Phonon transport in two dimensional nano-membranes with periodic variations in acoustic properties, a.k.a. phononic crystals, has drawn tremendous interests recently due to their novel properties and potential applications in thermal energy conversion. Recent experiments have demonstrated drastically lower thermal conductivity than what one would expect from the Boltzmann transport equations (BTE) that describe phonon transport as particle diffusion. To understand the intriguing behavior, we used a partially coherent picture to model thermal transport in 2D phononic crystals. In this model, phonons with mean free paths longer than the characteristic size of the phononic crystals are treated as coherent waves. The finite difference time domain method is utilized to simulate the wave behavior and to obtain the phonon dispersion relations in phononic crystals. On the other hand, phonons with mean free paths shorter than the characteristic size are considered particles and are treated by BTE after taking the diffusive boundary scattering into account. Our result shows that the thermal conductivity reduces as the characteristic sizes decrease due to both the zone folding effect and the diffusive boundary scattering, which is consistent with the recent experimental results.

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