**4. Summary and Outlook**

In the current review, we address selected applications of our recent implementations of quantum transport methodologies in low-dimensional materials. Hereby, we highlight the possibility to perform systematic investigations with atomic resolution, thus addressing material-specific problems for designing potential (nano)phononic devices.

We combined the NEGF formalism with the DFTB methodology to address quantum ballistic transport in various low-dimensional materials with atomistic resolution. This computational approach is implemented as a tool in the DFTB+ software. Although these systems may also be tractable using classical molecular dynamics, extensive parameterizations may be required to study different material combinations (here, machine learning approaches may be of interest). It is therefore more suitable to use the NEGF-DFTB approach, where the chemistry of the problem is naturally included in the first-principle calculation of the Hessian matrix. We showed that 2D puckered materials display strong thermal anisotropy due to their atomic structure, thus transporting heat preferably along the zigzag direction (higher phonon group velocity). As a next application, the influence of BN concentration and defect distribution on the thermal transport of BNC heteronanotubes was considered. Independently of the specific spatial BN distribution, the phonon transmission of pristine (6,6)-CNT is reduced at high frequencies after increasing the BN concentration. As a last application, we demonstrated that the vibrational features of molecular junctions can be exploited in conjunction with an appropriate choice of nanoscale thermal baths to implement a molecule-based phononic filter. This model offers the possibility of engineering different phonon filters based on the rich molecular chemical space. These three reviewed studies clearly demonstrate the potential of the PHONON tool to investigate nanoscale ballistic phonon transport.

In the last section, we present an atomistic method combining time-dependent NEGF with a first-principle based modeling to address phonon dynamics in nanoscale systems. The method is based on solving the equation of motion of the phonon density matrix with an efficient auxiliary-mode approach. The approach was applied to study thermal transport in the transient regime of a 1D chain, providing results in agreemen<sup>t</sup> with the Landauer formalism. By using density-functional theory to obtain the force constants and coupling matrices, the phonon dynamics of small molecular junctions was considered. Although the presented study is based on a Drude regularization of the spectral density, realistic scenarios can be easily addressed. This computational approach builds one of the first attempts to deal with time-dependent quantum phonon transport and it will allow studying various topical questions such as heat pumping, on a fully atomistic basis.

We are, however, not ye<sup>t</sup> able to address physical effects such as thermal rectification from a fully quantum picture. Although rectification can be induced by structural asymmetries, phonon–phonon interactions play a dominant role, too. The latter are also crucial when dealing with phonon transport at high temperatures. An implementation combining NEGF with first-principles requires, besides computing the dynamical matrix as the basic input, third and fourth order anharmonic coefficients as well [76,78]. They contribute additional self-energies in the Green's functions of the scattering region, and involve convolutions in frequency space of two-and three phonon Green's functions. As a result, the problem needs to be solved self-consistently, thus considerably increasing the computational effort.

Another issue is the inclusion of electron–phonon coupling in the description of heat transport. Although it has already been implemented within the NEGF approach to address electronic transport [154–158], there are not many atomistic-based studies related to their impact on phonon transport. Since the interaction with the electronic system will provide an additional energy exchange channel, it will be of interest to elucidate how some of the effects discussed in this review as well as in other investigations, such as thermal rectification and phonon filtering, will be modified by the inclusion of electron–phonon interactions.

**Author Contributions:** L.M.S. wrote the original draft. R.G. wrote and reviewed the final version. A.P., A.C., and G.C. critically revised the manuscript by providing inspiring comments.

**Funding:** This research was funded by the Deutscher Akademischer Austauschdienst (DAAD) within its doctoral programme scholarship. This work was also partly supported by the German Research Foundation (DFG) within the Cluster of Excellence "Center for Advancing Electronics Dresden".

**Acknowledgments:** The authors acknowledge very fruitful discussions with Arezoo Dianat, Vladimiro Mujica, and Alvaro Rodriguez Mendez. We acknowledge the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for computational resources.

**Conflicts of Interest:** The authors declare no conflict of interest.
