Ab initio and semiempirical modeling of excitons and trions in monolayer ${\mathrm{TiS}}_3$

We explore the electronic and the optical properties of monolayer ${\mathrm{TiS}}_3$, which shows in-plane anisotropy and is composed of a chain-like structure along one of the lattice directions. Together with its robust direct band gap, which changes very slightly with stacking order and with the thickness of the sample, the anisotropic physical properties of ${\mathrm{TiS}}_3$ make the material very attractive for various device applications. In this study, we present a detailed investigation on the effect of the crystal anisotropy on the excitons and the trions of the ${\mathrm{TiS}}_3$ monolayer. We use many-body perturbation theory to calculate the absorption spectrum of anisotropic ${\mathrm{TiS}}_3$ monolayer by solving the Bethe-Salpeter equation. In parallel, we implement and use a Wannier-Mott model for the excitons that takes into account the anisotropic effective masses and Coulomb screening, which are obtained from ab initio calculations. This model is then extended for the investigation of trion states of monolayer ${\mathrm{TiS}}_3$. Our calculations indicate that the absorption spectrum of monolayer ${\mathrm{TiS}}_3$ drastically depends on the polarization of the incoming light, which excites different excitons with distinct binding energies. In addition, the binding energies of positively and the negatively charged trions are observed to be distinct and they exhibit an anisotropic probability density distribution.

Figure 1

The optimized geometric structure of monolayer ${\mathrm{TiS}}_3$ [(a) and (b)]; the dark and the light blue balls represent the S atoms and the red ones represent the Ti atoms. The S atoms on different surfaces are colored differently for clarity, and the numbers in the figure are the bond lengths. The quasiparticle band structure of ${\mathrm{TiS}}_3$ monolayer (c); the Brillouin zone (BZ) is shown in the inset of the figure. The in-plane $({\mathbf{a}}_1$ and ${\mathbf{a}}_2)$ and out-of-plane $\left({\mathbf{a}}_3\right)$ lattice vectors are also shown in the figure.

Figure 2

Calculated absorption spectra of ${\mathrm{TiS}}_3$ monolayer for different light polarization directions (direction of the electric field vector of light), in-plane $({\mathbf{a}}_1$ and ${\mathbf{a}}_2)$ and out-of-plane $\left({\mathbf{a}}_3\right)$, and the contributed states to the particular excitonic peak.

Figure 3

Energy spectra of $X_2$ (red) and $X_3$ (blue) exciton states in monolayer ${\mathrm{TiS}}_3$ in (a) the suspended case and (b) deposited over a ${\mathrm{SiO}}_2$ substrate, along with the symmetry corresponding to each state, as calculated by the effective mass approach.

Figure 4

Binding energies of neutral $\left(X_i\right)$, positively $(T_i^{+}$, open symbols), and negatively $(T_i^{-}$, full symbols) charged excitons $\left(i=2,3\right)$ in monolayer ${\mathrm{TiS}}_3$ as a function of the relative permittivity of the substrate medium.

Figure 5

Correlation function $g_{e\text{−}h}$ between the first electron and hole for (a,c) positively and (b,d) negatively charged trions. The upper (lower) row is for $T_{2\left(3\right)}^{\pm }$ trions.

Figure 6

Correlation function $g_{c\text{−}c.m.}$ between the third charge and the exciton center of mass for (a,c) positively and (b,d) negatively charged trions. The upper (lower) row is for $T_{2\left(3\right)}^{\pm }$ trions.