Controlling hypersonic surface acoustic waves is crucial for advanced phononic devices such as high-frequency filters, sensors, and quantum computing components. While periodic phononic crystals enable precise bandgap engineering, their ability to suppress acoustic waves is limited to specific frequency ranges. Here, we experimentally demonstrate the control of surface acoustic waves using a hyperuniform arrangement of gold nanopillars on a lithium niobate layer. The hyperuniform structure exhibits characteristics of both random and ordered systems, leading to an overall reduction in acoustic transmission and the formation of bandgap-like regions where phonon propagation is strongly suppressed. We further demonstrate effective waveguiding by incorporating linear and S-shaped waveguides into the hyperuniform pattern. Both simulations and experiments confirm high transmission through the waveguides at frequencies within the bandgaps, demonstrating the flexibility of hyperuniform structures to support waveguides of complex shapes. These findings provide a novel approach to overcoming the limitations of traditional phononic crystals and advancing acoustic technologies in applications such as mechanical quantum computing and smartphone filters.

Mathematically inspired structure design has emerged as a powerful approach for tailoring material properties, especially in nanoscale thermal transport, with promising applications both within this field and beyond. By employing mathematical principles, based on number theory, such as periodicity and quasi-periodic organizations, researchers have developed advanced structures with unique thermal behaviours. Although periodic phononic crystals have been extensively explored, various structural design methods based on alternative mathematical sequences have gained attention in recent years. This review provides an in-depth overview of these mathematical frameworks, focusing on nanoscale thermal transport. We examine key mathematical sequences, their foundational principles, and analyze the influence of thermal behavior, highlighting recent advancements in this field. Looking ahead, further exploration of mathematical sequences offers significant potential for the development of next-generation materials with tailored, multi-functional properties suited to diverse technological applications.

Phonon coherence elucidates the propagation and interaction of phonon quantum states within superlattice, unveiling the wave-like nature and collective behaviors of phonons. Taking MoSe$_2$/WSe$_2$ lateral heterostructures as a model system, we demonstrate that the intricate interplay between wave-like and particle-like phonons, previously observed in perfect superlattice only, also occurs in disordered superlattice. By employing molecular dynamics simulation based on a highly accurate and efficient machine-learned potential constructed herein, we observe a non-monotonic dependence of the lattice thermal conductivity on the interface density in both perfect and disordered superlattice, with a global minimum occurring at relatively higher interface density for disordered superlattice. The counter-intuitive phonon coherence contribution can be characterized by the lagged self-similarity of the structural sequences in the disordered superlattice. Our findings extend the realm of coherent phonon transport from perfect superlattice to more general structures, which offers more flexibility in tuning thermal transport in superlattices.

The development of emerging technologies, such as quantum computing and semiconductor electronics, emphasizes the growing significance of thermal management at cryogenic temperatures. Herein, by designing isotope interfaces based on the Golomb ruler, we achieved effective suppression of the phonon thermal transport of cryogenic graphene. The pronounced disordering of the Golomb ruler sequence results in the stronger suppression of thermal transport compared to other sequences with the same isotope doping ratio. We demonstrated that the Golomb ruler-based isotope interfaces have strong scattering and confinement effects on phonon transport via extensive molecular dynamics simulations combined with wave packet analysis, with a proper correction for the missing quantum statistics. This work provides a new stream for the design of thermal transport suppression under cryogenic conditions and is expected to expand to other fields.