Chiral phonons in the indirect optical transition of a ${\mathrm{MoS}}_2/{\mathrm{WS}}_2$ heterostructure

Since experimental verification of the predicted chiral phonon (CP), CPs have become a rising topic. The previous observation was a two-step process, which identified the CP by intervalley transfer of valley holes created from direct optical transition. It is highly desirable to simplify the creation and detection of CPs. Moreover, while indirect optical transition in transition metal dichalcogenide heterostructures has been observed, it is important to promote the understanding of the interplay between the phonon, electron, and photon in the indirect transition. Here, we take the ${\mathrm{MoS}}_2$/${\mathrm{WS}}_2$ heterostructure as a prototype and study CPs in the indirect optical transition. By first-principles calculations, the heterostructure with an indirect electronic gap exhibits CPs at the high-symmetry $\pm \mathit{K}$ points of the phonon Brillouin zone, which have sizable circular polarization and discrete pseudoangular momentum. CPs with distinct chiralities can be selectively and directly generated in indirect optical transitions by tuning the optical chirality and frequency. These findings provide microscopic understanding of the indirect optical transition by interacting chiral fermions and chiral bosons and enable more convenient detection and control of CPs.

Figure 1

The geometric, electronic, and phononic properties of the ${\mathrm{MoS}}_2$/${\mathrm{WS}}_2$ heterostructure. (a) Top view and (b) side view of the geometric structure. The dashed box denotes a unit cell. Mo, W, and S atoms are colored in red, purple, and green, respectively. Four different S atoms in the unit cell are marked by S1–S4. $d$ is the interlayer distance in (b). (c) Electronic band structure with the fat-band representation denoting the contributions from two monolayers. The red and blue states are spin-up and spin-down ones, respectively. (d) Phonon dispersion with phonon circular polarization denoted by a color gradient on each band.

Figure 2

Lattice relative vibrations within a unit cell for phonon modes at $\mathit{K}$. The circular vibration is represented by the ellipse with the black arrow. The red arrow denotes the linear vibration along the $z$ axis. The size of the ellipse and the length of the red arrow are proportional to the amplitudes of the circular and linear vibrations, respectively. The direction and position of each black arrow denote the chirality and initial phase of the circular vibration, respectively. The phonon modes are arranged in order of increasing energy. The group index of each mode is given in the parentheses after the mode number. Besides the sublattice relative vibrations in a unit cell, the Bloch phase factor of the phonon wave function contributes to phase shifts between unit cells. The nonlocal phase difference is given in Fig. S2 of the SM [40].

Figure 3

Schematic depiction of the chiral-phonon-assisted indirect optical transition under (a) right-circularly and (b) left-circularly polarized optical fields. $l_v$, $l_c$, and $l_{ph}$ are the PAM of the VBM, CBM, and phonon, respectively, with the valley index in parentheses.

Figure 4

The effects from the applied strains on the electronic and phonon properties. (a) The electronic band structures and (b) phonon densities of states under in-plane biaxial tensile strain ${\epsilon }_{xy}$ and vertical compressive strain ${\epsilon }_z$.