2.3.1. 2D Orthorhombic Materials

The effect of anisotropic atomic structure on the phonon transport of two-dimensional puckered materials was studied by Medrano Sandonas et al. [117]. From this new family of 2D materials [122–125], three representative members, phosphorene, arsenene, and SnS monolayers, which display the main features of this family, were studied. The unit cell of these materials is composed by four atoms, as depicted in Figure 3. Each atom is pyramidally bonded to three neighboring atoms of the same type (phosphorene and arsenene, for homoatomic) or of different type (tin sulfide (SnS), for heteroatomic) forming a puckered-like honeycomb lattice. As shown in Table 1, the lattice constants computed with the DFTB approach quantitatively agree (error ≤ 5%) with those obtained at the full DFT level by other authors for all three materials.

We used a standard approach to compute the phonon band structure [80]. This consists in diagonalizing the dynamical matrix at selected k-points, after obtaining them through a Fourier transformation of the real-space force constants (this method is also part of the PHONON tool). Due to the absence of imaginary frequencies, all studied systems can be considered as mechanically stable (see the three lower panels of Figure 3). The acoustic branches display the typical dispersion of 2D materials: longitudinal (LA) and transversal (TA) acoustic branches show linear dispersion as *q* (wave vector) approaches the Γ point, while out-of-plane ZA branches show on the other hand a quadratic dispersion as a result of the rapid decay of transversal forces. The behavior of the dispersion relation for homoatomic puckered materials is almost identical, except for the maximum frequency of the optical modes, which is a consequence of the difference in mass between As ( ∼75 u) and P ( ∼31 u). We also remark that the phonon dispersion for P and As computed with DFTB agrees quite well with DFT results[117]. Only for SnS monolayer the high frequency optical modes are shifted upwards.

**Figure 3.** Phonon dispersion for homo-and heteroatomic two-dimensional puckered materials: (**a**) phosphorene; (**b**) arsenene; and (**c**) tin sulfide (SnS) monolayer. We also show the atomistic view of the two-dimensional materials, highlighting the zigzag (ZZ) and armchair (AC) transport directions. The figure is reproduced with permission from Ref. [117]. Copyright 2016 American Chemical Society.

**Table 1.** Calculated lattice constants of two-dimensional puckered materials along zigzag (ZZ) and armchair (AC) directions. For comparison, the lattice constants from other published theoretical studies are also given. In general, the DFTB lattice parameters agree quite well with those calculated by using DFT method, error ≤ 5%. The table is reproduced with permission from Ref. [117]. Copyright 2016 American Chemical Society.


Furthermore, based on the group velocities values obtained for ZZ (Γ → *X*) and AC (Γ → *Y*) transport directions, we may expect that these materials will display strong anisotropy in their thermal transport. Indeed, the group velocities for the longitudinal acoustic (LA) branch in phosphorene were found to be 8.35 km/s and 4.74 km/s along the Γ-X (ZZ) and Γ-Y (AC) directions, respectively, comparable to DFT results [127,131,132]. The values for arsenene, 5.01 km/s for ZZ and 2.71 km/s for AC, are also in good agreemen<sup>t</sup> with those in Ref. [129]. The SnS monolayer displayed group velocities of 6.48 km/s (ZZ) and 2.14 km/s (AC). We note that thermal anisotropy has only been reported for phosphorene [126,132] and arsenene [128,129], but not for SnS monolayers. Accordingly, the largest anisotropy in the thermal conductance was found in SnS monolayers due to the dominant contribution of acoustic modes to thermal transport [117].
