Quantum valley Hall effect and perfect valley filter based on photonic analogs of transitional metal dichalcogenides

We consider theoretically staggered honeycomb lattices for photons which can be viewed as photonic analogs of transitional metal dichalcogenide (TMD) monolayers. We propose a simple realization of a photonic quantum valley Hall effect (QVHE) at the interface between two inverted lattices. This results in the formation of valley-polarized propagating modes whose existence relies on the difference between the valley Chern numbers, an analog of the ${\mathbb{Z}}_2$ topological invariant. We show that the magnitude of the photonic spin-orbit coupling based on the energy splitting between TE and TM photonic modes allows to control the number and propagation direction of these interface modes. Finally, we consider the interface between a staggered and a regular honeycomb lattice subject to a nonzero Zeeman field, therefore showing quantum anomalous Hall effect (QAHE). In such a case, the topologically protected one-way modes of the QAHE become valley-polarized and the system behaves as a perfect valley filter.

Figure 1

(a) Cavity micropillar (artificial atom) scheme. (b) Zigzag interface between two TMD analog lattices with opposite A-B organization giving rise to zero lines modes and quantum valley Hall effect with a proper choice of parameters.

Figure 2

(a) Ribbon dispersion [colors represent localization on the interface (red) and on the edges of the structure (blue)]. (b) Corresponding in-gap interface wave-function absolute values projection on the transverse ($x$) direction for $K$ (solid black) and $K^{\prime }$ (dashed red) valleys. [Staggered potential: ${\mathrm{\Delta }}_{\mathrm{AB}}\left(r\right)=-{\mathrm{\Delta }}_{\mathrm{AB}}\left(l\right)=0.1J$.]

Figure 3

(a) and (b) Ribbon dispersions with TE-TM SOC: (a) $\delta J=0.2J$ and (b) $0.8J$. Colors represent localization on the interface (red) and on the edges of the structure (blue). (c) and (d) Corresponding absolute values of the interface wave functions in $K$ (solid black and solid blue) and $K^{\prime }$ (dashed red and dashed yellow) valleys. [Staggered potential: ${\mathrm{\Delta }}_{\mathrm{AB}}\left(r\right)=-{\mathrm{\Delta }}_{\mathrm{AB}}\left(l\right)=0.1J$.]

Figure 4

(a) Scheme of the structure, (b) ribbon dispersion [colors represent localization on the interface (red) and on the left edge of the structure (blue)]. (c) and (d) Wave-function projections on the transverse ($x$) direction for the states highlighted by the circles in (b): (c) edge (d) interface. [Parameters: ${\mathrm{\Delta }}_z\left(l\right)={\mathrm{\Delta }}_{\mathrm{AB}}\left(r\right)=0.1J, \delta J=0.2J, {\mathrm{\Delta }}_z\left(r\right)={\mathrm{\Delta }}_{\mathrm{AB}}\left(l\right)=0$.]

Figure 5

Dispersion of the bulk TMD (a) and the TMD/TMD interface states in its gap (b). The boundaries of allowed and forbidden bands (the gap) are marked by dashed white lines.

Figure 6

Behavior at the ${120}^{\circ }$ corners of polygonal interface (a) QVH/QVH and (b) QVH/QAH. Red arrows show the propagation direction.

Figure 7

Conversion of the interface states at the ${60}^{\circ }$ corners of polygonal interface (a) QVH/QVH and (b) QVH/QAH. Red arrows show the propagation direction.

Figure 8

Sensitivity of the QVH interface states to the perturbations: backscattering (a) on a localized defect (b) at the boundary with the “vacuum.” Red arrows show the propagation direction.

Figure 9

Berry curvature around $K$ point with trigonal warping contribution [Eq. (A2)]. [Parameters: ${\mathrm{\Delta }}_{\mathrm{AB}}\left(r\right)=0.1J, \delta J=0.2J$.]