Optical emission from light-like and particle-like excitons in monolayer transition metal dichalcogenides

Several monolayer transition metal dichalcogenides (TMDs) are direct band gap semiconductors and potentially efficient emitters in light emitting devices. Photons are emitted when strongly bound excitons decay radiatively, and accurate models of such excitons are important for a full understanding of the emission. Importantly, photons are emitted in directions uniquely determined by the exciton center of mass momentum and with lifetimes determined by the exciton transition matrix element. The exciton band structures of two-dimensional hexagonal materials, including TMDs, are highly unusual with coexisting particle- and light-like bands. The latter is nonanalytic with emission selection rules essentially opposite to the particle-like states, but has been ignored in analyses of TMD light emission so far. In the present work, we analyze the temperature and angular dependence of light emission from both exciton species and point out several important consequences of the unique exciton band structure. Within a first-principles density-functional-theory$+$Bethe-Salpeter-equation framework, we compute exciton band structures and optical matrix elements for the important TMDs ${\mathrm{MoS}}_2$, $\mathrm{Mo}{\mathrm{Se}}_2$, ${\mathrm{WS}}_2$, and $\mathrm{W}{\mathrm{Se}}_2$. At low temperature, only the particle-like band is populated and our results agree with previous work. However, at slightly elevated temperatures, a significant population of the light-like band leads to modified angular emission patterns and lifetimes. Clear experimental fingerprints are predicted and explained by a simple four-state model incorporating spin-orbit as well as intervalley exchange coupling.

Figure 1

Schematic of photon emission by ensembles of light-like (blue) and particle-like (red) excitons. Photon and exciton momenta are shown as wiggly and straight arrows, respectively. At low temperature, particle-like excitons dominate, whereas nearly identical populations are expected at high temperature. Insets to the right show their energy dispersions relative to the $Q=0$ exciton energy.

Figure 2

Exciton band structure for the two lowest bright excitons vs $\mathrm{Q}$ with exciton transition matrix elements $\mathrm{P}$ shown by arrows. Horizontal and vertical arrows indicate matrix elements parallel and perpendicular to $\mathrm{Q}$, respectively. Bands as a function of $\mathrm{Q}$ in the entire irreducible Brillouin zone for ${\mathrm{MoS}}_2$ (a) and (b). Particle-like (red) and light-like (blue) bands in ${\mathrm{MoS}}_2$, $\mathrm{Mo}{\mathrm{Se}}_2$, ${\mathrm{WS}}_2$, $\mathrm{W}{\mathrm{Se}}_2$, and two examples of screened ${\mathrm{MoS}}_2$ as a function of $\mathrm{Q}$ along the $\mathrm{\Gamma }\to M$ high symmetry line (c). The dielectric constant $\varepsilon$ describes either the screening from a ${\mathrm{SiO}}_2$ substrate ($\varepsilon =1.55$) or from encapsulation in hBN ($\varepsilon =4.5$).

Figure 3

Imaginary part of the susceptibility calculated from the transition matrix elements (blue) compared to experiment by Hsu et al. (red) [28] for TMDs on ${\mathrm{SiO}}_2$ ($\varepsilon =1.55$). An energy dependent broadening of $\hslash \mathrm{\Gamma }\propto \hslash \omega -E_A$ is applied, with the energy of the first bright exciton $E_A$ indicated by dotted lines. The parameter $d$ is the material thickness used to transform between computed 2D and measured 3D results.

Figure 4

(a) Calculated radiative lifetimes for four TMDs as a function of temperature. (b) Relative difference between lifetimes obtained from the particle-like band alone ${\tau }_p$ and the full calculation $\tau$. (c) Normalized emission $\gamma (T,\theta )/\gamma (300\text{K},\theta )$ at temperatures $T=3\mathrm{K}$ and $T=30\mathrm{K}$ as a function of the emission angle $\theta$. (d) Emission rate $\gamma (T,\theta )$ as a function of $\theta$ for low and high temperature. Here, the characteristic energy of light-like excitons $\hslash V\omega /c$ is 0.973 meV and 1.111 meV for ${\mathrm{MoS}}_2$ and $\mathrm{W}{\mathrm{Se}}_2$, respectively.

Figure 5

(a) Calculated lifetimes for screened and unscreened ${\mathrm{MoS}}_2$ as a function of temperature. (b) Normalized emission $\gamma (T,\theta )/\gamma (300\text{K},\theta )$ at $T=3\mathrm{K}$ and $T=30\mathrm{K}$ as a function of the emission angle $\theta$. The characteristic energy of light-like excitons in hBN encapsulated ${\mathrm{MoS}}_2$ with $\varepsilon =4.5$ is $\hslash V\omega \sqrt{\varepsilon }/c=0.661\mathrm{meV}$.

Figure 6

Schematic of the hybridization between $K$ (blue) and $K^{\prime }$ (red) excitons as a result of exchange for a center of mass momentum $\mathrm{Q}=Q\widehat{\mathrm{x}}$. Thick and thin dashed lines indicate large and small contributions, respectively, to the light-like (purple) and particle-like (pink) hybrids. The lower part shows exciton transition matrix elements for the light- and particle-like states oriented parallel and perpendicular to the center of mass momentum $\mathrm{Q}$, respectively. The quantity $2{\mathrm{\Delta }}_{SO}^{*}$ is the effective SO split between the two states given as ${\mathrm{\Delta }}_{SO}^{*}=\sqrt{{\mathrm{\Delta }}_{SO}^2+V_0^2}$.

Figure 7

Comparison of exciton dispersions in ${\mathrm{MoS}}_2$ in the analytical model (solid lines) and the full BSE solution (dots). Both light- and particle-like bands are shown and the range of the light cone $Q=\omega /c$ is indicated by vertical dotted lines.