Frequency-dependent substrate screening of excitons in atomically thin transition metal dichalcogenide semiconductors

Atomically thin layers of transition metal dichalcogenides (TMDCs) exhibit exceptionally strong Coulomb interaction between charge carriers due to the two-dimensional carrier confinement in connection with weak dielectric screening. The van der Waals nature of interlayer coupling makes it easy to integrate TMDC layers into heterostructures with different dielectric or metallic substrates. This allows to tailor electronic and optical properties of these materials, as Coulomb interaction inside atomically thin layers is very susceptible to screening by the environment. Here, we theoretically investigate dynamical screening effects in TMDCs due to bulk substrates doped with carriers over a large density range, thereby offering three-dimensional plasmons as a tunable degree of freedom. We report a wide compensation of renormalization effects leading to a spectrally more stable exciton than predicted for static substrate screening, even if plasmons and excitons are in resonance. We also find a nontrivial dependence of the single-particle band gap on substrate doping density due to dynamical screening. Our investigation provides microscopic insight into the mechanisms that allow for manipulations of TMDC excitons by means of arbitrary plasmonic environments on the nanoscale.

Figure 1

Schematic of the heterostructure composed of a single TMDC layer placed on a three-dimensional substrate with dielectric function ${\varepsilon }_{\mathbf{q}}^{\mathrm{s}}\left(\omega \right)$. The substrate is assumed to be electrically contacted to tune its dynamical dielectric properties.

Figure 2

(a) Heterostructure dielectric function in the static limit. (b) Resonance frequency ${\omega }_{1,\mathbf{q}}$ of the loss function (imaginary part of the inverse heterostructure dielectric function). Both quantities are given for ${\mathrm{MoS}}_2$ on a substrate with ${\varepsilon }_{\infty }=2$ and $m_{\mathrm{s}}=0.4m_0$ at different carrier-doping densities $n$.

Figure 3

Band-gap renormalization and the $1s$-exciton position relative to the unrenormalized band gap for ${\mathrm{MoS}}_2$ on a substrate at $T=300\mathrm{K}$ with ${\varepsilon }_{\infty }=2, m_{\mathrm{s}}=0.4m_0$, and quasiparticle broadening $\mathrm{\Gamma }=10\mathrm{meV}$, shown for different carrier-doping densities. The horizontal line shows the band-gap renormalization due to inner-shell electrons in the substrate and TMDC layer. Results using the full dynamical dielectric function and a static approximation are compared.

Figure 4

The $1s$-exciton position relative to the unrenormalized band gap for ${\mathrm{MoS}}_2$ on a substrate at different temperatures with ${\varepsilon }_{\infty }=2, m_{\mathrm{s}}=0.4m_0$, and quasiparticle broadening $\mathrm{\Gamma }=10\mathrm{meV}$, shown for different carrier-doping densities. The vertical dashed lines show the density where the loss function resonance at zero momentum becomes resonant with the exciton binding energy for all three temperatures. The resonance shifts to larger densities with increasing temperature. For comparison, the result obtained from the static approximation of the dielectric function is shown.

Figure 5

Band-gap renormalization and the $1s$-exciton position relative to the unrenormalized band gap for ${\mathrm{MoS}}_2$ on a substrate at $T=300\mathrm{K}$ with $m_{\mathrm{s}}=0.4m_0$ and quasiparticle broadening $\mathrm{\Gamma }=10\mathrm{meV}$, shown for different carrier-doping densities and dielectric constants ${\varepsilon }_{\infty }$.

Figure 6

Band-gap renormalization and the $1s$-exciton position relative to the unrenormalized band gap for ${\mathrm{MoS}}_2$ on a substrate at $T=300\mathrm{K}$ with ${\varepsilon }_{\infty }=2$ and quasiparticle broadening $\mathrm{\Gamma }=10\mathrm{meV}$, shown for different carrier-doping densities and effective masses $m_{\mathrm{s}}$.

Figure 7

Band-gap renormalization and the $1s$-exciton position relative to the unrenormalized band gap for ${\mathrm{MoS}}_2$ on a substrate at $T=300\mathrm{K}$ with ${\varepsilon }_{\infty }=2$ and $m_{\mathrm{s}}=0.4m_0$, shown for different carrier-doping densities and quasiparticle broadenings $\mathrm{\Gamma }$.

Figure 8

Band-gap renormalization and the $1s$-exciton position relative to the unrenormalized band gap for ${\mathrm{MoS}}_2$ on a substrate with ${\varepsilon }_{\infty }=2, m_{\mathrm{s}}=0.4m_0$, and $\mathrm{\Gamma }=10\mathrm{meV}$, shown for different carrier-doping densities and temperatures $T$.