Fig. 2

Band structure of the DSP crystal with breaking of ΓK-mirror symmetry and topological property of its first band. a Calculated band structure of the DSP crystal along the special directions of the FBZ when ΔR = ±0.5 mm (pattern A or B). Blue dashed lines represent the light line and only the guided part of the band structure is shown. The light yellow shaded region represents the full bandgap. b Berry curvature of the first band for pattern A along the line \({k_y} = 2{\it{\pi }}/\left( {\sqrt 3 a} \right)\) in the reciprocal space from numerical calculations (open circles) and effective Hamiltonian model near the K (blue solid line) or K′ (red solid line) valley. c Berry curvature for pattern A around a K′ valley as a function of δk for different polar angles θ (open symbols). The red solid line represents the isotropic result from the effective Hamiltonian model. The inset shows the definition of δk and θ in the reciprocal space. d Valley Chern numbers summarized for pattern A and pattern B