2.2.1. Electronic Structure

Test DFT calculations at LDA level were performed on example of the bulk 2H-MoS2 and fcc-PbS crystals (see Section 3). 2H-MoS2 was found to be a semiconductor with direct band gap 2.18 eV and indirect Ƚĺ½KȽ gap of 0.59 eV. The valence band is composed mainly of S3*p*-states, about ~3.5–6.0 eV below the Fermi level. The top of the valence band and the bottom of the conduction band are dominated by Mo4*d*-states. In the electronic structure of fcc-PbS the occupied Pb6*s* band is present between 6.5 and 9.5 eV below the Fermi level. The three predominately S3*p* bands are located between í6 and 0 eV. The fundamental band gap of 0.11 eV is direct at the L point with the conduction band composed predominately of Pb6*p* states. While the values of the band gaps are also typically underestimated in LDA approach, the band structures are in full accordance with other theoretical and experimental results [21,22].

Pb atoms intercalate into the interlayer space of the MoS2 host lattice (Pb*y*/MoS2 intercalates).

In general, the band structure of MoS2 and the corresponding picture of the densities of states (DOS) are essentially not perturbed by substitutional doping of the Mo sublattice by a single Pb atom (Figure 5a). A single level separates from the bottom of the conduction band into the band gap of pristine MoS2, while the Fermi energy shifts downwards the top of valence band. In addition, a new level emerges at 1 eV below the bottom of the valence band of doped MoS2. It is composed mainly of Pb6*s* states, which do not overlap with the S3*p* band and do not contribute to the chemical bonding of the system. Thus, like in the case of PbS compound, the chemical bonding of Pb atom with S atoms is released due to the depopulation of only Pb6*p* states, which can be found at 5–6 eV above the Fermi energy. All these features are evidence for a p-type semiconducting character of the Pb-doped MoS2 and nominal 2+ oxidation state of Pb. Such oxidation state and the arising electron deficiency of MoS2 electronic structure should lead to destabilization of the chemical environment of the Pb-doped MoS2 lattice. The map of the differential electron densities for this system supports this deduction and demonstrates a vanishing electron density and weakening of the bonds near the Pb atom compared to the electron density nearby the Mo atoms (Figure 5a).

Another possible process, which could occur during the Pb-promoted growth of MoS2 nanotubes, is an intercalation of MoS2 lattice by Pb atoms. The intercalation by a single Pb atom also does not affect the band structure essentially (Figure 5b). Yet, in this case the Fermi level shifts upwards into the conduction band. The localized Pb6*s* states occur deeper at 2 eV below the bottom of valence band. Pb6*p* states of the intercalating Pb atom are more delocalized within the conduction band, than in the aforementioned case of Pb doping. Thus, a Pb*y*/MoS2 intercalate should behave as an n-type semiconductor. The map of differential electron density for this system does not reveal any essential covalent bond formation with S atoms and serves as an evidence for the non-amicable environment of the intercalating Pb atoms.

**Figure 5.** (**a**) Band structure; electronic partial densities of states (DOS) for Pb and differential electron density map for 2H-MoS2 crystal doped by single Pb atom; (**b**) Band structure, DOS for Pb and differential electron density map for 2H-MoS2 crystal intercalated by single Pb atom. LDA DFT calculations.

2.2.2. Estimation of the Stability Limit for Lead Atoms in the MoS2 Lattices

Both doping and intercalation by single Pb atoms lead to a destabilization of the MoS2 electronic system. A weakening of the chemical bonding within the MoS2 lattice due to electron deficiency of the valence band or occupation of anti-bonding Mo4*d*-states, are observed, respectively. A stabilization of the lattice is not favored also by the sterical strain induced after the difference in the radii between Mo and Pb atoms. e.g., optimized lattice parameters for substitutionally Pb-doped MoS2 lattice reveal a slight increase of the lattice parameter *a* from 3.12 Å to 3.15 Å due to the single Pb impurity atom and up to 3.19 Å due to the "cluster" of four Pb atoms. The interlayer distance is not considerably affected in both cases and is ~0.05 Å smaller, than in the pure 2H-MoS2. The intercalation of MoS2 lattice by individual Pb atoms has opposite effect: the *a* lattice parameter is increased by ~0.01 Å, while the interlayer distance is getting larger by ~0.4 Å. These trends agree well with the large atomic radius of Pb atoms. Indeed, the calculated metal-sulfur bond lengths in the bulk of PbS and MoS2 are: 2.94 Å for Pb-S and 2.40 Å for Mo-S, while the length of the covalent Pb-S bond within Pb-doped MoS2 lattice is 2.65 Å.

Furthermore, the influence of the concentration and ordering of the impurity atoms on the thermodynamic stability of doped Mo1í*x*Pb*x*S2 and intercalated Pb*y*/MoS2 phases can be considered. To characterize the stability of Mo1í*x*Pb*x*S2 and the Pb*y*/MoS2 phases, the cohesion energies *E*coh were calculated for a set of 4 × 4 × 1 2H-MoS2 supercells modified by 1–4 Mo atoms (Table 1). In agreement with the picture of the electronic structure, the absolute values of *E*coh for all modified systems vanish with the growing content of Pb, which is an evidence for the weakening of chemical bonding in both doped and intercalated MoS2 compared to the pristine MoS2. Noteworthy, the cohesion energies of the solid solutions as well as intercalates vary almost in the same order of magnitude, and the competing formation of both phases during the Pb-promoted growth of MoS2 nanotubes can be anticipated.

The exact growth mechanism and the main compounds participating in the growth of MoS2 nanotubes have still to be found. Yet, a first insight in this process is possible by the consideration of some model reactions. The formation energies ǻ*E* for Mo1*íx*Pb*x*S2 and Pby/MoS2 phases from MoS2

and the corresponding compounds were estimated using calculated change in the total energies of the next formal reactions:

$$\text{x}(\text{l}-\text{x})\,\text{MoS}\,\text{z} + \text{xPbS} + \text{xS} = \text{Mo} \,\text{l}-\text{xPbS}\,\text{z} \tag{1}$$

$$\text{Pbb} + \text{MoS}\_2 = \text{Pb}\_2 / \text{MoS}\_2 \tag{2}$$

Both reactions have been found to be highly endothermic (Table 2). Concerning the calculated values of ǻ*E*, a strong tendency for the phase separation of Pb-modified MoS2 lattice into a mixture of simple binary sulfides and simple elements may be contemplated. These theoretical observations are in agreement with the experimental finding of the time-depended Pb content in fabricated MoS2 samples.


**Table 2.** Cohesion energies *E*coh and energies of formation ǻ*E* of Mo1í*x*Pb*x*S2 solid solutions and Pb*y*/MoS2 intercalates concerning reactions (1) and (2), as a function of the content and the arrangement of Pb atoms within model supercells. LDA DFT calculations.

Noteworthy, the formation energies for the model supercells with "associated" (neighboring) Pb atoms in both Pb-doped and Pb-intercalated MoS2 are considerably lower, than for those supercells, where all Pb atoms are separated. e.g., the interaction between two Pb atoms separated by

the distance ~6.3 Å in Mo1í*x*Pb*x*S2 solid solutions is almost absent (ǻ*E* for D2b and D2c isomers are close to that of D1, Table 1). The same tendency can be obtained in Pb*y*/MoS2 intercalates, yet, with a higher range of interaction between Pb atoms (ǻ*E* for I2b isomer is still less, than for I1, Table 1). Thus, the solid solutions of Mo1í*x*Pb*x*S2 should be more stable than Pb*y*/MoS2 intercalates, since the coalescence of intercalated Pb atoms is more favorable and Pby/MoS2 intercalates might exist in the narrower part of the phase diagram at a lower Pb content, than Mo1í*x*Pb*x*S2 solid solutions.

Concerning the analysis of calculated formation energies for the case of single doping or intercalating Pb atoms, the formation of Mo1í*x*Pb*x*S2 solid solutions is more likely, than the formation of Pb*y*/MoS2 intercalates. As well, the values of formation energies allow an estimation of the limit of the lead solubility at certain temperature T at thermodynamic equilibrium. As example, we consider roughly the possible content within Mo1í*x*Pb*x*S2 compounds after reaction (1). The preference of one of two states in a chemical reaction is determined by the free energy change ǻ*F* = ǻ*U* í *T*ǻ*S*, where ǻ*U* is the change of internal energy, ǻ*S* is the change of entropy. The condition corresponding to the phase separation is ǻ*F* 0.

In the first approximation, ǻS can be defined as the configurational entropy of randomly distributed Pb atoms in the metal sublattice of Mo1í*x*Pb*x*S2. From the theory of ideal solutions it follows, that

$$
\Delta S = R \left( \ln x + \frac{\left(1 - x\right)}{x} \ln \left(1 - x\right) \right) \tag{3}
$$

where ǻ*S* is expressed per mole of Pb. ǻ*U* can be approximated as the energy of reaction (1) (ǻ*U* = ǻ*E*). The estimation of the free energy change for the phase separation within Mo1í*x*Pb*x*S2 solid solutions with a fortiori low *x* imply the use of the energy for the formation of single Pb atom within MoS2 lattice, *i.e.*, ǻ*E* = 4.65 eV or 448.7 kJ/Pb-mol (isomer D1, Table 1).

The results of this approach based on the formal reaction (1) are visualized in Figure 6. They reveal that the substitution of Pb atom instead of the Mo atom within the MoS2 lattice is a quite rare event and the content of Pb at thermodynamic equilibrium would be around *x* = 10<sup>í</sup><sup>7</sup> at the temperatures ~3000 K. In this manner, the experimentally fabricated MoS2 nanotube samples with primary Pb content of *x* = 0.12, which were formed during the extreme reaction conditions of the sun-light driven evaporation, are thermodynamically unstable and should show a high urge towards a phase separation.

**Figure 6.** Calculated free energy ǻ*F* of the phase separation within Mo1í*x*Pb*x*S2 solid solutions depending on the temperature and Pb content (*x*).
