Multiphonon Raman scattering mediated by the exciton states in monolayer transition metal chalcogenides

In the frame of Huang-Rhys's model, we study multiple orders of Raman scattering in monolayer transition metal chalcogenides (TMDC) based on the intrinsic excitons coupled with three types of phonon modes, including longitudinal optical (LO) phonon, surface optical (SO) phonon, and longitudinal acoustic (LA) phonon. We find that multiphonon structures display multiple order overtones at equal interval energies of $\hslash {\omega }_{\text{LO}}$ and $\hslash {\omega }_{\text{SO}}$, respectively, and intensities of overtones depend on the obtained values of Huang-Rhys factor, which can be modulated by the strength of exciton-optical phonons coupling, cutoff wave vector of optical phonon modes, and large exciton Bohr radius. Moreover, we propose the combined multiphonon processes of $p\hslash {\omega }_{\text{LO}}+m\hslash {\omega }_{\text{SO}}$ and $p\hslash {\omega }_{\text{LO}}\left(m\hslash {\omega }_{\text{SO}}\right)+\hslash {\omega }_{\text{LA}}$. These combined processes with weaker intensity could be used to analyze the fine structure of multiphonon Raman spectra in experiments. Our theoretical results indicate that the strong exciton effect in monolayer TMDC provides an excellent platform for exploring multiphonon Raman scattering.

Figure 1

The schematic diagram of multiphonon Raman scattering mediated by the exciton state. The whole process can be divided into the following steps: (1) an exciton is created by the absorption of the incident photon ($\hslash {\omega }_i$), (2) this exciton relaxes into lower energy states with successive emission of optical phonons ($\hslash {\omega }_{\text{LO}}$ or $\hslash {\omega }_{\text{SO}}$), and (3) the radiative recombination of the exciton with the emission of the scattered photon ($\hslash {\omega }_s^n=\hslash {\omega }_i-n\hslash {\omega }_{\text{SO(LO)}}$). A series of exciton Rydberg states are denoted by the principal quantum number $l=1,2,...$.

Figure 2

(a) Huang-Rhys (HR) factor of the LO phonon mode as functions of the exciton Bohr radius ($a_B$) and cutoff wave vectors $k_c$ at the polarization parameter ${\eta }_0=0.1$ in monolayer ${\mathrm{MoS}}_2$. (b) HR factor of the LO phonon mode as functions of $a_B$ and ${\eta }_0$. (c) The dependences of HR factor on $a_B$ and $k_c$ for the SO phonon mode in monolayer ${\mathrm{MoS}}_2$ on ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate at the internal distance $z_0=0.3$ nm. (d) The dependences of HR factor on $a_B$ and $z_0$ for the SO phonon mode. (e) The relations of HR factor with $a_B$ and $k_c$ for the LA phonon mode at fixed deformation potential constant $D=8$ eV. (f) The relations of HR factor with $a_B$ and $D$ for the LA phonon mode.

Figure 3

(a) Multiple LO phonons Raman scattering for different polarization parameters at ${\mathrm{a}}_B=3$ nm. (b) Multiple LO phonons Raman scattering for different exciton Bohr radius at ${\eta }_0=0.1$ nm. (c) Multiple SO phonons Raman scattering in monolayer ${\mathrm{MoS}}_2$ laying on different polar substrates at ${\mathrm{a}}_B=3$ nm and at ${\mathrm{z}}_0=0.3$ nm. (d) Multiple SO phonons Raman scattering in monolayer ${\mathrm{MoS}}_2$ on ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate at different internal distances and at ${\mathrm{a}}_B=3$ nm.

Figure 4

In the structure of ${\mathrm{MoS}}_2$/${\mathrm{Al}}_2{\mathrm{O}}_3$, (a) multiphonon Raman scattering (MRS) of $p\hslash {\omega }_{\text{LO}}$ and combined $p\hslash {\omega }_{\text{LO}}+\hslash {\omega }_{\text{LA}}$ processes, (b) MRS of the combined $p\hslash {\omega }_{\text{LO}}+m\hslash {\omega }_{\text{LO}}$ processes, and (c) MRS of $m\hslash {\omega }_{\text{SO}}$ and combined $m\hslash {\omega }_{\text{SO}}+\hslash {\omega }_{\text{LA}}$ processes, where several parameters $D=8$ eV, ${\eta }_0=0.1$, $T=77$ K, and $z_0=0.3$ nm are fixed.

Figure 5

(a) Intensity of peaks for the processes of $\hslash {\omega }_{\text{SO}}$ ($\hslash {\omega }_{\text{LO}}$) and $2\hslash {\omega }_{\text{SO}}$ ($2\hslash {\omega }_{\text{LO}}$) as a function of temperature. (b) Intensity of peaks for the combined processes of $\hslash {\omega }_{\text{SO}}+\hslash {\omega }_{\text{LA}}$ ($\hslash {\omega }_{\text{LO}}+\hslash {\omega }_{\text{LA}}$) and $2\hslash {\omega }_{\text{SO}}+\hslash {\omega }_{\text{LA}}$ ($2\hslash {\omega }_{\text{LO}}+\hslash {\omega }_{\text{LA}}$) as a function of temperature in monolayer ${\mathrm{MoS}}_2$ on ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate, where several parameters $D=8$ eV, ${\eta }_0=0.1$, and $z_0=0.3$ nm are fixed.