Semi-Dirac Transport and Anisotropic Localization in Polariton Honeycomb Lattices

Compression dramatically changes the transport and localization properties of graphene. This is intimately related to the change of symmetry of the Dirac cone when the particle hopping is different along different directions of the lattice. In particular, for a critical compression, a semi-Dirac cone is formed with massless and massive dispersions along perpendicular directions. Here we show direct evidence of the highly anisotropic transport of polaritons in a honeycomb lattice of coupled micropillars implementing a semi-Dirac cone. If we optically induce a vacancylike defect in the lattice, we observe an anisotropically localized polariton distribution in a single sublattice, a consequence of the semi-Dirac dispersion. Our work opens up new horizons for the study of transport and localization in lattices with chiral symmetry and exotic Dirac dispersions.

Figure 1

Dirac cone merging and semi-Dirac dispersion. (a1) Electron microscopy image of a polariton honeycomb lattice. Red and blue circles demarcate $A$ and $B$ sublattices. Yellow and white lines denote hopping among horizontal and diagonal nearest neighbors ($t^{\prime }$ and $t$, respectively). (a2)–(a3) Sketch of the Brillouin zones in momentum space for $\beta =1$ and $\beta \ge 2$. Middle and bottom rows of columns (b)–(d) are measured polariton photoluminescence intensity in momentum space for different values of $\beta$. Each image is normalized to its maximum intensity. (b2),(c2),(d2) The emission along $k_y$ for $k_x=2\pi /3a$ [line 1 in (a2) and (a3)], while (b3),(c3),(d3) exhibit the measurements along $k_x$ for $k_y=-4\pi /3\sqrt{3}a$ (line 2), and for $k_y=-6\pi /3\sqrt{3}a$ (line 3). The white continuous and dashed lines are fits to the lower and upper tight-binding bands [Eq. (1)]. $E_0=1589.2\mathrm{meV}$; $a=2.4\mu \text{m}$.

Figure 2

Transport at the Semi-Dirac point. (a1) and (b1) show the photoluminescence intensity in real space at the energy of the Dirac point ($E_0=1589.2\mathrm{meV}$) for $\beta =1$ and $\beta =2$, respectively. Each image is normalized to its maximum intensity. (a2)–(b2) The measured intensity along $x$ (circles) and $y$ (triangles) directions for $\beta =1$ (a2) and $\beta =2$ (b2), extracted from the dashed boxes in (a1) and (b1). Lines are exponential decay fits. (a3) and (a4) The measured propagation lengths (dots), at several energies, along the $y$ and $x$ directions for $\beta =1$. (b3) and (b4) Same for $\beta =2$. Solid and dashed lines display the theoretical propagation lengths [Eq. (2)] corresponding to the solid and dashed bands in Figs. 1, 1, 1, 1. The vertical line depicts the Dirac-point energy $E_0$.

Figure 3

All-optical analog of a vacancy localization in semi-Dirac graphene. (a) and (b) The measured photoluminescence intensity in real space at the energy of the Dirac point when a single pillar is pumped (demarcated by a circle) in an $A$ pillar (a) and in a $B$ pillar (c). Insets show the reflected pump spot when the beam block is removed from the central region. (b) and (d) The polariton distribution calculated from Eq. (3) when a single $A$ and $B$ pillar, respectively, is pumped at $E_0$ energy. Hexagons depict the underlying lattice.