Stability of charged sulfur vacancies in 2D and bulk ${\mathrm{MoS}}_2$ from plane-wave density functional theory with electrostatic corrections

Two-dimensional (2D) semiconducting transition metal dichalcogenides such as ${\mathrm{MoS}}_2$ have attracted extensive research interests for potential applications in optoelectronics, spintronics, photovoltaics, and catalysis. To harness the potential of these materials for electronic devices requires a better understanding of how defects control the carrier concentration, character, and mobility. Utilizing a correction scheme developed by Freysoldt and Neugebauer to ensure the appropriate electrostatic boundary conditions for charged defects in 2D materials, we perform density functional theory calculations to compute formation energies and charge transition levels associated with sulfur vacancies in monolayer and layered bulk ${\mathrm{MoS}}_2$. We investigate the convergence of these defect properties with respect to vacuum spacing, in-plane supercell dimensions, and different levels of theory. We also analyze the electronic structures of the defects in different charge states to gain insights into the effect of defects on bonding and magnetism. We predict that both vacancy structures undergo a Jahn-Teller distortion, which helps stabilize the sulfur vacancy in the $-1$ charged state.

Figure 1

Uncorrected (open symbols) and corrected (filled symbols) formation energies for S vacancies in (a) monolayer ${\mathrm{MoS}}_2$ and (b) layered bulk ${\mathrm{MoS}}_2$ in different charge states, calculated with $\text{SCAN}+\text{rVV10}$. The Fermi level is set to the valence band maximum. In the top plot for S vacancies in monolayer ${\mathrm{MoS}}_2$, the multiple data points corresponding to each supercell size indicate the energies evaluated in supercells with different vacuum spacings of 10, 15, and 20 Å. The uncorrected energies diverge with increasing vacuum spacing and also exhibit a strong dependence on in-plane supercell size. The corrected energies are well converged across all supercell and vacuum sizes in all cases except the $+1$ S vacancy in layered bulk ${\mathrm{MoS}}_2$, for which the correction does not work due to the delocalized nature of the defect state (see text).

Figure 2

Formation energy of the S vacancy in (a) monolayer ${\mathrm{MoS}}_2$ and (b) layered bulk ${\mathrm{MoS}}_2$ as a function of Fermi level position, calculated with PBE (orange) and $\text{SCAN}+\text{rVV10}$ (blue). The valence and conduction band edge positions calculated with respect to vacuum level are indicated by the vertical dashed lines. The slopes of each segment of the formation energy plots are indicated, and correspond to the most stable charge state for the defect over that range of Fermi energies. The points at which the slopes change are the charge transition levels (CTLs) and are marked by black vertical lines. In all cases, the neutral S vacancy is predicted to be most stable across most of the gap, and only the $0/-1$ charge transition level is predicted to be within the band gap, close to the conduction band minimum.

Figure 3

Side view of the total charge density distribution of the extra hole around the positively charged S vacancy in layered bulk ${\mathrm{MoS}}_2$. Mo atoms are depicted in purple and S in yellow, the position of the S vacancy within the $4\times 4$ supercell is marked by the green circle, and the red (blue) indicate the positive (negative) $0.001e/r_{\mathrm{Bohr}}^3$ charge isosurfaces. The charge density is completely delocalized not only within the layer containing the S vacancy, but also on adjacent layers.

Figure 4

(a) Total charge and (b) spin density distributions of the extra hole or electron(s) around the charged S vacancy in monolayer and layered bulk ${\mathrm{MoS}}_2$. Mo atoms are depicted in purple and S in yellow, the position of the S vacancy within each $4\times 4$ supercell is marked by the green circles, and the red (blue) indicate the positive (negative) charge and spin isosurfaces. The symmetry is broken in the $-1$ charge state in both the monolayer and layered bulk systems.

Figure 5

Projected density of states for the pristine monolayer ${\mathrm{MoS}}_2$ and the S vacancy in monolayer ${\mathrm{MoS}}_2$ in the neutral (${\mathrm{V}}_{\text{S}}^0$), $+1$ (${\mathrm{V}}_{\text{S}}^{1+}$), and $-1$ (${\mathrm{V}}_{\text{S}}^{1-}$) charge states. The density of states is projected onto the $d$ orbitals of the three Mo atoms directly adjacent to the S vacancy. The defect state in the band gap has primarily $d_{x^2-y^2}$ (dark blue) and $d_{xy}$ (light blue) character. In the $-1$ charge state, the degeneracy between these two orbitals is broken, with the additional electron occupying a state with dominant $d_{x^2-y^2}$ character. The insets illustrate the electronic orbitals corresponding to the defect states of interest.

Figure 6

Projected density of states for the S vacancy in layered bulk ${\mathrm{MoS}}_2$ in the neutral (${\mathrm{V}}_{\text{S}}^0$), $+1$ (${\mathrm{V}}_{\text{S}}^{1+}$), $-1$ (${\mathrm{V}}_{\text{S}}^{1-}$), and $-2$ (${\mathrm{V}}_{\text{S}}^{2-}$) charge states. The density of states is projected onto the $d$ orbitals of the three Mo atoms directly adjacent to the S vacancy. Similar to the S vacancy in the monolayer, the degeneracy between $d_{x^2-y^2}$ (dark blue) and $d_{xy}$ (light blue) orbitals is broken in the $-1$ charge state. The degeneracy is restored in the $-2$ charge state, for which the parallel spin configuration is lower in energy. The insets show the electronic orbitals corresponding to the defect states of interest.