Probing the Band Structure of Topological Silicon Photonic Lattices in the Visible Spectrum

We study two-dimensional hexagonal photonic lattices of silicon Mie resonators with a topological optical band structure in the visible spectral range. We use 30 keV electrons focused to nanoscale spots to map the local optical density of states in topological photonic lattices with deeply subwavelength resolution. By slightly shrinking or expanding the unit cell, we form hexagonal superstructures and observe the opening of a band gap and a splitting of the double-degenerate Dirac cones, which correspond to topologically trivial and nontrivial phases. Optical transmission spectroscopy shows evidence of topological edge states at the domain walls between topological and trivial lattices.

Figure 1

Hexagonal photonic lattice and CL measurements. (a) Schematic of unit cell of expanded and shrunken lattices composed of Si hexagons composed of Si pillars on a 10-nm-thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane, where $a_0$ is the lattice constant, $d$ is the diameter of the Si pillars, $h$ is the pillar height, and $r$ is the distance of the Si pillars from the center of the unit cell. (b) Excitation geometries: the 30 keV electron beam is incident through the hole in the parabolic mirror and light emitted by the excited lattice is collected by a parabolic mirror; (top) spectral detection using a spectrometer, (bottom) angle-resolved detection using a CCD imaging camera. (c),(d) SEM images of the fabricated lattice. (e), (f) Cathodoluminescence spectra averaged over a single Mie scatterer (left, with measured mode profiles at the two most intense peaks shown as insets) and averaged over a single Mie scatterer in a hexagonal lattice (right, with inset of geometry).

Figure 2

Measured density of states and dispersion of hexagonal photonic lattice. (a) SEM image and CL maps at 426, 664, and 750 nm (bandwidth 10 nm). (b) Angular CL emission distributions $(k_{\perp }/k_0)$ for positions A in the SEM image at 650, 600, 550, 500, and 450 nm, and for position B at 600 nm, where $k_{\perp }$ is the in-plane wave vector and $k_0=\omega /c$ ($k=0$ directed towards mirror hole). The spacing between the overlaid concentric contours represents the measurement bandwidth of 40 nm. The black spots in the center correspond to the $600\mu \mathrm{m}$-diameter hole in the parabolic mirror that spans 6.9° and where no light is collected.

Figure 3

Measured and simulated photonic band structure of trivial and topological lattices. (Top) shrunk lattice; bottom: expanded lattice. SEM images (a) and calculated and measured dispersion of the triangular lattice with six silicon pillars for the unit cell, along $\mathrm{\Gamma }\text{−}K$ (b) and $\mathrm{\Gamma }\text{−}M$ (c) directions (calculated dispersion overlaid over the measurements as dashed curves). The blue cones on either side of the calculated band structures correspond to modes outside the light cone. Data are taken with the tilted sample holder. The black vertical band in the CL data corresponds to the angular range of the hole in the parabolic mirror, where no data are collected. For the band structures, quality factor was encoded into the color of the bands.

Figure 4

Edge states between shrunk and expanded hexagonal photonic lattices. (a) Edge geometry. (b) SEM image of edge region. (c) Numerically calculated band structure, red lines indicate edge states. (d) Optical transmission spectra taken in bulk and edge regions, black dashed lines indicate the band gap.