Fig. 1. (a) Schematic of the platform: graphene over a metagate. Left: lateral view of the entire structure and right: unit cell geometry of the metagate. (b) Metagate-induced Fermi energy EF(x,y) profile inside graphene (degenerate case: r1=r2=120nm). Solid (dashed) line: unit cell for nondegenerate (degenerate) cases. (c) BZ and its high-symmetry points. Solid (dashed) hexagons: BZ boundaries for the nondegenerate (degenerate) cases. (d) Unfolded and (e) artificially folded BZs. Parameters: a=600nm, h1=15nm, h2=h3=10nm, and V0=1V throughout this paper.

Fig. 2. (a) Schematic of the TB model on a Kekulé lattice. Shaded triangular segment of a metagate maps onto a lattice site of the TB model. (b) PBS calculated for tin=1.1t/tout=0.9t (left) and tin=0.9t/tout=1.1t (right). Color: ratio between the projection amplitudes onto the dipole-like orbitals |p1,2⟩ and onto the quadrupole-like orbitals |d1,2⟩. (c) PBS calculated for a structure interfacing the two configurations from (b) along the zigzag-oriented domain wall. Bandgap-crossing purple lines: dispersion curves for the edge states propagating along the domain wall.

Fig. 3. (a) Plasmonic band structures calculated for r1=90nm/r2=150nm (left: topologically trivial domain) and r1=160nm/r2=110nm (right: topologically nontrivial domain). Line color: dipole-to-quadrupole ratios (see text). (b) Color: AC surface charge density in graphene associated with the GPPs at the Γ-point (below the bandgaps). Parameters: same as in Fig. 1.

Fig. 4. (a) Projection of the PBS of the GPPs supported by a graphene nanoribbon gated by a metagate comprised of the two domains shown in Fig. 3. Each domain contains 18 unit cells, zigzag-type domain wall runs parallel to the x-direction, the radius of peripheral holes at the domain wall: r¯=130nm. Black thin lines: x-projected bulk bands; red (blue) lines and circles: forward (backward)-propagating chiral edge states. (b) Metagate-induced Fermi energy profile on graphene near the domain boundary (white dotted line). (c) AC plasma charge density profiles for the forward- and backward-propagating edge states. 2-D color code: the magnitude and phase of the charge density. The gray arrows: guiding to the eye for the directions of the phase increase. (d) Reflection-free propagation of topological edge modes along the domain wall with sharp corners. The forward (backward) mode on the left (right) is excited at f=4.05THz with a surface current density (green arrows) J+=x^+iy^ (J−=x^−iy^). Color plots are absolute squares of the out-of-plane electric field. Parameters: same as in Figs. 1 and 3.

Fig. 5. Designer interfaces between topologically distinct PTIs enable nanoscale waveguides and cavities. (a) Left: a uniform domain wall configuration used in Fig. 4 for supporting bandgap-spanning chiral edge modes. Right: staggered domain wall for supporting cavity-like corner modes. Small icons above the arrow denote the 1-D band structures for each domain wall type. (b) Left: gapping of the edge states supported by the staggered domain wall (r1e=165nm, r2e=120nm, r1s=100nm, and r2s=155nm) enables corner states. Right: DOS for the topological (expanded) domain (10×10 unit cells) imbedded into a trivial (shrunken) domain. Inset: the field profile of a midgap corner state. (c) A TB model qualitatively mimics the imbedded structure with a staggered domain wall. Small circles: sublattice sites, color-coded links: hopping amplitudes. (d) Same as (b), but for the TB model shown in (c).

Fig. 6. Shifts of plasmonic resonance frequencies due to nonlocal responses; eigenfrequencies are evaluated at Γ above and below topological bandgap, using σpheno[n] in Eq. (4) for n=0 (local Drude limit) and n=1,2 (nonlocal corrections up to the first or the second order). (a) With V0=1V, the bandgap is located around 4 THz and its size is about 0.3 THz. (b) With V0=0.05V, the bandgap is located around 2 THz and its size is about 0.15 THz.