Fig. 1: Time-reversal symmetry breaking in charge-doped PZM phononic crystals.

a Schematic of the phononic crystal design. Within each hexagonal unit cell, a snowflake-shaped white region is etched away. The geometrical parameters (a₀ = 2.3 μm, r₀ = 1 μm, h₀ = 0.5 μm) are kept the same for panels. The band structure is varied when the bulk material changes from (b) a pure elastic material to (c) a charge neutral PZM material and to (d) a PZM material with a uniform doping charge density q. A Chern insulator is achieved in d, where bandgaps are opened. The bandgap size variation with charge doping density q at different PZM coupling strength d₀ (e) between band I/II at the Γ point, and (f) between band II/III at the K point. d₀ is the magnitude of nonzero elements of d_m, normalized by β₀ = 5.9 nm/A, and q₀ = 232.5 C/mm³.