Strong light matter coupling in two-dimensional atomic crystals

Transcription

1 SUPPLEMENTARY INFORMATION DOI: /NPHOTON Strong light matter coupling in two-dimensional atomic crystals Xiaoze Liu 1, 2, Tal Galfsky 1, 2, Zheng Sun 1, 2, Fengnian Xia 3, Erh-chen Lin 4, Yi-Hsien Lee 4, Stéphane Kéna-Cohen 5 and Vinod M. Menon 1,2 1 Department of Physics, City College of New York, New York, New York, USA 2 Department of Physics, Queens College & Graduate Centre, City University of New York, New York, New York, USA 2 Department of Electrical Engineering, Yale University, New Haven, Connecticut, USA 3 Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan 4 Department of Engineering Physics, École Polytechnique de Montréal, Montréal, Quebec, Canada vmenon@ccny.cuny.edu NATURE PHOTONICS 1

2 Figure S1: Photoluminescence (PL) and differential reflectivity spectra of the MoS 2 monolayer. Figure S2: Simulated angle-resolved reflectivity spectrum and energy dispersion of the microcavity. Figure S3. Angle-resolved reflectivity spectra and dispersion at another spot showing strong coupling at room temperature and low temperature. Figure S4. Temperature-dependent PL of bare MoS 2 monolayers and reflectivity of microcavity at low temperature. Figure S5. Geometry of Finite Element Method (FEM). Figure S6. Colour map of angle-resolved PL for the MoS 2 microcavity. Figure S7. Simulation for PL and reflectivity at various detunings. Figure S8. Schematic of angle-resolved spectroscopy setup and photo of the sample. 2

3 PL intensity (a. u.) R/R ex A ex B Energy (ev) Figure S1. Photoluminescence (PL) and differential reflectivity spectra of the MoS 2 monolayer. The differential reflectivity spectrum of monolayer MoS 2 clearly shows two prominent absorption peaks at ev and ev, identified as A and B excitons, respectively 1. The PL spectrum shows one dominant peak at ev resulting from the direct bandgap transition of exciton A (ex A ) 1. These features of absorption and PL differentiates monolayer MoS 2 to bulk MoS 2 due to the transition of indirect bandgap to direct bandgap when it transitions from bulk to monolayer 1,2. 3

4 Figure S2. Simulated angle-resolved reflectivity spectrum and energy dispersion of the microcavity. Calculated reflectivity contour map for the MC, colour gradient represents the reflectivity. Also shown in the same plot is the energy versus angle dispersion, extracted from the angle-resolved reflectivity spectra. The black solid circles are the polariton mode energies obtained from the reflectivity spectra, the horizontal blue dashed line represents the bare ex A energy, the blue dashed curve represents the cavity modes and the two blue solid curves correspond to theoretical fit of the polariton branches via a coupled-oscillator model. The simulation shows good agreements with the measured data. Given that we only observed coupling to ex A only, the dielectric function of monolayer MoS 2 is modelled by a Lorentzian oscillator: f ( ) b i ex 4

5 Where is the ex A energy (1.87eV from the differential reflectivity spectrum in Figure S1), ex is the linewidth of the exciton transition (60 mev from the PL spectrum), b is the background dielectric function, and f is the oscillator strength. b and f are adjustable parameters. With these assumptions, the predicted absorption coefficient (at ev) is 10 6 cm -1, which is consistent with reported data 1. Refractive index of monolayer are derived from this model and applied to the transfer matrix method with a thickness of 0.75 nm 2,3 to simulate the MC angle-resolved reflectivity colour map as in Figure 3a. 5

6 Reflectivity Energy (ev) a TM ex A RT b 30 o mev Cavity mode ex A o Angle (deg) Energy (ev) Figure S3. Angle-resolved reflectivity spectra and dispersion for different detuning showing strong coupling at room temperature. a, Angle-resolved reflectivity spectra at another spot showing anticrossing modes at room temperature (RT). The angle range is also from 7.5 to 30, the red dashed line represent the ex A, the red curves trace the reflectivity modes. The spectra are similar to Fig. 2, but with different detunings. b, The reflectivity dispersions at RT. The detuning here is -56 mev. In contrast, the detuning in Fig. S2 is -40meV. Using the coupled oscillator model, the Rabi splittings are obtained as 53±4 mev. 6

7 Reflectivity Energy (ev) PL intensity (a. u.) PL intensity (a. u.) a b RT RT 150 K 150 K ex A 77 K ex A 77 K 10 K 10 K Energy (ev) c TM T=10K ex A d Energy (ev) 30 o mev Cavity mode ex A Angle (deg) 7.5 o Energy (ev) Figure S4. Temperature-dependent PL of bare MoS 2 monolayers and reflectivity of microcavity at low temperature. Temperature-dependent PL on a, the as-grown CVD monolayer MoS 2 samples on SiO 2 /Si substrate and b, transferred monolayer MoS 2 samples on DBR/glass substrate. The PL peaks of as-grown MoS 2 sample blue-shift as the temperature 7

8 decreases. However, when the monolayer MoS 2 is transferred to DBR/glass substrate, the PL peaks show barely any temperature-dependence. This is attributed to the difference between the thermal expansion coefficients of MoS 2 ( /K) 4 and CVD-grown SiO 2 ( /K) 5 which causes compressive stress at the interface. This stress could prevent effective thermal conductance at the interface as well as modify the optical properties. A similar effect (reduction in temperature dependence) was observed when monolayer MoS 2 was covered with dielectric oxides 6. Also, graphene has been shown to have similar behavior 7. c, Angle-resolved reflectivity spectra of the microcavity at the same spot of Figure S3 at cryostat temperature of 10 K. The red dashed line represent the ex A, the red curves trace the reflectivity modes. Since the PL does not shift with temperature, the reflectivity data is similar to the room temperature data as in Figure S3, barely showing temperature dependence. d, Since the transferred MoS 2 samples on DBR/glass substrates do not show temperature-dependence, the reflectivity dispersion is fitted with exciton A energy at room temperature using the coupled oscillator model. A Rabi splitting of 52±4 mev is obtained from this fitting. 8

9 Figure S5. Geometry of Finite Element Method (FEM). Finite Element Method (FEM) simulations were performed with commercial software COMSOL. Simulations for emission from the cavity were performed in the frequency domain for MoS 2 emission spectrum. The monolayer of MoS 2 is modeled as a 0.8nm thin dielectric layer with optical constants n and k which have been determined from absorption measurements of monolayer MoS 2 using Kramers- Kronig relation. An electric dipole is placed in the center of the MoS 2 layer with a dipole moment oriented parallel to the layer since in the 2D monolayer the dipoles are confined to the in-plane of the layers 8. 9

10 Figure S6. Color map of angle-resolved PL for the MoS 2 microcavity. Simulated color map for a horizontally aligned dipole emission from the MoS 2 microcavity showing modified emission dispersion as seen in the experiment. The black circles with error bars highlight the extracted experimental PL peak positions. For reference, the horizontal dashed line represents the ex A energy, the dashed curve represents the cavity modes and the two solid curves correspond to theoretical fit of the polariton modes. Inset shows the emission pattern from the same horizontal dipole in the absence of the cavity where the MoS 2 is sandwiched between two SiO 2 layers. 10

11 a b c d e f Figure S7. Simulation for PL and reflectivity at various detunings. a, b, c are simulations using COMSOL for angle resolved PL at various detunings, and e, f, g are simulations using transfer matrix method for angle resolved reflectivity various detunings. The dashed lines at all the figures represent ex A. At a and c, simulations are for detuning of around -10 mev; at b and e, detuning is around -25 mev; at c and f, the detuning is around -40 mev. At all the detunings, both PL and reflectivity show consistent anti-crossing features, indicating strong coupling regime all achieved. For the PL, the intensity maximum is also shifting to larger angles as the detuning becomes larger due to horizontal dipole emission coupled to different cavity modes. Specifically, the PL maximum at detuning of -25 mev is around 25, then it moves to around 30 at detuning of -40 mev, and then it is maximized in the range of Thus detuning is an important 11

12 parameter determining the orientation of the angular distribution of emission intensity for such microcavities. 12

13 a b Figure S8. Schematic of angle-resolved spectroscopy setup and photo of the sample. a, Schematic of angle-resolved spectroscopy setup. The measured angle range can be from 7.5 to 45. The lower limit is a physical constraint when the arms of the goniometer touch each other, the upper limit is due to the geometry of the cryostat window. b, Photo of the sample, showing the size of the whole microcavity is around 25mm 25mm, and the active spots are sparse around the centre area 5mm 5mm. 13

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