Quantum Mechanical Rippling of a ${\mathrm{MoS}}_2$ Monolayer Controlled by Interlayer Bilayer Coupling

Nanoscale corrugations are of great importance in determining the physical properties of two-dimensional crystals. However, the mechanical behavior of atomically thin films under strain is not fully understood. In this Letter, we show a layer-dependent mechanical response of molybdenum disulfide (${\mathrm{MoS}}_2$) subject to atomistic-precision strain induced by $2H$-bilayer island epitaxy. Dimensional crossover in the mechanical properties is evidenced by the formation of star-shaped nanoripple arrays in the first monolayer, while rippling instability is completely suppressed in the bilayer. Microscopic-level quantum mechanical simulations reveal that the nanoscale rippling is realized by the twisting of neighboring Mo—S bonds without modifying the chemical bond length, and thus invalidates the classical continuum mechanics. The formation of nanoripple arrays significantly changes the electronic and nanotribological properties of monolayer ${\mathrm{MoS}}_2$. Our results suggest that quantum mechanical behavior is not unique for $s{p}^2$ bonding but general for atomic membranes under strain.

Figure 1

Atomistic origin of 1D NRAs in ${\mathrm{MoS}}_2$ ML induced by centrosymmetric $2H$-BL growth. (a) Characteristic FFM image of 1D NRAs in ${\mathrm{MoS}}_2$ ML. Normal force setpoint: 1 nN. (b),(c) DFT simulation of $2H$-BL and $3R$-BL growth on a fully relaxed ML, respectively. The $2H$ BL induces compressive strain at the ML-BL boundaries, in contrast to tensile strain by the $3R$ BL. Lattice changes are in Å.

Figure 2

Peak force-dependent FMAFM of 1D NRAs. (a) Star-shaped 1D NRAs formed along the three angle-bisector directions (indicated by the black dashed lines) of the ML due to the $2H$-BL growth. (b) Zoom in over the dashed square in (a). The black line is the position for the cross section in Fig. 3. (c)–(e) Nanoripple morphology at 500 pN, 1 nN, and 4 nN peak force, respectively. Color scale: 0–2.4 nm for (a) and (b), and 0–2 nm for (c)–(e) from black to white.

Figure 3

Quantum mechanical simulation of 1D NRAs formation in ${\mathrm{MoS}}_2$ ML. (a) Corrugations of 1D NRAs compared with nonrippled ${\mathrm{MoS}}_2$ ML. (b) Atomistic simulation of a 1D NRA showing amplitude and wavelength matching the experiments. Inset shows the Mo—S bond lengths at the maximum amplitude location. The top (bottom) S atoms are located in the convex (concave) region of the ripple.

Figure 4

(a),(b) Raman characterizations of 1D NRAs formed in twinned crystals. The average strain intensity is estimated as $\sim 1\%$. (c),(d) PL measurements of 1D NRAs. The measured shift in photon energy peaks agrees with the Raman results.