Plasmonic eigenmodes in individual and bow-tie graphene nanotriangles

In classical electrodynamics, nanostructured graphene is commonly modeled by the computationally demanding problem of a three-dimensional conducting film of atomic-scale thickness. Here, we propose an efficient alternative two-dimensional electrostatic approach where all calculation procedures are restricted to the graphene sheet. Furthermore, to explore possible quantum effects, we perform tight-binding calculations, adopting a random-phase approximation. We investigate multiple plasmon modes in 20 nm equilateral triangles of graphene, treating the optical response classically as well as quantum mechanically. Compared to the classical plasmonic spectrum which is “blind” to the edge termination, we find that the quantum plasmon frequencies exhibit blueshifts in the case of armchair edge termination of the underlying atomic lattice, while redshifts are found for zigzag edges. Furthermore, we find spectral features in the zigzag case which are associated with electronic edge states not present for armchair termination. Merging pairs of triangles into dimers, plasmon hybridization leads to energy splitting that appears strongest in classical calculations while splitting is lower for armchair edges and even more reduced for zigzag edges. Our various results illustrate a surprising phenomenon: Even 20 nm large graphene structures clearly exhibit quantum plasmonic features due to atomic-scale details in the edge termination.


2D Meshes in homogenous nanostructured graphene
We use the open source code Gmsh 1 to generate the triangular meshes employed in our classical calculations. In Figure S1 we show a simple example of such a mesh, extending a triangular graphene nanostructure. In the practical computations, we obtain satisfactory convergence and very good resolutions with about K = 3000 vertices (about J = 6000 elements). Due to the electrostatic scaling law, 2,3 a fixed mesh configuration can be used in the calculations of differently sized triangles. Regarding symmetry, we construct the bow-tie triangles by simply copying the mesh from one triangle to the mirrored triangle, allowing a potential speed up of convergence.

f n in different nanostructures
The parameter f n = iσ(ω n )/(4πε s Lω n ), introduced below Eq. (2), which gives a dimensionless number for the eigenresonances -independent of chosen conductivity profile -is solely dependent on the choice of geometry and characteristic feature length L. In Table 1 we list calculated −f n in different relevant geometries, indicating also the choice of L for each geometry.  The values of f n in disks were studied in Ref. 6 using a highly accurate semi-analytical approach. For comparison, and to assess the accuracy of our numerical approach, we compare the value for the dipolar mode and find less than 1% relative deviation. Since the disk-results converge slower compared with the other geometries considered here (cf. the challenge of approximating a curved boundary with a triangular mesh), we expect the values in Table 1 to have less than roughly 1% relative deviation from exact values throughout.
Taking only the Drude term for the graphene surface conductivity [see Eq. (13) in main text], we have illustrating that the plasmonic frequency scales as

indicating its tunability
with Fermi energy ϵ f , feature length L, and substrate properties via ε s . Finally, the resonance line width is proportional to the relaxation loss τ −1 (associated with a finite carrier mobility).

Construction of armchair and zigzag triangles
To construct the triangles with different edge terminations, we fix one of the vertices as the origin, arrange the carbon atoms in the bottom edge with different configurations, and fill the upper layers one by one with a decreasing width. Finally, we delete the single-bonded carbon atoms. Thus for a typical side length, the constructed triangles may not satisfy the rotation symmetry, but this can be realized by reducing or adding a few carbon atoms. We show in Figure S2 the nanotriangles with two different edge terminations. The direct diagonalization of the tight-binding Hamiltonian matrix yields the singleparticle energy states ϵ j and the corresponding wave functions ψ j (r). The local density of states are computed through

Electronic edge states
with f j the fermi distribution, and the energy density of states are where the factor 2 accounts for the spin degeneracy, γ is a very small damping factor (chosen as γ = 0.006eV in the computation), and ∆ is the surface area. The numerical results are shown in Figure S3, where we can see that the electrons are almost uniformly distributed on the structure with armchair edges, whereas non-uniform distributions are found for zigzag edges, with very large densities occurring at these edges. The consequence arises from an enormous amount of zero-energy states shown in the inset, which is essentially lacking in the structures with armchair edges. 4 These electronic edge states extremely affect the plasmon modes in both the resonance energy and damping. 5,6

Calculations on graphene hexagons
The methods introduced in this article are very general for the calculations of different geometries. As an additional example, we will discuss the plasmon modes in graphene hexagons which are recently produced in experiments. 7,8 Using the parameters given in the main text, we show in Figure S4    We also extract the mode patterns from the quantum calculations as shown in Figure S6.
Similar to triangles, the plasmon modes exhibit blueshift in hexagons with armchair edges, but redshift in those with zigzag edges. The dipole modes are very similar in three cases, although due to the practical construction the two dipolar modes for two polarizations in zigzag hexagons have different energy. This may be due to the the relatively smaller perimeter to area ratio, being 3 times larger in triangles than that in hexagons. The differences in some higher order modes, for example the modes (7) and (8) in armchair hexagons and the modes (6) and (7) in zigzag hexagons, can be clearly observed.
We show in Figure S7 the eigenvalue loss spectrum in bow-tie graphene hexagons. The energy splitting is large in both homogenous and armchair hexagons bow-ties, also evident in a pronounced energy splitting for the y-polarized dipole modes, because of the relatively stronger coupling here which is almost absent in triangles. In zigzag hexagons, the cou-pling is much weaker, and even the x−polarized dipole does not exhibit two significantly hybridized peaks. In Figure S7b, mode (1) (corresponding to the bonding dipole mode along x−direction) has an apparent peak, while the mode (2) (antibonding) is very close to modes (3) and (4) at current numerical precision. Except for the coupling strength, the hybridization procedure in bow-tie hexagons is then similar to those in bow-tie triangles. Figure 5 of the main text can thus be generalized to describe the plasmon hybridization in bow-tie graphene hexagons. Finally, the 12 lowest-energy plasmon modes in bow-tie homogenous graphene hexagons are shown in Figure S8. In both armchair and zigzag bow-tie graphene hexagons, the mode patterns can be easily constructed from Figure S6, and thus will not be presented here.