*2.2. Density Functional Tight-Binding*

The main quantities to obtain the quantum phonon transport properties by using the NEGF formalism are the mass-reduced force constant matrix in each region *Kα* (*α* = *L*, *C*, *R*) and the coupling matrices of the left and right bath to the central region, *VLC* and *<sup>V</sup>CR*, respectively [80]. The accuracy of the results depends on the reliability of these quantities to catch the atomistic features of the system. Density-functional theory (DFT) is nowadays the main computational approach used in chemistry and physics to perform quantitative studies on molecules and materials due to its favorable accuracy-to-computational-time ratio [85]. The strong increase in accuracy coming from the development of gradient corrected and hybrid functionals such as PBE [86] and B3LYP [87], which compensate deficiencies of older approximations, has largely contributed to a further increase in popularity. However, hybrid functionals are computationally demanding, limiting DFT to a maximum of a few hundreds of atoms, depending on the chemical species. Classical force fields appear as a reasonable solution to this problem but, in many cases, they suffer of limited transferability and do not yield any information on the electronic structure.

Semiempirical methods appear as another option to DFT, conceptually lying between empirical force fields and first principle approaches, allowing for the treatment of thousands of atoms [88]. These methods can be understood as approximations to more accurate methods (full DFT or Hartree–Fock), but including empirical parameters that are fitted to reproduce reference data. One example of a semiempirical method, which is used in the present work, is the density functional tight-binding (DFTB) approach [89–91]. Here, the basic electronic parameters (Slater–Koster parameters) are consistently obtained from full DFT-based calculations for atom pairs, while the repulsive part of the electronic energy is fitted by means of splines. Based on it, the Hamiltonian and Overlap matrices of a specific system can be decomposed into pair interactions (not only between nearest-neighbors) yielding a generalized tight-binding Hamiltonian. Many studies have been carried out by using the DFTB method, including transport properties of 2D materials [92–94], stability and mechanical properties [95], vibrational signatures [96], computation of molecular absorption spectra [97], and of charge transfer excitation energies [98] (see recent review papers [99–101] for additional topics).

Three different DFTB models have been proposed up to now, which are derived by expanding the DFT total energy functional around a reference density *ρ*0 to first, second, and third order, respectively [101]. The choice of the DFTB model depends on the system under study. The non self-consistent DFTB method (or DFTB0) is more appropriate for systems with negligible charge transfer between atoms (typically homonuclear systems or those involving atoms of similar electronegativity as

in hydrocarbons [102]). Ionic systems with large inter-atomic charge transfer can also be treated with this method [103]. On the other hand, in systems where a delicate charge balance is crucial such as biological and organic molecules [104,105], a self-consistent charge treatment is required (DFTB2 and DFTB3) [101,106]. Based on the advantages of DFTB for accurately describing large systems involving few thousand of atoms, the force constant matrices of the studied systems are numerically obtained by applying a finite difference method to ge<sup>t</sup> the second derivatives of the total energy with respect to the atomic displacements (implemented in the DFTB+ software) [80]. These matrices can also be obtained by density functional perturbation theory, which in the case of DFTB reduces to analytic expressions involving derivatives of only two-center matrix elements [107].

### *2.3. Application of the DFTB-Based PHONON Tool*

From electron transport studies, it is well-known that transport properties of nanoscale systems can be tailored by varying different control parameters. This can include covalent or non-covalent chemistry [108,109], atomic doping [35,110], topological defects [111,112], quantum confinement [113], and mechanical strains [114,115], among others. Similarly, a major focus of research on phonon transport is to identify the major variables allowing for effectively tuning the heat transport properties of nanoscale materials. In this section, we review few of our previous research in this direction using the NEGF-DFTB method [116–121], which is already implemented as a tool in the DFTB+ code (for details of the PHONON tool, see [80]). We focus on 2D orthorhombic materials, BNC heteronanotubes, and phonon filter effects in molecular junctions.
