Giant Enhancement of Defect-Bound Exciton Luminescence and

May 31, 2017 - ... in PL intensity whereas other TMDs experience an increase;(18, 19) this ...... G.; Maier , S. A.; Yang , J. K. W.; Qiu , C. W.; Wee...
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Letter pubs.acs.org/NanoLett

Giant Enhancement of Defect-Bound Exciton Luminescence and Suppression of Band-Edge Luminescence in Monolayer WSe2−Ag Plasmonic Hybrid Structures Alex D. Johnson, Fei Cheng, Yutsung Tsai, and Chih-Kang Shih* Department of Physics, University of Texas at Austin, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: We have investigated how the photoluminescence (PL) of WSe2 is modified when coupled to Ag plasmonic structures at low temperature. Chemical vapor deposition (CVD) grown monolayer WSe2 flakes were transferred onto a Ag film and a Ag nanotriangle array that had a 1.5 nm Al2O3 capping layer. Using low-temperature (7.5 K) micro-PL mapping, we simultaneously observed enhancement of the defect-bound exciton emission and quenching of the band edge exciton emission when the WSe2 was on a plasmonic structure. The enhancement of the defectbound exciton emission was significant with enhancement factors of up to ∼200 for WSe2 on the nanotriangle array when compared to WSe2 on a 1.5 nm Al2O3 capped Si substrate with a 300 nm SiO2 layer. The giant enhancement of the luminescence from the defect-bound excitons is understood in terms of the Purcell effect and increased light absorption. In contrast, the surprising result of luminescence quenching of the bright exciton state on the same plasmonic nanostructure is due to a rather unique electronic structure of WSe2: the existence of a dark state below the bright exciton state. KEYWORDS: Monolayer WSe2, photoluminescence, surface plasmon polaritons, dark exciton, defect-bound exciton, transition metal dichalcogenide

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candidate to investigate how light emission properties are modified when coupled to plasmonic resonators, especially at low temperature, as we report here. Depending on the nature of the excitonic states, we found totally different behavior when coupled to the plasmonic structures at low temperatures: the band-edge exciton states are strongly quenched while the defect-bound exciton states are greatly enhanced when coupled to the same plamonic platform. We found that the presence of a low-lying dark state causes indirect quenching of the exciton and trion emission but not the defect-bound exciton emission when WSe2 is placed on a substrate that facilitates energy transfer, such as a metallic structure. Because the defect-bound states are unaffected by this quenching, we are able to show that with a well-designed nanostructure the luminesce of the defectbound exciton states can be enhanced by ∼200 fold at low temperature, making them extremely efficient light emitters. The Ag nanotriangle arrays were created using the convenient and easily scalable fabrication technique of colloidal lithography as described below.30,31 We deposited a monolayer of 1 μm in diameter polystyrene spheres onto a Si substrate that had a 300 nm thick oxide layer. We then used thermal evaporation to deposit 45 nm of Ag. The spheres were

onolayer transition metal dichalcogenides (TMDs) have recently attracted great interest due to their novel electronic and optical properties.1,2 In particular, it was found that at the monolayer limit, TMDs exhibit orders of magnitude higher light emission efficiency than their multilayer counterparts. Recently, a tremendous effort has been put forth in exploring TMD-plasmonic coupling to enhance light emission efficiencies even further. While many studies have been done on MoS23−12 and WS213−16 plasmonic hybrid structures, little work has focused on WSe2,17 which leaves a large gap in our understanding of TMD-plasmonic hybrid structures because, out of the commonly studied TMDs, WSe2 stands out as having behavior that differs from the others. When brought to low temperatures, WSe2 has a decrease in PL intensity whereas other TMDs experience an increase;18,19 this can be explained by the existence of a dark exciton state that is lower in energy than the band-edge bright transition.19,20 This dark state likely comes from a conduction band minimum (CBM) that is either located at the Q-point instead of the K-point21 or a CBM that has an antiparallel spin to the valence band maximum.22−25 Furthermore, defect-bound exciton emission is more prominent in WSe2, making it the preferred TMD material for the study of quantum emitters.26−29 These rich light-emission phenomena associated with the different varieties of excitonic states, including the unusual temperature dependence of light emission efficiency, and the abundance of defect-bound exciton states, make WSe2 an ideal © XXXX American Chemical Society

Received: April 1, 2017 Revised: May 26, 2017 Published: May 31, 2017 A

DOI: 10.1021/acs.nanolett.7b01364 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. (a) Optical image of the sample showing islands of plain Ag film (tan regions) surrounded by the Ag nanotriangle array with a lighter SiO2 region at the bottom. The inset shows a higher magnification optical image of the region where the Ag nanotriangle array and the SiO2 region meet; the transferred WSe2 flakes can be seen. (b) SEM image of a Ag nanotriangle array. The nanotriangles have an edge length of ∼150 nm. (c) Drawing of the sample showing the three different regions.

removed, leaving behind an array of Ag nanotriangles where a compact monolayer of spheres were located, areas of Ag film (which will be referred to as islands) where there were no spheres, and bare SiO2/Si substrate where multiple layers of spheres were deposited, as can be seen in Figure 1a. Atomic layer deposition (ALD) was then immediately used to deposit 1.5 nm of Al2O3 to act as a spacer between the WSe2 and the Ag, and to prevent oxidation of the Ag. WSe2 flakes grown by chemical vapor deposition (CVD) were transferred onto this structure using a wet transfer method. Because of the capability of both CVD and colloidal lithography to create wafer scale samples, we were left with many WSe2 flakes on all three areas. Figure 1b shows that the nanotriangle array was comprised of many highly uniform Ag nanotriangles. Because the flakes in all three regions of the sample were from the same CVD growth, transferred under the same conditions at the same time, and are in contact with the same 1.5 nm Al2O3 layer, this fabrication method provided us with an ideal platform for observing the effects of plasmonic coupling. A differential reflectance spectrum along with a calculated scattering cross section in Figure 2a shows that these nanotriangles had a plasmonic resonance centered at 645 nm. This is mainly from single nanotriangle plasmonic modes since the nanotriangles are spaced too far apart (∼300 nm) to form bowtie resonators. There were strong, tightly confined plasmonic modes at the corners of the nanotriangle that were capable of greatly enhancing light-matter interaction in extremely localized areas, as seen in Figure 2b,c. Because the nanotriangles were formed in an array that spans a large area, they are the ideal platform for TMD-plasmonic hybrid structures because due to the presence of defects and grain boundaries it is important to be able to observe the effects of plasmonic coupling over all areas of a flake and to study multiple flakes in order to gain full insight into the effects of the plasmonic structure on the emission of the TMD.

Figure 2. (a) Differential reflection spectrum of the Ag nanoantenna array (blue line) and the calculated scattering cross section (green circles). Simulated plasmonic modes of a single nanotriange for (b) yand (c) x-electric field polarization, showing strong tightly confined modes at the corners of the nanotriangle. For our array, the nanotriangles are spaced too far apart for ensemble plasmonic coupling, so the plasmonic resonance is predominately determined by individual nanotriangles. Simulations were done using COMSOL for a triangle with a 150 nm edge length and 45 nm thickness. The images were taken 5 nm above the Ag/oxide interface for a 640 nm resonance.

When WSe2 is on the Ag island or the Ag nanotriangle array, strong enhancement of light matter interaction is expected. This can lead to PL enhancement or quenching depending on how the plasmonic structures modify the radiative and nonradiative relaxation rates of the excitons within WSe2. Figure 3a shows PL spectra for WSe2 on all three regions (Ag B

DOI: 10.1021/acs.nanolett.7b01364 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 3. (a) Low-temperature (10 K) PL spectra for WSe2 on the Ag nanotriangle array, a Ag island, and the SiO2 region taken with an excitation power density of 3800 W/cm2. The inset shows zoom in spectra of the exciton emission. These spectra were collected with a wider spectrometer slit in order to collect PL from a larger area and observe the average behavior. (b) Low-temperature (10 K) defect band emission enhancement factor as a function of power density for excitation power densities ranging from 380 to 40 000 W/cm2. The enhancement factor for WSe2 on the array is a factor of 10 larger than what is plotted. (c) Low-temperature (7.5 K) power-dependent PL spectra of WSe2 on a Ag island with excitation powers densities ranging from 70 to 57 000 W/cm2. The 57 000 W/cm2 spectrum is reduced by a factor of 3.5. A dotted vertical line at 740.7 nm shows that the PL peaks do not shift with increasing excitation power. (d) Energy diagram of the bright, dark, and defect-bound exciton states with relaxation rates given by ΓB, ΓD, and Γd, respectively. Phonon scattering between the bright and dark states (γup and γdown) is shown in the diagram, however phonon scattering occurs between all of the states, allowing for a temperature dependent distribution of the exciton populations between the states. For all spectra, the WSe2 was excited with a 532 nm laser focused to a 1.5 μm in diameter area.

on the nanotriangle array was ∼20, when taking into consideration that the nanotriangles only occupy about onetenth of the excitation area the normalized enhancement factor was ∼200. For WSe2 on the nanotriangle array, the flakes are draped over the nanotriangles, which may cause them to experience strain. Strain has been shown to affect the electronic structure and the localization of excitons in WSe2.38−40 Though it is difficult to say precisely how strain affects the system, we know from observing similar behavior from WSe2 on a Ag island (where there would be much less strain) and the nanotriangle array that strain is not responsible for the difference between the band edge and defect-bound exciton emission. Figure 3c shows power-dependent PL spectra for WSe2 on a Ag island. It is seen that the emission is comprised of individual defect emission peaks that do not shift in wavelength at higher excitation powers, demonstrating that laser heating41 and hot electron doping42 are not occurring. On the other hand, the quenching of bright exciton emission presents an apparent puzzle. Because they are coupled to the same plasmonic structure, why would the interaction with the metallic nanostructure simultaneously boost the defect band emission and quench the band edge emission? We have noted that WSe2 is inherently different from the other TMDs. Temperature-dependent studies of WSe2 have shown that unlike the other TMDs WSe2 shows a decrease in its PL intensity at low temperatures.18−20 This behavior has been attributed to the existence of low lying optically dark exciton states that deplete the population of the higher energy states.19,20 These dark states are said to arise from the spin orbit splitting at the K-point, producing low-lying K−K and K− K′ dark exciton states;20,43 however, comprehensive scanning tunneling spectroscopy studies have shown that the conduction

triangle arrays, Ag films and 300 nm SiO2) of the sample. For WSe2 on 300 nm SiO2 (which was coated with 1.5 nm of Al2O3 to eliminate the influence of substrate dependent nonradiative recombination32 for a more accurate comparison with the plasmonic structures), the PL spectrum shows both band edge exciton emission (∼705 nm) and defect-bound exciton emission (>720 nm). We studied samples from several different CVD growths, and for WSe2 that was unintentionally doped trion emission was observed at ∼715 nm. The emission we observed is blueshifted compared to work done studying WSe2 on SiO2.29 This can be explained by the difference in substrate. There is 1.5 nm of Al2O3 over all regions of our sample and so the WSe2 is in direct contact with Al2O3, not SiO2. This is an important distinction because SiO2 has been shown to redshift emission for monolayer TMDs.32,33 When WSe2 is on the plasmonic structures (either on the Ag film or on Ag nanotriangle array) strong enhancement of light emission in the wavelength region related to defect band emission is observed. On the other hand, for the band edge exciton emission strong quenching is observed (inset of Figure 3a). The enhancement of the defect-bound exciton emission is similar to what is commonly seen in other emitter-plasmonic hybrid structures and is due to the Purcell effect34−36 and increased light absorption.37 Figure 3b shows the enhancement factor for WSe2 on both plasmonic structures as a function of excitation power density, revealing that maximum enhancement occurs for lower excitation powers. The enhancement factor was calculated by integrating the defect band emission intensity (720−800 nm) for WSe2 on the Ag structures and dividing it by the integrated defect band emission of WSe2 on the SiO2 region. The enhancement factor for WSe2 on the Ag island was found to be ∼8. Though the overall increase in the PL intensity C

DOI: 10.1021/acs.nanolett.7b01364 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 4. (a) PL intensity map of the defect band emission (720−780 nm) of a WSe2 flake on the SiO2 region. (b) SEM image of the residual Ag nanoparticles that the flake in (a) is resting upon overlaid on the PL intensity map of (a). (c) PL intensity map of the defect band emission of WSe2 flakes on the Ag nanotriangle array. (d) SEM image the nanoantenna array that the flake in (c) is resting upon overlaid on the PL intensity map of (c) demonstrating the size and location of the nanotriangles as compared to the flake. (e) PL spectra that were taken from the areas marked by the dashed circle and white arrow in (a,c). These flakes were from a different CVD growth than those in Figure 3a which is why the ratio of the exciton, trion, and defect band emission is different, however the PL enhancement seen is comparable between the samples. For the PL mapping, an excitation power density of 30 000 W/cm2 was used and the spectra were collected at 7.5 K. The WSe2 was excited with a 532 nm laser was focused to 0.4 μm in diameter area.

band Q point of WSe2 is lower in energy than the K-point,21 creating an additional low-lying K−Q dark exciton state that is not seen in the other TMDs which should be taken into consideration. Figure 3d shows a qualitative energy diagram that includes one of the dark exciton states, the bright exciton state, and a collection of defect states that make up the defect band. Electron−phonon scattering between the different exciton states creates a temperature-dependent population distribution of the excitons between these states. At low temperatures, excitons primarily populate the lowest energy state, which for WSe2 is a dark state, and this is what is responsible for the temperature-dependent difference in the PL of WSe2. Now consider that the plasmonic nanostructure, which enhances the radiative efficiency of the defect-bound exciton states, will also enhance the energy relaxation rate of the dark state. By enhancing the relaxation rate of the dark state, the population of both the dark and bright states will significantly reduce due to the effective scattering between the two states, forcing a quenching effect on the bright exciton emission. The defect states that are lower in energy than the dark states do not experience this quenching effect and are therefore able to experience enhancement from coupling to the plasmonic structure. This mechanism accounts for the novel phenomenon observed where the bright exciton emission is quenched and the defect band emission is enhanced. Further discussion of this mechanism can be found in the Supporting Information. There, we show that at room temperature quenching of the bright exciton emission for WSe2 on the plasmonic structures is not observed, supporting the idea that a temperature-dependent population distribution between the states facilitates the quenching Additionally, we include a lowtemperature study of WSe2 on Si with only a native oxide layer. This study reveals that when the relaxation rate of the dark state is increased, an energy cutoff at 1.696 ± 0.004 eV (731 ± 2 nm) forms above which all emission is quenched. Because this behavior can be achieved by increasing the nonradiative relaxation rate of the system without plasmonic coupling, it demonstrates that the difference in behavior between the

different exciton states is not due to energy transfer to the plasmonic structure via absorption44 and is instead a manifestation of a dark state. To gain further insight on how local field enhancement manifests, we carried out micro-PL mapping of WSe2 flakes in all three regions at low temperature (7.5 K). When the modes of the plasmonic structure spatially overlap with defects in the WSe2 dramatic PL enhancement is observed. Not surprisingly, this enhancement could only occur in areas where defects already existed and, therefore, there are only a few localized areas where enhancement is observed. Figure 4a shows a PL intensity map of the defect band emission (integrating the PL spectrum over 720 nm −780 nm) of a WSe2 flake in the SiO2 region. It can be seen that defect emission is more intense at the edges of the flake. There are a few small Ag particles under parts of the flake, the largest of which creates a slight enhancement of the defect band emission in the top center part of the flake. An overlay of the Ag nanoparticles with the PL mapping is shown in Figure 4b. Figure 4c shows a PL intensity map of the defect band emission of a WSe2 flake on the Ag nanotriangle array; an SEM image of the nanotriangle array is shown in Figure 4d. An order of magnitude enhancement can be seen in areas where the defects in the flake overlap with the plasmonic mode of one of the nanotriangles. As is the case with the flake on the SiO2 region, the defect band emission is more prominent at the edges of the flake. This shows that the strain from being draped over the nanotriangles is not creating additional defects, causing enhancement of defect emission, because the center of the flake is also draped over nanotriangles and this enhancement is not observed. Figure 4e shows the comparison of a PL spectrum acquired from the WSe2 flake on the SiO2 region with that acquired from the WSe2 flake on the nanotriangle array. Sharp peaks can be seen in the PL spectrum for the flake on the array. These sharp peaks occur because the defect band is comprised of emission from many defect states, and because the plasmonic modes of the nanotriangle are localized they only enhance a small number of these defect D

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(5) Lee, K. C. J.; Chen, Y. H.; Lin, H. Y.; Cheng, C. C.; Chen, P. Y.; Wu, T. Y.; Shih, M. H.; Wei, K. H.; Li, L. J.; Chang, C. W. Sci. Rep. 2015, 5, 9. (6) Sobhani, A.; Lauchner, A.; Najmaei, S.; Ayala-Orozco, C.; Wen, F. F.; Lou, J.; Halas, N. J. Appl. Phys. Lett. 2014, 104 (3), 031112. (7) Liu, D.; Yu, L.; Xiong, X.; Yang, L.; Li, Y.; Li, M.; Li, H. O.; Cao, G.; Xiao, M.; Xiang, B.; Min, C. J.; Guo, G. C.; Ren, X. F.; Guo, G. P. Opt. Express 2016, 24 (24), 27554−27562. (8) Yu, Y.; Ji, Z. H.; Zu, S.; Du, B. W.; Kang, Y. M.; Li, Z. W.; Zhou, Z. K.; Shi, K. B.; Fang, Z. Y. Adv. Funct. Mater. 2016, 26 (35), 6394− 6401. (9) Liu, W. J.; Lee, B.; Naylor, C. H.; Ee, H. S.; Park, J.; Johnson, A. T. C.; Agarwal, R. Nano Lett. 2016, 16 (2), 1262−1269. (10) Akselrod, G. M.; Ming, T.; Argyropoulos, C.; Hoang, T. B.; Lin, Y. X.; Ling, X.; Smith, D. R.; Kong, J.; Mikkelsen, M. H. Nano Lett. 2015, 15 (5), 3578−3584. (11) Lee, B.; Park, J.; Han, G. H.; Ee, H. S.; Naylor, C. H.; Liu, W. J.; Johnson, A. T. C.; Agarwal, R. Nano Lett. 2015, 15 (5), 3646−3653. (12) Lin, J. D.; Li, H.; Zhang, H.; Chen, W. Appl. Phys. Lett. 2013, 102 (20), 031112. (13) Boulesbaa, A.; Babicheva, V. E.; Wang, K.; Kravchenko, II; Lin, M. W.; Mahjouri-Samani, M.; Jacobs, C. B.; Puretzky, A. A.; Xiao, K.; Ivanov, I.; Rouleau, C. M.; Geohegan, D. B. ACS Photonics 2016, 3 (12), 2389−2395. (14) Mulpur, P.; Yadavilli, S.; Rao, A. M.; Kamisetti, V.; Podila, R. ACS Sens. 2016, 1 (6), 826−833. (15) Kern, J.; Trugler, A.; Niehues, I.; Ewering, J.; Schmidt, R.; Schneider, R.; Najmaei, S.; George, A.; Zhang, J.; Lou, J.; Hohenester, U.; de Vasconcellos, S. M.; Bratschitscht, R. ACS Photonics 2015, 2 (9), 1260−1265. (16) Choi, S. Y.; Yip, C. T.; Li, G. C.; Lei, D. Y.; Fung, K. H.; Yu, S. F.; Hao, J. AIP Adv. 2015, 5 (6), 067148. (17) Wang, Z.; Dong, Z. G.; Gu, Y. H.; Chang, Y. H.; Zhang, L.; Li, L. J.; Zhao, W. J.; Eda, G.; Zhang, W. J.; Grinblat, G.; Maier, S. A.; Yang, J. K. W.; Qiu, C. W.; Wee, A. T. S. Nat. Commun. 2016, 7, 11283. (18) Wang, G.; Robert, C.; Suslu, A.; Chen, B.; Yang, S. J.; Alamdari, S.; Gerber, I. C.; Amand, T.; Marie, X.; Tongay, S.; Urbaszek, B. Nat. Commun. 2015, 6, 10110. (19) Arora, A.; Koperski, M.; Nogajewski, K.; Marcus, J.; Faugeras, C.; Potemski, M. Nanoscale 2015, 7 (23), 10421−10429. (20) Zhang, X. X.; You, Y. M.; Zhao, S. Y. F.; Heinz, T. F. Phys. Rev. Lett. 2015, 115 (25), 6. (21) Zhang, C. D.; Chen, Y. X.; Johnson, A.; Li, M. Y.; Li, L. J.; Mende, P. C.; Feenstra, R. M.; Shih, C. K. Nano Lett. 2015, 15 (10), 6494−6500. (22) Kormanyos, A.; Burkard, G.; Gmitra, M.; Fabian, J.; Zolyomi, V.; Drummond, N. D.; Fal’ko, V. 2D Mater. 2015, 2 (2), 022001. (23) Kormanyos, A.; Zolyomi, V.; Drummond, N. D.; Burkard, G. Phys. Rev. X 2014, 4 (1), 16. (24) Kosmider, K.; Gonzalez, J. W.; Fernandez-Rossier, J. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88 (24), 7. (25) Liu, G. B.; Shan, W. Y.; Yao, Y. G.; Yao, W.; Xiao, D. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88 (8), 10. (26) He, Y. M.; Clark, G.; Schaibley, J. R.; He, Y.; Chen, M. C.; Wei, Y. J.; Ding, X.; Zhang, Q.; Yao, W.; Xu, X. D.; Lu, C. Y.; Pan, J. W. Nat. Nanotechnol. 2015, 10 (6), 497−502. (27) Tonndorf, P.; Schmidt, R.; Schneider, R.; Kern, J.; Buscema, M.; Steele, G. A.; Castellanos-Gomez, A.; van der Zant, H. S. J.; de Vasconcellos, S. M.; Bratschitsch, R. Optica 2015, 2 (4), 347−352. (28) Koperski, M.; Nogajewski, K.; Arora, A.; Cherkez, V.; Mallet, P.; Veuillen, J. Y.; Marcus, J.; Kossacki, P.; Potemski, M. Nat. Nanotechnol. 2015, 10 (6), 503−506. (29) Srivastava, A.; Sidler, M.; Allain, A. V.; Lembke, D. S.; Kis, A.; Imamoglu, A. Nat. Nanotechnol. 2015, 10 (6), 491−496. (30) Fischer, U.; Zingsheim, H. P. J. Vac. Sci. Technol. 1981, 19 (4), 881. (31) Deckman, H.; Dunsmuir, J. H. Appl. Phys. Lett. 1982, 41, 377.

states. This results in emission peaks with narrower inhomogeneous line widths of ∼2 nm. In summary, by using low-temperature micro-PL mapping of monolayer WSe2 on Ag plasmonic structures we observed a dramatic enhancement of the defect band emission and quenching of the exciton and trion emission. The difference in response between the band edge and defect band excitons is caused by indirect quenching of the bright state through a lowlying dark state via nonradiative energy transfer to the Ag structure. The enhancement of the defect band emission when the WSe2 is on the Ag nanotriangle array is significant, with an enhancement factor of ∼200 in comparison to WSe2 on the SiO2 region. In addition, a narrower inhomogeneous line width can be seen for the PL of WSe2 on the Ag nanotriangle array. The easy scalability of the fabrication techniques used to create the WSe2−Ag nanotriangle array hybrid structure, combined with the resulting enhancement makes this system a valuable tool for the future study of quantum emitters in WSe2.



ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b01364. Includes additional discussion of how the dark exciton state quenches the band edge bright exciton emission when WSe2 in on a substrate that increases the nonradiative recombination rate of the excitons; it also includes room-temperature PL spectra of WSe2 on the three different regions of the sample (PDF)



AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Alex D. Johnson: 0000-0003-1219-1169 Yutsung Tsai: 0000-0001-5473-9028 Chih-Kang Shih: 0000-0003-2734-7023 Author Contributions

A.J. carried out the PL measurements. F.C. fabricated the nanotriangle array. Y.T. preformed the CVD growth of the monolayer WSe2. C.K.S advised on the experiment and provided input on the data analysis. A. J. and C.K.S. wrote the paper with input from the other coauthors. Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS This work was partially supported by the Welch Foundation (F-1672) and by the National Science Foundation (DMR1306878, ECCS-1408302, and EFMA-1542747).



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