Collective resonance in helical superstructures of gold nanorods

Chiroptical responses of helical superstructures are determined by collective behaviors of the individual building blocks. In this paper, we present a full theoretical description of the collective resonance in superstructures. We use the gold nanorods as individual building blocks and arrange them helically along an axis in an end-to-end fashion. Numerical simulations on single unit cells reveal that the plasmonic coupling between the nanorods produces hybridized resonances, whose intensity is strongly dependent on the excitation light with left- or right-handed circular polarizations (LCP or RCP). A node-mode criterion is proposed on the basis of the microscopic mechanism, which successfully explains the difference between LCP and RCP. We further demonstrate, by repeating the unit cell from 1 to infinity along the helical axis, the multiple hybridized resonances gradually evolve and merge into a single collective resonance, whose energy is also dependent on LCP and RCP. An analytical description is provided for the collective resonance of the helical superstructure on the basis of the coupled dipole approximation method. Our theory shows that $n$ collective resonance modes are present in the helical superstructure with the unit cell consisting of $n$ nanorods. Strikingly, only one resonance can be excited by the incident light with certain circular polarization. We propose a universal selection rule for such selective excitation of the collective resonances by analyzing the symmetry of the helical superstructures. The insights provided in this work may shed light on future designs and fabrications of helical superstructures using plasmonic building blocks.

Figure 1

Schematic illustration of gold nanorod helical superstructures with lateral and longitudinal arrangements. The unit cell consists of two side-by-side or end-to-end assembled nanorods, which exhibit chiroptical response under the excitation of CPL. The unit cell consisting of $n$ nanorods is repeated in the $z$ direction to form a helical superstructure. The bottom of the figure shows that the $n$ nanorods form a circle with an in-plane angle ${\phi }_n^0=\frac{2\pi }{n}$ between adjacent nanorods when projected to the $x-y$ plane.

Figure 2

FEM-simulated chiroptical responses of the single unit cell consisting of three nanorods. The extinction spectra of the single unit cell excited by LCP (blue line) and RCP (red line) exhibit three hybridized resonance peaks located at 1.40, 1.64, and $1.82\mathrm{eV}$, respectively. The middle panel of the figure demonstrates the charge distribution profiles of the helical structure at the three hybridized energies with the LCP and RCP excitations. The number of antibonding and bonding nodes, $n_a$ and $n_b$, are counted and compared at the three hybridized resonance energies. The green dotted line on the left panel of the figure indicates the opening position of a single spiral structure, and the thickness of the arrow indicates the strength of the oscillation mode of the dipole. In our simulations, the diameter and length of the gold nanorods were set at 12 and 14 nm, respectively. The refractive index of the ambient medium was set at 1.33. The gap distance between adjacent nanorods is 1 nm, and the helix angle is ${30}^{\circ }$.

Figure 3

FEM-simulated chiroptical responses of the single unit cell consisting of four nanorods. The extinction spectra of the single unit cell excited by LCP (blue line) and RCP (red line) exhibit four hybridized resonance peaks located at 1.34, 1.54, 1.72, and $1.82\mathrm{eV}$, respectively. The charge distribution profiles of the helical structure are plotted at the four resonance energies for the LCP and RCP excitations. The number of antibonding and bonding nodes, $n_a$ and $n_b$, are counted and compared at the four resonance energies.

Figure 4

The chiroptical responses of two successive unit cells (the single unit cell consisting of three nanorods) from FEM simulation. (a) The extinction spectra of the helical structure under the excitation of incident light with LCP (blue line) and RCP (red line) show six hybridized peaks are located at 1.31, 1.43, 1.56, 1.68, 1.79, and $1.84\mathrm{eV}$, respectively. (b) LCP and RCP charge distribution profiles of dextrorotatory and levorotatory nanostructures at 1.56, 1.68, and $1.79\mathrm{eV}$. The numbers of nodes $(n_a,n_b)$ at these energies are (3,2), (2,3), and (1,4), respectively. (c) CD spectra of the dextrorotatory and levorotatory helical structures.

Figure 5

Investigation of the gap-dependent chiroptical responses by FEM simulation. (a) Gap distance-dependent extinction spectra of the helical structures (the same configuration as Fig. 4) excited by LCP (blue line) and RCP (red line). The gap distance is increased from 1 to $20\mathrm{nm}$. (b) Corresponding CD spectra of the helical structures. The hybridized peaks almost disappear at the gap distance at $20\mathrm{nm}$, which indicates that only the dipolar oscillation of individual nanorods is present in the helical structures at large gap distance.

Figure 6

Investigation of the chiroptical responses on the number of the unit cell by FEM simulation. Left panel: evolution of the LCP and RCP extinction spectra of the helical structures when the unit cell is repeated along the helical axis and forms a superstructure. The number of the unit cell is 1, 2, 3, 4, 5, and infinity from the top to the bottom. In this case, one of the unit cells consisting of three nanorods (the same configuration as Fig. 2). Right panel: evolution of corresponding CD spectra. Please note that the hybridized peaks tend to merge into a single peak when the number of the unit cells continues to increase. For helical structure containing five cells, the hybridized peaks almost disappear. For an infinite helical superstructure, a standalone single peak can be observed at $1.66\left(1.80\right)\mathrm{eV}$ for RCP (LCP).

Figure 7

Model of helical dipoles used in the CDA calculation. Each nanorod in the helical structure is modeled as a dipole located on the helix and orients along its tangent with a certain gap.

Figure 8

CDA calculation of the collective resonance modes. In our calculations, each gold nanorod is approximated as a prolate ellipsoid. The major and minor radii of the ellipsoid were set as 6.1 and $22.265\mathrm{nm}$, respectively. The vertical distance between the nearest ellipsoid in all the structures is set at 30 nm. (a) Dispersion contours of collective resonance modes in infinite helical superstructures. The investigated helical superstructures with $n=3,4,5$, and 6 are presented from left to right. The dispersion curves with azimuthal index $m$ are plotted as the dashed lines in different colors. The light cone is shown as the white dashed line. The selection rule determines the active mode under the excitation of LCP or RCP. (b) Corresponding extinction spectra under the excitation of LCP (blue) and RCP (green).