Essential role of momentum-forbidden dark exciton in the energy transfer responses of monolayer transition-metal-dichalcogenides

We present a theoretical investigation of exciton-mediated Förster resonant energy transfers (FRET’s) from photoexcited quantum dots (QD’s) to transition-metal dichalcogenide monolayers (TMD-ML’s), implemented by the quantum theory of FRET on the base of ﬁrst-principles-calculated exciton ﬁne structures. With the enhanced electron-hole Coulomb interactions, atomically thin TMD-ML’s are shown to serve as an exceptional platform for FRET that are mediated purely by excitons and take full advantage of the superior excitonic properties. Remarkably, the energy-transfer responses of atomically thin TMD-ML’s are shown to be dictated by the momentum-forbidden dark excitons rather than the commonly recognized bright ones. Speciﬁcally, the longitudinal dark exciton states following the exchange-driven light-like linear band


Introduction
Förster resonant energy transfer (FRET) is a non-radiative electromagnetic process that, via non-contact near field coupling, enables the transfer of excitation energy from a photoexcited object (donor) to another unexcited one (acceptor) in a close proximity (typically apart by a distance of few to tens of nm). [1][2][3][4][5] Since FRET is not a radiative process mediated by real photons, the rate of FRET is not restricted by the slow spontaneous radiative decay of donor and finite optical cross section of acceptor. Inspired by efficient excitonic energy transfer in botanical photosynthesis revealed by early studies, 6-10 the FRET mediated with exciton is considered as an alternative route for grading up solid-state photo-voltaic devices by taking advantage of superior oscillator strength of exciton to enhance the light-generation and -harvesting. 11-17 Nowadays, numerous researches have been persistently conducted to investigate various emergent opto-electronic materials that are potentially advantageous for the realization of exciton-based FRET.  Atomically thin transition-metal dichalcogenides (TMD's) are known as excellent optoelectronic materials, in possession of ultra-strong light-matter interaction and pronounced excitonic properties. [40][41][42][43][44][45][46][47][48][49] Due to weak Coulomb screening in low-dimensionality, the interparticle Coulomb interactions in the 2D materials are greatly enhanced and the photo-excited electron-hole (e-h) pairs in TMD-ML's form tightly bound excitons with giant binding energy over hundreds of meV. [50][51][52][53] In fact, TMD monolayers (TMD-ML's) serve as an exceptional and suitable platform for studying exciton-mediated FRET's because the giant binding energy (≳ 10 2 meV) of exciton well separates the low-lying exciton states spectrally apart from the free e-h pair continuum spectra. Hence, over the wide range of resonance energy (> 10 2 meV) the FRET from photo-excited donors to TMD-acceptors can be mediated only by excitons to take the full advantage of the superior excitonic properties of TMD-ML.
The latest experiments have evidenced the FRET's of TMD-ML's to be extraordinarily fast and show the near unity transfer efficiencies 18,25,26,[54][55][56] that far exceeds the limited FRET efficiencies of ordinary bulk materials or mesoscopic systems. 57,58 Interestingly, the observed ultra-fast FRET of TMD-ML's cannot be accounted for only in terms of bright excitons (BX's) but speculated to be related to optically inactive dark excitons (DX's). 56 However, theoretically, the physics of dark exciton in the FRET's of TMD-ML's or generic low-dimensional systems remain rarely investigated so far. 39 In this work, we study the energy transfer from donors of nanocrystal quantum dots (QD's) to an acceptor of MoS 2 monolayer (MoS 2 -ML), one of the best known TMD-ML's, as schematically illustrated in Fig.1(c). According to the violated optical selection rules, optically inactive excitons are classified as so-called spin-forbidden (SFDX) 59-64 and momentumforbidden dark exciton (MFDX). 63,[65][66][67][68] We show that the intra-valley MFDX's play an essential role in the resonant energy transfer processes of TMD-ML's. In fact, the number of the MFDX states over the entire reciprocal space is tremendously large, much more than that of bright ones that count only the states in the narrow reciprocal area of light-cone where Q < q c = 2π λ ≈ 10µm −1 for MoS 2 -ML. 69,70 Specifically, the longitudinal MFDX states of the exchange-resulting light-like linear exciton bands of TMD-ML's play the key role to grade up the efficiency and robustness of FRET against the inhomogeneity of QD-donor ensembles. Taking into account the pronounced excitonic effects, the donor-acceptor-distance dependence (d) of the dark-exciton-dictated FRET of TMD-ML's is found not to follow the classically predicted d −4 -power-law and, beyond the broad expectation, actually cannot reflect the dimensionality of FRET system.

Valley-characteristic band structures of TMD-ML's
The band structure of transition-metal-dichalcogenide monolayer (TMD-ML) is characterized by two distinctive valleys located at the K and K ′ corner points of the first Brillouin zone where the direct band gaps in the visible light range are opened. 40,71 As the time-reversal counterparts, the optical transitions between the conduction and valence band extrema in the K-and K ′ -valleys follow the opposite optical helicity, allowing for the valley-selective photo-excitation by using polarized lights. [72][73][74] The first-principles-calculated spin and valleycharacteristic band structure of a MoS 2 -ML is shown in Fig.S1 of Supporting Information (SI) or can be found in many published papers of literature, e.g. Refs. [ 71,[75][76][77][78][79]. Throughout this work, we focus on the spin-allowed transitions of a MoS 2 -ML from the topmost valence band to the spin-like conduction band in the same valley (K or K ′ ), i.e. the spin-allowed intra-valley A exciton, 53,80-83 which satisfies the spin selection rule for FRET.

Valley-split exciton band structures of TMD-ML's
Due to enhanced Coulomb interaction in 2D materials, photo-excited electron-hole pairs in a TMD-ML form tightly bound excitons featured with extraordinarily high binding energy over hundreds of meV and giant transition dipoles. As compared with the excitons in bulks or mesoscopic quantum wells, the binding energy of exciton in a TMD-ML is two orders of magnitude larger, and the resulting Bohr radius (merely 1-2nm) of exciton is one order of magnitude smaller. 84,85 Such a small Bohr radius of exciton in a TMD-ML leads to the giant dipole moment of exciton and superiorly significant light-matter interaction. 41,44,46,86 Moreover, the enhanced electron-hole Coulomb interactions in valley-characteristic TMD-ML's gives rise also to efficient inter-valley excitonic couplings that, via the intrinsic eh exchange interaction, intermix the spin-allowed exciton states of distinct valleys. 63,[87][88][89]   (c) and (d): Schematics of the Förster energy transfer systems composed of (c) a semiconductor quantum dot (QD) and an atomically thin TMD-ML, and (d) a QD and a thick mesoscopic quantum well. In the case of (c), the FRET is fully exciton mediated because of large binding energy (E X b > 10 2 meV) of exciton that well separates the spectra of exciton and non-interacting free e-h pairs. By contrast, in the case of (d), the excitation energy of the QD yet likely falls in the energy regime of continuum spectrum of e-h pairs since the binding energy of exciton in a mesoscopic quantum well is typically only few meV. 5 momentum Q in the K-valley to that in the K ′ -valley. Figure 1 Based on the quasi-particle band structure, an exciton state of TMD-ML in the exciton band S with the center-of-mass (CoM) momentum Q can be written as the superposition of numerous e-h configurations with the momentum Q, which reads are the creation (annihilation) operator of electron of energy c,k+Q in conduction band c carrying quasi-particle wavevector k + Q and hole of energy − v,k in valence band v carrying quasi-particle wavevector −k for TMD-ML, respectively, |GS〉 denotes the ground state, A vc S,Q (k) is the amplitude of the e-h configuration c † c,k+Qĥ † v,−k |GS〉, and Ω A is the total area of the TMD-ML sample. Numerically, the eigen energy, E X S,Q , of an exciton state ψ X S,Q can be calculated by solving the Bethe-Salpeter equation (BSE), 63,89-94 which reads where the kernel of e-h Coulomb interaction Throughout this work, we consider a free-standing MoS 2 -ML, whose non-local dielectric function is evaluated by solving the Poisson's equation. 95 shows the Q-dependent dipole moment of the upper-band [lower-band] BX states over the momentum space. Note that the exciton dipole of light-like (particle-like) band D X +(−),Q is longitudinal (transverse) with respect to wave vector Q.
2M X and the upper linear band is where E  87 and a X B ≡ (4πε 0 /e 2 ) · 21m 0 is the reduced mass of exciton, ε TMD = ε MoS 2 = 4.6 is the effective dielectric constant of MoS 2 -ML. 52,70,102,103 Fitting to the BSE-calculated exciton dispersion, γ = 101meV · nm is determined, in consistency with the exciton band splitting reported by Refs. [ 69,70,88,104]. Accordingly, the transition dipole moment of exciton for MoS 2 -ML is estimated to be 149(e) and the magnitude of the dipole moment of single particle transition is also estimated to be d 0 = 0.108(e · nm). With the small a X B , the value of γ for the tightly bound BX of TMD-ML is so large such that Eq. (3),

Excton-mediated energy transfer from QD's to TMD-ML's
According to Fermi's golden rule, the rate of the FRET from a QD with the excitation energy, ε X 0 , to a TMD-ML is evaluated by where the delta function ensures the energy conservation and the matrix element of the Förster coupling from the lowest exciton of QD reads where O ′ = (0, 0, d) is the central position of QD measured from the origin point, O = (0, 0, 0), in the plane of TMD-ML, is the pair-density of exciton transition in the TMD-ML, and ε b is the effective dielectric constant of dielectric environment. Here, we neglect the weak Coulomb correlation for strongly confining QD's and consider the exciton wave function of QD as the direct product of φ v * 0 (r) and φ c 0 (r). Taking the electrical dipole approximation (See SI for derivation details), we can show that where is the Q-component of the Föster field induced by the dipole moment of the QD, D X QD ≡ 〈φ v 0 |er|φ c 0 〉. Throughout this work, we consider Al 2 O 3 as the spacer material between CdSe QD's with |D X QD | = 0.44 e · nm and a MoS 2 -ML and take ε b = ε Al 2 O 3 = 9.2 (See Table 1). According to Eqs.(6) and (7), the matrix element of Föster coupling, M S,Q , as a function of Q is maximum as Without losing the generality, we consider D X QD = D X QDx for elongated CdSe QD's throughout this work. Figure 3(a) presents the numerically calculated vectorial Q-components of the Föster field, ε Q , over the Q-space. As expected, one sees that the maximal Q-

Excitation-energy dependence of dark-exciton-dictated FRET
Substituting the M S,Q of Eq. (6) into Eq.(4) and taking into account all the coupled longitudinal exciton states, one can formulate the total rate of FRET as a function of excitation energy, ε X 0 , as where the averaged Föster coupling for the exciton states with the energy E X +,Q = ε X 0 is defined by and derived as where is the magnitude of the wave vectors of the resonant exciton states to the excitation energy of the QD. Note that the maximum Förster coupling critically occurs as Q R = Q 0 , leading to where ∆ε X 0 ≡ ε X 0 − E X 0 measures the energy difference between the excitation energy of QD and the lowest exction energy, E X 0 . This indicates that the size of QD that determines ε X 0 and the distance between QD and TMD-ML, d, act as two tuning knobs related by γ for optimizing the energy transfer rate.
For tightly bound exciton with a B ∼ 1−2nm in a MoS 2 -ML, the value of γ = 100meV · nm is significantly large and, to achieve the maximal FRET, the excitation energy of QD should be properly higher than the BX states to match the energy of the longitudinal exciton states with Q = Q 0 . In Fig.2(a), we see that ∆E X Q 0 ,0 ≡ E X +,Q 0 − E X 0 ≈ 25meV (≈ 22meV) for the QD-TMD system with d = 8nm according to the model-predicted (BSEcalculated) exciton band structure of MoS 2 -ML. One also find in Fig.3(b) that M +,Q persist high for the finite momentum exciton states over a wide Q-regime where Q ∈ {Q 0 /2, 2Q 0 }, corresponding the energy range, ∆E X , according to the model-predicted (BSE-calculated) upper exciton band. This indicates, due to the large γ of TMD-ML, the robustness of efficient FRET's from QD's to TMD-ML's against the variation of the excitation energies of QD's over a wide range of ∆E X 2Q 0 ,Q 0 /2 . By contrast, in mesoscopic quasi-2D systems, e.g. epitaxial semiconductor quantum wells, excitons are weakly bound and possess the larger Bohr radius a B ∼ 10nm, 106 leading to much smaller γ ∼ few meV·nm. 107 With the nearly vanishing γ (leading to small ∆ε X 0 ), both BX bands of a quantum well remain weakly dispersive, and the excitation energies of QD's must be critically little higher than that of BX to maximize the FRET rate, i.e. ∆ε X 0 ≳ 0.  Fig.4(a)). The large Bohr radius of weakly bound exciton suppresses the e-h exchange interaction and retains the degenerate weakly dispersive exciton bands. As one sees in Fig.4(a), the rate of FRET between the QD and the quantum well with the same distance One notes that the rates of the FRET of the TMD-ML remain significantly high over a wide range of excitation energy, from tens of meV to over 60meV, in the regime of MFDX. Black dashed curve: The model-predicted rate of exciton-mediated FRET between a QD and a mesoscopic quantum well with the same d = 8nm, where exciton in the quantum well is weakly bound with negligible e-h exchange interaction. (b) Blue (red dashed curve) curve: The BSE-calculated (model-predicted) averaged Föster coupling for the all resonantly coupled exciton states,ξ + , as a function of ∆ε X 0 . The maximum Föster coupling occurs as Q 0 = 1/d, corresponding to the BSE-calculated (model-predicted) energy of the exciton states, ∆E X 2Q 0 ,Q 0 /2 ≡ E X +,2Q 0 − E X +,Q 0 /2 = 22meV (38meV).(c) Blue (red dashed) curve: The BSE-calculated (model-predicted) density of states (DOS) of exciton upper band ρ + (∆ε X 0 ).
15 the excitation energy of QD critically falls in the narrow energy range, ∆ε X 0 ∼ 0 − 10meV, slightly above the exciton band edge of QW (See the black curve in Fig.4(a)). Once the excitation energy of QD is out of the narrow energy range (∆ε X 0 > 10meV), the FRET rate from a QD to a mesoscopic QW drops to almost vanishing.
Distance-dependence of dark-exciton-dictated FRET From Eq.(10), it is shown that the calculated rate of the exciton-mediated FRET from a QD to a TMD-ML by the quantum theory yields an exponential dependence on the QD-TMD distance, Γ DA ∝ e −2Q R d , rather than the d −4 -power-law (Γ DA ∝ d −4 ) as predicted by the classical FRET model for 0D-2D systems. 36,[108][109][110][111][112][113][114][115] This is because, in the quantum regime, the CoM motion of an exciton in a 2D system is described by an spatially extended planewave function, instead of a localized point dipole as considered in the classical model. 111,116 Practically, the wave aspect of exciton in the ideal quantum regime might be suppressed by introducing a variation of exciton momentum to the 2D exciton system. This situation occurs, e.g., in the FRET from an inhomogeneous QD-layer where the unequal QD's with various excitation energies simultaneously couple the different exciton states of TMD-ML with various CoM momenta. To confirm the scenario, we consider an inhomogeneous QDlayer where the excitation energies of QD's are considered to follow a Gaussian distribution over energy, f (ε X 0 ;ε X 0 , σ ε ) = 1/ 2πσ 2 ε exp − ε X 0 −ε X 0 2 /2σ 2 ε , whereε X 0 is the average excitation of QD-ensemble and σ ε is the parameter of energy deviation defined by For simplicity, we consider that each QD's in the ensemble have the same distance to the TMD-ML and neglect the energy transfer between QD's. Accordingly, the average rate of energy transfer from a photo-excited inhomogeneous QD-layer to a TMD-ML  Figure 5: Distance dependence of the FRET to a MoS 2 -ML from an inhomogeneous QD layer whose excitation energy distribution is modeled by the Gaussian function parametrized by the average excitation energy,ε X 0 , and energy variation, σ ε . (a) Red curve: the rate of FRET between a QD and a MoS 2 -ML with d = 8nm, as a function of excitation energy of QD. Blue curve: the Gaussian distribution of excitation energy for an inhomogeneous QD layer with the average energy coincident with the energy of the maximum FRET rate,ε X 0 − E X 0 = 55meV, and the energy variation, σ ε = 20meV. (b) [(c)] Blue curves: The linear [logarithm] plot of the QD-TMD distance dependence of the rate of the FRET between a MoS 2 -ML and an inhomogeneous QD layer with σ ε = 20meV. For comparison, the distance dependences of the FRET for the same QD-TMD system but with σ ε = 0meV and σ ε ≫ 100meV are presented by the magenta and green dashed lines, respectively. (d)-(f) consider the FRET between a TMD ML and a QD-layer system with the lower average excitation energy,ε X 0 −E X 0 = 25meV. (g)-(i) consider the FRET between a TMD ML and a QD-layer system with the higher average excitation energy,ε X 0 − E X 0 = 100meV.
in Fig.5(b) and (c), respectively. As inferred from Eq.(9), the calculated rate of FRET from the homogeneous QD-layer, similar to a single QD, does show a multi-exponential decay with increasing d,Γ DA ∝ e −Q R d , where Q R is the wave vector of the resonantly coupled exciton states, not following the classically predicted distance power law (Γ DA ∝ d −4 ).
Next, we consider inhomogeneous QD-layers with the sameε X 0 − E X 0 = 55meV but finite energy variations, σ ε ∕ = 0. As a model test, we first set a extremely large energy variation, σ ε ≫ 100meV, to mimic and simulate the classical FRET. As expected, we indeed retain the d-dependence ofΓ DA that perfectly follows the classically predicted power-law, 36,[108][109][110][111][112][113][114][115] for the FRET from the extremely inhomogeneous QD-layer. For realistic simulation, one might take the moderate energy variation like σ ε = 20meV, with which the distance dependence ofΓ DA yet shows a multi-power law behavior as shown by blue solid curves in Fig.5(c). Carefully examining Fig.5(c), one finds that the rate of the FRET between a TMD-ML and QD's with σ ε = 20meV exhibits the weaker (higher) d-dependence than the d −4 -power law in the short (long) distance regime where d < 10nm (d > 10nm). From our studies, the d-dependence of dark-exciton-mediated FRET of a TMD-ML is realized not to certainly follow the d −4 -power-law as classically predicted but show a multiexponential decay that is substantially affected by extrinsic factors such as the inhomogeneity of materials. This accounts for the often-seen ambiguity of the d-power-law-fitting of the measured FRET rates reported by many existing experiments. 18,21,27,29,33,36,38

Conclusion
In summary, we present a theoretical investigation of exciton-mediated Förster resonant energy transfers from photoexcited quantum dots to monolayer TMD's. In contrast to conventional semiconductor bulk or mesoscopic nano-materials, excitons in atomically thin TMD-ML's are tightly bound by the extraordinarily strong Coulomb interaction, leading to large transition dipole moments and unusual exchange-induced exciton band dispersions.
The large dipole moment of exciton directly enhances the FRET rate of a TMD-ML to be faster than the FRET of mesoscopic quantum wells by several times. Moreover, our studies point out the FRET of a TMD-ML is dictated by momentum-forbidden dark excitons, rather than the commonly expected bright excitons. The momentum-forbidden dark exciton states of TMD-ML that follow the unusual linear exciton band dispersion are shown to substantially grade up the efficiency and robustness of FRET of TMD-ML against the inhomogeneity of donor ensembles. We also find that the dark-exciton-dictated FRET's of TMD-ML's no longer follow the classically predicted d −4 -power-law dependence on the donor-acceptor-distance. Hence, the measured distance power-law dependences of the FRET's of TMD-ML's actually cannot reflect the dimensionality of donor-acceptor system. Our studies reveal the essential role of momentum-forbidden dark exciton in the FRET of generic 2D materials and suggest the prospect of dark-exciton-based near-field technology using 2D materials.

Data Availability
The data supporting this study's findings are available from the corresponding author upon reasonable request.