Impact of dark excitons on Förster-type resonant energy transfer between dye molecules and atomically thin semiconductors

Interfaces of dye molecules and two-dimensional transition metal dichalcogenides (TMDCs) combine strong molecular dipole excitations with high carrier mobilities in semiconductors. Förster type energy transfer is one key mechanism for the coupling between both constituents. We report microscopic calculations of a spectrally resolved Förster induced transition rate from dye molecules to a TMDC layer. Our approach is based on microscopic Bloch equations which are solved self-consistently together with Maxwell's equations. This approach allows to incorporate the dielectric environment of a TMDC semiconductor, sandwiched between donor molecules and a substrate. Our analysis reveals transfer rates in the meV range for typical dye molecules in closely stacked structures, with a nontrivial dependence of the Förster rate on the molecular transition energy resulting from unique signatures of dark, momentum forbidden TMDC excitons.

Figure 1

(a) Schematic illustration: HIOS of a sheet of dye molecules above one or two layers of TMDC, (b) scheme of the Förster energy transfer from DCM molecules to the ${\mathrm{MoS}}_2$ TMDC in momentum space, and (c) sketch of the dielectric environment in real space.

Figure 2

Förster induced transition rate for the energy transfer from optically pumped molecular excitons into the TMDC monolayer, plotted (a) over the detuning $\mathrm{\Delta }$ between the transition energy of the donor and the 1s-resonance energy in the acceptor for different molecule-TMDC distances $R$. Clearly, the excitonic states are visible in the spectrally resolved FRET. The plateau between A and B resonance result from the momentum dark excitons above the bright 1s exciton, thus the Förster rate clearly does not resemble the absorption (grey line), which only displays the bright exciton resonances. (b) Rate plotted over the distance $R$ between donors (DCM molecules) and acceptor (${\mathrm{MoS}}_2$ monolayer), for selected values of detuning $\mathrm{\Delta }$. (c) Förster rate as a two parameter plot of $\mathrm{\Delta }$ and $R$. Dashed lines depict the respective positions of the 1d plots [(a) and (b)]. The inset shows the Maximum of the rate with respect to distance $R$ and detuning $\mathrm{\Delta }$. For parameters cf. Appendix pp8.

Figure 3

Comparison of Förster transition rates between TMDC monolayer (ML, dashed lines, from Fig. 2) and bilayer (BL, full lines) substrates, for two different distances $R=1$ and 2 nm. Although in the BL the number of acceptor states is doubled, the rate is significantly smaller than for the ML, due to strongly increased screening. Besides, the continuum is energetically closer to the s1 resonance, as the screening yields smaller exciton binding energies in both layers.

Figure 4

Calculated (a) reflection, (b) transmission, and (c) absorption of the DCM-${\mathrm{MoS}}_2$ stack, as a function of the detuning $\mathrm{\Delta }$ relative to the energetically lowest 1s A exciton transition energy for a sweep of four different molecular transition energies below and above the pristine semiconductor 1s A resonances (additionally depicted in gray). The detuning $\mathrm{\Delta }$ ranges from $-0.15$ to $0.15\mathrm{eV}$. The linear optical response of the underlying TMDC layer was subtracted for better visibility of the effect, and is depicted in gray. This is justified due to the small $\chi$, cf. Appendix pp7. For comparison, we illustrate with dashed lines the pristine molecular spectra without TMDC substrate.

Figure 5

Absorption for a Gaussian distribution of transition energies (pink) in the molecules, compared to the case for pristine molecules (dashed purple). As for the single molecular case, a slight overall redshift can be recognized, but clearly the effect of the Förster process above the excitonic resonances (gray) can hardly be seen. The Gaussian distribution of tranistion energies is depicted in dashed gray lines. It is evident that linear spectroscopy of molecular ensembles does not provide access to the spectrally resolved Förster rate.

Figure 6

Near resonance optical excitation. Only those excitons of the molecules with transition energies similar to the driving laser frequency are excited, which leads to luminescence only in a small spectral range, compare Fig. 7. After recombination, two scenarios are possible: a creation of a photon which is detectable as PL signal, or a Förster transfer to the TMDC bilayer, where the excitation will vanish to the momentum-indirect intervalley exciton state and thus not be visible in the PL detection.

Figure 7

(a) Photoluminescence (PL) of a sheet of molecules with a Gaussian distribution of transition energies ${\mathrm{\Omega }}_{\mathbf{q}}$ around the lowest 1s A resonance of the semiconductor. The laser energy ${\omega }_L$ is sweeped over the molecular resonances, with each Lorentzian peak referring to a different laser excitation energy, plotted in one picture but in different colors from red to blue, accordingly, see key on the right side of the plot. This plot is for small nonradiative dephasing ${\gamma }^{\mu \ell }=1\mathrm{meV}$, for better visibility of the different plots. The effect is however also clearly visible at room temperature. (b) Photoluminescence excitation (PLE) spectroscopy; for each laser excitation energy ${\omega }_L$, we show the PL signal integrated over the whole emission frequency range ${\mathrm{\Omega }}_{\mathbf{q}}$. The dashed line shows the PLE of the pristine molecules without the TMDC substrate. The linear response of the TMDC excitons is depicted in gray to show the connection of the effect to the energetic resonances in the semiconductor. Both plots make visible, that above the excitonic resonances, the additional decay channels cause dips in the luminescence, as less excitons can recombine radiatively. (c) Förster rate (with distance $R=1\mathrm{nm}$) and density of states (DOS) of molecular transition energies for comparison.

Figure 8

Sketch for the setup with far off-resonant driving. The laser is creating excitons in excited molecular levels. Nonradiative processes lead to a population of the LUMO. There, the recombination takes place, producing either a photon, which can be detected in the PL measurement, or exciting a semiconductur exciton in the TMDC via Förster coupling.

Figure 9

PL spectrum for off resonant driving for different values for the distance $R$ between donor (dye molecules) and acceptor (TMDC). Again the dashed line gives the PL of the pristine molecules without a TMDC substrate, and the excitonic resonances of the TMDC are again depicted in gray for better orientation. The result is very similar to the near-resonant case (Fig. 7). This suggests a good experimental accessibility of the Förster process via luminescence experiments, as the signatures proof robust towards different driving scenarios.

Figure 10

Finite angles between the molecular dipole and the TMDC plane for the example of a monolayer ${\mathrm{MoS}}_2$. The angle causes an overall increase of the Förster rate, but not a change in the momentum selection rules.

Figure 11

Analytical Förster rate as computed in the main part, Eq. (17), vs imaginary part of the numerical solution of the full equation [Eq. (16)] for different nonradiative dephasing rates ${\gamma }^{\mu \ell }$, related to temperatures up to room temperature [35].

Figure 12

Log-log plot of the Förster rate. In the limit of long distances, the rate in Eq. (16) gives a power-law dependence as suggested also by previous works. This accounts for detunings that only allow transfer into bound states as well as for transfer into the unbound exciton continuum.

Figure 13

The full absorption can in good approximation be divided into the part of the TMDC and the Molecular part, as it is done in Fig. 4. Dashed lines show the absorption of the molecule for different molecular transition energies, full lines indicate the respective full spectra, while the pristine TMDC absorption is depicted in gray.