Exciton states and absorption spectra in freestanding monolayer transition metal dichalcogenides: A variationally optimized diagonalization method

The comprehensive analyses of exciton states and absorption spectra from freestanding monolayer transition metal dichalcogenides such as ${\mathrm{MoS}}_2, {\mathrm{MoSe}}_2, {\mathrm{MoTe}}_2, {\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$ are theoretically presented in the Wannier−Mott exciton theory. We propose an efficient and simple exact diagonalization method based on a variationally optimized two-dimensional (2D) modified Laguerre basis set, which has been successfully applied to calculate both the ground and excited states. The convergence of the method as a function of the basis set size was analyzed in detail. The modified Laguerre basis functions were very closely concentrated in space, and they led to a rather rapid convergence. Moreover, the convergence rate was faster under variational optimization. Our results regarding the exciton binding energies of a ground state with even a small number of basis states are in excellent agreement with those of the density-function theory calculations in the literature [I. Kylänpää and H.-P. Komsa, Phys. Rev. B 92, 205418 (2015)]. The exciton states appeared in an anomalous energy-level ordering of the orbital angular momentum caused by the dielectric screening effect. The exciton absorption spectra exhibited strong absorption lines of the nonhydrogenic exciton Rydberg series, which was dominated by the s-type exciton states. There were two prominent low-energy peaks of the A and B excitons, and the A exciton was slightly higher in intensity than the B exciton, which is in good agreement with the experimental observations. In general, our presented exact diagonalization method based on a variationally optimized 2D modified Laguerre basis set has proven to be applicable to more general cases of screened potentials in 2D materials and atomic physics, as well as providing a simple approach for investigating exciton properties.

Figure 1

The screened Coulomb potential $V\left(r\right)$ as a function of $r$ in freestanding monolayer TMDCs with different screening lengths $r_0$. The cases of $r_0=0$ and $r_0\to \infty$ are shown for comparison.

Figure 2

(a) Relative error of the exciton energy with respect to the reference values ($N=22$) and (b) the variational parameters $\varsigma$ vs the modified Laguerre basis-set size $N$.

Figure 3

Exciton binding energy $E_b$ of the A 1s exciton versus (a) the screening length $r_0$ and (b) the exciton reduced mass $\mu$ in freestanding monolayer TMDCs.

Figure 4

Normalized radial wave functions $R_{nl}\left(r\right)$ of the A exciton states vs $r$ in a freestanding monolayer ${\mathrm{MoS}}_2$.

Figure 5

Exciton absorption spectra $\alpha \left(\omega \right)$ of a circularly polarized light as a function of the photon energy for various freestanding monolayer TMDCs.