Opto-Electronic Advances  2019, Vol. 2 Issue (5): 190008      DOI: 10.29026/oea.2019.190008     
Large Rabi splitting obtained in Ag-WS2 strong-coupling heterostructure with optical microcavity at room temperature
Bowen Li1, Shuai Zu1, Zhepeng Zhang2, Liheng Zheng1, Qiao Jiang1, Bowen Du1, Yang Luo1, Yongji Gong3, Yanfeng Zhang2, Feng Lin1, Bo Shen1, Xing Zhu1, Pulickel M. Ajayan3, Zheyu Fang1,4     
1. School of Physics, State Key Laboratory for Mesoscopic Physics, Academy for Advanced Interdisciplinary Studies, and Nano-optoelectronics Frontier Center of Ministry of Education, Peking University, Beijing 100871, China;
2. Department of Materials Science and Engineering, College of Engineering, Peking University, Beijing 100871, China;
3. Department of Materials Science and NanoEngineering, Rice University, 6100 Main Street, Houston, TX 77005, USA;
4. Collaborative Innovation Center of Quantum Matter, Beijing 100871, China


Strong light-matter coupling is crucial for the modern physics, including semiconductor optoelectronics, plasmonics, ultrafast optics, and quantum electrodynamics, etc. When the coherent exchange rate of energy between light and matter is higher than the decay rate, it reaches the strong coupling regime, where two hybrid modes with different energies instead of original independent eigenstates are formed, known as the vacuum Rabi splitting1. The strong coupling can result in the formation of half-light, half-matter bosonic quasiparticles called polaritons, which provides a number of possibilities in fascinating advances such as the low-threshold lasing2, Bose-Einstein condensation3-5, chemical reactivity tuning6-8, and optical spin switching9, etc., and has been widely demonstrated in solid-states, like GaAs quantum dots10, GaN and ZnO wide-bandgap semiconductors11, 12, and organic polymer materials13. However, either the small binding energy of excitons or the strong localization effect of disordered potentials14, makes traditional semiconductors hardly realize strong-coupled polaritonic eigenstates at ambient conditions.

Two-dimensional transition metal dichalcogenides (TMDs) have generated broad interests owing to their unique optical and electronic properties15, 16. Sharp exciton resonances and large exciton binding energies make them excellent candidates for both fundamental and application studies17. The monolayer of TMDs gives rise to strongly confined excitons with Bohr radii of ~1 nm and large oscillator strength18. Embedding TMD monolayers in an optical microcavity can realize the strong light-matter coupling and generate polaritonic eigenstates, which creates opportunities for engineering the exciton-polariton interaction19-21. Though TMD-based polaritons inherit the strong nonlinear interaction and low effective mass properties from the constituent exciton and photon components, most of strong-coupling eigenstates with small Rabi splitting energy were observed in a cryogenic condition and required sophisticated measurement techniques, which impedes the development of practical polaritonic devices.

Surface plasmons (SPs), collective electron oscillations of the metallic nanostructure, can tightly confine the incident light into nanoscale22, which results in a strongly enhanced local field with ultrasmall mode volume and hence provides an advantaged platform for strong coupling investigations23-28. In the past decades, many attentions have been paid to plasmon-TMD-exciton systems for studying light-matter interactions, such as plasmon-induced resonance energy transfer29, 30, tunable photoluminance by plasmon31-37 and Fano resonance38-40. The strong plasmon-TMD-exciton coupling also has been investigated41-48. However, due to the intrinsic loss of the metal, the quality factor and cooperativity of plasmonic nanocavities are lower than those of conventional optical cavities, and therefore the reliable plasmon-exciton strong coupling has seldom been reported at room temperature. Despite of the Ohmic loss, the strong electric field enhancement and extremely small resonance mode volume49 indeed make the plasmonic nanostructure an ideal candidate for the strong light-matter coupling investigation.

Here, we report for the first time the observation of strong coupled plasmon-exciton polaritonic hybrid states by embedding Ag-WS2 heterostructure in an optical microcavity under ambient conditions. We demonstrate the hybrid states of optical cavity and plasmonic nanostructure can be further coupled with the exciton of WS2 monolayers, and generate a new kind of quasiparticle with part-plasmon, part-exciton and part-light. The resulted plasmon-exciton polaritonic states can push the Rabi splitting up to 300 meV at the strong coupling regime under the room temperature. Experimental measurements are confirmed by simulations and calculations that based on a three-oscillator model. We believe that the proposed configuration can achieve strong light-matter coupling with a large Rabi-splitting for various flexible device applications based on ultrathin materials in the future.

Results and discussion

The schematic of Ag-WS2 heterostructure is depicted in Fig. 1(a). WS2 monolayers were first transferred to the prepared substrate that consists of a 100 nm Ag mirror and a 30 nm MgF2 spacer. Then Ag nanodisks were fabricated on the WS2 monolayer by using E-beam lithography (EBL) and the following lift-off process. Figure 1(b) shows the scanning electron microscopy (SEM) image of Ag nanodisks with a diameter of 90 nm and thickness of 30 nm on top of WS2 monolayers. Ag was chosen because of its strong plasmon resonance and the relatively low dissipation in the visible range. Photoluminescence (PL) and Raman spectra were measured (See Fig. S1 in the Supplementary Information), which confirms that the WS2 used in the experiment is indeed a monolayer. In Fig. 1(c), we can see the Ag plasmon resonance can be tuned to 610 nm with the designed Ag-MgF2 substrate, and overlaps with the A-exciton absorption peak of WS2 monolayers to realize the plasmon-exciton coupling.

Fig. 1 Characterization of the Ag-WS2 heterostructure. (a) Schematic of the Ag-WS2 heterostructure, where Ag nanodisks were fabricated on WS2 monolayers by EBL. The resonance wavelength of the Ag disk is designed to be resonant with the A-exciton in WS2 monolayers. (b) SEM image of the Ag-WS2 heterostructure. The diameter of Ag disk is 90 nm. Scale bar: 150 nm. (c) Normalized reflectivity spectra of pure WS2 monolayers (red line) and bare Ag nanodisks (blue line). The reflectivity spectrum of WS2 monolayers displays a deep dip at ~610 nm, which is consist with the plasmon resonance of Ag nanodisk with diameter of 90 nm. (d) Reflectivity spectra of the Ag-WS2 heterostructure with the diameter of Ag nanodisk increased from 85 to 95 nm.

To systematically investigate the exciton-plasmon interaction of this Ag-WS2 heterostructure, a series of Ag nanodisks with different diameters were fabricated on the WS2 monolayers. Reflectivity spectra of bare Ag nanodisks are shown in Fig. S2. The plasmon resonance of Ag nanodisks can be gradually red-shifted from 540 to 700 nm with the diameter increasing, which is in good agreement with finite-difference time-domain (FDTD) simulation results (See Fig. S3 in the Supplementary Information). Figure 1(d) shows reflectivity spectra of Ag-WS2 heterostructures with the Ag disk diameter changed from 85 to 95 nm. The coupling strength in Ag-WS2 heterostructure is determined by the resonance overlap between the plasmon and excitons. For the nanodisk with diameter of 90 nm, the plasmon resonance is close to the WS2 exciton energy, an obvious Rabi splitting can be observed, and two prominent resonance modes are identified as the lower (LPB) and upper plexciton branch (UPB). The Rabi splitting (Ω) between the UPB and LPB on resonance was measured as 85 meV. By comparing this value with the full width at half maximum (FWHM) of the plasmon (γpl≈320 meV) and exciton (γX≈50 meV) modes, the obtained splitting energy (Ω≈85 meV) is unsatisfied with the strong coupling criterion1, where Ω < (γpl+γX)/2, implying that this plasmon-exciton hybridization is still in the weak coupling regime. The size-dependent dispersion of the Ag-WS2 heterostructure was also calculated by using FDTD solutions. A similar Rabi splitting was observed with the plasmon resonance of the Ag nanodisk getting closer to the exciton of WS2 monolayers at 610 nm, which is consistent well with experimental measurements. (See Fig. S4 in the Supplementary Information).

In order to enhance the plasmon-exciton coupling, and eventually facilitate entering the strong coupling regime, an optical microcavity was manufactured. As shown in Fig. 2(a), by further depositing a 155 nm thickness MgF2 layer and a 20 nm top Ag film onto the fabricated Ag-WS2 heterostructure, we successfully obtained an optical microcavity with the Ag-WS2 heterostructure embedded in MgF2 spacer. Figure 2(b) is the reflection mapping of our cavity sample, and its different layered structures are clearly color distinguished. From the cross section of SEM, the top Ag film, MgF2 spacer, bottom Ag layer, and embedded Ag nanodisks can be detailed characterized, as shown in Fig. 2(c).

Fig. 2 Strong coupling in the Ag-WS2 heterostructure with an optical microcavity. (a) Schematic of the optical microcavity with an embedded Ag-WS2 heterostructure. The microcavity is manufactured in a sandwich structure with a 100 nm-thick Ag layer at the bottom, a 185 nmthick MgF2 layer in the middle, and a 20 nm-thick Ag layer on the top. (b) The reflectivity spectral mapping of the Ag-WS2 heterostructure with optical microcavity. Here the diameter of Ag nanodisk is 95 nm. Scale bar: 10 μm. (c) Cross section view of the Ag-WS2 heterostructure with optical microcavity at a tilted angle 52°, scale bar is 150 nm. (d) Normal-incidence reflectivity spectra of the Ag nanodisk with different sizes directly embedded in the optical microcavity. The Ag nanodisks couple with the optical microcavity, which generates two new hybrid modes with the bandwidth smaller than plasmon resonances of bare Ag nanodisks. (e) Normal-incidence reflectivity spectra of the Ag-WS2 heterostructure with optical microcavity. Three different hybridized modes, as the upper branch, middle branch, and lower branch are observed in reflectivity spectra, which are indicated by black solid lines. Vertical gray dashed lines respectively represents the WS2 A-exciton energy and the resonance energy of bare optical microcavity. (f, g) Expanded views of reflectivity spectral features of the heterostructure with the Ag nanodisk diameter of 110 nm (f) and 95 nm (g).

The optical microcavity was carefully designed to couple well with excitons and plasmons. And the reflectivity spectrum of an empty microcavity was first measured (Fig. S5(a) (solid line) in the Supplementary Information), where a sharp reflection dip centered at 650 nm was achieved, which is resulted from the excitation of the 1st-order of optical cavity mode50. This cavity mode was confirmed with the FDTD simulation (Fig. S5(b) (dotted line) in the Supplementary Information), where the electric field intensity distribution (E/Ein)2 on the xoz plane manifests a huge cavity resonance (Fig. S5(b) in the Supplementary Information).

In order to explore the physics origin of strong light-matter coupling, the same optical microcavity with embedded Ag nanodisks (without the WS2 monolayers) was also investigated. The direct plasmon-photon coupling in this Ag-cavity structure leads to two hybridized modes with a maximum bandwidth of 200 meV as shown in Fig. 2(d). The plasmon-photon coupling strength can be effectively tuned with different sized Ag nanodisks. In comparison with the pure plasmon mode of Ag nanodisk, the Ag-cavity structure shows a higher electric field enhancement and a narrower resonance bandwidth51, which provides a better option for the further strong coupling study.

Figure 2(e) gives reflectivity spectra of optical microcavity with the Ag-WS2 heterostructure embedded in the MgF2 layer. Three different hybridized modes, as the upper branch, middle branch, and lower branch were generated. By increasing the size of Ag nanodisk, the upper branch gradually approaches the A-exciton of the WS2 monolayers, the middle branch shifts away from the A-exciton and gradually approaches the microcavity mode, while the lower branch shifts away from the A-exciton mode. Black solid lines in Fig. 2(e) trace the dispersion of three branches, and anticrossing trends are found for the Ag disk size changed from 90 to 110 nm, which indicates the strong coupling happened in this Ag-WS2 heterostructure that associated with the optical microcavity. To highlight these three branches, the spectrum for 110 nm and 95 nm in Fig. 2(e) are expanded and shown in Figs. 2(f) and 2(g), respectively. When the diameter of Ag nanodisk is 110 nm, the weak coupling was observed, only two hybrid modes can be distinguished. However, for diameter of 95 nm, three pronounced hybridized modes generated by the strong coupling effect corresponding to the upper, middle, and lower branches, which can be clearly observed as shown in Fig. 2(g).

In order to clearly see the mode revolution from the week coupling to strong coupling, FDTD simulations were performed on the same optical microcavity with the bare Ag nanodisk and Ag-WS2 heterostructure, respectively. The hybridization of the Ag plasmon and cavity mode can be easily distinguished in Fig. 3(a) when the disk size changed from 60 to 130 nm. All resonance dips of measured reflectivity spectra (Fig. 2(d)) correspond to the generated hybrid modes in Fig. 3(a), and the spectral resonance shifting reflects the size-dependent dispersion of each hybrid mode. When the Ag nanodisk changed to the Ag-WS2 heterostructure, the size-dependent dispersion mapping shows upper, middle and lower branches, which has a good agreement with measured resonances (Fig. 2(e)) as shown in Fig. 3(b), where the original hybridized states in Fig. 3(a) are coupled with the A-exciton of WS2 monolayers, and achieve a strong Rabi splitting around 610 nm. In comparison with the Ag nanodisk without microcavity, the electric field of the Ag nanodisk at resonance wavelength 610 nm is enhanced almost 4 times when the microcavity is implemented, as shown in Fig. 3(c), which creates a feasible condition for the plasmon-exciton strong coupling. By extracting the simulated reflectivity for disks with size of 95 nm and 110 nm, as denoted by dashed lines in Fig. 3(b), one can see the output spectra (Fig. 3(d)) present an excellent agreement with the measured reflectivity as Figs. 2(f) and 2(g). Due to the fabrication error of microcavity and the difference of dielectric function between experiment and simulation, the observed splitting energy between cavity and plasmon is smaller than simulation results, which results in a redshift in simulated low branch compared to the experimental one.

Fig. 3 FDTD simulation results. (a) Simulated reflectivity spectra of Ag nanodisk embedded in an optical microcavity. White dashed lines represent the resonance wavelength of the empty microcavity (~650 nm). (b) Normalized reflectivity spectra of Ag-WS2 heterostructure with the optical microcavity. Three different energy branches are emerged because of the strong coupling among the optical microcavity, surface plasmons and A-exciton of WS2 monolayers. (c) Electric field intensity distributions (E/Ein)2 on the xoy plane of Ag nanodisk with and without microcavity. (d) Expanded views of the simulated reflectivity spectral feature for disk size of 110 nm (top line) and 95 nm (bottom line) in (b), respectively.

The observed strong coupling of the Ag-WS2 heterostructure with optical microcavity can be further described and analyzed by using coupled oscillator model. Fig. 4(a) is the schematic of plasmon-exciton-cavity three-coupled oscillator model. In the experiment, because the optical microcavity mode is designed far away from the A-exciton of WS2 monolayers, only plasmon-exciton and plasmon-cavity oscillators are connected, and there is no direct coupling between the exciton and cavity photon to generate traditional polaritons.

Fig. 4 Anticrossing behavior of the strong plasmon-exciton-cavity coupling. (a) The three-coupled harmonic oscillator model, which includes the surface plasmons, A-exciton, and microcavity mode as three oscillators. (b) Energies of reflectivity dips as a function of the nanodisk diameter extracted from the reflectivity spectrum. Red dots with error bars show energies obtained from the reflectivity spectrum. The horizontal black dashed lines respectively represent the A-exciton and the microcavity resonant energy. The black slanted short-dashed line represents plasmon resonance mode. Three green solid curves correspond to theoretical fits of hybrid branches based on the three-coupled oscillator model. The error bar represents the standard error of a set of measurements. (c) Hopfield coefficients for plasmon, exciton, and microcavity contributions to upper, middle, and lower hybrid states as a function of diameter, calculated using the three-coupled oscillator model, which provide the weighting of each constituent. (d) Reflectivity spectral linewidths of the upper, middle, and lower branch modes as a function of nanodisk diameter.

The Hamiltonian of this three-coupled system can be presented by a matrix as Eq. (1).

$ \mathop H\limits^ \wedge = \hbar \left[{\begin{array}{*{20}{c}} {{U_{\rm pl}}-{\rm{i}}\frac{{{\gamma _{\rm pl}}}}{2}}&{{g_{\rm MC}}}&{{g_{\rm X}}}\\ {{g_{\rm MC}}}&{{U_{\rm MC}}-{\rm{i}}\frac{{{\gamma _{\rm MC}}}}{2}}&0\\ {{g_{\rm X}}}&0&{{U_{\rm X}}-{\rm{i}}\frac{{{\gamma _{\rm X}}}}{2}} \end{array}} \right], $ (1)

where γpl, γX and γMC are the linewidths of plasmon, exciton, and microcavity modes, Upl, UX, and UMC are the resonance energies of each mode, while gX and gMC represent plasmon-exciton and plasmon-microcavity interaction constants. In three-oscillator model, the eigenstates of Hamiltonian correspond to the three hybrid branches, the eigenvalues of Hamiltonian correspond to the Hopfield coefficients, and the modular square of Hopfield coefficient represents the proportion of uncoupled states in hybrid state. The diagonalization of these Hamiltonians yields three eigenstates and Hopfield coefficients which present the contribution of plasmons, excitons and microcavity modes to each of the state52. With the eigenfrequencies of upper, middle, and lower branches that obtained from experimental reflectivity spectra, the Rabi splitting of anticrossing curves can be fitted with the three-coupled oscillator model. By adjusting the parameters gMC and gX, and then solving the Hamiltonian, the obtained eigenstates of Hamiltonian can be fitted well with the experimental results.

Figure 4(b) depicts resulting dispersion curves of the Ag-WS2 heterostructure that was embedded in the optical microcavity. Three resonance bands, corresponding to the upper, middle, and lower hybrid states are observed. The extracted values from the fitting give gMC≈90 meV and gX≈60 meV. The splitting between each of two branches depends on both plasmon-exciton (gX) and plasmon-microcavity (gMC) coupling strengths, as well as the resonant frequency and linewidths of all contributing parts. Thus, the existed analytical expression cannot obtain simple analytical expressions of the eigenfrequencies in this three-coupled oscillator. In order to evaluate the Rabi splitting of the system and find the trigger condition of strong coupling, the weight of each contribution should be estimated.

As discussed above, the wave function of each hybrid branch is an admixture contribution from the plasmon, cavity and exciton, as $\left| \psi \right\rangle = {\alpha _{{\rm{pl}}}}\left| {{\rm{plasmon}}} \right\rangle + {\alpha _{{\rm{MC}}}}\left| {{\rm{cavity}}} \right\rangle + {\alpha _{\rm{X}}}\left| {{\rm{exciton}}} \right\rangle $. In order to quantify this hybridization, Hopfield coefficients for three branches were calculated as shown in Fig. 4(c), where we can see that each branch consisted of part-plasmon, part-exciton, and part-cavity modes. For the middle branch, a strong hybridization between A-exciton and plasmon is induced by the coupling of cavity mode. A state with comparable weight of plasmon, cavity and A-exciton (|αpl|2=0.24, |αMC|2=|αX|2= 0.38) is obtained at the nanodisk diameter of 99 nm, where all three oscillators intensely participate in the plasmon-exciton-cavity interaction. Therefore, the Ag nanodisk with diameter of 99 nm is considered as the best structure to realize the strongest interaction among these three oscillators, where the corresponding Rabi splitting value Ω≈300 meV is obtained.

On the other side, the linewidths of each hybrid branch also consist of the contributed plasmon, exciton, and cavity modes, and can be written as γi=αplγpl+αMCγMC+αXγX (i=upper, middle, lower). By putting the linewidths (γpl, γMC and γX) into above formula, the relationship of Ag nanodisk size with the linewidths of three hybrid states can be calculated as shown in Fig. 4(d). In the strongest coupling regime as the Ag nanodisk diameter of 99 nm, the linewidths of upper, middle, and lower branches are γUpper= 150 meV, γMiddle= 150 meV, and γLower= 220 meV. Compared to the strong coupling criterion in two coupled oscillator model, the requirement for strong coupling of three coupled oscillators that according to the contribution of each component (plasmon, exciton, and cavity modes), can be written as Ω > Wplγpl+WMCγMC+WXγX (see the detailed derivation in Supplementary Information), where Wpl, WMC, and WX are the weight of each component in strong plasmon-exciton-cavity coupling, while γpl, γMC and γX are the linewidths of plasmon, exciton, and microcavity modes. After calculating above parameters (Supplementary Information), the obtained minimal splitting between upper and lower branches of 300 meV, satisfies the proposed criterion condition, which demonstrates that the strong plasmon-exciton polaritonic hybrid state is successfully generated by combining the Ag-WS2 heterostructure with the optical microcavity.


In summary, we have successfully shown that a large Rabi splitting can be realized under ambient conditions with a high-quality optical microcavity that participated in the plasmon-exciton coupling. The generated plasmon-exciton polaritonic hybrid states can achieve a corresponding Rabi splitting as 300 meV. By calculating the linewidths and Hopfield coefficient of hybridized modes using harmonic oscillator model, the criteria of the strong coupling for three oscillators was proposed for the first time, and the corresponding large Rabi splitting was achieved in the strong coupling regime of the Ag-WS2 heterostructure that was embedded in the optical microscopy. The proposed three-oscillator configuration provides an effective way to develop the strong coupling effect for the practical applications of plexciton polaritonic device at room temperature based on low dimensional materials.

Experimental section

Growth of WS2 monolayers: Low-pressure chemical vapor deposition (LPCVD) method was employed to grow WS2 monolayers on the sapphire substrate (c-plane of α-Al2O3). Sulfur and tungsten trioxide powders were used as the S and W sources, respectively. A mixture flow of Ar (60 sccm) and H2 (10 sccm) was used as the carrier gas. During the CVD growth processes, the sulfur powder was placed on the upstream zone and heated to 100 ℃. Tungsten trioxide powder and substrate were placed on the middle zone of the tube furnace and heated to 887 ℃ with 35 min, and then maintained 60 min for the deposition of the WS2 monolayers. The pressure of the tube furnace was pumped to ~80 Pa. After the growth of WS2, the temperatures were naturally cooled down to room temperature.

Sample fabrication: The substrate was prepared by depositing 100 nm Ag and 30 nm MgF2 successively, which could obviously enhance the optical absorption of WS2 monolayers (Fig. S6, Supplementary Information). The as-grown WS2 monolayers were transferred onto the prepared Ag-MgF2 substrate by a PMMA-assisted method. Then, the designed Ag nanodisks were directly fabricated on WS2 monolayers by E-beam lithography, and there is no spacer between the Ag nanodisk array and the WS2 monolayer. The periods of the Ag nanodisk arrays were set at 230 nm, which makes the surface plasmonic lattice resonances far away from the A-exciton resonance wavelength. Positive resist (MircoChem PMMA A2 950) was spin-coated on the prepared substrate at 3000 rpm for 51 s and then heated on a hot plate for 5 min. The designed structure was drawn by nanometer pattern generation system (NPGS) and exposed by scanning electron microscope (FEI Quanta 450 FEG). A 30 nm Ag film was evaporated by E-beam evaporation system (DE Technology Inc, DE400DH) then lifted off after soaking in acetone for 2 h. Finally, by depositing 155 nm MgF2 and 20 nm Ag, the Ag-WS2 heterostructure with the optical microcavity was achieved.

Simulations and optical measurements: The commercial FDTD solution was used to simulate the optical characterization of strong coupling system. Ag nanodisks with diameter increased from 60 to 130 nm were used to acquire the anti-crossing line shape and the thickness of Ag (up), MgF2 and Ag (bottom) for microcavity analysis was chosen as 20 nm, 185 nm and 100 nm. The refractive index for MgF2 was set as a constant for 1.38. The minimum mesh size for the WS2 monolayers was 0.05 nm and the optical constants for WS2 monolayers were taken from literature and were used without further modification53. A non-polarized plane wave with wavelength from 400 to 900 nm was applied as an input source. In the simulation of Fig. 3(c), the monitor of FDTD solutions is located at the interface between the Ag nanodisk and the WS2 monolayer. In the experiment, the reflectivity spectral mapping was obtained by a commercial microscopic imaging system (HIS V3, CytoViva Co.), and the objective lens is 50× magnifications. The reflectivity spectra of the Ag-WS2 heterostructure with and without microcavity were extracted from the mapping images. The whole experiment was operated at room temperature.


This work is supported by the National Key Research and Development Program of China (Grant No. 2017YFA0205700), National Basic Research Program of China (Grant No. 2015CB932403, 2017YFA0206000), National Natural Science Foundation of China (Grant Nos. 11674012, 61521004, 21790364, 61422501, and 11374023), Beijing Natural Science Foundation (Z180011, and L140007), and Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 201420), National Program for Support of Top-notch Young Professionals (Grant No. W02070003).

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information for this paper is available at https://doi.org/10.29026/oea.2019.190008

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