Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons

Two-dimensional lattices of coupled micropillars etched in a planar semiconductor microcavity offer a workbench to engineer the band structure of polaritons. We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene. Using energy-resolved photoluminescence, we directly observe Dirac cones, around which the dynamics of polaritons is described by the Dirac equation for massless particles. At higher energies, we observe $p$ orbital bands, one of them with the nondispersive character of a flatband. The realization of this structure which holds massless, massive, and infinitely massive particles opens the route towards studies of the interplay of dispersion, interactions, and frustration in a novel and controlled environment.

Figure 1

(a) Scanning electron microscope image of a corner of the microstructure. One hexagon of pillars is underlined with blue disks. The dark arrows show the growth axis of the cavity. The overlap between pillars is sketched in (b). (c) First Bz. (d) Measured momentum space energy resolved photoluminescence at $k_x=-2\pi /3a$ [line 0 in Fig. 2], under nonresonant low-power excitation. (e) Sketch of the real space distribution of $S$ and $P$ modes in a single pillar.

Figure 2

(a) Measured photoluminescence intensity in momentum space at the energy of the Dirac points [dotted line in Fig. 1]. (b) Spectrally resolved far field emission along line 1 in (a). The black line is a fit to Eq. (1). (c) Same as (b) along line 2 in (a). (d) Spectrally resolved far field emission along line 3 in (a), passing through the first Bz.

Figure 3

(a) Photoluminescence emission in momentum space above the condensation threshold. The black solid/dashed line shows the first/second Bz. (b) Real space image of the condensed state. (c) Interference pattern above condensation threshold. The position of six pillars is underlined with gray disks. (d) Total emitted intensity (black line) and blueshift of the polariton emission at the top of the ${\pi }^{*}$ band (blue dashed line) as a function of excitation intensity $I_0$ or power $P_0$.

Figure 4

(a) Zoom on the four $P$ bands shown in Fig. 1. (b) Real space image integrated in energy over the flatband, marked with two arrows in (a). The centers of six pillars are shown in white disks. The kagome geometry of the emission lobes is underlined with white lines. (c) and (d): calculated energy dispersion at $k_x=-2\pi /3a$ [same direction as in (a)] from Eq. (3) for $t_{\parallel }=-1$, $t_{\perp }=0$, and (c) $t_{\perp }=0.2\mathrm{meV}$ (d). For each state, the color scale indicates the relative brightness of its emission at energy $E$ and wave vector $k_y$, as predicted by the Fourier transform of the spatial wave function.