Both high-efficient thermoelectric materials and thermal insulating coatings requiring low thermal conductivities, layered materials and superlattices prove to be an efficient multiscale material design for such requirements. The interfaces are artificially introduced to scatter thermal phonons, thus hindering thermal transport. Very recently, it has been found that interface modulation can further reduce the thermal conductivity. All of the recent advances originate from highly demanding numerical computations. An efficient estimate of the thermal properties is important for fast and/or high-throughput calculations. In this article, the phonon transport on layered material is studied theoretically for general purposes, based on the fact that long-wavelength phonons contribute dominantly in general. According to the Debye hypothesis, the classical wave equation can describe phonon transport very well. This fact has been very recently used to model phonon transport carbon nanotubes, which justifies the applicability of continuum mechanics for nanomaterials. Furthermore, Kronig and Penny have solved the electron transport on periodic lattices. In a very similar way, for the periodic layered materials and superlattices, with Floquet and linear attenuation theory, the wave equations with and without damping are solved analytically. The wave equation decouples to Helmholtz equations in each direction with periodic excitation functions. In this paper, we propose to model the phonon transport by using Matthew-Hill equation, with which we can obtain the phonon spectrum (i.e. phonon dispersion relation). The proposed theory is justified by two-dimensional (2D) graphene/hexagon boron nitride superlattice and three-dimensional (3D) silicon/germanium superlattices. Like the carbon nanotube cases, using this continuum-mechanics method, we can reproduce the previous numerical results very quickly compared with using published molecular dynamics and density functional theory The effects of interface modulation and phonon localization are shown over full phase space, which further enables the calculating of both high and low bounds of thermal conductivity for all possible superlattices and layered materials. In order to model real interfaces, with considering possible mixing and transition due to other mechanisms, we use the smooth transition function, which is further modeled via sinusoidal series. Very interestingly, interface grading is shown to erase band gaps and delocalize modes. This fact has been seldom reported and can be helpful for designing real materials. Likewise, we take phonon damping (equivalent to inter-phonon scattering) into account by adding damping into the wave equation. It is observed that phonon damping smears the originally sharp boundaries of phonon phase space. In this way, evanescent phonons and transporting phonons can be treated simultaneously on the same footing. The proposed method can be used for modeling the efficient and general thermal materials.
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