Phonon transport in 2D phononic crystals has attracted attention due to the possibility to control (a) the heat flow, i.e., achieving directional phonon transport, that is of relevance for diode-like behaviour, and (b) controlling the thermal conductivity over orders of magnitude, which is of interest for thermoelectric applications. In recent years the most important research questions have been: (i) to what degree the thermal transport depends on its nature (ballistic or diffusive)? and consensus is emerging that at room temperature phonon transport is diffusive. (ii) In a patterned suspended membrane what is the role of the porosity and surface roughness? . Published results suggest that it is the roughness and the surface to volume ratio of the holes (pores) making up the phononic crystal.  (iii) What is the role of spatial  and phase disorder, such as native oxides and amorphous shells , in 2D phononic crystals? The latter still being an open question where the roles of filling factors and geometry are being discussed. To understand these issues, it is paramount to be aware of the various length scales, the mean-free path distribution, which is strongly temperature dependent, the suspended membrane thickness and its surface roughness. Furthermore, the boundary conditions, the interplay of diffusive (scattering relevant) and specular (phase relevant) transport, are crucial for the data treatment and comparison to models. Finally, the measurements methods need to be carefully compared, as assumptions on all of the above will make the analysis rather intricate and comparison meaningless.
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