bonding LSPR modes are bright modes in heterodimers, enabling a larger range of wavelengths over which heterodimer plasmonic systems could be used in applications such as surfaceenhanced spectroscopy. According to the PHT, the plasmonic modes on monomeric particles couple to give rise to two new hybrid dimeric states, namely bonding (lower energy) and anti-bonding (higher energy) states. The λmax values corresponding to the bonding modes are expected to be red shifted with respect to the monomeric Au peaks and those of the anti-bonding modes are expected to be blue shifted with respect to monomeric Ag peaks (Figure 3). The computed spectral positions of the hybrid states of dimers for AR=2.5 and AR=3.0 are in agreement with the expected trends from PHT. However, for AR=1.5 and AR=2.0, in some cases, the λmax values corresponding to the anti-bonding modes are red shifted with respect to the monomeric Ag peaks. A similar observation had earlier been made in the case of Ag-Au nanosphere heterodimers and was attributed to the coupling of the silver LSPR with the quasicontinuum of interband absorption of gold. 26 The effect of variation in gap size for a chosen nanorod heterodimer system (AR=2.5) and the effect of variation in AR of the constituent nanorods for a chosen gap size (g=3 nm) on plasmon coupling are also shown in Figure 3. One of the notable features of the simulated spectra (Figure S2 in SI) is the significant redistribution of spectral weights of the extinction peaks as a function of g and AR. It is apparent from the figure that the contribution of the bonding mode (on the red side in the spectra) to the total extinction increases with decrease in gap size and increase in AR. In order to quantify this, we deconvoluted the computed extinction spectra by employing Lorentzian fits and estimated the percentage contributions of the bonding and the antibonding modes to the extinction spectra. The results are presented in Table S3 in SI. The anti-bonding mode makes a significant contribution to the extinction spectra of heterodimers with AR=2.0, which

10Environment ACS Paragon Plus

Page 11 of 19

decreases as one goes to heterodimers with AR=2.5 and AR=3.0. This suggests that, in surfaceenhanced spectroscopy experiments employing Ag-Au heterodimers as the plasmonic system, nanorods with lower ARs could be employed when the wavelength of interest is in the 450-500 nm region. In contrast, heterodimers with large AR could be used for probing the systems in the 600-800 nm region. The Ag-Au nanorod heterodimer system investigated herein therefore offers flexibility in terms of operating wavelength range for surface-enhanced spectroscopy experiments. The contributions of the bonding and the anti-bonding modes to the total extinction can be obtained from experimentally measured optical extinction spectra. We hope that such measurements will soon be performed on Ag-Au nanorod heterodimer systems.

0.16 V (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.08

3

6

9

12

15

18

3.0 2.5 2.0

Figure 4. The variation in plasmon coupling strengths estimated from the quantum two-state model for Ag-Au nanorod heterodimers as a function of the gap size and the aspect ratio of the constituent nanorods. Subsequently, we employ the quantum two-state model29 to estimate the plasmonic coupling strengths in the dimers and to provide a complete description of the hybrid states in terms of the monomeric states. Let EAg and EAu represent the energies of the monomeric plasmonic states. If the monomeric states are coupled by a coupling energy of strength V, the energies of the hybrid states are given by the eigenvalues of the Hamiltonian matrix 11Environment ACS Paragon Plus


𝐻=(

𝐸𝐴𝑔 𝑉

Page 12 of 19

𝑉 ), 𝐸𝐴𝑢

(1)

and can be shown as 𝐸± =

1 1 2 (𝐸𝐴𝑔 + 𝐸𝐴𝑢 ) ± √(𝐸𝐴𝑔 − 𝐸𝐴𝑢 ) + 4𝑉 2. 2 2

(2)

The E- and E+ refer to the energies of the bonding and anti-bonding hybrid states, respectively. The above equation could be recast to express V in terms of the energies of the unhybridized and the hybrid states as 2

𝑉=

√(𝐸+ − 𝐸− )2 − (𝐸𝐴𝑔 − 𝐸𝐴𝑢 ) 2

.

(3)

From the extinction spectra calculated using the Q-S as well as the FDTD methods, it is therefore possible to estimate the plasmon coupling strengths for various dimer systems. The estimation of coupling strengths in interacting chemical systems is an interesting theoretical problem. In case of molecular systems, one often evaluates the matrix elements of the wave functions of the interacting systems with the Coulombic term to arrive at the coupling strengths.45 However, in plasmonic systems, in view of their large system size, one often resorts to arguments based on classical electrostatics to estimate the plasmon coupling strengths. For instance, Mulvaney et al. have employed the dipolar approximation for estimating the interaction strengths between plasmonic particles in linear nanosphere oligomers.46 The coupling strengths could be estimated from the polarizabilities (which are in turn related to the dipole moments) or from phenomenological approaches by considering a range of coupling values and monitoring the variation in the plasmon resonances with the variation in coupling strengths. Herein, we use the polarizabilities to compute the extinction spectra, from which the plasmon coupling strengths are estimated. Henceforth, in view of their higher accuracy, energies of the unhybridized states and the hybrid states computed using the FDTD method are employed for


Page 13 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the analysis of the plasmon coupling strengths. The energies of the hybridized plasmonic resonances and the computed plasmon coupling strengths for Ag-Au nanorod heterodimers of various ARs as a function of gap size are presented in Table S4 in SI and summarized in Figure 4. A systematic variation of the coupling strengths can be seen as a function of the gap size as well as the ARs of the constituent nanorods. The coupling strengths are clearly found to decrease with increase in gap size for all the dimer systems. The coupling strengths increase with increase in AR of the constituent nanorods. Though plasmon coupling in nanorod dimers has been a topic of study in the last few years, 13, 24, 42 to the best of our knowledge, there have been no estimates of the plasmon coupling strengths so far in the literature. Our findings indicate that the lessons learnt from quantum mechanical modeling of twolevel systems and MOT can be seamlessly adapted to describe plasmonic states and to estimate strengths of plasmon coupling in hybrid systems. Indeed, the two-state model was recently employed by researchers to describe plasmon-exciton hybrid states.36, 39, 47 Furthermore, we have carried out a similar analysis to characterize the hybrid states and estimate the coupling strengths in Ag and Au nanorod homodimers. In Figures S4 and S5 of SI, we present the optical extinction features of homodimers. The λmax values and percent deviations of the Q-S results from the FDTD results are compiled in Tables S5 and S6 in SI. Compared to the heterodimers, the plasmonic interactions in homodimers are strong, resulting in larger shifts of the hybridized levels from the monomeric unhybridized levels. On comparing the plasmon coupling strengths of homodimers (Table S7 in SI) and heterodimers (Table S4 in SI), we find that the plasmon coupling is overall very strong in Ag homodimers, when compared to Au homodimers and Ag-Au heterodimers. Though the plasmon coupling in Ag-Au nanorod heterodimers is not as strong as in case of Ag homodimers, the feasibility of employing the heterodimers across a wider spectral region (in view of both the bonding and the



anti-bonding modes being bright) in surface-enhanced spectroscopy experiments provides them the edge over homodimers. Ag Au

(a)

(b)

50

50

3.0 2.5 2.0

3

6

9

12

15

18

3

6

9

12

15

18

% contribution

100 100

% contribution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 19

3.0 2.5 2.0

Figure 5. The variation in the relative contributions of the monomeric Ag and Au nanorods toward (a) the E- and (b) the E+ hybrid dimeric states as a function of the gap size and the aspect ratio of the constituent nanorods. Our efforts thus far have been to calculate the hybrid plasmonic energy levels and to estimate the plasmon coupling strengths in Ag-Au nanorod heterodimers. The case of plasmonic heterodimers indeed represents a classic example of a problem wherein the analogy to molecular orbital description can be propelled further by quantitatively assessing the relative contributions of the unhybridized monomeric plasmonic states to the hybrid dimeric plasmonic states. The solutions to the two-level system could be arrived at by solving the matrix equation: 𝐸𝐴𝑔 − 𝐸± ( 𝑉

𝑉 𝐶𝐴𝑔 0 ) ( ) = ( ). 𝐸𝐴𝑢 − 𝐸± 𝐶𝐴𝑢 0

(4)

On substituting the computed hybrid plasmonic energies into the above matrix equation and by 2 2 employing the normalization condition (𝐶𝐴𝑔 + 𝐶𝐴𝑢 = 1) for the combining coefficients, it is possible

to solve for the coefficients corresponding to the bonding as well as the anti-bonding plasmonic modes of the heterodimers. The squares of the coefficients represent the fractional contributions of the monomeric states to the hybrid states. This is indeed in direct analogy to a molecular orbital calculation 14Environment ACS Paragon Plus

Page 15 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


for a molecule like HF, that one encounters in quantum chemistry. We therefore adopt this approach to compute the contributions of the Ag and the Au monomeric nanorods to the bonding and the antibonding modes of the Ag-Au heterodimers for varying gap sizes and ARs. The results are presented in Table S8 in SI and summarized in Figure 5. The contributions from the monomeric Ag states to the bonding modes of the heterodimers increase with increase in AR and decrease in gap size. This is in direct correlation with the trend of increased coupling strengths for large AR and small g and can be attributed to the increased propensities for mixing of the monomeric states in case of nanorods of large AR and dimers of small g. In the case of anti-bonding modes of heterodimers, opposite trend is found. Thus, we see that, using the formalisms of MOT, meaningful interpretations about plasmonic molecules can be arrived at, thereby expanding the scope of PHT in nanoplasmonics. CONCLUSIONS In conclusion, we have employed Ag-Au nanorod heterodimers as a model system for deciphering the optical extinction features of plasmonic molecules in terms of the bonding and the anti-bonding combinations of the monomeric states. The contributions of the bonding and the anti-bonding states to the total extinction are quantified by deconvoluting the extinction spectra computed using the FDTD simulations. Further, the contributions of the monomeric states toward the hybrid states in heterodimers and the strengths of plasmon coupling are estimated by employing a quantum two-state model, demonstrating the applicability of the formalisms of the molecular orbital theory in the context of the plasmon hybridization theory. Ag-Au nanorod heterodimer system exhibits excellent tunability in localized surface plasmon resonance features, one of the current day challenges in nanoplasmonics.

ASSOCIATED CONTENT



Page 16 of 19

Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Details of the quasi-static approximation, supplementary figures and tables (PDF) AUTHOR INFORMATION Corresponding Author *[email protected] ACKNOWLEDGMENT The authors are grateful to Prof. K. George Thomas for many useful discussions. The authors acknowledge IISER-TVM for computational facilities. RSS acknowledges the Department of Science and Technology (DST Nanomission Project; SR/NM/NS-23/2016), Government of India for financial support. HMA and RT acknowledge the Department of Science and Technology (DST) and Council of Scientific and Industrial Research (CSIR), Government of India, respectively for the fellowships. The authors thank P. P. Rafeeque for graphical support. REFERENCES (1) Sun, Y.; Xia, Y., Shape-Controlled Synthesis of Gold and Silver Nanoparticles. Science 2002, 298, 2176-2179. (2) Jain, P. K.; Huang, X.; El-Sayed, I. H.; El-Sayed, M. A., Noble Metals on the Nanoscale: Optical and Photothermal Properties and Some Applications in Imaging, Sensing, Biology, and Medicine. Acc. Chem. Res. 2008, 41, 1578-1586. (3) Ueno, K.; Misawa, H., Surface Plasmon-Enhanced Photochemical Reactions. J. Photochem. Photobiol. C: Photochem. Rev. 2013, 15, 31-52. (4) Mayer, K. M.; Hafner, J. H., Localized Surface Plasmon Resonance Sensors. Chem. Rev. 2011, 111, 3828-3857. (5) Chen, T.; Wang, H.; Chen, G.; Wang, Y.; Feng, Y.; Teo, W. S.; Wu, T.; Chen, H., HotspotInduced Transformation of Surface-Enhanced Raman Scattering Fingerprints. ACS Nano 2010, 4, 3087-3094. (6) Prodan, E.; Nordlander, P., Plasmon Hybridization in Spherical Nanoparticles. J. Chem. Phys. 2004, 120, 5444-5454. (7) Stiles, P. L.; Dieringer, J. A.; Shah, N. C.; Duyne, R. P. V., Surface-Enhanced Raman Spectroscopy. Annu. Rev. Anal. Chem. 2008, 1, 601-626.


16

Page 17 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(8) Le, F.; Brandl, D. W.; Urzhumov, Y. A.; Wang, H.; Kundu, J.; Halas, N. J.; Aizpurua, J.; Nordlander, P., Metallic Nanoparticle Arrays: A Common Substrate for Both Surface-Enhanced Raman Scattering and Surface-Enhanced Infrared Absorption. ACS Nano 2008, 2, 707-718. (9) Zhang, J.; Fu, Y.; Chowdhury, M. H.; Lakowicz, J. R., Metal-Enhanced Single-Molecule Fluorescence on Silver Particle Monomer and Dimer: Coupling Effect between Metal Particles. Nano Lett. 2007, 7, 2101-2107. (10) Nie, S.; Emory, S. R., Probing Single Molecules and Single Nanoparticles by SurfaceEnhanced Raman Scattering. Science 1997, 275, 1102-1106. (11) Guerrero-Martínez, A.; Auguié, B.; Alonso-Gómez, J. L.; Džolić, Z.; Gómez-Graña, S.; Žinić, M.; Cid, M. M.; Liz-Marzán, L. M., Intense Optical Activity from Three-Dimensional Chiral Ordering of Plasmonic Nanoantennas. Angew. Chem. Int. Ed. 2011, 50, 5499-5503. (12) Thomas, R.; Swathi, R. S., Organization of Metal Nanoparticles for Surface-Enhanced Spectroscopy: A Difference in Size Matters. J. Phys. Chem. C 2012, 116, 21982-21991. (13) Kumar, J.; Thomas, K. G., Surface-Enhanced Raman Spectroscopy: Investigations at the Nanorod Edges and Dimer Junctions. J. Phys. Chem. Lett. 2011, 2, 610-615. (14) Funston, A. M.; Novo, C.; Davis, T. J.; Mulvaney, P., Plasmon Coupling of Gold Nanorods at Short Distances and in Different Geometries. Nano Lett. 2009, 9, 1651-1658. (15) Hao, E.; Schatz, G. C., Electromagnetic Fields around Silver Nanoparticles and Dimers. J. Chem. Phys. 2004, 116, 357-366. (16) Xing, H.; Wang, Z.; Xu, Z.; Wong, N. Y.; Xiang, Y.; Liu, G. L.; Lu, Y., DNA-Directed Assembly of Asymmetric Nanoclusters Using Janus Nanoparticles. ACS Nano 2012, 6, 802-809. (17) Tabor, C.; Van Haute, D.; El-Sayed, M. A., Effect of Orientation on Plasmonic Coupling between Gold Nanorods. ACS Nano 2009, 3, 3670-3678. (18) Chuntonov, L.; Haran, G., Trimeric Plasmonic Molecules: The Role of Symmetry. Nano Lett. 2011, 11, 2440-2445. (19) Ausloos, M.; Clippe, P.; Lucas, A. A., Infrared Active Modes in Large Clusters of Spheres. Physical Review B 1978, 18, 7176-7185. (20) Jain, P. K.; Eustis, S.; El-Sayed, M. A., Plasmon Coupling in Nanorod Assemblies: Optical Absorption, Discrete Dipole Approximation Simulation, and Exciton-Coupling Model. J. Phys. Chem. B 2006, 110, 18243-18253. (21) Chen, F. Y.; Alemu, N.; Johnston, R. L., Collective Plasmon Modes in a Compositionally Asymmetric Nanoparticle Dimer. AIP Adv. 2011, 1, 032134. (22) Encina, E. R.; Coronado, E. A., Near Field Enhancement in Ag Au Nanospheres Heterodimers. J. Phys. Chem. C 2011, 115, 15908-15914. (23) Roopak, S.; Kumar Pathak, N.; Sharma, R.; Ji, A.; Pathak, H.; Sharma, R. P., Numerical Simulation of Extinction Spectra of Plasmonically Coupled Nanospheres Using Discrete Dipole Approximation: Influence of Compositional Asymmetry. Plasmonics 2016, 11, 1603-1612. (24) Slaughter, L. S.; Wu, Y.; Willingham, B. A.; Nordlander, P.; Link, S., Effects of Symmetry Breaking and Conductive Contact on the Plasmon Coupling in Gold Nanorod Dimers. ACS Nano 2010, 4, 4657-4666. (25) Huang, C.-p.; Yin, X.-g.; Kong, L.-b.; Zhu, Y.-y., Interactions of Nanorod Particles in the Strong Coupling Regime. J. Phys. Chem. C 2010, 114, 21123-21131. (26) Sheikholeslami, S.; Jun, Y.-w.; Jain, P. K.; Alivisatos, A. P., Coupling of Optical Resonances in a Compositionally Asymmetric Plasmonic Nanoparticle Dimer. Nano Lett. 2010, 10, 26552660.


17


Page 18 of 19

(27) Goeken, K. L.; Subramaniam, V.; Gill, R., Enhancing Spectral Shifts of Plasmon-Coupled Noble Metal Nanoparticles for Sensing Applications. Phys. Chem. Chem. Phys. 2015, 17, 422427. (28) Quillin, S. C.; Cherqui, C.; Montoni, N. P.; Li, G.; Camden, J. P.; Masiello, D. J., Imaging Plasmon Hybridization in Metal Nanoparticle Aggregates with Electron Energy-Loss Spectroscopy. J. Phys. Chem. C 2016, 120, 20852-20859. (29) Atkins, P. W.; Friedman, R., Molecular Quantum Mechanics, 5th Ed.; Oxford University Press: Oxford; New York, 2011. (30) Zhu, J.; Li, J.-J.; Zhao, J.-W., Tuning the Plasmon Band Number of Aluminum Nanorod within the Ultraviolet-Visible Region by Gold Coating. Phys. Plasmas 2014, 21, 112108. (31) Link, S.; Mohamed, M. B.; El-Sayed, M. A., Simulation of the Optical Absorption Spectra of Gold Nanorods as a Function of Their Aspect Ratio and the Effect of the Medium Dielectric Constant. J. Phys. Chem. B 1999, 103, 3073-3077. (32) Kreibig, U.; Vollmer, M., Optical Properties of Metal Clusters; Springer: New York, 1995. (33) Ni, W.; Kou, X.; Yang, Z.; Wang, J., Tailoring Longitudinal Surface Plasmon Wavelengths, Scattering and Absorption Cross Sections of Gold Nanorods. ACS Nano 2008, 2, 677-686. (34) Dai, H.; Zhang, L.; Wang, Z.; Wang, X.; Zhang, J.; Gong, H.; Han, J.-B.; Han, Y., Linear and Nonlinear Optical Properties of Silver-Coated Gold Nanorods. J. Phys. Chem. C 2017, 121, 12358-12364. (35) Bernath, P. F., Spectra of Atoms and Molecules, 2nd Ed.; Oxford University Press: New York, 2005. (36) Ebbesen, T. W., Hybrid Light–Matter States in a Molecular and Material Science Perspective. Acc. Chem. Res. 2016, 49, 2403-2412. (37) Scarani, V., Quantum Computing. Am. J. Phys. 2012, 66, 956-960. (38) Novotny, L., Strong Coupling, Energy Splitting, and Level Crossings: A Classical Perspective. Am. J. Phys. 2010, 78, 1199-1202. (39) Thomas, R.; Thomas, A.; Pullanchery, S.; Joseph, L.; Somasundaran, S. M.; Swathi, R. S.; Gray, S. K.; Thomas, K. G., Plexcitons: The Role of Oscillator Strengths and Spectral Widths in Determining Strong Coupling. ACS Nano 2018, 12, 402-415. (40) Biswas, S.; Duan, J.; Nepal, D.; Park, K.; Pachter, R.; Vaia, R. A., Plasmon-Induced Transparency in the Visible Region Via Self-Assembled Gold Nanorod Heterodimers. Nano Lett. 2013, 13, 6287-6291. (41) Thakkar, N.; Cherqui, C.; Masiello, D. J., Quantum Beats from Entangled Localized Surface Plasmons. ACS Photonics 2015, 2, 157-164. (42) Shao, L.; Woo, K. C.; Chen, H. J.; Jin, Z.; Wang, J. F.; Lin, H. Q., Angle- and EnergyResolved Plasmon Coupling in Gold Nanorod Dimers. ACS Nano 2010, 4, 3053-3062. (43) Mathematica, Version 9.0, Wolfram Research, Inc.: Champaign, Illinois, 2012. (44) Fdtd Solutions, Lumerical Solutions, Inc.: Vancouver, Canada, 2003. (45) May, V.; Kuhn, O., Charge and Energy Transfer Dynamics in Molecular Systems; WileyVCH: Weinheim, 2004. (46) Barrow, S. J.; Funston, A. M.; Gómez, D. E.; Davis, T. J.; Mulvaney, P., Surface Plasmon Resonances in Strongly Coupled Gold Nanosphere Chains from Monomer to Hexamer. Nano Lett. 2011, 11, 4180-4187. (47) Fofang, N. T.; Park, T.-H.; Neumann, O.; Mirin, N. A.; Nordlander, P.; Halas, N. J., Plexcitonic Nanoparticles: Plasmon−Exciton Coupling in Nanoshell−J-Aggregate Complexes. Nano Lett. 2008, 8, 3481-3487.


18

Page 19 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic

σ* 1s

H 2p

Ag

F σ

Au

Plasmon Hybridization Theory

Molecular Orbital Theory


19

Overwhelming Analogies between Plasmon Hybridization Theory and

Recommend Documents