Correlation between structure, phonon spectra, thermal expansion, and thermomechanics of single-layer ${\mathrm{MoS}}_2$

Using first-principles simulation, the correlation between structure, phonon spectra, thermal expansion, and thermomechanics of single-layer ${\mathrm{MoS}}_2$ is established. The laminar structure results in the low-dimension ZA mode with a parabolic dispersion and negative Grüneisen constants ($\gamma$), while the nonorthogonal covalent Mo–S bonds (or intralayer thickness) result in the interatom and interdirection vibrational hybridizations, which tend to increase $\gamma$. There is a negative-positive crossover in thermal expansion coefficient at 20 K, because of the competition between the modes with negative and positive $\gamma$. Although the phononic activation at finite temperatures has a stiffening effect on the bulk modulus, the dominant effect from thermal expansion softens the lattice upon heating. The intralayer thickness results in the similarity between the thermal expansions of SL and bulk ${\mathrm{MoS}}_2$. Our numerical results explicitly support that the experimentally measured thermal shifts of the Raman modes are dominated by multiphonon scattering, but not thermal expansion.

Figure 1

(a) Top and side views of SL ${\mathrm{MoS}}_2$, where the lattice constant, bond length, thickness (in Å), and the Mo-S-Mo and S-Mo-S angles (in degrees) are labeled. (b) The differential electron density $\left(\Delta Q\right)$ (isovalues: $\pm 0.0675$ $e/AA{}^3)$ in real space and its $z$-direction projection $\left(\Delta Q_z\right)$.

Figure 2

(a) Phonon dispersions and PhDOS, (b), (c) fat bands and projected PhDOSs, and (d) Grüneisen constants of SL ${\mathrm{MoS}}_2$. In (a), the experimental frequencies of the optical modes ($A_{2u},A_{1g},E_{2g},E_{1g}$) at the $\Gamma$ point [19, 45, 46, 47, 48, 49, 50] are shown in the left panel, and the detailed PhDOS at low frequencies are shown in the inset of the right panel. In (d), the dispersion of $\gamma$(ZA) is fully shown in the inset.

Figure 3

Temperature dependence of the (a) lattice constant, (b) thermal expansion coefficient, (c) isovolume heat capacity, and (d) two-dimensional bulk modulus of SL ${\mathrm{MoS}}_2$. In (a), the available experimental measurements on bulk ${\mathrm{MoS}}_2$ [53, 54] are shown. In (b), the detailed variation of $\alpha$ at low temperature is shown in the inset. In (d), the bulk modulus with the phononic contribution excluded ($B_{2\text{D}}^{*}$) is shown, and the linear slopes of the curves are indicated.