Topologically guided slow Lamb waves via height-modulated valley Hall heterostructures

Topological valley-Hall phononic crystals offer robust platforms for broadband, backscattering-immune acoustic waveguiding. Here, we introduce a silicon-based phononic heterostructure that supports an extended valley-locked Lamb wave state (EVLS)---a topologically protected guided mode confined within a quasi-two-dimensional Dirac semimetal channel between two valley-Hall domains. A key innovation is the introduction of a geometric tuning parameter---the pillar height difference $dh$ ---which enables precise control of the EVLS dispersion. By increasing $dh$, we systematically flatten the band and reduce the group velocity, achieving slow-wave transport while preserving topological immunity. The EVLS arises from hybridization of valley-polarized edge modes at the opposite interfaces and inherits their topological character, consistent with previously studied quantum valley Hall systems. Full three-dimensional simulations confirm that the EVLS supports broadband, slow wave propagation with suppressed backscattering under moderate disorders and bends. The guided Lamb wave operates in the gigahertz regime, and the frequency-scalable design enables large signal delay within a compact footprint while remaining compatible with complementary metal-oxide semiconductor/micro-electromechanical systems fabrication. This work establishes a general design strategy that unifies dispersion engineering and topological robustness, offering a scalable route toward GHz-frequency, dispersion-engineered, robust wave transport in chip-scale phononic systems.