1 Supporting information Optically Pumped Two-Dimensional MoS2 ...
Supporting information Optically Pumped Two-Dimensional MoS2 Lasers Operating at Room-Temperature Omid Salehzadeh, Mehrdad Djavid, Nhung Hong Tran, Ishiang Shih, and Zetian Mi* Department of Electrical and Computer Engineering, McGill University 3480 University Street, Montreal, Quebec H3A 0E9, Canada *
Correspondence to: E-mail: [email protected]; Phone: 1-514-398-7114
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Contents 1. Population inversion .................................................................................................................................. 3 2. Auger effect: 1L versus 4L........................................................................................................................ 4 3. Microdisk and microsphere whispering gallery modes (WGMs) ............................................................. 5 3a. Microdisk WGMs ................................................................................................................................ 5 3b. Microsphere WGMs ............................................................................................................................ 8 4. Finite-difference time-domain (FDTD) simulation................................................................................... 9 4a. Microsphere at the center of microdisk ............................................................................................... 9 4b. Mode volume ..................................................................................................................................... 10 4c. Optical confinement factor ................................................................................................................ 11 5. Fabrication of microdisk and microsphere cavities ................................................................................. 11 6. Calculation of WGM peak position using Mie’s theory ......................................................................... 13 7. Lasing characteristics of the second order TE mode ............................................................................... 14 8. Rate equation analysis ............................................................................................................................. 14 9. Threshold gain ......................................................................................................................................... 16 10. Gain and differential gain calculations .................................................................................................. 17 11. Excitonic effects .................................................................................................................................... 18 12. Relation between carrier concentration and laser power....................................................................... 19 13. References ............................................................................................................................................. 20
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1. Population inversion Population inversion will occur if the Fermi levels diffuse into the corresponding bands. After combining Eqns. 9 and 10 in ref. 1, Fermi functions (fc and fv) in a 2D structure, considering quantum confinement effect, are given by: 1 exp
exp
exp
1 expa
ħ
!
1
ħ !
1
(1a) (1b)
where mc and mv are the electron and hole effective masses, respectively, mr is the reduced mass, N is the carrier concentration and w is the well width. Figure S1a shows fc – fv as a function of photon energy for carrier concentrations in the range of 1 × 1018 cm-3 to 2.0 × 1019 cm-3 and a well width of 2.8 nm. It is seen that population inversion is achieved for carrier concentrations larger than 5 × 1018 cm-3 where fc – fv has a positive region above the bandgap energy (1.8 eV). The electronic bandgap was determined from room temperature photoluminescence (PL) measurements, shown in Fig. S1b.
Figure S1: Population inversion. a) Plot of fc-fv as a function of photon energy for MoS2 with a thickness of 2.8 nm for carrier concentrations of 1 × 1018 cm-3 to 2.0 × 1019 cm-3 increasing from left to right with an increment of 5.0 × 1018 cm-3. b) RT-PL measurement on a MoS2 flake with a thickness of 2.8 nm. The arrow in (a) shows the approximate position of MoS2 bandgap determined from the PL spectrum.
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2. Auger effect: 1L versus 4L Recently, we reported an unusually strong Auger effect in monolayer MoS2 resulting in efficiency droop at high carrier concentrations (excitation power larger than 0.1 mW).2 Here, we used the model outlined in ref. [2] to evaluate the effect of Auger recombination in as-exfoliated 4-layer MoS2. The result shown in Fig. S2 clearly indicates much weaker efficiency droop in 4L MoS2 at high carrier concentrations compared to monolayer samples. Using the model described in ref. 2, A/B0.5 and C/B1.5 was determined and summarized in Table S1, where A, B, and C represent the nonradiative ShockleyRead-Hall, radiative bimolecular, and Auger recombination coefficients, respectively. We found a similar value of A/B0.5 for 1L and 4L samples indicating that any reduction in surface recombination in 4L sample was compensated by a reduction of its radiative recombination coefficient. However, the value of C/B1.5 was 2 orders of magnitude smaller for 4L compared to that of 1L MoS2. This indicates a much weaker Auger effect in 4L compared to 1L MoS2. Assuming Shockley Read Hall recombination coefficient of A = 2 × 109 s-1, absolute values of Auger (C) and radiative (B) recombination coefficient were estimated to be 1.1 × 1028 cm6·s-1 and 8.7 × 10-10 cm3·s-1 for 4L MoS2, respectively. The results were compared to monolayer data in Table S1. Our observed dependence of excitonic recombination coefficients on material thickness is similar to the results reported for WS2.3
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Figure S2: Auger effect in 1L and 4L MoS2. Plot of the relative external quantum efficiency (IPL/G) as a function of generation rate (G). The solid lines are the fit to data using the model described in ref. 2. Table S1: Summary of the fitting parameters (A/B0.5 and C/B1.5) and estimated values of A, B and C coefficients in 1L and 4L MoS2. 1L 4L
A/B0.5 5.8 ×1013 6.8 ×1013
C/B1.5 3.0 ×10-13 4.3 ×10-15
A (s-1) 1.0 ×1010 2.0 ×109
B (cm3·s-1) 3.0 ×10-8 8.7 ×10-10
C (cm6·s-1) 1.6 ×10-24 1.1 ×10-28
3. Microdisk and microsphere whispering gallery modes (WGMs) 3a. Microdisk WGMs Over 30 deviced were fabricated via electron beam lithography and hunderds via optical lithography. Room temperature micro-PL (514 nm excitation wavelength and 50× objective) was used to study the fabricated microdisks covered with MoS2 or WSe2 flakes with various thicknesses. The laser beam was focused close to the edge of the disk. For monolayer MoS2, the onset of WGM was about 0.8 mW where the device was physically damaged due to laser induced heating and the results could not be reproduced (see Fig. S3). For bilayer materials, weak WGMs were observed at pump powers about 1 mW accompanied with a large red shift due to the laser-induced heating effect, shown in Figs. S4a and b. In
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the free-standing disk geometry, the laser-induced heating is significant due the poor heat dissipation at relatively high excitation power. The samples were damaged at higher powers and therefore systematic studies of the microdisk devices were not feasible. For 4L samples, the results were similar to 2L samples and the pump power could be increased up to 2 mW.
Figure S3: Monolayer microdisk devices. a) Optical microscope (OM) image of a mechanically exfoliated MoS2 on SiO2/Si substrate. The monolayer region is indicated by a dotted loop. b) OM image of the final device fabricated via standard EBL process. c) OM image of a monolayer device after PL measurements at excitation power of 0.8 mW showing that the device was damaged during measurement. d) PL spectrum obtained from a monolayer MoS2 device at excitation power of 0.8 mW. The scale bar is 15 µm.
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Figure S4: Whispering gallery modes from microdisk devices. RT-micro-PL spectra of a microdisk covered by a) 2-layer MoS2 flake and b) 2-layer WSe2 flake using excitation power of 0.1 mW and 1.0 mW. The redshift under high excitation power is due to heating effect. Also, we compared the emission spectra of a 2-layer and a 10-layer MoS2 devices, shown in Fig. S5. As expected indirect transitions, centered at ~ 840 nm, dominated for the thicker MoS2 flake, with a much weaker intensity compared to the 2-layer MoS2 device, resulting in WGMs centered around 830 nm wavelength with Q-factors less than 200. We systematically observed similarly enhanced spontaneous emission from other thick flakes at high excitation powers with peak emissions in the range of 800 to 900 nm (8L and 11 L). However, in the case of the thick MoS2 flakes sitting on an infinite Si/SiO2 substrate, the overall PL emission was very weak. The underlying mechanism for the observed enhanced emission at high excitation powers from thick flakes sitting on SiOx microdisk is not yet clear. The high energy peak in 10L MoS2 and strong emission in 2L MoS2 originate from hot-electron luminescence along ±K points of the Brillioun zone. The observed change in the spontaneous emission and WGMs peak positions with changing the active material and the thickness of the flakes confirm that the observed WGMs originate from the 2D material and not from the SiOx. It is well known that nanoscale Si crystals embedded in SiOx may result in a broad emission in the visible range.4,5 However, the background emission from the SiOx disk (without the presence of 2D materials) was more than two orders of magnitude weaker than the emission from the 2D materails, shown in Fig. S5.
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Figure S5: Effect of the thickness of MoS2 flake. RT-micro-PL spectra of microdisks covered by 2layer and 10-layer MoS2 flakes using an excitation power of 1.0 mW. For comparison, the background emission from SiO2 disk is also plotted. 3b. Microsphere WGMs The WGM characteristics of the silica microspheres sitting on MoS2/SiO2/Si substrate and WSe2/SiO2/Si substrate are shown in Figs. S6a and b, respectively. Detailed measurements on MoS2 sample indicated a clear enhancement of the WGMs with increasing the pump power. The observed trend in Fig. S6a is characteristics of laser devices with a lasing threshold of ~ 0.8 mW. The Q-factors of both devices were poor, in the range of 200 to 400, due to optical loss into the higher index Si substrate. The observed change in the spontaneous emission and WGMs peak positions of WSe2 device compared to MoS2 device confirm that the observed WGMs originate from the 2D material and not from the silica microsphere. In addition, the background emission from the microsphere (without 2D materials) was at least 50 times weaker than the emission from 2D materials, shown in Fig. S6b.
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Figure S6: Whispering gallery modes from microsphere devices. RT-micro-PL spectra of a microsphere sitting on a a) multi-layer MoS2 flake and b) multi-layer WSe2 flake. Excitation power for the WSe2 device was 1.5 mW. For comparison, the background emission from silica microsphere is also plotted in (b). The inset in (a) shows the change of the integrated intensity a WGM (wavelength of 760 nm) with changing the excitation power. 4. Finite-difference time-domain (FDTD) simulation 4a. Microsphere at the center of microdisk For a microsphere sitting at the middle of a MoS2/SiO2 microdisk, schematically shown in Fig. S7a, there is a large optical loss into the SiO2 disk as shown in Fig. S7b. In this configuration, WGMs do not form inside the microdisk and there is no constructive coupling between the microsphere and the microdisk resulting in a poor Q-factor, unlike the case for a microsphere sitting at the rim of the disk. The maximum intensities were observed at the poles of the microsphere, shown in Fig. S7c, and the field distribution was similar on x-z and y-z planes.
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Figure S7: FDTD simulation of a microsphere sitting at the middle of microdisk. a) Schematic configuration of the microsphere and microdisk. Electric distribution b) inside the microdisk (on the x-y plane) and c) inside the microsphere and microdisk (on the y-z plane). 4b. Mode volume The mode volume was calculated numerically via 3D FDTD simulation using:
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"
# $%,',( |*,+,,| -. /0%|$%,',( |*,+,,| |
(2)
where n(x,y,z) and E(x,y,z) are the refractive index and the electric field inside the hybrid cavity. The integral in the numerator is over the entire volume of the hybrid cavity. We obtained an average value of Vm = 0.15 ± 0.05 µm3 in the wavelength range of 630 nm to 700 nm. 4c. Optical confinement factor Optical confinement factor (Γ) was calculated numerically via 3D FDTD simulations, using: 1
# $%,',( |*,+,,| -.
23 #43 $%,',( |*,+,,| -.
(3)
where integral in the numerator and denominator are over the active region of MoS2 and entire cavity volume, respectively. The active region of MoS2 was defined by a disk with a thickness of 2.8 nm and radius of 0.5 µm with its center located at the microsphere/microdisk contact point. From such an analysis, the optical confinement factor was estimated to be ~ 1%. Using monolayer MoS2, confinement factor would reduce to ~ 0.15%. 5. Fabrication of microdisk and microsphere cavities Figure S8a shows an example of a mechanically exfoliated MoS2 flake (using scotch tape) with a thickness of 1.7 nm on a Si substrate coated by a 340 nm thermal SiO2 layer. The thickness of the flake was determined from atomic force microscopy measurements as shown in the inset of Fig. S8a. After coating the sample with 350 nm PMMA A2, standard electron beam lithography (EBL) was used to define an annular opening with inner and outer diameters of 15 µm and 25 µm, respectively. XeF2 gas was used to remove the exposed MoS2 followed by reactive ion etching (RIE) using CF4/CHF3/Ar to selectively remove the oxide, shown in Fig. S8b. Then, Si was etched isotropically using XeF2 to form free standing SiO2 disks supported by a Si post at the middle of the disk, shown in Fig. S8c. Finally, the PMMA was removed using a solvent resist remover. A scanning electron microscopy (SEM) image of the final microdisk device is shown in Fig. S8d. We also used optical lithography to fabricate large arrays of microdisks covered with MoS2 flakes, shown in Fig. S8e. For the devices fabricated using photolithography, the processing was similar to the EBL based process except that the substrates (2” wafers) containing the exfoliated 2D materials were first spin-coated by Shipley 1813 photoresist and then an array of microdisks with diameters of 15 µm was created after UV exposure and development process. In the final step, after removing the resist using a solvent stripper, the sample was exposed to O2 plasma for 1 min to remove the resist residues (RF power of 50 W and chamber pressure of 100 mTorr). Recently, it was found, theoretically and experimentally, that multilayer MoS2 could become direct bandgap after treatment with O2 plasma accompanied with more than an order of magnitude enhancement
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in PL intensity.6 Under our experimental conditions, we only observed significantly enhanced PL intensity after O2 plasma treatment. The silica microspheres suspended in methanol solution, purchased from Cosphereic Company, were dispersed on SiO2/Si substrates containing mechanically exfoliated MoS2 or WSe2 flakes. Then, the sample was inspected by optical microscope and the flakes located beneath the isolated spheres were identified for photoluminescence (PL) measurements, shown in Fig. 8f.
Figure S8: Fabrication of microdisk and microsphere cavities. (a) Optical microscope (OM) image of a mechanically exfoliated MoS2 on SiO2/Si substrate. The inset in (a) shows an AFM image of the edge of the MoS2 flake. (b) OM image of the sample after defining an annular opening using standard EBL process and removing the oxide using RIE process. (c) Optical and (d) SEM images of the sample after etching the Si isotropically using XeF2. (e) OM image of an array of microdisks formed using optical lithography. (f) OM image of a silica microsphere sitting on a MoS2 flake. The scale bars are 15 µm.
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6. Calculation of WGM peak position using Mie’s theory According to Mie’s theory6, the positions of WGMs, with angular mode number of l confined in a microsphere with radius of R (3.85 µm) and refractive index of nc (~1.5 for silica) surrounded by a medium with refractive index of n0 (= 1 for air), are the roots of: F(λ ) = (ηα + l/R)× jl(kncR) - kns × jl+1(kncR) (4a) where η = 1 for TE and (ns/n0)2 for TM modes and: β = [l(l+1)] 0.5/R
(4b)
α = (β2 - k2n02)0.5
(4c)
k = 2π/λ
(5d)
l = 2πRneff/λ
(5e)
where neff (~1.43, see the main manuscript) is the effective refractive index of WGMs. An example of such a calculation is shown in Fig. S9 for the TE mode with l = 51. The first root at 658 nm from right corresponds to n = 1 radial mode. Similarly, the position of the second order mode is determined by the second root from right and so on.
Figure S9: Calculation of WGM peak position using Mie’s theory. Plot of F(λ) versus wavelength for angular mode number of l = 51. The roots indicate the positions of the WGMs.
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7. Lasing characteristics of the second order TE mode Figure S10 shows the variation of the integrated intensity of the TE245 mode with pump power indicating an abrupt change of the slope above ~ 8 µW accompanied with a clear spectral narrowing from 0.39 ± 0.02 nm to 0.27 ± 0.02 nm. These characteristics confirms the achievement of lasing.
Figure S10: Lasing characteristics of the second order TE mode of a MoS2 device. Plot of the variation of integrated intensity (left axis, solid symbols) and FWHM (right axis, open symbols) of TE245 mode as a function of pump power. Error bars are standard deviations of 10 measurements. 8. Rate equation analysis Rate equation analysis was carried out to fit the experimental integrated PL intensity versus pump power data to determine the β-factor of the coupled microsphere/microdisk device. Carrier density in the active region (N) and photon density in the cavity mode (P) under incident optical pumping of Pin are described by7,8: - -5 -7 -5
7
6 ħ:89. = ? @= A 1BC D=E ;
7
1BC D=E F G>= ?
(5a) (5b)
;
where η and ħωp are the fraction of the absorbed pump power and the incident photon energy, respectively. A, B and C are the surface recombination velocity, radiative bimolecular recombination coefficient and nonradiative Auger recombination coefficient, respectively. Γ is confinement factor. vg = c/neff is the group velocity. neff is the effective refractive index ~1.43. τp is photon lifetime. Logarithmic modal gain was assumed:7,8 LM
D DH ln
N LM
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where Ntr is the transparency carrier density and g0 was determined as a fitting parameter. Ns is a shift to force the natural logarithm to be finite at N = 0. The theoretical curves (See Fig. S11) fit the experimental data with the best fitting parameters summarized in Table S2.
Table S2: Summary of the fitting parameters of the rate equation analysis. TE51 TE52 TE57 TE58
A(s-1) 2.20 × 109 2.10× 109 1.95× 109 1.94× 109
B (s-1.cm3) 4.0 × 10-10 4.0 × 10-10 4.0 × 10-10 4.0 × 10-10
C (s-1.cm6) 5.0 × 10-29 5.0 × 10-29 5.0 × 10-29 5.0 × 10-29
β 0.67 0.68 0.60 0.61
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Ntr (cm-3) 1.2 × 1018 1.2 × 1018 1.3 × 1018 1.3 × 1018
Ns (cm-3) 6 × 1016 6 × 1016 6 × 1016 6 × 1016
g0 (cm-1) 3900 3900 3900 3900
η 0.2 0.2 0.2 0.2
Figure S11: Rate equation analysis. Plot of the integrated intensity versus pump power of mode a) TE151, b) TE152, c) TE157 and d) TE158 along with the fitting curves using different values of β. Error bars represent the standard deviation of 10 measurements.
9. Threshold gain In a WGM cavity, the total round-trip loss (δc) can be determined from the experimentally obtained value of the Q-factor, as follows. The energy stored in the cavity decays exponentially with time according to I = I0 e-t/τ, where τ = Q/ω is the cavity photon lifetime and ω=2πc/λ. After a time equal to one round-trip (tr = neffLc/c, Lc and neff are the optical path length and effective refractive index of the cavity, respectively), the energy will reduce to:
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O OH P
Q9RSS TU
V
(6a)
From Eq. 6a, one can define the total round-trip loss as: W
?$RSS
(6b)
XY
Lasing threshold is achieved when the cavity loss (δc) is compensated by the material gain (Γg), giving threshold gain (gth) of: D5Z
?$RSS
(6c)
[YX
10. Gain and differential gain calculations Taking into account the homogeneous broadening and only the contribution from the first subband, the linear gain (g) in a quantum-well can be expressed by1: D
g ? 〈` 〉 \ # ħ ^ ħ h
]
ħ e-f d89 ħ f Lc e d89
b b! c
(7)
where E is the energy of light, Ech is the transition energy, Eg (= 1.8 eV) is the bandgap energy, w (=2.8 nm) is the well width, mr (= 0.2 m0)9 is the reduced electron-hole effective mass, µ is the permeability, ε is the dielectric constant, and τin (~ 2 ps)10 is the intraband relaxation time. Dielectric constant was calculated using an average MoS2 refractive index11 of nact = 5.5. is the transition dipole moment given by: mħ
h
< ` ? > 1.5? ? f
±
o9 p o9 2o4oh p
(8)
?
where plus and minus signs correspond to TE and TM polarizations, respectively, and Ecn is the quantization energy given by: ħ
$ ?
(9)
To calculate the differential gain, the peak gain was determined numerically for a given carrier concentration and then dg/dn was calculated. As shown in Fig. S12a, the peak gain is expected to be as large as 7700 cm-1 at carrier concentration of 1020 cm-3. The differential gain was calculated to be 3.5 × 10-16 cm2 at the threshold power (Fig. S12b).
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Figure S12: Gain and differential gain analysis. (a) Calculated room-temperature material gain spectra of 4L MoS2 for carrier concentrations of 1 × 1019 cm-3 to 5 × 1019 cm-3. Overlaid in (a) is the calculated threshold gain of TE152 (short dashed line) and TE245 (long dashed line) modes. (b) Plot of the calculated maximum attainable gain and differential gain as a function of carrier concentration. 11. Excitonic effects Excitonic effects are expected in monolayer TMDCs where the layer thickness is less than Bohr radius.13-15 However, such effects are extremely sensitive to the sample preparation and environment dielectric screening. For example, according to ref. 15, the binding energy of monolayer MoS2 is about 870 meV in vacuum and 450 meV with one surface in contact with SiO2.15 In our case, the samples were sandwiched between two SiO2 surfaces and therefore the binding energy of a monolayer may be reduced to ~200 meV for monolayer MoS2. Furthermore, the thickness of 4L MoS2 is above the Bohr radius and therefore confinement effects and consequently the excitonic effects may not be significant. For further clarification, we carried out detailed temperature dependent PL measurements on 4L MoS2 prepared via mechanical exfoliation on Si/SiO2 substrate. Variations of the integrated PL intensity with temperature are plotted in Fig. S13 (for mode A). The observed quenching of the PL intensity can be well fitted using an Arrhenius-type formula, O q OH /s1 >PtuH /vqw with a single activation energy of E0 = 55 ± 5 meV. As expected, this value is smaller than that of a 1L MoS2 (~ 100 meV).2 Therefore, based on our previous discussions, we can conclude that the binding energy of 4L MoS2 sandwich between two SiO2 surfaces to be about 27 meV in this study. Therefore, excitonic effects should be negligible in our experiments.
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Figure S13: Exciton binding energy of 4L MoS2. Plot of the integrated intensity (mode A) as a function of (kT)-1. The solid line is a fit to the data as discussed in the text.
12. Relation between carrier concentration and laser power The required laser power (P) to create carrier concentration of N can be calculated by: E
xM;yN Zz{ |}
~
xM;yN Zz{
(10)
|} F
where G is the generation rate, τ is the recombination lifetime, α ~ 0.1/layer = 1.5 × 106 cm-1 is the absorption coefficient of MoS2,12 Aspot = 7 × 10-10 cm-2 (considering the lensing effect from the microsphere) is the laser spot size, and R = 0.4 is the reflection coefficient of MoS2 and νL is the incident laser frequency. Considering an estimated overall lifetime of 0.3 ns, using Eq. 10, the estimated Nth of 3× 1019 cm-3 translates to Pth of ~ 20 µW. The estimated lifetime was obtained using the radiative and nonradiative recombination coefficients obtained in section 2. Our estimated lifetime is consistent with the experimental work of Shi et al. on multilayer MoS2 (~ 0.3 ns).10
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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Asada, M.; Kameyama, A.; Suematsu, Y. IEEE J. Quantum Electron. 1984, 20, 745-753. Salehzadeh, O.; Tran, N. H.; Liu, X.; Shih, I.; Mi, Z. Nano Lett. 2014, 14, 4125-4130. Yuan, L; Huang, Libai.Nanoscale 2015, 7, 7402-7408. Min, K. S., Shcheglov, K. V., Yang, C. M., Atwater, H.A., Brongersma, M. L., & Polman, A. Appl. Phys. Lett. 1996, 69, 2033-2035 Makimura, T., Uematsu, H., Okada, Y., Murakami, K. J. Phys: Conference Series 2007, 59, 466469. Mie, G. Beitrage zur Optik truber Medien. Ann. Phys. 1908, 25, 377-445. Coldren, L. A., Corzine S. W., Masanovic, M. L. Diode Lasers and Photonic Integrated Circuits. Second edition, Wiley, 2012. Chen, R.; Tran, T.-T. D. ; Ng, K. W. ; Ko, W. S. ; Chuang, L. C. ; Sedgwik, F. G.; ChangHasnain, C. Nature Photon. 2011, 5, 170-175. Peelaers, H.; Van de Walle, C. G. Phys. Rev. B 2012, 86, 241401-1 – 24401-5. Shi, H., Yan, R., Bertolazzi, S., Brivio, J., Gao, B., Kis, A., Jena, D., Xing, H. G., Huang, L. ACS Nano 2013, 7, 1072-1080 . Castellanos-Gomez, A.; Agrait, N.; Rubio-Bollinger, G. Appl. Phys. Lett. 2010, 96, 213116-1 – 213116-3. Mak, K. F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T. F. Phys. Rev. Lett. 2010, 105, 136805-1 – 1306805-5. Shi, H.; Pan, H.; Zhnag, Y.-W.; Yakobson B. I. Phys. Rev. B 2013, 87, 155304. Qiu, D. Y., Felipe, H., Louie, S. G. Phys. Rev. Lett. 2013, 111, 216805-1 - 216805-5. Berghäuser, G., Malic, E. Phys. Rev. B 2014, 89, 125309-1 – 125309-6.
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Correspondence to: E-mail: [email protected]; Phone: 1-514-398-7114
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Contents 1. Population inversion .................................................................................................................................. 3 2. Auger effect: 1L versus 4L........................................................................................................................ 4 3. Microdisk and microsphere whispering gallery modes (WGMs) ............................................................. 5 3a. Microdisk WGMs ................................................................................................................................ 5 3b. Microsphere WGMs ............................................................................................................................ 8 4. Finite-difference time-domain (FDTD) simulation................................................................................... 9 4a. Microsphere at the center of microdisk ............................................................................................... 9 4b. Mode volume ..................................................................................................................................... 10 4c. Optical confinement factor ................................................................................................................ 11 5. Fabrication of microdisk and microsphere cavities ................................................................................. 11 6. Calculation of WGM peak position using Mie’s theory ......................................................................... 13 7. Lasing characteristics of the second order TE mode ............................................................................... 14 8. Rate equation analysis ............................................................................................................................. 14 9. Threshold gain ......................................................................................................................................... 16 10. Gain and differential gain calculations .................................................................................................. 17 11. Excitonic effects .................................................................................................................................... 18 12. Relation between carrier concentration and laser power....................................................................... 19 13. References ............................................................................................................................................. 20
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1. Population inversion Population inversion will occur if the Fermi levels diffuse into the corresponding bands. After combining Eqns. 9 and 10 in ref. 1, Fermi functions (fc and fv) in a 2D structure, considering quantum confinement effect, are given by: 1 exp
exp
exp
1 expa
ħ
!
1
ħ !
1
(1a) (1b)
where mc and mv are the electron and hole effective masses, respectively, mr is the reduced mass, N is the carrier concentration and w is the well width. Figure S1a shows fc – fv as a function of photon energy for carrier concentrations in the range of 1 × 1018 cm-3 to 2.0 × 1019 cm-3 and a well width of 2.8 nm. It is seen that population inversion is achieved for carrier concentrations larger than 5 × 1018 cm-3 where fc – fv has a positive region above the bandgap energy (1.8 eV). The electronic bandgap was determined from room temperature photoluminescence (PL) measurements, shown in Fig. S1b.
Figure S1: Population inversion. a) Plot of fc-fv as a function of photon energy for MoS2 with a thickness of 2.8 nm for carrier concentrations of 1 × 1018 cm-3 to 2.0 × 1019 cm-3 increasing from left to right with an increment of 5.0 × 1018 cm-3. b) RT-PL measurement on a MoS2 flake with a thickness of 2.8 nm. The arrow in (a) shows the approximate position of MoS2 bandgap determined from the PL spectrum.
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2. Auger effect: 1L versus 4L Recently, we reported an unusually strong Auger effect in monolayer MoS2 resulting in efficiency droop at high carrier concentrations (excitation power larger than 0.1 mW).2 Here, we used the model outlined in ref. [2] to evaluate the effect of Auger recombination in as-exfoliated 4-layer MoS2. The result shown in Fig. S2 clearly indicates much weaker efficiency droop in 4L MoS2 at high carrier concentrations compared to monolayer samples. Using the model described in ref. 2, A/B0.5 and C/B1.5 was determined and summarized in Table S1, where A, B, and C represent the nonradiative ShockleyRead-Hall, radiative bimolecular, and Auger recombination coefficients, respectively. We found a similar value of A/B0.5 for 1L and 4L samples indicating that any reduction in surface recombination in 4L sample was compensated by a reduction of its radiative recombination coefficient. However, the value of C/B1.5 was 2 orders of magnitude smaller for 4L compared to that of 1L MoS2. This indicates a much weaker Auger effect in 4L compared to 1L MoS2. Assuming Shockley Read Hall recombination coefficient of A = 2 × 109 s-1, absolute values of Auger (C) and radiative (B) recombination coefficient were estimated to be 1.1 × 1028 cm6·s-1 and 8.7 × 10-10 cm3·s-1 for 4L MoS2, respectively. The results were compared to monolayer data in Table S1. Our observed dependence of excitonic recombination coefficients on material thickness is similar to the results reported for WS2.3
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Figure S2: Auger effect in 1L and 4L MoS2. Plot of the relative external quantum efficiency (IPL/G) as a function of generation rate (G). The solid lines are the fit to data using the model described in ref. 2. Table S1: Summary of the fitting parameters (A/B0.5 and C/B1.5) and estimated values of A, B and C coefficients in 1L and 4L MoS2. 1L 4L
A/B0.5 5.8 ×1013 6.8 ×1013
C/B1.5 3.0 ×10-13 4.3 ×10-15
A (s-1) 1.0 ×1010 2.0 ×109
B (cm3·s-1) 3.0 ×10-8 8.7 ×10-10
C (cm6·s-1) 1.6 ×10-24 1.1 ×10-28
3. Microdisk and microsphere whispering gallery modes (WGMs) 3a. Microdisk WGMs Over 30 deviced were fabricated via electron beam lithography and hunderds via optical lithography. Room temperature micro-PL (514 nm excitation wavelength and 50× objective) was used to study the fabricated microdisks covered with MoS2 or WSe2 flakes with various thicknesses. The laser beam was focused close to the edge of the disk. For monolayer MoS2, the onset of WGM was about 0.8 mW where the device was physically damaged due to laser induced heating and the results could not be reproduced (see Fig. S3). For bilayer materials, weak WGMs were observed at pump powers about 1 mW accompanied with a large red shift due to the laser-induced heating effect, shown in Figs. S4a and b. In
5
the free-standing disk geometry, the laser-induced heating is significant due the poor heat dissipation at relatively high excitation power. The samples were damaged at higher powers and therefore systematic studies of the microdisk devices were not feasible. For 4L samples, the results were similar to 2L samples and the pump power could be increased up to 2 mW.
Figure S3: Monolayer microdisk devices. a) Optical microscope (OM) image of a mechanically exfoliated MoS2 on SiO2/Si substrate. The monolayer region is indicated by a dotted loop. b) OM image of the final device fabricated via standard EBL process. c) OM image of a monolayer device after PL measurements at excitation power of 0.8 mW showing that the device was damaged during measurement. d) PL spectrum obtained from a monolayer MoS2 device at excitation power of 0.8 mW. The scale bar is 15 µm.
6
Figure S4: Whispering gallery modes from microdisk devices. RT-micro-PL spectra of a microdisk covered by a) 2-layer MoS2 flake and b) 2-layer WSe2 flake using excitation power of 0.1 mW and 1.0 mW. The redshift under high excitation power is due to heating effect. Also, we compared the emission spectra of a 2-layer and a 10-layer MoS2 devices, shown in Fig. S5. As expected indirect transitions, centered at ~ 840 nm, dominated for the thicker MoS2 flake, with a much weaker intensity compared to the 2-layer MoS2 device, resulting in WGMs centered around 830 nm wavelength with Q-factors less than 200. We systematically observed similarly enhanced spontaneous emission from other thick flakes at high excitation powers with peak emissions in the range of 800 to 900 nm (8L and 11 L). However, in the case of the thick MoS2 flakes sitting on an infinite Si/SiO2 substrate, the overall PL emission was very weak. The underlying mechanism for the observed enhanced emission at high excitation powers from thick flakes sitting on SiOx microdisk is not yet clear. The high energy peak in 10L MoS2 and strong emission in 2L MoS2 originate from hot-electron luminescence along ±K points of the Brillioun zone. The observed change in the spontaneous emission and WGMs peak positions with changing the active material and the thickness of the flakes confirm that the observed WGMs originate from the 2D material and not from the SiOx. It is well known that nanoscale Si crystals embedded in SiOx may result in a broad emission in the visible range.4,5 However, the background emission from the SiOx disk (without the presence of 2D materials) was more than two orders of magnitude weaker than the emission from the 2D materails, shown in Fig. S5.
7
Figure S5: Effect of the thickness of MoS2 flake. RT-micro-PL spectra of microdisks covered by 2layer and 10-layer MoS2 flakes using an excitation power of 1.0 mW. For comparison, the background emission from SiO2 disk is also plotted. 3b. Microsphere WGMs The WGM characteristics of the silica microspheres sitting on MoS2/SiO2/Si substrate and WSe2/SiO2/Si substrate are shown in Figs. S6a and b, respectively. Detailed measurements on MoS2 sample indicated a clear enhancement of the WGMs with increasing the pump power. The observed trend in Fig. S6a is characteristics of laser devices with a lasing threshold of ~ 0.8 mW. The Q-factors of both devices were poor, in the range of 200 to 400, due to optical loss into the higher index Si substrate. The observed change in the spontaneous emission and WGMs peak positions of WSe2 device compared to MoS2 device confirm that the observed WGMs originate from the 2D material and not from the silica microsphere. In addition, the background emission from the microsphere (without 2D materials) was at least 50 times weaker than the emission from 2D materials, shown in Fig. S6b.
8
Figure S6: Whispering gallery modes from microsphere devices. RT-micro-PL spectra of a microsphere sitting on a a) multi-layer MoS2 flake and b) multi-layer WSe2 flake. Excitation power for the WSe2 device was 1.5 mW. For comparison, the background emission from silica microsphere is also plotted in (b). The inset in (a) shows the change of the integrated intensity a WGM (wavelength of 760 nm) with changing the excitation power. 4. Finite-difference time-domain (FDTD) simulation 4a. Microsphere at the center of microdisk For a microsphere sitting at the middle of a MoS2/SiO2 microdisk, schematically shown in Fig. S7a, there is a large optical loss into the SiO2 disk as shown in Fig. S7b. In this configuration, WGMs do not form inside the microdisk and there is no constructive coupling between the microsphere and the microdisk resulting in a poor Q-factor, unlike the case for a microsphere sitting at the rim of the disk. The maximum intensities were observed at the poles of the microsphere, shown in Fig. S7c, and the field distribution was similar on x-z and y-z planes.
9
Figure S7: FDTD simulation of a microsphere sitting at the middle of microdisk. a) Schematic configuration of the microsphere and microdisk. Electric distribution b) inside the microdisk (on the x-y plane) and c) inside the microsphere and microdisk (on the y-z plane). 4b. Mode volume The mode volume was calculated numerically via 3D FDTD simulation using:
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"
# $%,',( |*,+,,| -. /0%|$%,',( |*,+,,| |
(2)
where n(x,y,z) and E(x,y,z) are the refractive index and the electric field inside the hybrid cavity. The integral in the numerator is over the entire volume of the hybrid cavity. We obtained an average value of Vm = 0.15 ± 0.05 µm3 in the wavelength range of 630 nm to 700 nm. 4c. Optical confinement factor Optical confinement factor (Γ) was calculated numerically via 3D FDTD simulations, using: 1
# $%,',( |*,+,,| -.
23 #43 $%,',( |*,+,,| -.
(3)
where integral in the numerator and denominator are over the active region of MoS2 and entire cavity volume, respectively. The active region of MoS2 was defined by a disk with a thickness of 2.8 nm and radius of 0.5 µm with its center located at the microsphere/microdisk contact point. From such an analysis, the optical confinement factor was estimated to be ~ 1%. Using monolayer MoS2, confinement factor would reduce to ~ 0.15%. 5. Fabrication of microdisk and microsphere cavities Figure S8a shows an example of a mechanically exfoliated MoS2 flake (using scotch tape) with a thickness of 1.7 nm on a Si substrate coated by a 340 nm thermal SiO2 layer. The thickness of the flake was determined from atomic force microscopy measurements as shown in the inset of Fig. S8a. After coating the sample with 350 nm PMMA A2, standard electron beam lithography (EBL) was used to define an annular opening with inner and outer diameters of 15 µm and 25 µm, respectively. XeF2 gas was used to remove the exposed MoS2 followed by reactive ion etching (RIE) using CF4/CHF3/Ar to selectively remove the oxide, shown in Fig. S8b. Then, Si was etched isotropically using XeF2 to form free standing SiO2 disks supported by a Si post at the middle of the disk, shown in Fig. S8c. Finally, the PMMA was removed using a solvent resist remover. A scanning electron microscopy (SEM) image of the final microdisk device is shown in Fig. S8d. We also used optical lithography to fabricate large arrays of microdisks covered with MoS2 flakes, shown in Fig. S8e. For the devices fabricated using photolithography, the processing was similar to the EBL based process except that the substrates (2” wafers) containing the exfoliated 2D materials were first spin-coated by Shipley 1813 photoresist and then an array of microdisks with diameters of 15 µm was created after UV exposure and development process. In the final step, after removing the resist using a solvent stripper, the sample was exposed to O2 plasma for 1 min to remove the resist residues (RF power of 50 W and chamber pressure of 100 mTorr). Recently, it was found, theoretically and experimentally, that multilayer MoS2 could become direct bandgap after treatment with O2 plasma accompanied with more than an order of magnitude enhancement
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in PL intensity.6 Under our experimental conditions, we only observed significantly enhanced PL intensity after O2 plasma treatment. The silica microspheres suspended in methanol solution, purchased from Cosphereic Company, were dispersed on SiO2/Si substrates containing mechanically exfoliated MoS2 or WSe2 flakes. Then, the sample was inspected by optical microscope and the flakes located beneath the isolated spheres were identified for photoluminescence (PL) measurements, shown in Fig. 8f.
Figure S8: Fabrication of microdisk and microsphere cavities. (a) Optical microscope (OM) image of a mechanically exfoliated MoS2 on SiO2/Si substrate. The inset in (a) shows an AFM image of the edge of the MoS2 flake. (b) OM image of the sample after defining an annular opening using standard EBL process and removing the oxide using RIE process. (c) Optical and (d) SEM images of the sample after etching the Si isotropically using XeF2. (e) OM image of an array of microdisks formed using optical lithography. (f) OM image of a silica microsphere sitting on a MoS2 flake. The scale bars are 15 µm.
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6. Calculation of WGM peak position using Mie’s theory According to Mie’s theory6, the positions of WGMs, with angular mode number of l confined in a microsphere with radius of R (3.85 µm) and refractive index of nc (~1.5 for silica) surrounded by a medium with refractive index of n0 (= 1 for air), are the roots of: F(λ ) = (ηα + l/R)× jl(kncR) - kns × jl+1(kncR) (4a) where η = 1 for TE and (ns/n0)2 for TM modes and: β = [l(l+1)] 0.5/R
(4b)
α = (β2 - k2n02)0.5
(4c)
k = 2π/λ
(5d)
l = 2πRneff/λ
(5e)
where neff (~1.43, see the main manuscript) is the effective refractive index of WGMs. An example of such a calculation is shown in Fig. S9 for the TE mode with l = 51. The first root at 658 nm from right corresponds to n = 1 radial mode. Similarly, the position of the second order mode is determined by the second root from right and so on.
Figure S9: Calculation of WGM peak position using Mie’s theory. Plot of F(λ) versus wavelength for angular mode number of l = 51. The roots indicate the positions of the WGMs.
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7. Lasing characteristics of the second order TE mode Figure S10 shows the variation of the integrated intensity of the TE245 mode with pump power indicating an abrupt change of the slope above ~ 8 µW accompanied with a clear spectral narrowing from 0.39 ± 0.02 nm to 0.27 ± 0.02 nm. These characteristics confirms the achievement of lasing.
Figure S10: Lasing characteristics of the second order TE mode of a MoS2 device. Plot of the variation of integrated intensity (left axis, solid symbols) and FWHM (right axis, open symbols) of TE245 mode as a function of pump power. Error bars are standard deviations of 10 measurements. 8. Rate equation analysis Rate equation analysis was carried out to fit the experimental integrated PL intensity versus pump power data to determine the β-factor of the coupled microsphere/microdisk device. Carrier density in the active region (N) and photon density in the cavity mode (P) under incident optical pumping of Pin are described by7,8: - -5 -7 -5
7
6 ħ:89. = ? @= A 1BC D=E ;
7
1BC D=E F G>= ?
(5a) (5b)
;
where η and ħωp are the fraction of the absorbed pump power and the incident photon energy, respectively. A, B and C are the surface recombination velocity, radiative bimolecular recombination coefficient and nonradiative Auger recombination coefficient, respectively. Γ is confinement factor. vg = c/neff is the group velocity. neff is the effective refractive index ~1.43. τp is photon lifetime. Logarithmic modal gain was assumed:7,8 LM
D DH ln
N LM
14
where Ntr is the transparency carrier density and g0 was determined as a fitting parameter. Ns is a shift to force the natural logarithm to be finite at N = 0. The theoretical curves (See Fig. S11) fit the experimental data with the best fitting parameters summarized in Table S2.
Table S2: Summary of the fitting parameters of the rate equation analysis. TE51 TE52 TE57 TE58
A(s-1) 2.20 × 109 2.10× 109 1.95× 109 1.94× 109
B (s-1.cm3) 4.0 × 10-10 4.0 × 10-10 4.0 × 10-10 4.0 × 10-10
C (s-1.cm6) 5.0 × 10-29 5.0 × 10-29 5.0 × 10-29 5.0 × 10-29
β 0.67 0.68 0.60 0.61
15
Ntr (cm-3) 1.2 × 1018 1.2 × 1018 1.3 × 1018 1.3 × 1018
Ns (cm-3) 6 × 1016 6 × 1016 6 × 1016 6 × 1016
g0 (cm-1) 3900 3900 3900 3900
η 0.2 0.2 0.2 0.2
Figure S11: Rate equation analysis. Plot of the integrated intensity versus pump power of mode a) TE151, b) TE152, c) TE157 and d) TE158 along with the fitting curves using different values of β. Error bars represent the standard deviation of 10 measurements.
9. Threshold gain In a WGM cavity, the total round-trip loss (δc) can be determined from the experimentally obtained value of the Q-factor, as follows. The energy stored in the cavity decays exponentially with time according to I = I0 e-t/τ, where τ = Q/ω is the cavity photon lifetime and ω=2πc/λ. After a time equal to one round-trip (tr = neffLc/c, Lc and neff are the optical path length and effective refractive index of the cavity, respectively), the energy will reduce to:
16
O OH P
Q9RSS TU
V
(6a)
From Eq. 6a, one can define the total round-trip loss as: W
?$RSS
(6b)
XY
Lasing threshold is achieved when the cavity loss (δc) is compensated by the material gain (Γg), giving threshold gain (gth) of: D5Z
?$RSS
(6c)
[YX
10. Gain and differential gain calculations Taking into account the homogeneous broadening and only the contribution from the first subband, the linear gain (g) in a quantum-well can be expressed by1: D
g ? 〈` 〉 \ # ħ ^ ħ h
]
ħ e-f d89 ħ f Lc e d89
b b! c
(7)
where E is the energy of light, Ech is the transition energy, Eg (= 1.8 eV) is the bandgap energy, w (=2.8 nm) is the well width, mr (= 0.2 m0)9 is the reduced electron-hole effective mass, µ is the permeability, ε is the dielectric constant, and τin (~ 2 ps)10 is the intraband relaxation time. Dielectric constant was calculated using an average MoS2 refractive index11 of nact = 5.5. is the transition dipole moment given by: mħ
h
< ` ? > 1.5? ? f
±
o9 p o9 2o4oh p
(8)
?
where plus and minus signs correspond to TE and TM polarizations, respectively, and Ecn is the quantization energy given by: ħ
$ ?
(9)
To calculate the differential gain, the peak gain was determined numerically for a given carrier concentration and then dg/dn was calculated. As shown in Fig. S12a, the peak gain is expected to be as large as 7700 cm-1 at carrier concentration of 1020 cm-3. The differential gain was calculated to be 3.5 × 10-16 cm2 at the threshold power (Fig. S12b).
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Figure S12: Gain and differential gain analysis. (a) Calculated room-temperature material gain spectra of 4L MoS2 for carrier concentrations of 1 × 1019 cm-3 to 5 × 1019 cm-3. Overlaid in (a) is the calculated threshold gain of TE152 (short dashed line) and TE245 (long dashed line) modes. (b) Plot of the calculated maximum attainable gain and differential gain as a function of carrier concentration. 11. Excitonic effects Excitonic effects are expected in monolayer TMDCs where the layer thickness is less than Bohr radius.13-15 However, such effects are extremely sensitive to the sample preparation and environment dielectric screening. For example, according to ref. 15, the binding energy of monolayer MoS2 is about 870 meV in vacuum and 450 meV with one surface in contact with SiO2.15 In our case, the samples were sandwiched between two SiO2 surfaces and therefore the binding energy of a monolayer may be reduced to ~200 meV for monolayer MoS2. Furthermore, the thickness of 4L MoS2 is above the Bohr radius and therefore confinement effects and consequently the excitonic effects may not be significant. For further clarification, we carried out detailed temperature dependent PL measurements on 4L MoS2 prepared via mechanical exfoliation on Si/SiO2 substrate. Variations of the integrated PL intensity with temperature are plotted in Fig. S13 (for mode A). The observed quenching of the PL intensity can be well fitted using an Arrhenius-type formula, O q OH /s1 >PtuH /vqw with a single activation energy of E0 = 55 ± 5 meV. As expected, this value is smaller than that of a 1L MoS2 (~ 100 meV).2 Therefore, based on our previous discussions, we can conclude that the binding energy of 4L MoS2 sandwich between two SiO2 surfaces to be about 27 meV in this study. Therefore, excitonic effects should be negligible in our experiments.
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Figure S13: Exciton binding energy of 4L MoS2. Plot of the integrated intensity (mode A) as a function of (kT)-1. The solid line is a fit to the data as discussed in the text.
12. Relation between carrier concentration and laser power The required laser power (P) to create carrier concentration of N can be calculated by: E
xM;yN Zz{ |}
~
xM;yN Zz{
(10)
|} F
where G is the generation rate, τ is the recombination lifetime, α ~ 0.1/layer = 1.5 × 106 cm-1 is the absorption coefficient of MoS2,12 Aspot = 7 × 10-10 cm-2 (considering the lensing effect from the microsphere) is the laser spot size, and R = 0.4 is the reflection coefficient of MoS2 and νL is the incident laser frequency. Considering an estimated overall lifetime of 0.3 ns, using Eq. 10, the estimated Nth of 3× 1019 cm-3 translates to Pth of ~ 20 µW. The estimated lifetime was obtained using the radiative and nonradiative recombination coefficients obtained in section 2. Our estimated lifetime is consistent with the experimental work of Shi et al. on multilayer MoS2 (~ 0.3 ns).10
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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Asada, M.; Kameyama, A.; Suematsu, Y. IEEE J. Quantum Electron. 1984, 20, 745-753. Salehzadeh, O.; Tran, N. H.; Liu, X.; Shih, I.; Mi, Z. Nano Lett. 2014, 14, 4125-4130. Yuan, L; Huang, Libai.Nanoscale 2015, 7, 7402-7408. Min, K. S., Shcheglov, K. V., Yang, C. M., Atwater, H.A., Brongersma, M. L., & Polman, A. Appl. Phys. Lett. 1996, 69, 2033-2035 Makimura, T., Uematsu, H., Okada, Y., Murakami, K. J. Phys: Conference Series 2007, 59, 466469. Mie, G. Beitrage zur Optik truber Medien. Ann. Phys. 1908, 25, 377-445. Coldren, L. A., Corzine S. W., Masanovic, M. L. Diode Lasers and Photonic Integrated Circuits. Second edition, Wiley, 2012. Chen, R.; Tran, T.-T. D. ; Ng, K. W. ; Ko, W. S. ; Chuang, L. C. ; Sedgwik, F. G.; ChangHasnain, C. Nature Photon. 2011, 5, 170-175. Peelaers, H.; Van de Walle, C. G. Phys. Rev. B 2012, 86, 241401-1 – 24401-5. Shi, H., Yan, R., Bertolazzi, S., Brivio, J., Gao, B., Kis, A., Jena, D., Xing, H. G., Huang, L. ACS Nano 2013, 7, 1072-1080 . Castellanos-Gomez, A.; Agrait, N.; Rubio-Bollinger, G. Appl. Phys. Lett. 2010, 96, 213116-1 – 213116-3. Mak, K. F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T. F. Phys. Rev. Lett. 2010, 105, 136805-1 – 1306805-5. Shi, H.; Pan, H.; Zhnag, Y.-W.; Yakobson B. I. Phys. Rev. B 2013, 87, 155304. Qiu, D. Y., Felipe, H., Louie, S. G. Phys. Rev. Lett. 2013, 111, 216805-1 - 216805-5. Berghäuser, G., Malic, E. Phys. Rev. B 2014, 89, 125309-1 – 125309-6.
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