Charged excitons in two-dimensional transition metal dichalcogenides: Semiclassical calculation of Berry curvature effects

We theoretically study the role of the Berry curvature on neutral and charged excitons in two-dimensional transition metal dichalcogenides. The Berry curvature arises due to a strong coupling between the conduction and valence bands in these materials that can to great extent be described within the model of massive Dirac fermions. The Berry curvature lifts the degeneracy of exciton states with opposite angular momentum. Using an electronic interaction that accounts for nonlocal screening effects, we find a Berry-curvature induced splitting of $\sim 17\mathrm{meV}$ between the $2p_{-}$ and $2p_{+}$ exciton states in ${\mathrm{WS}}_2$. Furthermore, we calculate the trion binding energies in ${\mathrm{WS}}_2$ and ${\mathrm{WSe}}_2$ for a large variety of screening lengths and different dielectric constants for the environment. Our approach indicates the prominent role played by the Berry curvature along with nonlocal electronic interactions in the understanding of the energy spectra of neutral and charged excitons in transition metal dichalcogenides and in the interpretation of their optical properties.

Figure 1

Color surface plot of the exciton binding energy in monolayer ${\mathrm{WS}}_2$, exposed to the air and deposed on thick dielectric substrate. Calculations are performed over a range of possible material screening lengths ${\lambda }_s$ and varying dielectric constant ${\varepsilon }_{\mathrm{sub}}$.

Figure 2

Theoretical excitonic spectrum reported for ${\mathrm{WS}}_2$ monolayer deposed on the ${\mathrm{SiO}}_2$ substrate ${\varepsilon }_{\mathrm{sub}}=2.1$ and exposed to the air ${\varepsilon }_{\mathrm{vac}}=1$ for the screening length ${\lambda }_s=28\AA$ and $a_B=5\AA$. The results are obtained by numerical diagonalization of the Hamiltonian given in Eq. (15), using the nonlocal screening potential and taking into account the Berry curvature correction. We show a mixing and splitting of $np$ and $nd$ states due to the Berry curvature.

Figure 3

Spectrum of a ${\mathrm{WS}}_2$ monolayer showing the effect of Berry curvature correction on exciton X and trion T energies. Inset shows the energies of low-lying trion states ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$, considering the ground exciton state ${\mathrm{E}}_{1s}$. The parameters used are ${\lambda }_s=28\AA , {\varepsilon }_{\mathrm{sub}}=2.1$, and $m_h=m_e=0.34m_0$.