Coupled electric dipole model for a Su-Schrieffer-Heeger chain of optically resonant coreshell nanoparticles

Abstract

Coreshell nanoparticles can combine optical features of different materials in a single nanostructure, which makes them interesting for many applications from biomedicine to energy harvesting. On the other hand, periodic arrays of plasmonic nanoparticles can exhibit topological phenomena such as topological edge states. Here we study periodic chains of Si@Ag coreshell nanoparticles. For this task, we combine the hybridization of surface plasmonic modes in complex nanostructures with the coupled electric dipole formalism employed for modelling the scattering of light by periodic arrays of small plasmonic nanoparticles. We propose treating coreshell nanoparticle as several coupled electric dipoles instead of just one and show how this is a more appropriate framework to study arrays of coreshell nanoparticles, which allows to build a one-to-one connection between the resonant modes of the nanoparticles and the dispersion bands of the system. Within this formalism, we show that a Su-Schrieffer-Heeger (SSH) chain of coreshell Si@Ag nanoparticles host multiple topological edge states pinned at the resonant frequencies of the coreshell nanoparticles.

Figures (8)

• Figure 1: Coreshell polarizability: (a) Localized surface plasmon at a metallic sphere of permittivity $\varepsilon(\omega)$ and radius $a$ in host medium of permittivity $\varepsilon_B$. The external field $E_\textrm{inc}$ induces a dipole $\textbf{p}$. The shaded pink zone represents the oscillating free electrons. (b) Localized surface plasmon at metallic void $\varepsilon(\omega)$ and radius $a$ and permittivity of core $\varepsilon_c$ (c) Localized plasmonic modes at coreshell nanoparticle of radius $a$ and $b$ and core and host permittivity $\varepsilon_B$ and $\varepsilon_c$ (d) and (e) Decomposition of coreshell nanoparticle as the hybridization of the surface sphere and void modes with dipoles $\textbf{p}_a$ and $\textbf{p}_b$.
• Figure 2: Coupling between surface modes in coreshell nanoparticle: (a) Coupling and Rabi splitting between coreshell modes. The purple and gray lines represent the coreshell modes, while the blue and pink are the bare void and sphere modes $\omega_a$ and $\omega_b$. The black arrows mark the Rabi splitting, $\hbar\Omega = 0.21$ in this case. We fix $a=30$ nm and $b = 5nm$. Same as (a) but for $b = 20\; \textrm{nm}$. In this case $\hbar\Omega = 1.74$ eV. (c) Optical extinction $\sigma_{ext}(\omega)$ of coreshell nanoparticle : The pink and blue lines represent the optical extinction of a silver sphere on air and Si sphere on silver medium. The purple line is the optical extinction of the coreshell nanoparticle, while the pink and bluea aere the partial optical extinctions at each surface. For all the plots we used the parameters $\varepsilon_{c} = 11.7, \varepsilon_B = 1, \varepsilon_\infty = 5, \hbar\omega = 8.9~\textrm{eV}, \gamma = 0.0366~\textrm{eV}$Vial2005.
• Figure 3: SSH chain of coreshell nanoparticles: The SSH chain is a periodic 1D array with two particles per unit cell, with staggered couplings due to alternating distances. The unit cell is delimited by the dashed gray line. The distance between the particles in the unit cell is $\beta d^2$ where $\beta$ is a dimensionless parameter between 0 and 2, while the distance between first neighbours from adjoint cells is $(2-\beta) d/¨2$. The couplings between the particles are mediated by the Green's dipole-dipole function $\overleftrightarrow{\textbf{G}}(\omega,\textbf{R})$. (b) Mapping of the coreshell nanoparticle chain to a bilayer SSH chain. The solid, dashed and dotted lines represent the intraparticle coupling $g_{ab}$, the intracell coupling, $G_1$ and the intercell coupling, $G_2$.
• Figure 4: Dispersion bands of SSH: In panel (a) we represent the bands around the upper coreshell resonance frequency $\hbar\omega_\textrm{cs}^+ \simeq 3.71$eV and in panel (b) the same for the lower resonance frequency $\omega_\textrm{cs}^- \simeq 1.95$eV. Purple and gray lines represent the dispersion bands for transversal and longitudinal modes. The parameters used are the same as in previous figures and $d = 200$nm and $\beta = 1.2$.
• Figure 5: Topological invariants. (a) Zak phase of the dispersion bands of the SSH chain, depending on $\beta$. The jump from 0 to 1 at $\beta = 1$ signals the topological transition. We used the same parameters as in the previous figures. (b) Winding number $W$ of $\mathcal{G}_0(q)$ for $\beta = 0.8, 1, 1.2$. The rest of parameters is the same as in previous figures. We plot the trajectories $[g_1(q), g_2(q)]$. For $\beta < 1$ (pink line), the trajectory don't enclose the origin, so the winding number is $W=0$, indicating a trivial phase. At $\beta = 1$ (blue line), the trajectory touches the origin. For $\beta > 1$ (black line), the trajectory winds once around the origin, $W=1$, indicating a non-trivial phase.