Realization of a universal topological waveguide by tuning adiabatic geometry

TL;DR

This work addresses the limitation of armchair boundaries in quantum valley Hall–based topological waveguides by implementing topological adiabatic geometry to suppress valley mixing. Using numerical models and MEMS-based experiments, the authors show that widening the domain-wall profile of the armchair boundary restores valley-protected modes across the bulk band gap and enables robust 90°, 120°, and 150° bends through phase matching with zigzag and bridge boundaries. The main contributions include a quantitative demonstration that adiabatic domain walls reduce inter-valley scattering, a demonstrated high-transmission armchair waveguide across the bulk gap, and the establishment of a universal design framework for topological waveguides applicable to a range of wave phenomena. The findings have broad implications for designing robust, multifunctional topological devices in photonic, acoustic, elastic, and diffusive systems.

Abstract

Quantum valley Hall-based topological phases have been attracting attention across diverse fields as a robust platform for wave guidance due to their high compatibility with engineering frameworks. Combining three representative boundary types enables topological waveguides with flexible designability and enhanced functionality. However, one of the three, namely the armchair boundary, has long been limited by inter-valley scattering, resulting in weak topological protection and severely restricting its use in practical devices. This long-standing constraint is a major barrier to realizing broadly applicable topological waveguide systems. Here, to address this challenge toward a broadly applicable design framework for topological waveguides, we experimentally demonstrate that topological adiabatic geometry implemented in a micro electromechanical system suppresses valley mixing. We found that the adiabaticity enhances immunity to defects and increases the transmission efficiency of the armchair boundary. As the adiabaticity increases, topological protection is recovered over an increasingly broad portion of the bulk band gap, extending from low to high frequencies. Furthermore, we show that the recovery of protection in the adiabatic armchair boundary enables waves to propagate through 90^° and 150^°-bent waveguides by coupling with other interface geometries. Suppressing valley mixing via adiabaticity paves the way for a universal design framework for topological waveguides and for restoring robust topological characteristics across a wide range of wave phenomena.

Figures (4)

• Figure 1: Tunable adiabatic armchair boundaries and dispersion diagrams.a Schematics of quantum valley-Hall (QVH)-based unit cell and supercell having an armchair boundary. In a numerical model of the supercell, we align 40 unit cells along the $y$-axis and assume the infinite periodic condition along the $x$-axis. b Dispersion diagrams of the QVH-based symmetry (black) and asymmetry (red) unit cell. The solid and open plots represent the out-of-plane and in-plane modes, respectively. c Side length $L_1$ of the triangular plate as a function of the $y$-axis coordinate for $\lambda=0.01$ µ m (circles), $\lambda=50$ µ m (crosses), and $\lambda=100$ µ m (triangles). Dispersion diagrams of the supercells having armchair boundary for (d) $\lambda=0.01$ µ m and (e) $\lambda=100$ µ m. The bulk band gap (shaded gray) and inner band gap (shaded green) depend on the asymmetric and adiabatic factors, respectively. f Bandwidth of the inner band gap $\Delta f$ as a function of adiabatic factor $\lambda$. The circle plot and dashed line represent the numerical and theoretical results, respectively.
• Figure 2: Transmission efficiency and immunity to a defect of armchair boundary.a The top view of a numerical model of a straight armchair waveguide (broken green line). The red hexagon and yellow rhombus represent the excitation point and position of the defect, respectively. In the waveguide, the input energy is calculated by summing kinetic energy at the right input (RI, magenta) and left input (LI, cyan) ports. The output energy of the rightward and leftward elastic waves is calculated at the right output (RO, red) and left output (LO, blue) ports, respectively. b The enlarged view of the excitation point. In this schematic, the three small sub-lattices are excited with a phase difference of $2\pi/3$ in a clockwise direction to excite the rightward wave. The transmission efficiencies of the straight armchair waveguides without the defect for (c) $\lambda=0.01$ and (d) $\lambda=100$ µ m. The solid red (dashed blue) line indicates the transmission efficiency from the input ports to the right (left) output port. The shaded gray and green regions represent the bulk and inner band gaps obtained by the dispersion diagrams of the unit cell. e The enlarged view of the sub-lattice defect in (a). The transmission efficiencies of the straight armchair waveguides with the defect for (f) $\lambda=0.01$ and (g) $\lambda=100$ µ m. h The displacement profiles on the $xz$-plane of the waveguides with the defect at 334 kHz for $\lambda=0.01$ and $\lambda=100$ µ m. The black dashed lines at the center and right of the waveguide represent the excitation point and defective point, respectively. i The bandwidth exceeding 70% transmission efficiency as function of $\lambda$. The open circles and crosses show the results without and with the defect, respectively.
• Figure 3: Experimental setup and the wave propagation of armchair-based 120$^{\circ}$-bent waveguides.a Top overview of the 120$^{\circ}$-bent armchair waveguide for $\lambda=100$ µ m and the schematic of the experimental setup to measure the vibration on the waveguide. The waveguide (dashed green line) is excited from the left edge of the waveguide by an excitation electrode on the back side of the substrate (dashed blue line). Here, a pre-amplifier and lock-in amplifier are described as PA and LA, respectively. The inset indicates the scanning electron microscope (SEM) image of the bending point in the waveguide (shaded magenta region). The 2D displacement profiles of the structures having the armchair waveguide for (b) $\lambda=100$ and (c) $\lambda=0.01$ µ m. The inset of (c) represents the SEM image around the bending point in the waveguide (shaded magenta line). d Transmission efficiencies of the armchair waveguides for $\lambda=100$ (red) and $\lambda=0.01$ (blue) µ m. The experimental results (symbols) are supported well by numerical results (dashed lines). The shaded gray region represents the bulk band gap obtained by the dispersion diagram of the supercell.
• Figure 4: Wave propagation through armchair boundary combined with other boundaries. The 2D displacement profiles of the zigzag waveguide combined with the armchair waveguide for (a) $\lambda=100$ and (b) $\lambda=0.01$ µ m. Each inset in both figures shows the scanning electron microscopic (SEM) image around the first bending point highlighted by blue squares. The shaded blue and magenta regions in the SEM images represent the zigzag and armchair boundaries, respectively. c Transmission efficiencies of the waveguides including the armchair boundary for $\lambda=100$ (red) and $\lambda=0.01$ (blue) µ m. Those experimental transmission efficiencies are calculated by the ratio of squared displacement at the input port (magenta rectangle in (a) and (b)) to the output port (cyan rectangle in (a) and (b)). The symbols and lines indicate the experimental and numerical results, respectively. The shaded gray region shows the common bulk band gap of the zigzag and armchair boundaries. The 2D displacement profiles of the bridge waveguide combined with the armchair waveguide for (d) $\lambda=100$ and (e) $\lambda=0.01$. In the inset of the SEM images, the shaded green region represents the bridge boundary. f Transmission efficiencies of the waveguides in (d) and (e) calculated similarly to the results in (c).