Fig. 1. (a) Schematic of the PhC supercell array with a thickness t = 0.5a. One supercell is highlighted by the dashed box. (b) The first two band structures for the trivial and nontrivial unit cells. The symbols “+” and “−” indicate even and odd parity of the modes at high-symmetry points, where green and blue colors represent trivial and nontrivial cases, respectively. Insets show unit cells containing four air holes arranged in compact (trivial) or expanded (nontrivial) configurations. (c) Evolution of the eigenfrequencies for the four topological corner (defect) modes at the Γ point as a function of the nontrivial region size m within the supercell. Here, m is an integer that implies the number of columns (rows) of nontrivial unit cells. Insets display top-view schematics of the supercell for m = 3 and m = 1. (d) Normalized H_z fields for the modes shown in (c), computed in the x–y midplane of the slab. The symbols indicate the corresponding eigenmodes in (c).

Fig. 2. (a) Iso-frequency surfaces with color-mapped Q-factors for the four defect modes in the PhC slab. (b) Band structures of the four defect modes along k_x at k_y = 0. (c) Analytical band structures based on the TB approximation.

Fig. 3. (a), (b) Transmission spectra of the PhC slab for the p- and s-polarized plane wave incidences at varying angles, respectively.

Fig. 4. (a), (b) Magnitudes of the force acting on the nanoparticle under p- and s-polarized wave incidences at variant angles, respectively. The inset shows the schematic of nanoparticle manipulation. The color bar is plotted on a logarithmic scale.

Fig. 5. (a) SHG power under normal incidence for the 2D material placed on the PhC slab and a bare substrate. The inset illustrates the schematic of the nonlinear enhancement mechanism. (b), (c) SHG peak positions and enhancement factors for the p-polarized and s-polarized wave incidences as a function of the incident angle.

Fig. 6. (a) Transmission spectra of the PhC slab, the 2D material (2DM), and the hybrid structure with the 2D material placed atop the PhC slab. (b), (c) Transmission spectra under p- and s-polarized wave incidences as a function of the incident angle.