Phase-controlled topological plasmons in 1D graphene nanoribbon array

In this Letter, we report on the phase-controlled topological plasmons in 1D graphene nanoribbons (GNRs) based on a Su-Schrieffer-Heeger (SSH) model variant. By considering the dipole-dipole mode interactions, we first study the normal SSH model by an effective Hamiltonian and calculate the Zak phase as a topological invariant, finding that it is nontrivial (trivial) when the coupling distance is bigger (smaller) than half the period. Then, we reveal that the edge modes with fields highly localized at only one side exist in the model with nontrivial topology and shows the robustness of strong field confinement and extreme frequency stability against in-plane and out-of-plane disorders. Finally, we introduce the offset SSH model variant by vertically offsetting one of the GNR in SSH unit, which allows us to greatly engineer both the width of topological gap and the number of topological windows. The underlying physics are uncovered by defining a parameter called phase difference, which reveals that the topological edge modes appear (disappear) generally near the positions where the inter-unit coupling strength is bigger (smaller) than the intra-unit coupling strength, and, more notably, the phase difference is around even (odd) multiple numbers of p, which is much different from the normal SSH model where the topological phase is simply affected by the resonator distance. In addition to opening up a possibility to explore the fundamental physics of topologically protected graphene plasmons, this work also offers potential applications of these concepts to design graphene-based plasmon devices with immunity to structural imperfections.