Orbital Edge States in a Photonic Honeycomb Lattice

We experimentally reveal the emergence of edge states in a photonic lattice with orbital bands. We use a two-dimensional honeycomb lattice of coupled micropillars whose bulk spectrum shows four gapless bands arising from the coupling of $p$-like photonic orbitals. We observe zero-energy edge states whose topological origin is similar to that of conventional edge states in graphene. Additionally, we report novel dispersive edge states in zigzag and armchair edges. The observations are reproduced by tight-binding and analytical calculations, which we extend to bearded edges. Our work shows the potentiality of coupled micropillars in elucidating some of the electronic properties of emergent two-dimensional materials with orbital bands.

Figure 1

(a) Scanning electron microscope image of a single micropillar. The red lines sketch the position of the quantum wells embedded in the cavity (depicted in green). (b) Corresponding momentum space spectra showing confined $s$ and $p$-modes. (c) Optical microscope photograph of the honeycomb lattice under study showing two types of edges. The inset shows a scanning electron microscope image. (d) Sketch of the $p_{x,y}$ orbital (blue and yellow, respectively) and their couplings along the link ($t_L$) and perpendicular to it ($t_T$). [(e) and (f)] Momentum space luminescence from the bulk (e) and the zigzag edge (f) for $k_x=4\pi /(3a)$. $K$ and $K$’ mark the positions of the Dirac cones. Yellow and black dashed lines surround photoluminescence from edge states. $E_0=1573\mathrm{meV}$ and $t_L=-1.2\mathrm{meV}$. (g) Tight-binding calculation of the band structure. Blue lines, bulk energy bands at $k_x=4\pi /(3a)$; red lines, edge states obtained for a nanoribbon with zigzag edges. All calculations are for $t_T=0$.

Figure 2

Calculated eigenmodes in the first Brillouin zone for a nanoribbon as a function of the wave vector $k_{\parallel }$ parallel to the edges, with (a) zigzag ($k_{\parallel }=k_y$), (b) bearded ($k_{\parallel }=k_y$), and (c) armchair ($k_{\parallel }=k_x$) terminations. Blue curves: bulk spectra for different values of the transverse momentum $k_{\perp }$. Red and green curves: edge states. For bearded edges, the inset shows the uncoupled $p_y$ orbitals that give rise to a pair of edge states spreading over all $k_{\parallel }$.

Figure 3

Real space emission from dispersive edge states in an armchair termination. [(a)–(c)] Measured photoluminescence when selecting the energies indicated with circles in Fig. 2. [(d)–(f)] Corresponding tight-binding eigenfunctions. A hexagonal lattice is sketched on top of the data to mark the position of the center of the micropillars.