Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene

The extraordinary electronic properties of Dirac materials, the two-dimensional partners of Weyl semimetals, arise from the linear crossings in their band structure. When the dispersion around the Dirac points is tilted, one can predict the emergence of intricate transport phenomena such as modified Klein tunneling, intrinsic anomalous Hall effects, and ferrimagnetism. However, Dirac materials are rare, particularly with tilted Dirac cones. Recently, artificial materials whose building blocks present orbital degrees of freedom have appeared as promising candidates for the engineering of exotic Dirac dispersions. Here we take advantage of the orbital structure of photonic resonators arranged in a honeycomb lattice to implement photonic lattices with semi-Dirac, tilted, and, most interestingly, type-III Dirac cones that combine flat and linear dispersions. Type-III Dirac cones emerge from the touching of a flat and a parabolic band when synthetic photonic strain is introduced in the lattice, and they possess a nontrivial topological charge. This photonic realization provides a recipe for the synthesis of orbital Dirac matter with unconventional transport properties and, in combination with polariton nonlinearities, opens the way to study Dirac superfluids in topological landscapes.

Popular Summary

In a wide range of 2D materials such as graphene, electrons behave as massless “Dirac particles” that propagate at remarkable speeds with very low resistance. These properties are very promising for the fabrication of ultralow-power microelectronic devices. However, these “Dirac materials” are rare and difficult to synthesize, and so there is a need for new strategies to engineer them. Here, we arrange photonic resonators in a honeycomb lattice to create a photonic material with many properties of Dirac materials, including the discovery of Dirac quasiparticles that are massless when propagating in one direction yet have infinite mass when moving in the orthogonal direction.

The properties of Dirac materials arise from crossings in the band structure at the Dirac points, where the valence band transitions to the conduction band. When the dispersion around these points is asymmetric, many striking phenomena have been predicted, including superconductivity at high temperatures, yet they are difficult to find in nature. Recently, artificial photonic materials have appeared as promising candidates for the engineering of such exotic dispersions.

We fabricate an asymmetric honeycomb lattice of photonic micron-sized resonators to implement a photonic material with exotic Dirac crossings in its band structure, and we study the localization and transport properties of photons in such materials.

Our photonic lattices provide a recipe for the synthesis of Dirac materials with remarkable transport properties.

Figure 1

Types of Dirac dispersions in two dimensions. (Top) Dispersions together with the zero-energy plane (grey). (Bottom, in red) Zero-energy Fermi surface. (a) Standard, type-I Dirac cones characterized by a linear dispersion in all directions in $k$ space and a pointlike Fermi surface. (b) Type-I tilted Dirac cone. (c) Type-III Dirac point (critically tilted), combining flat band and linear dispersions. Its Fermi surface is a line. (d) Type-II Dirac cone.

Figure 2

Honeycomb polariton lattice with orbital bands. (a) Scanning electron microscopy image of a single micropillar, the elementary building block of the honeycomb lattice. (b) Characteristic emission spectrum of a single micropillar, showing $s$, $p$, and $d$ discrete modes. The micropillar has been etched from the same wafer as the lattices studied in this work. (c) Scheme of the coupling of $p_x$ and $p_y$ orbitals in the honeycomb lattice with hoppings $t_L\gg t_T$. (d) Image of a honeycomb lattice of micropillars with homogeneous hoppings ($t_L=t_L^{\prime }$, $\beta =1$, see inset). The circles pinpoint the upper surface of the micropillars in two adjacent hexagons; the white arrows indicate the direction along which the coupling $t_L^{\prime }$ is modified. (e) Experimental photoluminescence of an unstrained lattice showing $s$ and $p$ bands.

Figure 3

Tilted Dirac cones in orbital graphene under strain. The central panels (a)–(d) show the measured polariton photoluminescence intensity as a function of $k_y$ for different values of $\beta$ (the color scale of each panel has been independently normalized to its maximum value). The plot is done for $k_x=2\pi /(d^{\prime }+d/2)$, dashed line in (e). Note that $d=2.40\mu \mathrm{m}$ for the unstrained coupling, while for the strained coupling, $d^{\prime }$ is $2.40\mu \mathrm{m}$, $2.60\mu \mathrm{m}$, $2.70\mu \mathrm{m}$, and $2.72\mu \mathrm{m}$, respectively, in (a)–(d). Here, $E_0=1.5687\mathrm{eV}$ and photon-exciton detuning $\delta =-10\mathrm{meV}$ at the energy minimum of the $p$ bands for panels (a), (b), and (d); $E_0=1.5780\mathrm{eV}$ and $\delta =-2\mathrm{meV}$ for panel (c); $t_L=-0.90\mathrm{meV}$ for panel (a), $-0.85\mathrm{meV}$ for panels (b) and (d), and $-0.65\mathrm{meV}$ for panel (c), which are obtained by fitting the measured spectra with a tight-binding Hamiltonian (fits are shown as white lines). The left inset of panel (c) shows the semi-Dirac dispersion measured along $k_x$ for the value of $k_y$ marked by a yellow dashed line in panel (c). The upper row in panels (a)–(d) depicts the calculated tight-binding bands for energies close to $E_0$ [orange dashed rectangle in (a)] and the winding around the Dirac points. The bottom row shows the tight-binding bands in the region at the bottom of the bands [green dashed rectangle in (a)] together with the windings. (e) Sketch of the Brillouin zones in momentum space.

Figure 4

Dirac cone tilt. Green dots: Absolute value of $v_y$ (a) and $v_{0y}$ (b) extracted from fits of Eq. (2) to the two experimental Dirac cones visible at low energies in Figs. 3 (see Appendix pp3). The plotted value is the average of $|v_y|$ and $|v_{0y}|$ extracted from the two Dirac points. Solid lines show the tight-binding result.

Figure 5

Type-III Dirac cones. (a) Measured photoluminescence intensity for $\beta =1$ (micropillar diameter: $2.60\mu \mathrm{m}$, $d=d^{\prime }=2.40\mu \mathrm{m}$) along $k_x$ for $k_y=-2\pi /(\sqrt{3}d)$ [line 1 in panel (b)]. Here, $E_0=1.5687\mathrm{eV}$ and $t_L=-0.85\mathrm{meV}$. (b) Emission in momentum space at $E_0$ for $\beta =1$, showing the usual Dirac points at the high-symmetry points in reciprocal space. Crosses indicate the position of the type-III Dirac points reported in panel (c). (c) Zoom of the low-energy section of the measured spectrum for a lattice with $\beta =1.5$ ($d=2.40\mu \mathrm{m}$, $d^{\prime }=2.10\mu \mathrm{m}$, $t_L=-0.78\mathrm{meV}$) showing the emergence of type-III Dirac cones combining flat and dispersive bands. (d) Measured dispersion along the $k_y$ direction for $k_x=-(0.5\times 2\pi )/3d$ [line 2 in panel (b)], crossing the Dirac point in the dashed rectangle in panel (c). (e) Dispersion obtained from the tight-binding model zoomed in the momentum space region marked with a dashed rectangle in panels (c) and (d). In panels (a), (c), and (d), the white lines show the tight-binding model. (f) Calculated dispersions for $k_y=-2\pi /(\sqrt{3}d)$ for different values of $\beta >1$. Type-III Dirac cones merge at $k_x=0$ for $\beta =2$.

Figure 6

Real-space wave functions of the flat band at the type-III Dirac cones. (a) Measured real-space photoluminescence intensity for $\beta =1.5$ at the energy of the type-III Dirac cone in Fig. 5 ($E=1.56752\mathrm{eV}$). The white arrow points to the emission observed in between the horizontal links. (b) Same for the top of the $s$ bands ($E=1.56577\mathrm{eV}$), for which the usual hexagonal emission is observed (here shown as a reference). The white dots mark the center of the micropillars. (c) Calculated eigenstate at the energy of the flat band at the $\mathrm{\Gamma }$ point with $t_T=0$. To obtain the real-space distribution, we account for the $p_x$ and $p_y$ shapes of the orbitals in each micropillar. The $\pm$ signs indicate the sign of the wave function in each lobe. In panels (a) and (c), the dashed rectangles mark the shape of a plaquette state. (d) Same as panel (c) with $t_T=-0.1t_L$. The arrows highlight one of the strained links.

Figure 7

Evolution of the winding plane around the tilted Dirac cones. The arrows indicate the direction of $\mathbf{h}(\mathbf{q})$ as a function of $\beta$ around the two Dirac cones (red and blue lines, respectively) emerging from the parabolic-flat-band touching.

Figure 8

(a) Measured dispersion along the $k_y$ direction for $k_x=2\pi /(d^{\prime }+d/2)$ for $\beta =0.45$ (same as Fig. 3). (b)  Data points extracted from the Dirac point marked in a dashed rectangle in panel (a). From linear fits to the points, we obtain $|v_{0y}|$ and $|v_y|$, plotted in Fig. 4.