Strain effects on the behavior of isolated and paired sulfur vacancy defects in monolayer ${\mathrm{MoS}}_2$

We investigate the behavior of sulfur vacancy defects, the most abundant type of intrinsic defect in monolayer ${\mathrm{MoS}}_2$, using first-principles calculations based on density functional theory. We consider the dependence of the isolated defect formation energy on the charge state and on uniaxial tensile and compressive strain up to 5%. We also consider the possibility of defect clustering by examining the formation energies of pairs of vacancies at various relative positions, and their dependence on charge state and strain. We find that there is no driving force for vacancy clustering, independent of strain in the material. The barrier for diffusion of S vacancies is larger than 1.9 eV in both charged and neutral states regardless of the presence of other nearby vacancies. We conclude that the formation of extended defects from S vacancies in planar monolayer ${\mathrm{MoS}}_2$ is hindered both thermodynamically and kinetically.

Figure 1

(a) Top view of square region of the supercell used for the calculations with the primitive unit cell shown in black solid lines, with the lattice constant $a=3.195\AA$. (b) and (c) Top and side views of the neighborhood of the S vacancy defect with the Wannier function corresponding to the $\sigma$ bond between the Mo and S atoms. Blue and red lobes are positive and negative isosurfaces of the Wannier function.

Figure 2

(a) Magnitude of atomic displacements around the isolated S vacancy in monolayer ${\mathrm{MoS}}_2$, $|\mathrm{\Delta }\vec{R}|$, as a function of distance $d_{xy}$ between the atom and the vacancy in the $xy$ plane. (b) Isosurfaces of the defect states $|{\phi }_x\left(\vec{r}\right){|}^2$ and $|{\phi }_y\left(\vec{r}\right){|}^2$; the nodal planes are shown as dashed lines in each case. (c) Energy levels of the defect states ${\phi }_y$ (blue), ${\phi }_x$ (red), and thermodynamic charge transition level (black) in the band gap associated with a single S vacancy: their position within the gap range as a function of uniaxial strain, relative to the valance band maximum of the unstrained monolayer ${\mathrm{MoS}}_2$, ${\varepsilon }_{\mathrm{VBM}}^{\left(0\right)}$. (d) Formation energy of isolated S vacancy as a function of uniaxial strain.

Figure 3

Clustering of S vacancies. (a) Defect wave functions for the pairs of S vacancies in the $x$ direction for various relative positions; black circles highlight the vacancy positions. (b) Formation energies $E_f\left(q\right)$ for vacancy pairs in charge states $q=0,-1,-2$ along the $x$ direction for 0% (solid line, filled circles) and 5% (dashed line, open squares) tensile strain along the $y$ direction. (c) Difference in formation energies of a pair of $V_{\mathrm{S}}$ relative to the energy of two isolated S vacancies $\mathrm{\Delta }{E}_f\left(q\right)$.

Figure 4

Energy barriers for S vacancy diffusion: (a) Geometric structures along the NEB path for isolated vacancy in neutral state. (b) Diffusion barrier as a function of strain applied along the $y$ direction. (c) Minimal energy path for the diffusion of an isolated vacancy (solid line) for the neutral (blue, $q=0$) and charged (red, $q=-1$) states.

Figure 5

The expected photoluminescence spectrum peak (black line) from ${\mathrm{MoS}}_2$ monolayer containing S vacancy defects in regions with no strain (blue dashed line) and regions with $-2$% (red dashed line) and $+2$% (green dashed line); we have assumed a peak width of 60 meV, typical for a free-standing monolayer.