Research ArticleAPPLIED PHYSICS

Reconfigurable Floquet elastodynamic topological insulator based on synthetic angular momentum bias

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Science Advances  17 Jul 2020:
Vol. 6, no. 29, eaba8656
DOI: 10.1126/sciadv.aba8656
  • Fig. 1 Time-reversal symmetry breaking in a phononic crystal.

    (A) Schematic of the time-dependent phononic crystal formed by hexagonal unit cells with PLA (h0 = 0.5 mm thick) as the host layer and attached circular piezoelectric (PZT) patches with thickness of h1 = 0.5 mm (d1 = 7 mm in diameter). (B) The elastic modulus of each PZT patch is periodically modulated in time in a rotating manner (with uniform handedness throughout the lattice) through connections to external circuits. (C) The shunted circuits, composed of two resistors (R1 and R2 in blue), one capacitor (C0 in red), and an operational amplifier, provide a negative capacitance C′ = −(R2C0)/R1, which is altered in time by a switch that turns on/off at the frequency f0 = ω0/2π with the function G(ϕ)=sgn(cos(ω0t)+ϕ))1. The resistor (R0) parallel to the capacitor prevents saturation of the capacitor. (D) Comparison between the band structures along the irreducible Brillouin zone perimeters in the absence (red dashed curves with a Dirac point at ≈84.5 kHz) and in the presence (blue solid curves with modulation at f0 = 50 kHz, δC = 2.5 nF, and δE/E0 = 0.22) of modulation. (E) Comparison between the band structures along the irreducible Brillouin zone perimeters in the absence (red dashed curves with a Dirac point at ≈84.5 kHz) and in the presence (blue solid curves with optimal modulation at f0 = 50 kHz, δC = 2 nF, and δE/E0 = 0.31) of modulation. The time modulation has the effect of folding the band structure along the frequency axis and lifting the degeneracy at the K and Γ points, opening a complete bandgap with topological protection. Note S1 provides the complete band structure of the proposed system, showing folded frequencies in the vertical direction.

  • Fig. 2 Numerical computation of topologically protected edges states.

    (A) Quasi-band structure of a super cell composed of 1 × 7 arrays of unit cells, periodically repeated in the x direction and analyzed using Floquet boundary conditions. Gray areas depict bulk bands, and blue dashed lines depict the edge modes. (B) Schematic of the super cell and the corresponding wave distribution at frequency f = 83 kHz (marked with a green star), showing wave localization at the top edge. All displacements are normalized by the maximum deformation of the cell.

  • Fig. 3 Experimentally measured edge states.

    (A) Fabricated TI configured to exhibit an edge mode on free surfaces for a uniform right-handed synthetic angular momentum imparted by the spatiotemporal modulation. Each PZT is connected to an external circuit. (B) Experimentally measured transfer function (TF) at the two points marked with pink and green stars. The light blue areas show the topological bandgap. (C) Experimentally measured RMS displacement field excited by a source with frequency 89.5 kHz, documenting wave propagation along the free edges without backscattering. All displacements have been normalized to the amplitude of the input wave.

  • Fig. 4 Experimentally measured domain wall states.

    Experimentally measured RMS displacement field of the system excited by a source (marked with a green star) with frequency 89.5 kHz, documenting immune to backscattering wave propagation along (A) a horizontal domain wall and (B) a triangular-shaped domain wall, created by modulating different regions of the lattice with oppositely handed synthetic angular momentum. All displacements have been normalized to the amplitude of the input wave. For these interfaces, modulation of unit cells on one side of the yellow line is right handed, and on the other side is left handed.

Supplementary Materials

  • Supplementary Materials

    Reconfigurable Floquet elastodynamic topological insulator based on synthetic angular momentum bias

    Amir Darabi, Xiang Ni, Michael Leamy, Andrea Alù

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    • Notes S1 to S4
    • Figs. S1 to S7
    • References

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