Enhanced light-matter interaction in an atomically thin semiconductor coupled with dielectric nano-antennas

Unique structural and optical properties of atomically thin two-dimensional semiconducting transition metal dichalcogenides enable in principle their efficient coupling to photonic cavities having the optical mode volume close to or below the diffraction limit. Recently, it has become possible to make all-dielectric nano-cavities with reduced mode volumes and negligible non-radiative losses. Here, we realise low-loss high-refractive-index dielectric gallium phosphide (GaP) nano-antennas with small mode volumes coupled to atomic mono- and bilayers of WSe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2. We observe a photoluminescence enhancement exceeding 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{4}$$\end{document}4 compared with WSe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2 placed on planar GaP, and trace its origin to a combination of enhancement of the spontaneous emission rate, favourable modification of the photoluminescence directionality and enhanced optical excitation efficiency. A further effect of the coupling is observed in the photoluminescence polarisation dependence and in the Raman scattering signal enhancement exceeding 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{3}$$\end{document}3. Our findings reveal dielectric nano-antennas as a promising platform for engineering light-matter coupling in two-dimensional semiconductors.

Unique structural and optical properties of atomically thin two-dimensional semiconducting transition metal dichalcogenides enable in principle their efficient coupling to photonic cavities with optical modes volumes close to or below the diffraction limit.Recently, it has become possible to make all-dielectric nano-cavities with reduced mode volumes and negligible non-radiative losses.
Here, we realise low-loss high-refractive-index dielectric gallium phosphide (GaP) nano-antennas with small mode volumes coupled to atomic mono-and bilayers of WSe2.We observe a photoluminescence enhancement exceeding 10 4 compared with WSe2 placed on planar GaP, and trace its origin to a combination of enhancement of the spontaneous emission rate, favourable modification of the photoluminescence directionality and enhanced optical excitation efficiency.A further effect of the coupling is observed in the photoluminescence polarisation dependence and in the Raman scattering signal enhancement exceeding 10 3 .Our findings reveal dielectric nano-antennas as a promising platform for engineering light-matter coupling in two-dimensional semiconductors.
Monolayer semiconducting transition metal dichalcogenides (TMDs) [1] such as WSe 2 exhibit bright excitonic luminescence and strong absorption at room temperature with potential for photonic applications [2,3].An important property favouring integration in devices is their compatibility with a wide range of substrates.So far, photonic device demonstrations include monolayer TMDs coupled to nano-cavities in photonic crystals [4,5], nanobeam waveguides [6] and to microdisk cavities [7].TMD monolayers and van der Waals heterostructures [8] comprised of vertically stacked atomic layers of TMDs, hexagonal boron nitride and graphene have been integrated in Fabry-Perot microcavities [10,11].These devices provide photonic modes with relatively high quality factors, Q, in the range of 100s to 1000s.Despite the coupling to TMD monolayers via the relatively weak evanescent field, lasing in hybrid TMD-dielectric cavities has been observed [4,6,7].Moreover, the strong lightmatter interaction regime has been realized in optical microcavities [9][10][11][12] and photonic crystals [5], where atomic layers of 2D TMDs were placed at the anti-node of the photonic mode.Most of these devices relied on confining electromagnetic fields in diffraction-limited volumes, V eff > ∼ (λ/n) 3 , in order to increase the spontaneous emission rate by the Purcell enhancement factor F p , which scales as Q/V eff .Whereas the high Q has been readily realized, V eff provided by these structures is relatively large, leading to modest values of F p .Most of these devices also show a reduction of the light intensity compared with bare TMD monolayers, explained by the presence of fast non-radiative processes in the currently available TMDs, where the quantum yield is typically below 0.1% [13].
The effective volume of the optical mode can be reduced below the diffraction limit in plasmonic nanocavities and nano-antennas [14,15].By coupling semiconducting TMDs to such plasmonic structures, large photoluminescence (PL) enhancements [16][17][18][19][20][21], strong light-matter coupling [22][23][24], brightening of the dark excitonic states [25], and modification of optical properties of quantum light emitters [26,27] have been observed.In some of these reports, special care had to be taken to overcome optical losses in metallic plasmonic structures by introducing a few nm dielectric spacer separating the TMD layer [20,21,29].This turns out to be particularly important to suppress quenching for quantum light emitters [26,27].
Recently, it has also been shown that high-refractiveindex dielectric nano-antennas can provide confined optical modes with strongly reduced mode volumes [28,[30][31][32][33][34].The main advantages of such structures are low non-radiative losses induced in the coupled light emitting material [28,[30][31][32].Experimentally, strong fluorescence enhancement and radiative life-time shortening by a factor exceeding 20 has been shown for dye molecules coupled to GaP cylindrical nano-antennas (refractive index n > 3) [32].Furthermore, it has been shown that such nano-antennas can be designed in principle to provide Purcell enhancements of thousands [31].On the other hand, recently, modified directionality of PL was shown for monolayer MoS 2 coupled to a Si nanowire [33].Si nano-particles coupled to WS 2 were explored for possibilities to realise the strong light-matter interaction [34].Finally, multilayer TMDs themselves were used to fabricate high-index nanodisks, whose resonant response could be tuned over the visible and near-infrared ranges [35].
Here, we report large enhancements of the PL and Raman scattering intensity in mono-(1L) and bilayer (2L) WSe 2 placed on top of cylindrical GaP nano-antennas (Fig. 1a), compared with WSe 2 on the planar GaP.Upon illumination, the nano-antennas scatter radiation which is strongly confined around the pillar edges.This confinement effect is observed in a broad spectral range, overlapping with the optical response of both mono-and bi-layer WSe 2 .Our approach exploits the extreme ability of the atomically thin layers of TMDs to stretch and conform with the nano-structured surfaces and therefore favourably align themselves with the confined optical mode .Our interpretation of the observed PL enhancement shows that it arises from a combination of the Purcell enhancement, efficient redirection of the emitted PL in the space above the substrate, and larger absorption of light in the 2D layer related to the enhancement of the optical field in the confined mode.Similarly to experiment, our model shows increase of the PL enhancement as the pillar radius reduces.Supporting our interpretation, time-resolved measurements show the shortening of the PL life-time, which goes beyond our time resolution, only enabling a lower bound estimate of the PL decay rate enhancement by a factor of 6.We further confirm the photonic effect of the nano-antennas by demonstrating linearly polarised PL and enhanced Raman scattering in the coupled WSe 2 .Our findings show an effective approach to engineering light-matter coupling at the nanoscale, by exploiting the low-loss optical modes of the dielectric nano-antennas together with the unique mechanical and optical properties of 2D semiconductors.

Results
Coupling WSe 2 atomic layers with dielectric nano-antennas The calculated spatial distribution of the electric field around the cylindrical double-pillar GaP nano-antennas (referred to as dimers below) is shown in Figs.1b and c.Light at a pumping wavelength of 685 nm polarised linearly along (X) and perpendicular (Y ) to the line connecting the centres of the pillars is used.(|E|/|E 0 |) 2 is shown for the plane containing the top surface of the dimer, i.e. 200 nm above the planar substrate.Here E and E 0 are the electric field amplitudes of the wave scattered by the pillars and the normally incident wave, respectively.The two-dimensional distribution in the plane is shown as a colour-map.In the small volume within the dimer gap an enhancement is only observed for the X-polarisation [32,37] (see further details in Supplementary Note I).The graphs shown with white lines present the variation of (|E|/|E 0 |) 2 along the horizontal and vertical dotted lines, revealing strong maxima at the edges of the pillars.For the X-polarisation, at this height of 200 nm, these maxima are stronger than the (|E|/|E 0 |) 2 values in the gap.They are also stronger than the enhancement values at the pillar edges in the Y -polarisation.
Atomic force microscopy (AFM, Fig. 1d) shows that the transferred atomically thin layer of WSe 2 closely follows the shape of the dimer, thus strongly overlapping with the volume of the confined optical mode.As shown in Fig. 1e the spectral response of the nano-antenna is very broad, extending well into the visible and near-infra-red ranges, and fully overlapping with the PL spectra of both 1L and 2L-WSe 2 (see also Supplementary Note II).
Fig. 1f shows a PL image of 1L and 2L samples deposited on an array of nano-antennas and measured using an optical microscope.The image is recorded using the techniques of Ref. [38] with unpolarised white light illuminating the sample through a short-pass filter and a long-pass filter installed in the imaging path.Bright PL replicating the shape of the dimers is visible for both 1L (yellow) and 2L (purple), whereas the PL from WSe 2 on planar GaP is very weak (dark areas around the pillars).A comparison of the intensities in the PL and dark field microscopy images is shown in Fig. 1g, where the intensities are measured along the dotted line shown for one of the nano-antennas in Fig. 1f.The PL enhancement is observed most strongly around the edges and in the gap of the nano-antenna, where, as seen in the dark field profile, most of the light is scattered by the pillars.Further comparison of the PL, bright and dark field images is given in Supplementary Note III.
Photoluminescence enhancement factor We have additionally carried out detailed room temperature PL measurements in our micro-PL setup for 1L and 2L-WSe 2 placed on GaP nano-antennas.We use a laser with wavelength λ =685 nm, which is below the GaP absorption edge, and is absorbed only in the WSe 2 layer.Figs.2a and  b show PL spectra for 1L and 2L-WSe 2 coupled to GaP nano-antennas with r =300 nm and r =100 nm, respectively, and compare them with PL from the 2D layers placed on the planar GaP.Strong enhancement of PL intensity exceeding 50 times for WSe 2 placed on nanoantennas is observed.Lower PL intensity for 2L-WSe 2 is a consequence of its indirect bandgap, in contrast to the 1L-WSe 2 having a direct band-gap.The effect of strain present in WSe 2 placed on the nano-pillar is evident in the PL redshift for 1L and spectral modification for 2L samples.
We compare the observed PL intensity for WSe 2 coupled to the nano-antennas (I on ) to that of the uncoupled WSe 2 on planar GaP (I off ) by introducing the PL enhancement factor [15,19,39,40], EF , defined as: Here we take into account the considerable difference between the PL collection area A laser defined by the excitation laser spot of 3.5 µm diameter, and the geometrical area for a given dimer, A dimer = 2 × πr 2 .We expect that A dimer is larger than the actual area from where the enhanced PL is collected (mostly the edges of the pillars and the dimer gap).Thus, EF calculated in this way is expected to be a lower bound estimate of the observed effect.
Fig. 2c shows the EF values extracted from the experimental data for different nano-antennas.EF for 1L-WSe 2 exhibits an increase from 10 2 for the large r = 500 nm pillars to nearly 10 4 for r = 50 nm, whereas the variation is more pronounced for the 2L samples, where EF changes from ≈ 30 to 4 • 10 4 .Such large EF values are comparable with the highest reported in plasmonic/TMDs systems [19].
As we show below, the observed enhancement is the consequence of the interaction of WSe 2 with the optical mode of the nano-antenna.The extremely efficient overlap between the 2D layers and the optical mode field maxima are important for enhancing the interaction with the nano-antennas.Fig. 2c shows that there is a variation of EF between the antennas of the same size.There are several factors that can cause this.(1) Non-uniformity of the coupling between WSe 2 and the nano-antennas caused by a variety of factors such as local contamination from the polymer used for the WSe 2 transfer, local deformation of WSe 2 , local presence of water etc. (2) Non-uniformity of the structural properties of the nanoantennas.For example, the size of the gap may vary.The quality of etching may also vary, for example producing side-walls of the pillars, which are not perfectly vertical etc.
A smaller value of EF for the 2L sample for the nano-antennas with the large radii is probably due to its higher rigidity compared with 1L.As the radius becomes smaller the 2L conforms more closely with the shape of the nano-antenna, and in addition the increased strain in the crystal may lead to the indirect to direct bandgap crossover [41], yielding larger values of EF .

Comparison of experimental data with the model
In order to compare the results with our model (see Methods and Supplementary Note IV), we introduce an effective enhancement factor EF eff defined as the product of three factors [15]: Here γ exc /γ 0 exc is the ratio of the excitation rates at a wavelength λ exc , for an emitter coupled to the antenna (γ exc ) and placed on the planar substrate (γ 0 exc ).Their ratio would account for the enhancement of the incident radiation leading to stronger light absorption in WSe 2 .γ exc ∝ (|E|/|E 0 |) 2 , for which the spatial distribution is shown in in Figs.1b,c.We find that additional increase of the γ exc /γ 0 exc ratio arises from the reduction of γ 0 exc for an emitter placed 0.5 nm above the planar GaP substrate compared with that for an emitter in the free space.The dependence of γ exc /γ 0 exc on the pillar radius r is shown in Fig. 2d for γ exc calculated for an electric dipole placed 0.5 nm above the top surface of the pillars.The dipole is placed at the edge of one of the pillars just outside the gap.As shown in the inset in Fig. 2e and in Supplementary Note IV this is the position where the coupling to the optical mode of the dimer is maximized.The data in Fig. 2d-f, showing the individual contributions of the different terms to the overall enhancement in Eq.2, are calculated for this position of the dipole.
The second term describes the enhancement of the quantum yield (q/q 0 ) for an emitter at a wavelength λ em .This is achieved through the enhanced rate of spontaneous emission [40] described by the Purcell factor F p = γ r /γ 0 r , where γ r and γ 0 r are the rates of spontaneous emission for the emitter coupled to the antenna and placed on the planar GaP, respectively.In our model we consider the limit of a low quantum yield for the emitter, i.e. the non-radiative decay γ nr F p γ r , which leads to q/q 0 = F p (γ r + γ nr )/(F p γ r + γ nr ) ≈ F p .The dependence on r for this term is shown in Fig. 2e.
The third term η/η 0 describes the improved collection efficiency for WSe 2 PL on top of the nano-antennas (η) compared to planar GaP (η 0 ), as the emitted radiation is coupled to our detector using collection through numerical aperture NA=0.7 (see Supplementary Note V).The dependence on r of η/η 0 is shown in Fig. 2f.
Fig. 2g shows the calculated values of the effective enhancement factor EF eff taking into account the above three mechanisms [15,40] (see Methods and Supplementary Note IV).The dependences of EF eff and EF are in a good qualitative agreement, suggesting that our model captures the main contributing factors.
Polarisation dependent luminescence We find further evidence for the sensitivity of the WSe 2 coupling to the optical mode of the dimer in polarisation resolved PL measurements.The spatial asymmetry of the dimer nano-antenna and the enhanced field in the gap between the two pillars is expected to lead to a polarisationdependent response [32,37] as predicted by Figs.1b,c.Such behavior is found in PL in WSe 2 coupled to nanoantennas, as shown in Fig. 3a-c.Fig. 3a shows the case for a nano-antenna with r = 400 nm and a gap g = 65 nm.It is observed that PL is 20% stronger when excited with an X-polarised laser compared to the Y -polarised excitation (X and Y are selected as in Figs.1b,c).A similar modulation is observed in dimers with other values of r (see Supplementary Note VI).The origin of this behaviour is further revealed when considering touching pillars with no gap.Fig. 3b shows the dark-field microscope images of dimers without (top) and with (bottom) a gap.The graphs in Fig. 3b show the extracted scattering intensities along the line connecting the centres of the pillars, revealing the absence (left) or the presence (right) of the gap.Fig. 3c shows the integrated PL intensity for WSe 2 coupled to such dimers measured in a micro-PL set-up.In the dimer with a gap, the PL is modulated by 20% when varying the polarisation of excitation, exhibiting higher intensity for X-polarised laser, which is due to the excitation of the optical mode in the gap between the pillars.A negligible polarisation dependence is observed in the dimer with no defined gap, emphasizing nearly equal coupling of WSe 2 to the X-and Y -polarised optical modes.pretation in Fig. 2 demonstrates that partly the PL enhancement originates from the enhanced radiative decay rate in WSe 2 .We have been able to demonstrate this experimentally by measuring PL dynamics in WSe 2 .Fig. 4a shows the PL decay for a 1L-WSe 2 placed on the planar GaP (purple) compared to the emission when it is coupled to a nano-antenna with r = 200 nm (orange).The curves are measured using excitation with a 90 ps pulsed laser at 638 nm and detection with an avalanche photo-diode detector (see Methods).The corresponding instrument response function (IRF) is shown in Fig. 4a as a gray shaded area.For these measurements, we used a low excitation density (0.2 W/cm 2 ) to avoid nonradiative exciton-exciton annihilation, resulting in fast PL decay [2,42].For low pumping powers, single exponential decay is usually observed, reflecting the radiative recombination dynamics of the thermalised exciton population [43,44].Such behaviour is observed in Fig. 4a for the 1L-WSe 2 on the planar GaP, which shows a singleexponential PL decay with a lifetime τ GaP ≈ 1.3 ns.The PL decay measured for WSe 2 coupled to a dimer is dominated by a fast component with a lifetime τ dimer ≈ 0.2 ns.This is very similar to the fast component in the instrument response function (IRF) of τ IRF ≈ 0.16 ns.We thus conclude that the measured τ dimer ≈ 0.2 ns is a resolution-limited value.We interpret the shortening of the PL lifetime as a consequence of the Purcell enhancement of the radiative rate.A conservative lower-bound estimate of the Purcell enhancement factor is therefore ≈6.This is in a good agreement with the theoretical predictions, for which we should take into account the spatial variation of the Purcell enhancement in the WSe 2 coupled to the mode.Note that Fig. 2e only shows the values obtained for the optimum location of the dipole where the enhancement is maximised (see further details in Supplementary Note IV).

Radiative decay enhancement
Surface enhanced Raman scattering Further evidence for the efficient interaction of WSe 2 with the nanoantennas is obtained from the observation of the enhanced Raman scattering response.This effect, typically observed for nano-structured metals [14] and dielectric nano-particles [45], is related to the surface localization of the electromagnetic field.Both 1L and 2L-WSe 2 Raman spectra excited with a laser at 532 nm show a pronounced peak at 250 cm −1 composed of two degenerate modes, the in-plane E for 1L and E 1 2g for 2L WSe 2 , and out-ofplane A 1 for 1L and A 1g for 2L WSe 2 [46].This is shown in Figs.4b and c for 1L and 2L samples, respectively.Figs.4b,c further compare Raman spectra measured for WSe 2 on the planar GaP with the spectra collected on nano-antennas, and show a notable enhancement of the signal.While the overall Raman signal decreases with decreasing r, its relative strength per unit nano-antenna area is strongly enhanced.In analogy with the PL data, we define the Raman experimental enhancement factor, EF Raman (see Methods).The obtained EF Raman values increase when reducing the antenna radius, as for the PL enhancement, with values exceeding 10 3 for r = 50 nm (Fig. 4d).The observed Raman intensity enhance-ment is explained by the enhancement of the laser and the scattered light fields and a more efficient collection of the signal (see Eq.2).

Discussion
The reported enhancement in the PL emission and Raman signal intensities, and shortening of the radiative lifetime in 2D WSe 2 coupled to GaP nano-antennas, shows that nano-structured high-index dielectrics can be an efficient platform to engineer light-matter coupling on the nano-scale.Importantly, we show that the coupling to nano-antennas can be used to strongly enhance the quantum yield in TMDs via the enhancement of the radiative decay rate, emphasizing the potential of this approach for light-emitting devices applications.The PL enhancements, which we report are of the order or exceed those recently reported in TMDs coupled to metallic plasmonic nano-antennas, showing the viability of our approach in a broader nano-photonics context.Our approach could be further extended to arrays of dielectric nano-structures, or meta-surfaces, an emerging field of nano-photonics [36], and to the use of van der Waals nanophotonics structures made from multilayer TMDs [35], with potential in a wide range of applications, such as quantum optics, photovoltaics and imaging.The approaches for realisation of the Purcell effect in loss-less dielectrics demonstrated here will also be used in the field of strain-induced single photon emitters [47,48] for applications exploiting quantum light generation.

Sample fabrication
GaP nano-antennas arrays are fabricated with the topdown lithographic process following the procedure in Ref. [32].The fabricated pillars are 200 nm high and have radi of r = 50 nm, 100 nm, 200 nm, 300 nm, 400 nm, 500 nm.Monolayers and bilayers of WSe 2 were mechanically exfoliated from a bulk crystal (HQGraphene) onto a poly-methyldisyoloxane (PDMS) polymer stamp.The thin layers are identified via the PL imaging technique described in this work in Fig. 1 and Ref. [38].The stamp was then used for an all-dry transfer of the exfoliated materials on top of the GaP nano-antenna array, in an home-built transfer setup following Ref [49].

Photoluminescence imaging
The PL imaging technique [38] used in Fig. 1 is obtained in a modified commercial bright-field microscope (LV150N Nikon).The white light source is used both for imaging the sample and as the PL excitation source.To excite the sample a 550 nm short-pass filter is placed in the illumination beam path to remove the near-IR part of the emission spectrum.The white light is then directed in a large numerical aperture objective (Nikon x100 NA=0.9) and collected with the same objective.The PL signal produced by the TMDs is isolated with a 600 nm long-pass filter before reaching the colour microscope camera (DS-Vi1, Nikon) used to image the sample.

Photoluminescence spectroscopy
All PL spectra are acquired at room temperature in a micro-PL set-up with samples placed in vacuum.A 685 nm diode laser coupled to a single mode fibre is used as the excitation source.After passing through a 700 nm short-pass filter, the laser is focussed onto the sample through an infinity corrected objective (Mitutoyo 100x NA=0.7).For the data reported in Fig. 2, the average power entering the objective is 30 µW for monolayer PL and 120 µW for bilayer PL.The resulting laser spot has a radius of ≈ 3.5µm which is large enough to illuminate entire individual dimers, and kept constant for a uniform excitation of all the different nano-antennas sizes when determining the reported enhancement factor in Eqn.1.The dimers are separated by 10 µm, which allows optical measurement of an individual dimer.The emitted light is collected by the same objective and filtered with a 700 nm long pass filter before coupling into a spectrometer (Princeton Instruments SP2750) and detection with a high-sensitivity liquid nitrogen cooled charge-coupled device (CCD, Princeton Instruments PyLoN).For polarisation measurements, a Glan-Thompson linear polariser, followed by a half-wave plate mounted onto a motorized rotation stage, are inserted in the excitation path in order to control the linear polarisation angle of the laser source.No polarisation optics is used in the collection path.
Time-resolved spectroscopy PL decay of planar WSe 2 is measured by coupling light filtered by a spectrometer (≈4 nm bandwidth) into a multi-mode fibre directing light to an avalanche photodiode (APD, ID100-MMF50) with time-resolution of ∼40 ps.The signal from the APD is read using a photon counting card (SPC-130).A ≈90 ps pulsed diode laser (PicoQuant LDH) with wavelength 638 nm is used as the excitation source at a repetition rate of 80 MHz.Overall, the instrument response function (IRF) has a full width at half maximum of <200 ps.The PL decay curves are fitted with single and bi-exponential decay functions of the form y = y 0 + A 1 exp −t/τ1 +A 2 exp −t/τ2 .

Raman spectroscopy
Raman spectra are collected at room temperature in a micro-Raman set-up with samples placed in vacuum.A single mode 532 nm laser (Cobolt 04-01) is used, focussed on the sample through an infinity corrected objective (Mitutoyo 50x NA=0.55) with an average power of 200 µW before entering the objective.Background laser light is suppressed using three Optigrate Bragg filters, which allow Raman signal to be measured by the same spectrometer/CCD system as for PL spectroscopy.The Raman signals are analysed by fitting the two WSe 2 Raman peaks with Gaussian functions, following background subtraction.The 250 cm −1 peak intensities extracted in this way are used to calculate the Raman enhancement factor in Eq.3 given below.In this equation I fit Raman is the Raman intensity for WSe 2 coupled to a dimer and I fit ref is for the reference WSe 2 on the planar GaP.We determined experimentally that A laser = 16.8 µm 2 in the micro-Raman set-up.(3)

Simulations
The effective enhancement factor contributions are calculated with a finite-difference time-domain (FDTD) method using Lumerical FDTD solutions software.The near-field distribution is obtained by illuminating the structure with a plane wave, polarised along or perpendicular the dimer axis (the line connecting the centres of the pillars) and incident on the dimer perpendicular to the substrate from the vacuum side.The Purcell enhancement is calculated by exciting the structure with an oscillating electric dipole placed at different positions along the dimer axis at a vertical position 0.5 nm from the top surface of the GaP pillars, and compared with a dipole placed 0.5 nm above the planar surface of the GaP substrate.To model the photoluminescence in WSe 2 originating from the in-plane excitons, the dipole is placed parallel to the substrate surface.The dipole oscillates along the line connecting the centres of the pillars in order to couple efficiently to the gap mode.The Purcell enhancement is calculated as the decay rate γ of an emitter coupled to the GaP antenna, divided by the decay rate γ 0 of the same electric dipole emitter on the planar GaP substrate as described above.The decay rate enhancement γ/γ 0 corresponds to the enhancement of the rate of energy dissipation P/P 0 .The collection efficiency is calculated by exciting the structure with an electric dipole, as above, and integrating the radiation pattern within the numerical aperture of the objective used in the experiments (NA=0.7).The refractive index used for GaP is n=3.2.

2 FIG. 1 .
FIG. 1. Coupling of an atomically thin semiconductor to a dielectric nano-antenna (a) Schematic view of a GaP dimer nano-antenna with a height h = 200 nm, gap width g and pillar radius r. (b,c) Calculated relative intensity (|E|/|E0|) 2 of light at 685 nm scattered by a GaP dimer with r = 50 nm, h = 200 nm, and g = 65 nm.E (E0) is the electric field amplitude of the light scattered by (normally incident on) the dimer.The polarisation of the incident light is shown with arrows.(|E|/|E0|) 2 is calculated at the height of 200 nm corresponding to the top surface of the pillars.The inset shows the variation of (|E|/|E0|) 2 along the horizontal and vertical dotted lines.(d) Atomic force microscopy (AFM) image of a GaP nano-antenna (r = 500 nm) covered with a monolayer WSe2.(e) Simulated scattering (Qscatt) and absorption (Q abs ) efficiency integrated over numerical aperture NA=0.9 for a GaP nano-antenna (g = 35 nm, r = 100 nm) illuminated with a plane wave.Pink (red) dashed line Qscatt for light scattered upwards in air (downwards into the substrate).Room temperature PL spectra are shown for 1L-WSe2 (orange) and 2L-WSe2 (purple).(f) Optical microscope image showing PL from dimer nano-antennas array covered with 1L-and 2L-WSe2 illuminated with unpolarised white light (see Methods for details).Scale bar is 10 µm.The yellow lines show the boundaries between 1L-and 2L-WSe2.PL from 1L (2L) sample shows as false yellow (purple) in the image.(g) Intensity profiles extracted from a dark-field (black) and PL (yellow) microscope images for a dimer with r=500 nm measured along the dashed line in Fig.1e.The intensity profiles are overlaid with the schematic of a dimer.

FIG. 2 .
FIG. 2. Photoluminescence enhancement of mono-and bi-layer WSe2 coupled to GaP nano-antennas.(a,b) PL spectra for 1L-WSe2 (orange) and 2L-WSe2 (black) placed on top of GaP nano-antennas with r = 300 nm and 100 nm, respectively.Spectra in red are measured on 1L-(a) and 2L-WSe2 (b) placed on the planar GaP.Their intensity is multiplied by 10. (c) Experimental PL enhancement factor, EF , as a function of the antenna radius, for 1L-(orange) and 2L-WSe2 (black).(d-g) Results of simulations showing various parameters determining the observed PL enhancement in a dimer as a function of the pillar radius (see main text and Methods for further details).The values shown in the figures correspond to the behaviour of an oscillating electric dipole placed at the edge of the gap between the two pillars, 0.5 nm above the top of the pillar, and aligned along the line connecting the centres of the pillars [see inset in (e)].(d) Enhancement of the excitation, γexc/γ 0 exc , dependent on the electric field intensity at the antenna surface and on the planar GaP substrate.(e) Enhancement of the radiative recombination rate, γr/γ 0 r .Inset shows variation of this ratio as a function of the dipole position above the dimer gap.The values plotted in (d)-(g) are calculated for the dipole placed at the position where γr/γ 0 r reaches the maximuma.(f) Enhancement of the light collection efficiency, η/η 0 .(g) Dependence on r of the calculated effective enhancement factor, EF eff , defined as the product of the parameters in (d), (e) and (f) (see Eq.2 and the formula on the graph).

FIG. 3 .
FIG. 3. Polarisation properties of WSe2 coupled toGaP nano-antennas.(a) Photoluminescence spectra for 1L-WSe2 on a GaP nano-antenna (g = 65 nm, r = 400 nm), for a laser excitation linearly polarised along X-and Y-axes of the dimer (defined as in Fig.1).(b) Dark-field microscopy images and related scattered light intensity profiles for two GaP nano-antennas (r = 400 nm) covered with 1L-WSe2.The scattered light intensity profiles are taken along the long axes (Y) of the dimer as shown by the dotted lines.The top (bottom) images and left (right) graphs correspond to the dimers without (with) a gap.(c) Comparison of 1L-WSe2 PL intensity as a function of the orientation of linearly polarised excitation for the nano-antennas in Figure3b.Red (dark blue) shows the dependence for the dimer with (without) a gap.

FIG. 4 .
FIG.4.Time-resolved PL and Raman scattering signal.(a) PL decay traces for a WSe2 monolayer placed on a planar GaP substrate (purple) and coupled to a nano-antenna with r = 200 nm (yellow).Grey shows the instrument response function (IRF).The PL signal is measured with a 90 ps pulsed laser, under low excitation power of 0.2 W/cm 2 , to avoid exciton-exciton annihilation processes.Resolution-limited dynamics are measured for WSe2 coupled to a dimer.Dotted black lines show results of the fitting with exponential decay.(b) Raman scattering spectra for 1L-WSe2 placed on GaP nano-antennas with r= 50 nm (red) and 200 nm (yellow) and on the planar GaP (black).(c) Raman scattering spectra for 2L-WSe2 on a r=200 nm GaP nano-antenna (orange) and on the planar GaP (black).(d) Experimental Raman enhancement factor, EF Raman (see Eq.3 in Methods), obtained (after background subtraction) for the 250 cm −1 peak in 1L-and 2L-WSe2 coupled to nano-antennas with various r.