Three-Particle Complexes in Two-Dimensional Semiconductors

We evaluate binding energies of trions $X^{\pm }$, excitons bound by a donor or acceptor charge $X^{D(A)}$, and overcharged acceptors or donors in two-dimensional atomic crystals by mapping the three-body problem in two dimensions onto one particle in a three-dimensional potential treatable by a purposely developed boundary-matching-matrix method. We find that in monolayers of transition metal dichalcogenides the dissociation energy of $X^{\pm }$ is typically much larger than that of localized exciton complexes, so that trions are more resilient to heating, despite the fact that their recombination line in optics is less redshifted from the exciton line than the line of $X^{D(A)}$.

Figure 1

Binding energies $\tilde{\epsilon }$ of charged complexes $X^{A(D)}$, $X^{\pm }$, and $A^{+}$ and $D^{-}$ for various electron-hole mass ratios ${\mu }_e/{\mu }_h<1$ (for ${\mathrm{MoS}}_2$ and ${\mathrm{MoSe}}_2$, ${\mu }_e/{\mu }_h\approx 0.7$; for ${\mathrm{WS}}_2$ and ${\mathrm{WSe}}_2$, ${\mu }_e/{\mu }_h\approx 0.6$ [11, 15, 46, 47, 48, 49]). For trions, the results obtained by the newly developed method (diamonds) are compared to the binding energies determined using the diffusion Monte Carlo technique (crosses). Sketch: sequence of luminescence lines in TMDC spectra, including charged complexes as well as ground and first radiative excited states of the free exciton.

Figure 2

Eigenvalues $h_{\alpha }(r)$ of $\widehat{H}(r)$ in Eq. (6) for ${\theta }_1={\theta }_2=(\pi /2)$ and $L_{\text{max}}=30$ [53]. For $r\ll 1$, $h_{\alpha }$ are bunched by the angular momenta $l$, whereas for $r\gg 1$, $a$ and $s$ doublets correspond to the particle localization in the minima ${\mathbf{n}}_{1,2}$ with vanishing tunneling (note that $a$-$s$ crossings are allowed). The red dashed line marks the boundary $X_0^0$ of the continuum spectrum for the exciton and a free particle, and at $r\gg 1$, $h(r)\approx X_0^0-c/r^2$, determined by the 2D van der Waals attraction between the charged particle and the neutral exciton, which produces an infinite number of shallow bound states. Inset: color scale image of the potential $U$ in Eq. (3).

Figure 3

Eigenvalues ${\lambda }_{\alpha }$ of matching matrix $\widehat{\mathrm{\Lambda }}(R)$ evaluated numerically for $\epsilon ={\epsilon }_{\pm }$ slightly above (blue, ${\epsilon }_{-}$) and slightly below (red, ${\epsilon }_{+}$) the bound state energy of a trion with ${\mu }_e=0.7{\mu }_h$. The energy-sensitive highest eigenvalue is compared to the asymptotic of a logarithmic derivative in Eq. (9d) (green) calculated at the converged binding energy.