2. Computational Details

All calculations were carried out using density functional theory with the PBE (Perdew-Burke-Ernzerhof) exchange-correlation functional [40] as implemented in the Crystal09 code [41]. We have used an all electron triple-zeta valence basis set with one polarization function for sulphur atoms [42], a scalar-relativistic pseudopotential with 18 valence electrons for platinum atoms [43], Hay and Wadt effective core potentials with small core for palladium atoms [41], and the relativistic multi-electron pseudopotential with six valence electrons for selenium atoms [44]. We have employed helical boundary conditions, as implemented in the Crystal09 code [41], for the generation of the NT structures in order to reduce computational costs. Lattice vectors and atomic positions of MX<sup>2</sup> MLs and NTs were fully optimized. The tube diameters considered here are in the range of 10–50 Å, corresponding to chiral indices (10,0)–(32,0) and (6,6)–(24,24) for zigzag and armchair NTs, respectively. The shrinking factors of 16 for MLs and 8 for NTs were used, resulting in 30 and 5 k-points in the irreducible Brillouin zone, respectively, following the Monkhorst-Pack sampling [45]. Band structures were calculated along the high symmetry k-points using the M–Γ–K–M and X–Γ paths for MLs and NTs, respectively.

#### 3. Results and Discussion

The monolayered noble-metal chalcogenides, considered here as large diameter NT limits, were adopted in the 1T polymorph, with space group (P-3m1). Figure 1 shows the 1T geometry for a monolayered structures in the left side, the top and the side views of zigzag and armchair NTs are represented in the middle and the right parts, respectively. The optimized lattice parameters and bond lengths with respect to the tube diameter are shown in Figure 2. Increasing the tube diameter d, the lattice vectors of the tubes decrease for both zigzag and armchair chiralities. For (n,n) NTs, the lattice vectors correspond to those of the MLs; however, due to the curvature they enlarge as the diameters become smaller. Similar behaviour is observed for the bond lengths between metal and chalcogen atoms (M–X), nevertheless, the convergence to the ML limit is much slower. Generally, the bond lengths and lattice vectors of selenide NTs are larger than those of sulphide NTs, and the same holds for the comparison of the platinum over the palladium forms.

The M–X bond lengths can be divided into two types, the inner and the outer wall bond lengths, which are referred hereafter as M–X<sup>i</sup> and M–Xo, respectively. The M–X<sup>o</sup> (M–Xi) are longer (shorter) than the corresponding bond lengths found in the ML structures, and the deviations are more pronounced for armchair NTs than for zigzag NTs, particularly for small d. For diameters below 15 Å, the difference between outer and inner bond reaches 0.40 Å and 0.25 Å for armchair and zigzag NTs, respectively. These numbers strongly reduce to about 0.10 Å difference for diameters of at least 40 Å. Recently, we have reported that the bond lengths of MoS<sup>2</sup> and WS<sup>2</sup> NTs exhibit opposite behaviour, where the zigzag NTs have longer (shorter) M–X<sup>o</sup> (M–Xi) bond lengths than the armchair NTs [46].

Figure 1. 1T 2D (monolayer) and 1D (tubular) forms of noble metal chalcogenides. The unit cell of 2D systems is shown.

Figure 2. Lattice parameters and bond lengths *vs.* diameter of MX<sup>2</sup> NTs.

The stability of NTs can be expressed by the strain energy, EStrain, which is the difference of total energy per atom of the tube and the respective ML. Generally, the strain energy of NTs is correlated to the tube diameter through the archetypal relation <sup>E</sup>Strain <sup>∼</sup> <sup>1</sup>/d<sup>2</sup>. The calculated <sup>E</sup>Strain of MX<sup>2</sup> NTs with respect to their diameters (see Figure 3) follow the same dependence, where the strain energies decrease quadratically with the diameter and converge to the same value for all systems. The curves EStrain(d) were fitted to the equation EStrain = C/d<sup>2</sup> with correlation coefficients greater than 0.999 and the values of coefficients C are given in the Table 1 for each system. We note that strain energies of the PdX<sup>2</sup> NTs are smaller than for PtX<sup>2</sup> NTs for small diameters, this means that PdX<sup>2</sup> NTs are more stiff than PtX<sup>2</sup> NTs in that range. Furthermore, the strain energy of noble-metal chalcogenide tubes is the same for both zigzag and armchair chiralities, whereas for MoS<sup>2</sup> and WS<sup>2</sup> counterparts, armchair NTs are more stable than zigzag NTs, especially for large diameters [46]. In addition, the coefficients C and the strain energy values of the noble-metal chalcogenide NTs are smaller than those of MoS<sup>2</sup> and WS<sup>2</sup> NTs. This means that noble-metal chalcogenide NTs are more favorable and easier to form than MoS<sup>2</sup> and WS<sup>2</sup> NTs.

Figure 3. Strain energy *vs.* diameter of MX<sup>2</sup> nanotubes.


Table 1. Coefficients C ( in eV Å<sup>2</sup>) of the fitted curves EStrain = C/d<sup>2</sup>.

We have also investigated the electronic structure of these noble-metal chalcogenide systems. The MX<sup>2</sup> MLs are found to be semiconducting with indirect band gaps of 1.26 eV, 1.75 eV, 0.74 eV and 1.38 eV for PdS2, PtS2, PdSe2, PtSe2, respectively. This is in agreement with the results of Miró et al. [39] for these MX<sup>2</sup> MLs, where the obtained band gaps are 1.11 eV, 1.75 eV, 0.39 eV and 1.05 eV for PdS2, PtS2, PdSe<sup>2</sup> and PtSe2, respectively, using similar level of theory.

The band gaps Δ *versus* the tube diameter of all MX<sup>2</sup> NTs are plotted in Figure 4. Similar to MoS<sup>2</sup> and WS<sup>2</sup> NTs, these band gaps of MX<sup>2</sup> NTs increase with the diameter and approach the band gaps of their respective MLs. For small d, the band gaps of armchair NTs become larger than zigazag NTs for all systems. Substituting Pd with Pt causes an increase in Δ, while replacing S by Se decreases it. The band structures of noble-metal MX<sup>2</sup> MLs and NTs are depicted in Figure 5. Unlike MoS<sup>2</sup> and WS<sup>2</sup> MLs, where the band gaps are direct with values of 1.9 and 2.1 eV, respectively [47], the noble-metal MLs are all indirect bandgap semiconductors. This difference could be understood in the electronic configuration of the metal elements, as well as, in the 2H and 1T symmetries of MoS2/WS<sup>2</sup> and the noble-metal chalcogenides, respectively. The MX<sup>2</sup> chalcogenide NTs also exhibit indirect band gaps, for both zigazags and armchairs and for all systems, whereas MoS<sup>2</sup> and WS<sup>2</sup> NTs show direct and indirect band gaps for zigzag and armchair forms, respectively [46].

Figure 4. Band gap *vs.* diameter of MX<sup>2</sup> nanotubes.

Figure 5. Band structures of monolayers, (32,0), (18,18) nanotubes of MX<sup>2</sup> systems, respectively.
