Recent experiments have evidenced that effective Fourier models are unable to predict heat transfer at the nanoscale [1,2]. This has a big impact on electronic engineering that relies on this law to study the thermal behavior of their devices.
One of the last proposals to better describe thermal transport at the nanoscale has been phonon hydrodynamics. In this line, the Guyer-Krumhansl equation has been proposed as the required generalization of the Fourier law to describe heat transport at this scale . This equation is combined with the Kinetic Collective Model (KCM) to obtain the included parameters from ab initio calculations [4,5]. One of the main advantages of this approach is that its simplicity allows to obtain solutions for arbitrary geometries using Finite Element Methods. Therefore, this combination offers a full predictive framework to describe thermal conductivity in semiconductors in general geometries with characteristic sizes up to the order of the hydrodynamic characteristic length.
Validation of the model has been done with experimental data for different systems such as semiconductor porous membranes with different periodic alignments, thin membranes with different constrictions or 2D materials. In parallel, the tool offers also the interpretation of the results in terms of new phenomena like vorticity and viscosity, giving an insight on the reason for the reduction of the effective thermal conductivity at reduced scales.
 A. Ziabari et.al., Nat. Comm. 9, 255 (2018).
 K. M. Hoogeboom-Pot et. al., PNAS 112 16 4851 (2015)
 P. Torres et al. Phys Rev. Mat. 2, 076001 (2018)
 Y. Guo et. al., Phys. Rev. B 93, 035421 (2018)
 R.A. Guyer et. al. Phys. Rev. 2, 148 (1966)