Helical boundary modes from synthetic spin in a plasmonic lattice

Letter

Helical boundary modes from synthetic spin in a plasmonic lattice

Sang Hyun Park, Michael Sammon, E. J. Mele, and Tony Low

Phys. Rev. B 109, L161301 – Published 22 April 2024

Abstract

Artificial lattices have been used as a platform to extend the application of topological band theory beyond electronic systems. Here, using the two-dimensional Lieb lattice as a prototypical example, we show that an array of disks which each support localized plasmon modes gives rise to an analog of the quantum spin-Hall state enforced by a synthetic time-reversal symmetry. We find that the plasmonic modes naturally possess a synthetic spin degree of freedom which leads to a spin-dependent second-neighbor coupling mechanism mediated by interorbital coupling. This interaction introduces a nontrivial $Z_{2}$ topological order and gaps out the Bloch spectrum. A faithful mapping of the plasmonic system onto a tight-binding model is developed and shown to capture its essential topological signatures. Full wave numerical simulations of graphene disks arranged in a Lieb lattice confirm the existence of propagating helical boundary modes in the nontrivial band gap.

Received 29 May 2023
Revised 7 March 2024
Accepted 25 March 2024

DOI:https://doi.org/10.1103/PhysRevB.109.L161301

Physics Subject Headings (PhySH)

Photonics Plasmonics Plasmons Polaritons Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sang Hyun Park¹, Michael Sammon¹, E. J. Mele ², and Tony Low^1,*

¹Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA
²Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*tlow@umn.edu

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Vol. 109, Iss. 16 — 15 April 2024

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Images

Figure 1
Plasmonic band structure of graphene nanodisk arrays. (a) Schematic representation of an array of graphene disks configured into a Lieb lattice. (b) Electric potential of the first six plasmonic modes of an isolated graphene disk. (c) Top-down view of the plasmonic Lieb lattice with $d = 100$ nm and $a = 105$ nm. The dashed box shows the unit cell of the lattice. (d) Plasmonic band structure corresponding to the arrangement shown in (c). The red, black, and blue bands correspond to bands formed by the dipole, quadrupole, and hexapole modes, respectively. (e) Top-down view of the plasmonic lattice with overlapping disks ( $d = 100$ nm, $a = 85$ nm). Color coding of the bands is not applied since classification in terms of the multipole modes of the disk does not hold in this case. For all simulations, the graphene Fermi energy is set to $E_{F} = 0.5 eV$ and the substrate dielectric constant is $ε_{s} = 2.2$ .
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Figure 2
Orbital coupling between $ψ_{2}^{\pm}$ and $ψ_{3}^{\pm}$ . (a) Representation of nearest-neighbor hopping between $ψ_{3}^{+}$ and $ψ_{2}^{\pm}$ . Hoppings for $R = \pm a_{x}, \pm a_{y}$ are considered. The electric potential $Φ$ of the modes is shown. The plus and minus signs associated with each hopping parameter indicate its sign. (b) Hopping between $ψ_{3}^{-}$ and $ψ_{2}^{\pm}$ . All hopping parameters for (b) can be generated by rotation of the configuration in (a). (c) The allowed hopping transitions are represented on an energy level diagram. A second-order transition connecting $ψ_{2}^{+}$ to $ψ_{2}^{-}$ is possible via the $ψ_{3}^{\pm}$ modes.
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Figure 3
Tight-binding mapping of a plasmonic Lieb lattice. (a) Tight-binding model onto which the plasmonic crystal can be mapped. The red and blue arrows represent two paths through which the interaction given by Eq. (4) connects a $ψ_{2}^{+}$ state to a $ψ_{2}^{-}$ state. (b) Schematic representation of a graphene nanodisk Lieb lattice with $ψ_{2}^{+}$ and $ψ_{2}^{-}$ orbitals on each site. Parameters are equivalent to the setup in Figs. 1, 1, and 1 Band structure calculated from the tight-binding model given in Eq. (5) with $α = - 1$ and $α = - 1.2$ , respectively. For the $α = - 1$ case, the $Z_{2}$ topological invariant $ν$ for each band is also shown. For the $α = - 1.2$ case, topologically trivial second-neighbor hopping is included. (e) Full wave electromagnetic simulation of plasmonic band structure for the configuration shown in (b).
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Figure 4
Edge modes of the plasmonic Lieb lattice. (a), (d) Tight-binding band structure calculation for Lieb lattice in a ribbon geometry with $α = - 1$ . The edge modes are highlighted in red, while the bulk band projections are colored in gray. All edge modes are doubly degenerate. A more detailed view of the crossing point of (a) is given in (d). (b), (e) Tight-binding band structure for $α = - 1.2$ and with topologically trivial second-neighbor hopping included. (c), (f) Full wave numerical simulation for plasmonic Lieb lattice in ribbon geometry. Parameters are identical to those used for Figs. 1 and 3.
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Figure 5
Dipole launching the edge states of the plasmonic Lieb lattice. (a) Band structure of the plasmonic Lieb lattice in a ribbon geometry. Bands traversing the bulk gap are all doubly degenerate. (b) Schematic representation of the eigenstate corresponding to the black diamond in (a). The blue (red) shows projection onto state ${\tilde{ψ}}_{- 2}$ ( ${\tilde{ψ}}_{+ 2}$ ). Arrows indicate the group velocity of each edge mode. (c) Simulation of topological edge plasmon launched by circularly polarized dipole at frequency indicated in (a). Magnitude of the plasmon electric field is multiplied by a projection onto the ${\tilde{ψ}}_{\pm 2}$ angular momentum states. Dipole source is placed at the black arrow 5 nm above the graphene nanodisk.
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