Topological collective plasmons in bipartite chains of metallic nanoparticles

We study a bipartite linear chain constituted by spherical metallic nanoparticles, where each nanoparticle supports a localized surface plasmon. The near-field dipolar interaction between the localized surface plasmons gives rise to collective plasmons, which are extended over the whole nanoparticle array. We derive analytically the spectrum and the eigenstates of the collective plasmonic excitations. At the edge of the Brillouin zone, the spectrum is of a pseudorelativistic nature similar to that present in the electronic band structure of polyacetylene. We find the effective Dirac Hamiltonian for the collective plasmons and show that the corresponding spinor eigenstates represent one-dimensional Dirac-like massive bosonic excitations. Therefore, the plasmonic lattice exhibits similar effects to those found for electrons in one-dimensional Dirac materials, such as the ability for transmission with highly suppressed backscattering due to Klein tunneling. We also show that the system is governed by a nontrivial Zak phase, which predicts the manifestation of edge states in the chain. When two dimerized chains with different topological phases are connected, we find the appearance of the bosonic version of a Jackiw-Rebbi midgap state. We further investigate the radiative and nonradiative lifetimes of the collective plasmonic excitations and comment on the challenges for experimental realization of the topological effects found theoretically.

Figure 1

Sketch of a chain of $\mathcal{N}$ dimers, comprised of a pair of metallic nanoparticles denoted $A$ and $B$, each supporting localized surface plasmon resonances.

Figure 2

Collective plasmonic dispersion relation from Eq. (15) for the transverse (red and orange dashed lines, $\sigma =x=y$) and longitudinal (blue and cyan solid lines, $\sigma =z$) modes for both the upper ($\tau =+1$) and the lower ($\tau =-1$) bands. The undimerized case (thinner lines) with $d_1=d_2=3a$ and a typical dimerized case (thicker lines) with $d_1=3a$ and $d_2=3.5a$ are both displayed.

Figure 3

Plasmonic probability density corresponding to the midgap state at frequency ${\omega }_0$ (red dots, thin gray line: guide for the eye) at each site in a chain of 100 nanoparticles, where ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=2/3$. Inset: Sketch of a typical chain comprised of an even number of nanoparticles, where the loops depict the favored dimerization.

Figure 4

Plasmonic probability density corresponding to the midgap state at frequency ${\omega }_0$ at each site in a chain of 101 nanoparticles, for three different choices of the coupling constants. In the two symmetric setups ${\mathrm{\Omega }}_2^0/{\mathrm{\Omega }}_1^0={\mathrm{\Omega }}_1^{\pi }/{\mathrm{\Omega }}_2^{\pi }=2/3$, with the interface coupling constant being either ${\mathrm{\Omega }}_{\mathrm{i}}={\mathrm{\Omega }}_2^0$ (red dots) or ${\mathrm{\Omega }}_{\mathrm{i}}={\mathrm{\Omega }}_1^0$ (blue squares). In the asymmetric setup ${\mathrm{\Omega }}_2^0/{\mathrm{\Omega }}_1^0=2/3, {\mathrm{\Omega }}_1^{\pi }/{\mathrm{\Omega }}_2^{\pi }=1/6$, and ${\mathrm{\Omega }}_{\mathrm{i}}={\mathrm{\Omega }}_2^0$ (green triangles). The thin gray lines are guides for the eye.

Figure 5

Radiation damping decay rate from Eq. (33) as a function of momentum for the transverse (red and orange dashed lines, $\sigma =x=y$) and longitudinal (blue and cyan solid lines, $\sigma =z$) collective plasmonic modes for $q_{0}d=0.5$. The undimerized case (thinner lines) with $d_1=d_2=3a$ and a typical dimerized case (thicker lines) with $d_1=3.5a$ and $d_2=3a$ are both displayed. Only the results for the in-phase bands, i.e., the lower $\tau =-1$ band for the longitudinal polarization and the upper $\tau =+1$ band for the transverse modes, are displayed in the figure as the out-of-phase bands provide vanishingly small contributions.

Figure 6

Landau damping decay rate from Eq. (37) as a function of momentum for the transverse (red and orange dashed lines, $\sigma =x=y$) and longitudinal (blue and cyan solid lines, $\sigma =z$) collective plasmonic modes for $\hslash {\omega }_0/E_F=0.5$. The thinner (thicker) lines correspond to $d_1=d_2=3a$ ($d_1=3.5a$ and $d_2=3a$).

Figure 7

Sum of the radiative decay rate given by Eq. (33) and the Landau damping decay rate given by Eq. (37) as a function of both the scaled wave vector $q/q_0$ and the scaled nanoparticle radius $k_{\text{F}}a$, for a bipartite nanoparticle chain characterized by interparticle separations $d_1=3.5a$ and $d_2=3a$. The parameters in the figure are $\hslash {\omega }_0/E_{\mathrm{F}}=0.47$ and $q_0/k_{\mathrm{F}}=1.1\times {10}^{-3}$, corresponding to a chain composed of Ag nanoparticles. The plasmonic bands under consideration are (a) the upper transverse band $\sigma =x,y, \tau =1$, (b) the lower transverse band $\sigma =x,y, \tau =-1$, (c) the upper longitudinal band $\sigma =z, \tau =1$, and (d) the lower longitudinal band $\sigma =z, \tau =-1$.

Figure 8

Collective plasmon dispersion relation with dipole-dipole interaction including nearest (solid blue lines), next nearest (dashed green lines), third nearest (dotted orange lines), and fourth nearest (dash-dotted magenta lines) neighbors, see Eq. (A1). In the figure, $d_1=3a, d_2=3.5a$, and $\sigma =z$, corresponding to the longitudinal collective plasmon polarization.

Figure 9

Transmission probability though a system with parameters ${\omega }_0^{\mathrm{R}}/{\omega }_0^{\mathrm{L}}=1.2, d_1=3a, d_2=3.5a$, and $\sigma =z$. Inset: sketch of the spectrum of the quasiparticles. The excitations may be transmitted in three regimes (dotted lines) corresponding to: (I) electronlike to electronlike, (II) electronlike to holelike, and (III) holelike to holelike states.

Figure 10

Plasmonic probability density corresponding to the midgap state at frequency ${\omega }_0$ at each site in a chain of 101 nanoparticles, for ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=2/3$ (red dots) and ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=3/2$ (blue crosses). The thin gray lines are guides for the eye. Inset: Sketches of a typical chain comprised of an odd number of nanoparticles, where the loops depict the favored dimerization for ${\mathrm{\Omega }}_2>{\mathrm{\Omega }}_1$ (top) and ${\mathrm{\Omega }}_1>{\mathrm{\Omega }}_2$ (bottom).