{'fraction_non_alphanumeric': 0.03624465139692927, 'fraction_numerical': 0.01677993120228207, 'mean_word_length': 5.115956900974859, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Enhanced Excitation and Emission from 2D Transition Metal Dichalcogenides with All-Dielectric Nanoantennas

The recently emerged concept of all-dielectric nanophotonics based on optical Mie resonances in high-index dielectric nanoparticles has proven a promising pathway to boost light-matter interactions at the nanoscale. In this work, we discuss the opportunities enabled by the interaction of dielectric nanoresonators with 2D transition metal dichalcogenides (2D TMDCs), leading to weak and strong coupling regimes. We perform a comprehensive analysis of bright exciton photoluminescence (PL) enhancement from various 2D TMDCs, including WS2, MoS2, WSe2, and MoSe2 via their coupling to Mie resonances of a silicon nanoparticle. For each case, we find the system parameters corresponding to maximal PL enhancement taking into account excitation rate, Purcell factor, and radiation efficiency. We demonstrate numerically that all-dielectric Si nanoantennas can significantly enhance the PL intensity from 2D TMDC by a factor of hundred through precise optimization of the geometrical and material parameters. Our results may be useful for high-efficiency 2D TMDC-based optoelectronic, nanophotonic, and quantum optical devices.


Introduction
Single-layer transition metal dichalcogenides (TMDCs) described by the general chemical formula MX2 (M = Mo, W; X = S, Se) are two-dimensional (2D) semiconductors with energy bandgaps spannng the visible and near-infrared spectral region (1.0-2.1 eV), and thus raising great interest for photonics and optoelectronics applications [1][2][3][4][5][6][7][8][9]. They demonstrate enhancement of photoluminescence (PL) with decreasing number of layers, down to the atomic monolayer limit, which has been attributed to the transition from indirect to direct bandgap emission [9,10]. Because of weak dielectric screening, the optical properties of 2D TMDCs are governed by highly bound excitons, hydrogen-like excitation states formed by a negatively charged electron and a positively charged hole coupled through Coulomb attraction [3]. These excitations occur at the energy-degenerate K and K' endpoints of the hexagonal Brillouin zone and are characterized by large binding energies. For example, the binding energies of so-called bright (XA) excitons in 2D MoS2 [11], MoSe2 [12], WSe2 and WS2 [13,14] are in the range of 0.2 to 0.8 eV, well above the thermal energy (~50 meV). In addition, due to inversion symmetry breaking and strong spin-orbital coupling in 2D TMDCs, the electronic states of the two valleys have different chirality, which leads to valley-selective circular dichroism and thus can enable valleytronic applications [4,[15][16][17]. A variety of optical phenomena stem from the interaction of 2D TMDC with plasmonic (i.e., made of noble metals) nanoscale objects or nanoantennas, which have become a topic of extensive studies [9]. It has been demonstrated that the coupling of 2D TMDCs with plasmonic nanoantennas can lead to quantum yield enhancement of emission via Purcell effect [9,[18][19][20], formation of excitonpolaritons in strong coupling regime [9,[21][22][23], pronounced Fano resonances and induced optical transparency [23], and valley-selective response at room temperature [24]. However, all these effects suffer from high dissipative losses, inherent for metals at optical frequencies [25][26][27], which cause Joule heating of the structure. Since the response of excitonic systems is dramatically modified by temperature changes, heat generation is detrimental for 2D TMDCs. These losses are caused by the large free-carrier concentration ≈ 10 23 cm -3 in plasmonic materials. Moreover, hot-electron injection from plasmonic nanostructures to TMDCs may lead to phase change and unwanted doping of TMDCs, which are not desirable in many applications [28].
All-dielectric nanophotonics, based on Mie resonances in high-index and low-loss dielectric nanoparticles (cSi, GaP, Ge) [29,30], provides an interesting alternative to induce efficient and lowloss light-matter interactions at the nanoscale [31]. Although these nanoantennas are currently of great interest for spectroscopy, bioimaging and flat optics, only a small number of papers have been exploring their interaction with 2D TMDCs [32,33], and the optimal values of the key parameters governing this interaction remain unknown. In this work, we perform a comprehensive analysis of bright exciton photoluminescence (PL) enhancement from different 2D TMDCs via coupling to Mie resonances of silicon nanoparticle (SiNP). We restrict ourselves to the visible spectral range, being the most interesting from the practical viewpoint, and hence consider 2D WS2, MoS2, WSe2, and MoSe2, whose PL spectra span the range 600-800 nm. For each material, we define the system parameters (SiNP radius, excitation and emission wavelengths) corresponding to maximum PL enhancement, taking into account excitation rate, Purcell factor and radiation efficiency. We also analyze the power patterns of PL emission that allows an estimate of the emission towards different angles.

Results and Discussion
The system under study consists of 2D TMDC (WS2, MoS2, WSe2, and MoSe2) on a SiO2 substrate, corresponding to the typical geometry in experiments. We assume a Si nanoparticle (SiNP) of radius R placed on 2D TMDC ( Figure 1). In particular, we consider c-Si for the particle material, because of its large refractive index in the visible range and moderate losses [30]. SiNPs of spherical shape can be fabricated chemically [33] or with fs-laser ablation [34]. In [33], [32], it has been shown that spherical SiNPs and Si nanowires can be placed on 2D TMDCs (WS2 and MoS2, respectively) without damaging their crystalline structure, where an additional thin layer of h-BN can be used for protection or isolation of 2D TNDCs [35].
The optical response of 2D TMDC is governed by different exciton resonances. The crystal lattice inversion symmetry breaking combined with strong spin-orbit coupling results in a large valence band splitting at the K (K') points in the first Brillouin zone. This gives rise to two different valley excitons, XA (bright) and XB (dark) ones, which are associated with optical transitions from the upper and lower valence bands to the bottom of the conduction band, respectively. This causes the appearance of two resonances in the absorption and PL spectra of 2D TMDCs. Although, XB excitons play a crucial role in the exciton relaxation dynamics, their contribution to the absorption and PL emission is very weak, especially at a room temperature. For this work, the room temperature absorption and PL spectra of WS2, MoS2, WSe2, and MoSe2 have been extracted from Refs. [18,[36][37][38], Figures 2b and 3b, respectively.
In order to estimate the total PL intensity enhancement, we have to first consider the effect of excitation rate [31]. In the case of coherent excitons, the PL intensity PL I is proportional to the squared total dipole D moment of all excitons involved in the coupling with SiNP, i.e., Here, N is the number of excited excitons and d is the dipole moment of one exciton. However, for spontaneous emission, the assumption that the total PL intensity is the sum of the PL intensities from all excitons without interference, . We also note that the largest 2 || t E is expected at the MQ resonance, as it can be expected considering the vector E-field distribution profiles for selected modes presented in Figure 2c. In fact, in contrast to electric Mie modes, magnetic modes have a tangential component of electric field to the particle surface that coupled efficiently to the bright excitons of 2D TMDCs. For example, the electric dipole (ED) mode, polarized along the radial direction, does not contribute to 2 || t E (as well as the Purcell factor spectra, Figure 3). Next, although the MD mode has a tangential E-field component, its hot-spots are located at the particle poles with only moderate enhancement in the gap. This is different for the MQ mode. The MO and EO modes are localized more within the SiNP, which leads to higher losses and reduces 2 || t E . Excitons in 2D TMDC can interact with SiNP in two possible coupling regimes, i.e., weak and strong coupling. In Ref. [33] it was shown that the system under consideration (Figure 1) has an exciton dipole moment not large enough to induce strong coupling for a given loss rate in SiNP and air as the surrounding medium. Thus, in this regime the system response is accurately described by the Purcell factor [31]. The Purcell factor F quantitatively expresses the modification of the spontaneous emission rate  of the exciton induced by its interaction with the environment (nanoantenna), Here, 0  stands for the spontaneous emission rate in free-space. In steady-state under CW excitation, this quantity is simply equal to the enhancement of total emitted power t P , taking into account radiation to farfield rad P and local absorption abs P in the SiNP, , where 0 P stands for the emitted power in absence of the nanoantenna. Figure 3a shows the calculated total Purcell factor of a tangential dipole emitter placed in the gap between SiNP and silica substrate as a function of R and emission wavelength. Refs. [18,[36][37][38]).
In some circumstances a larger Purcell factor is achieved at the cost of high dissipative losses. Hence, in order to account for the effect of loss we calculate the radiation efficiency ( The resulting radiation enhancement (radiative Purcell factor) is presented in Figure 3c. This result shows that the radiative Purcell factor maximum can be adjusted with any 2D TMDC PL maxima Now we look for the maximum PL enhancement of XA excitons in 2D TMDCs, taking into account the excitation rate, Purcell factor and radiation efficiency considered above. As an example, let us consider the case of WS2. If we fix the excitation and PL emission wavelength to 615 nm and 633 nm, respectively, the calculated tangential electric field intensity and radiation enhancement for different SiNP radius are shown in Figure 4a (bottom). The product of these two curves provides the total enhancement achievable for a specific R, presented in Figure 4a Figure 4c. We can conclude that, for a given emission wavelength, the SiNP radius can be optimized to achieve directional emission towards the desired direction.  wavelength, as well as the radiation enhancement (blue curve) of a dipole source (at the PL maximum in the each specific case), and the total emission enhancement (|green curve) as a function of R for

Conclusions
In this paper, we have performed a comprehensive analysis of bright exciton photoluminescence enhancement from various 2D TMDCs, including WS2, MoS2, WSe2, and MoSe2 via coupling to Mie resonances of a silicon nanoparticle. We have demonstrated that all-dielectric Si nanoantennas can significantly enhance the intensity of the photoluminescence signal coming from 2D TMDC by two orders of magnitude, by thoughtful optimization of the system key parameters, SiNP radius, excitation and PL emission wavelengths. For each specific case, we have found the geometric parameters corresponding to maximal PL enhancement, taking into account the excitation rate, Purcell factor and radiation efficiency. We have also demonstrated that, in contrast to electric Mie modes, the magnetic modes have a tangential component of electric field to the particle surface that can couple efficiently to the bright excitons of 2D TMDCs. Finally, we have found that the MQ mode demonstrates the best performance for Si nanoantenna in the visible range, because it optimizes a tradeoff between tangential E-field component enhancement in the gap and mode confinement within the particle. Our results provide interesting insights into nanophotonic engineering of exciton emission, with opportunities for imaging, sensing and valleytronic applications.