Dirac-like Plasmons in Honeycomb Lattices of Metallic Nanoparticles

We consider a two-dimensional honeycomb lattice of metallic nanoparticles, each supporting a localized surface plasmon, and study the quantum properties of the collective plasmons resulting from the near-field dipolar interaction between the nanoparticles. We analytically investigate the dispersion, the effective Hamiltonian, and the eigenstates of the collective plasmons for an arbitrary orientation of the individual dipole moments. When the polarization points close to the normal to the plane, the spectrum presents Dirac cones, similar to those present in the electronic band structure of graphene. We derive the effective Dirac Hamiltonian for the collective plasmons and show that the corresponding spinor eigenstates represent Dirac-like massless bosonic excitations that present similar effects to electrons in graphene, such as a nontrivial Berry phase and the absence of backscattering off smooth inhomogeneities. We further discuss how one can manipulate the Dirac points in the Brillouin zone and open a gap in the collective plasmon dispersion by modifying the polarization of the localized surface plasmons, paving the way for a fully tunable plasmonic analogue of graphene.

Figure 1

(a) Honeycomb lattice with lattice constant $a$ and lattice vectors ${\mathbf{a}}_1=a(\sqrt{3},0)$ and ${\mathbf{a}}_2=a(\frac{\sqrt{3}}{2},\frac{3}{2})$. The three vectors ${\mathbf{e}}_1=a(0,-1)$, ${\mathbf{e}}_2=a(\frac{\sqrt{3}}{2},\frac{1}{2})$, and ${\mathbf{e}}_3=a(-\frac{\sqrt{3}}{2},\frac{1}{2})$ connect the $A$ and $B$ inequivalent lattice sites [dark gray (blue) and light gray (red) dots in the figure]. (b) First Brillouin zone in reciprocal space with primitive vectors ${\mathbf{b}}_1=\frac{2\pi }{3a}(\sqrt{3},-1)$ and ${\mathbf{b}}_2=\frac{4\pi }{3a}(0,1)$.

Figure 2

(a)–(c) Collective plasmon dispersion relation from Eq. (3) and (d)–(f) corresponding density of states for (a), (d) the out-of-plane mode ($\theta =0$), (b), (e) one in-plane mode ($\theta =\pi /2$), and (c), (f) $\theta =\arcsin \sqrt{1/3}$. (g) Polarization angles ($\theta$, $\phi$) for which the collective plasmon dispersion is gapless (dark blue regions) and gapped (white regions). (h) Collective plasmon dispersion along the $K^{\prime }\Gamma K$ direction ($q_y=0$) for different orientations $\theta$ of the dipoles. In the figure, $\phi =0$ and $\Omega /{\omega }_0=0.01$.