Controlling exciton many-body states by the electric-field effect in monolayer ${\mathrm{MoS}}_2$

We report magneto-optical spectroscopy of gated monolayer ${\mathrm{MoS}}_2$ in high magnetic fields up to $28\mathrm{T}$ and obtain new insights on the many-body interaction of neutral and charged excitons with the resident charges of distinct spin and valley texture. For neutral excitons at low electron doping, we observe a nonlinear valley Zeeman shift due to dipolar spin-interactions that depends sensitively on the local carrier concentration. As the Fermi energy increases to dominate over the other relevant energy scales in the system, the magneto-optical response depends on the occupation of the fully spin-polarized Landau levels (LL) in both $K/K^{\prime }$ valleys. This manifests itself in a many-body state. Our experiments demonstrate that the exciton in monolayer semiconductors is only a single particle boson close to charge neutrality. We find that away from charge neutrality it smoothly transitions into polaronic states with a distinct spin-valley flavor that is defined by the LL quantized spin and valley texture.

Figure 1

(a) Schematic of the dual-gate monolayer ${\mathrm{MoS}}_2$ vdW device in Faraday geometry. A monolayer ${\mathrm{MoS}}_2$ is sandwiched between hBN and contacted with thin graphite (FLG) electrodes. (b) Carrier concentration-dependent low-temperature ($T=5\mathrm{K}$) $\mu$-PL at $B=-28\mathrm{T}$. Dashed lines show the region of charge neutrality ($n\sim 0{\mathrm{cm}}^{-2}$). Arrows for $B=-28\mathrm{T}$ highlight the $+0K$ and $+1K$ LLs in the $X^{-}$. (c) Spin and valley exciton band structure at the $K$ and $K^{\prime }$ points of monolayer ${\mathrm{MoS}}_2$ for restored time-reversal symmetry ($B=0\mathrm{T}$). (d) A finite magnetic field ($B\ne 0\mathrm{T}$) breaks time-reversal symmetry inducing an electron spin imbalance ($N_{\uparrow }\ne N_{\downarrow }$). For positive magnetic field, the dipolar-spin-interaction is of intravalley-type, while for negative magnetic field of intervalley-type. (e) Calculated LL fan diagram. (f) Calculated degree of spin-polarization ${\eta }_s$ showing the density and magnetic field-dependent spin and valley texture in the conduction bands.

Figure 2

(a) and (b) Normalized $\mu$-PL spectra of $X^0$ for applied magnetic fields at $n=0{\mathrm{cm}}^{-2}$ ($n=1.45\times {10}^{12}{\mathrm{cm}}^{-2}$) from $-28\mathrm{T}$ to $28\mathrm{T}$. The solid lines are a guide to the eye. (c) Carrier density-dependent valley Zeeman shift $\mathrm{\Delta }{E}_{VZ}$ of the $X^0$ in the electron and hole charged regime.

Figure 3

(a) Carrier density-dependent $g$ factors of $X^0$ (black and gray data), the negatively (red and blue data) and positively (green data) charged excitons $X^{-}$ and $X^{+}$, and the $X^{\prime -}$ feature (yellow data). (b) Carrier density dependence of the $X^{\prime -}$ binding energy $E_B$. For $n_2<n<n_3{E}_B$ is linear with $E_F$ while it becomes sub-linear for $n>n_3$. For the highest carrier densities (regime IV), the quasi-particle excitation is strongly interacting and dissolving into a many-body state $X_{MB.}^{\prime }$. (c) Electron concentration dependence of the degree of global spin-polarization ${\eta }_s$ for $B=-20\mathrm{T}$ and $20\mathrm{T}$.