Reply to “Comment on ‘Ideal strength and phonon instability in single-layer ${\mathbf{MoS}}_2$’ ”

In the Comment, Cooper et al. pointed out a disagreement between two recent density functional theory (DFT) studies ([Li, Phys. Rev. B 85, 235407 (2012)] and [Cooper et al., Phys. Rev. B 87, 035423 (2013)]) on the uniaxial tensile stress as a function of the applied strain along the zigzag $\left(x\right)$ and armchair $\left(y\right)$ directions of single-layer ${\mathrm{MoS}}_2$. We show here that it is the in-plane atomic relaxation under the applied uniaxial tensile stress, rather than the errors in the execution of the DFT calculation, that leads to the discrepancy. Furthermore, through new analysis, we show that our original conclusion that the tensile strength of single-layer ${\mathrm{MoS}}_2$ is dictated by out-of-plane soft-mode phonon instability under the biaxial tension and the uniaxial tension along the armchair direction is still valid when the in-plane atomic relaxation is considered.

Figure 1

The effect of in-plane atomic relaxation on the computed uniaxial true stress vs engineering strain curves along (a) $x$ or zigzag and (b) $y$ or armchair directions. The * indicates the data extracted from Ref. [3], which were obtained based on the QE code and without involving in-plane atomic relaxation. Calculation using vasp (red square) reproduces the same result if atoms are unrelaxed in plane. When atoms are relaxed laterally, the calculated stresses using vasp (green diamond) and using QE [magenta triangle in (b)] coincide with those of Cooper et al. (blue triangles) using QE. The in-plane atomic relaxation also affects (c) the relaxed in-plane strain and (d) the relaxed out-of-plane strain.

Figure 2

The displacement of the S atoms along the armchair direction when the in-plane atomic relaxation is applied with respect to that without the in-plane atomic relaxation. The atomic positions of Mo in the unit cell between the relaxed and the unrelaxed strain paths are matched for comparison. The unit cell of single-layer ${\mathrm{MoS}}_2$ with the two basis vectors ${\vec{a}}_1$ and ${\vec{a}}_2$ is shown in the inset. Mo and S atoms are marked by green and yellow, respectively. A negative displacement indicates that the two S atoms (shown as one because one S atom sits on top of the other) within the unit cell move further away from the Mo atom, resulting in a larger Mo-S distance along $y$, compared to that of the unrelaxed case.

Figure 3

Calculated phonon dispersion along $M^{\prime }-\Gamma -M-K-\Gamma -K^{\prime }$ for single-layer ${\mathrm{MoS}}_2$ under uniaxial tension along the armchair $\left(y\right)$ direction at an engineering strain of ${\varepsilon }_{yy}=0.256$. The convention of the symmetry points in the first Brillouin zone is adopted from our previous paper [3]. The unstable phonon mode at $M^{\prime }$ leads to an out-of-plane displacement, which is of the same nature as identified in our previous paper [3].