  We study the effect of characteristic size variation on the phonon thermal
transport in crystalline GeTe for a wide range of temperatures using the
first-principles density-functional method coupled with the kinetic collective
model approach. The characteristic size dependence of phonon thermal transport
reveals an intriguing collective phonon transport regime, located in between
the ballistic and the diffusive transport regimes. Therefore, systematic
investigations have been carried out to describe the signatures of phonon
hydrodynamics via the competitive effects between grain size and temperature. A
characteristic nonlocal length associated with phonon hydrodynamics and a heat
wave propagation length has been extracted. The connections between phonon
hydrodynamics and these length scales are discussed in terms of the Knudsen
number. Further, the scaling relation of thermal conductivity as a function of
characteristic size in the intermediate size range emerges as a crucial
indicator of the strength of the hydrodynamic behavior. A ratio concerning
normal and resistive scattering rates has been employed to understand these
different scaling relations, which seems to control the strength and prominent
visibility of the collective phonon transport in GeTe. This systematic
investigation emphasizes the importance of the competitive effects between
temperature and characteristic size on phonon hydrodynamics in GeTe, which can
lead to a better understanding of the generic behavior and consequences of the
phonon hydrodynamics and its controlling parameters in low-thermal conductivity
materials.
