Phonons in twisted transition-metal dichalcogenide bilayers: Ultrasoft phasons and a transition from a superlubric to a pinned phase

The tunability of the interlayer coupling by twisting one layer with respect to another layer of two-dimensional materials provides a unique way to manipulate the phonons and related properties. We refer to this engineering of phononic properties as twistnonics. We study the effects of twisting on low-frequency shear modes (SMs) and layer breathing modes in a transition-metal dichalcogenide (TMD) bilayer using atomistic classical simulations. We show that these low-frequency modes are extremely sensitive to twisting and can be used to infer the twist angle. We find ultrasoft phason modes (frequency $\lesssim 1{\mathrm{cm}}^{-1}$, comparable to acoustic modes) for any nonzero twist, corresponding to an effective translation of the moiré lattice by relative displacement of the constituent layers in a nontrivial way. Unlike the acoustic modes, the velocity of the phason modes are quite sensitive to the twist angle. Also, high-frequency SMs appear for small twist angles, identical to those in stable bilayer TMD $(\theta =0^{\circ }$ or ${60}^{\circ })$, due to the overwhelming growth of stable stacking regions in relaxed twisted structures. Our study reveals the possibility of an intriguing $\theta$-dependent superlubric to pinning behavior and of the existence of ultrasoft modes in all two-dimensional materials.

Figure 1

Relaxed twisted bilayer of ${\mathrm{MoS}}_2$ and the coexistence of multiple high-symmetry stacking regions for (a) $\theta =2.9^{\circ }$ and (b) $\theta =57.1^{\circ }$. The high-symmetry stacking regions are marked with solid and dashed circles. Mo atoms of bottom (top) layer are depicted as large (small) in size and with faded (dark) red color. Similarly, we use blue for S atoms.

Figure 2

Evolution of the ILS landscape and its average (in angstroms) with twist angle in TBL ${\mathrm{MoS}}_2$. The in-plane $(x$ and $y)$ distances are in nanometers.

Figure 3

Inhomogeneity in the interlayer coupling in the MSL and the evolution of low-frequency modes. Here $A^{ij}$ (see the text for definition) is plotted for (a) untwisted and (b) twisted $\left(\theta =57.1^{\circ }\right)$ bilayers of the same dimensions for modes with $\omega <50{\mathrm{cm}}^{-1}$. (c) SM (green) [LBM (black)] frequencies at $T=0$ K marked as vertical lines with height proportional to $p^{\mathrm{SM}}\left(p^{\mathrm{LBM}}\right)$. Also shown are the power spectra of MVACF (with highest peak normalized to 1) at $T=300$ K. (d) SM and (e) LBM frequencies at $T=0$ K with the color bar indicating $p^{\mathrm{SM}}$ and $p^{\mathrm{LBM}}$.

Figure 4

Monotonic decrement of LBM frequencies of region I as $\theta \to 0^{\circ }$ in TBL ${\mathrm{MoS}}_2$ with the color bar indicating $p^{\mathrm{LBM}}$. Identical results are obtained near ${60}^{\circ }$ as well. The calculated LBM frequency for BL ${\mathrm{MoS}}_2$ is 43.$5{\mathrm{cm}}^{-1}$.

Figure 5

Exact normalized eigenvectors with the largest projection on the bilayer LBM $\left(p^{\mathrm{LBM}}\right)$ for several values of $\theta$. The eigenvectors for the top Mo layer are for (a) $\theta =1.9^{\circ }$, (b) $\theta =5^{\circ }$, and (c) $\theta =23.{48}^{\circ }$. The corresponding bottom layers move exactly in the opposite direction. The arrows indicate the direction and magnitude of the in-plane displacement (with the associated color bar). The out-of-plane displacement is shown as a field (colored, with associated color bar). Only at large twist angles does the LBM resemble that of BL ${\mathrm{MoS}}_2$.

Figure 6

Visualization of the eigenvectors at the $\mathrm{\Gamma }$ point corresponding to the ultrasoft shear modes for (a) and (b) $\theta =1.9^{\circ }$, (e) and (f) $\theta =5^{\circ }$, and (g) and (h) $\theta =23.{48}^{\circ }$, along with (c) and (d) the high-frequency SMs. The positions of the domain walls and point defects are the same as shown in Fig. 1. The arrows (gray colorbar) denote in-plane displacements (only for Mo atoms of the top layer, for clarity), whereas out-of-plane displacements are represented as a continuous field (colored). The in-plane displacements of the Mo atoms of the bottom layer are exactly opposite to that of the top layer.

Figure 7

Visualization of the eigenvectors at the $\mathrm{\Gamma }$ point corresponding to the shear modes for (a) and (b) $\theta =58.1^{\circ }$, (e) and (f) $\theta ={55}^{\circ }$, and (g) and (h) $\theta =36.{52}^{\circ }$, along with (c) and (d) the high-frequency SMs. The position of the domain walls and point defects are the same as shown in Fig. 1. The eigenvectors are represented in similar manner as in Fig. 6.

Figure 8

Dispersion of the low-frequency modes for several values of $\theta$. The $x$ axis represents momentum, in units of $\frac{4\pi }{\sqrt{3}{a}_{\mathrm{m}}}$, with $a_{\mathrm{m}}$ the moiré lattice constant. The LA, TA, and ZA (ultrasoft shear) modes are highlighted with blue solid (red dashed) lines. The insets show a close-up of dispersion for $q<0.005$ with $\omega <0.4{\mathrm{cm}}^{-1}$.

Figure 9

Twist-angle dependence of the velocity of the phonon modes for both acoustic modes (LA and TA) and ultrasoft phason modes computed from the linear region of phonon dispersion.

Figure 10

One-dimensional Frenkel-Kontorova model with (a) unit-cell commensuration, (b) large-scale commensuration, and (c) mostly unit-cell commensurate with discommensuration in between. The discommensuration is marked with a red line. Also shown is the development of a barrier against shearing as (d) $\theta \to 0^{\circ }$ and (e) $\theta \to {60}^{\circ }$. After relaxation of the twisted structures we slide the top ${\mathrm{MoS}}_2$ layer along the $x$ axis and calculate the total energy without further relaxation.

Figure 11

Evolution of low-frequency vibrational modes with twist angles for a twisted bilayer of ${\mathrm{MoSe}}_2$. The evolution of low-frequency modes as $\theta \to 0^{\circ }$ is similar to that as $\theta \to {60}^{\circ }$ and hence is not shown here.

Figure 12

Comparison of the calculated twist-angle dependence of low-frequency vibrational modes and Raman measurements [30] in TBL ${\mathrm{MoSe}}_2$. The experimental SMs (LBMs) are denoted by green stars (circles), whereas the computed SMs (LBMs) are denoted by variable red stars (circles). The color bar indicates $p^{\mathrm{SM}}$ and $p^{\mathrm{LBM}}$.

Figure 13

Twist-angle dependence difference in specific heat (in units of J/mol/K) of TBL ${\mathrm{MoS}}_2$ with respect to BL ${\mathrm{MoS}}_2$ (untwisted).