Monitoring Local Strain Vector in Atomic-layered MoSe 2 by. Second-Harmonic Generation

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1 Supporting information for Monitoring Local Strain Vector in Atomic-layered MoSe 2 by Second-Harmonic Generation Jing Liang, Jin Zhang, Zhenzhu Li, Hao Hong, Jinhuan Wang, Zhihong Zhang, Xu Zhou, Ruixi Qiao, Jiyu Xu, Peng Gao, Zhirong Liu, Zhongfan Liu, Zhipei Sun, *, Sheng Meng, *, Kaihui Liu, *, and Dapeng Yu, State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, School of Physics, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing , China Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing , China Centre for Nanochemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing , China Department of Electronics and Nanoengineering, Aalto University, Espoo 02150, Finland Department of Physics, Southern University of Science and Technology, Shenzhen , China * khliu@pku.edu.cn. Phone: * smeng@iphy.ac.cn. Phone: * zhipei.sun@aalto.fi. Phone: S1 / 18

2 Table of Contents: S1-7. Figure S1-7 S8. Reduced symmetry of monolayer MoSe2 under uniaxial strain. S9. Table S1 S10. Methods S2 / 18

3 S1. Strain dependence of PL peak position. PL Peak (ev) Slope: mev/ε Strain (%) Figure S1. Strain dependence of PL peak position. Line is the linear fit to the experimental data of Figure 1d in the text. S3 / 18

4 S2. SHG intensity and PL peak energy mapping of monolayer MoSe2. a b c MAX SHG Intensity (a.u.) d SHG (a.u.) =0 1.3% Wavelength (nm) = 0 = 1.3% e f = 0 = 1.3% MIN PL Peak Position (ev) Figure S2. SHG intensity and PL peak energy mapping of monolayer MoSe2. (a) Optical image of monolayer MoSe2 indicated by a dashed triangle. The scale bar is 5 μm. (b, c) SHG intensity mapping of monolayer MoSe2 without (b) and with 1.3% strain (c). The SHG intensity reduces by 67% after applying the strain. (d) SHG spectra of monolayer MoSe2 under different strain amplitudes. The corresponding PL peak energy mapping of monolayer MoSe2 without (e) and with (f) the strain. The PL peak energy redshifts 47 mev under the strain (f). S4 / 18

5 S3. Pattern evolution of SHG for 6 monolayer MoSe2 flakes under different strain directions. a b c d e f S5 / 18

6 Figure S3. Pattern evolution of SHG for 6 monolayer MoSe2 flakes under different strain directions. (a-f) Optical images of 6 monolayer MoSe2 flakes and their corresponding SHG patterns. Uniaxial tensile strain is fixed along horizontal direction. The scale bar is 10 μm. In the figure of SHG pattern, the color of orange, dark yellow, violet and dark cyan (from outer curves to inner ones) represents the strain amplitude of 0, 0.75%, 1.2% and 1.4% in sequence. S6 / 18

7 S4. Pattern evolution of SHG intensity for monolayer MoS2 under strain. a b c d e f g h Figure S4. Pattern evolution of SHG intensity for monolayer MoS2 under strain. (a,e) Optical image of monolayer MoS2 under strain ε along the horizontal direction. The scale bar is 5 μm. (b-d, f-h) Angle-dependent SHG intensity pattern when the polarization of generated SHG is parallel (b, f), perpendicular (c, g) to the incident polarization, or the total SHG (d, h) under strain. The color of orange, dark yellow, violet and dark cyan (from outer curves to inner ones) represents strain of 0, 0.7%, 1.0% and 1.3% in sequence for (b-d, f-h). S7 / 18

8 S5. The evolution of normalized SHG intensity of monolayer MoS2 with strain. Normalized SHG Strain (%) Figure S5. The evolution of normalized SHG intensity of monolayer MoS2 with strain. Line is the linear fit to the experimental data. The error bars represent the standard measurement error from experiments of 5 samples. S8 / 18

9 S6. Strain response of tensor elements. a b Parameters Strain (%) Parameters Strain (%) Figure S6. Strain response of tensor elements in d when strain is along armchair (a) or zigzag direction (b), respectively. The slope reflects the strain sensitivity of different components which is summarized in Table S1. S9 / 18

10 Intensity (a.u.) PL Peak (ev) S7. PL spectrum of bilayer MoSe2 under tensile strain. a 2 1.4% 1.2% b % 0.8% = Energy (ev) Slope: mev/ε Strain (%) Figure S7. PL spectrum of bilayer MoSe2 under tensile strain. (a) PL spectrum of bilayer MoSe2 shown in Figure 3b in the text under tensile strain up to 1.4%. (b) Strain dependence of PL peak position of bilayer MoSe2. Line is the linear fit to the experimental data and the slope is determined to be mev/ε. S10 / 18

11 S8. Reduced symmetry of monolayer MoSe2 under uniaxial strain. The polarization dependence of SHG could be understood by considering the nonlinear properties of the crystals point group. With incident electric field e ω = [ sin θ cos θ 0], where θ is the angle between incident laser polarization and armchair orientation, the generated SHG intensity could be expressed as I SHG = e 2ωd e ω 2 2 (1) where e 2ω = [ sin θ cos θ 0] ([cos θ sin θ 0]) for SHG signal I (I ) parallel (perpendicular) with incident laser polarization, e ω 2 = [sin 2 θ cos 2 θ sin θ cos θ], d is contracted notation of second-order susceptibility tensor 1. In strain free case, monolayer MoSe2 belongs to the D3h point group 2 with d matrix (D 3h ) = [ d 21 d d il 0 0 d ] (2) where d 21 = d 16 = d 22. Therefore the polarization-dependent SHG intensity is described as follows (D I 3h ) = d 22 cos 3 θ + d 21 sin 2 θcosθ + 2d 16 sin 2 θcosθ 2 = d 22 cos 3θ 2 I (D 3h ) = d 22 sinθcos 2 θ + d 21 sin 3 θ 2d 16 sinθcos 2 θ 2 = d 22 sin 3θ 2 (3) When uniaxial tensile strain ε a (ε z ) is applied to monolayer MoSe2, the resulted anisotropic lattice deformation can significantly modify crystal symmetry. The 3-fold (C 3 ) axis disappears, and only one 2-fold (C 2 ) axis which is parallel (perpendicular) to the ε a (ε z ) along armchair crystal S11 / 18

12 orientation survives. Therefore, the crystalline symmetry of monolayer MoSe2 is reduced from D3h to C2v point group with d matrix following by (C 2v ) = [ d 21 d 22 d d il 0 0 d ] (4) d Although two new tensor elements d 23, d 34 show up, they have no influence on the expression of SHG intensity since the pump photon is linearly polarized in a normal incidence configuration. (C The effective tensor elements are still d 16, d 21, d 22. On the basis of the same form between d 2v ) il (D and d 3h ) (C il, d 2v ) il is expected to be linearly dependent on the external strain perturbation as the strain is only on the percent level in our experiment. Under strain of ε a, (C d 2v ) (D 22 = d 3h ) 22 (1 + a 1 ε a ) (C d 2v ) (D 21 = d 3h ) 22 (1 + b 1 ε a ) (C d 2v ) (D 16 = d 3h ) 22 (1 + c 1 ε a ) (5) intensity: Putting new tensor elements into the formula above, we could get strain-dependent SHG (C I 2v ) (D = d 3h ) 22 (cos3θ + ε a (a 1 cos 3 θ b 1 sin 2 θcosθ 2c 1 sin 2 θcosθ)) 2 (C I 2v ) (D = d 3h ) 22 (sin3θ + ε a (a 1 sinθcos 2 θ b 1 sin 3 θ + 2c 1 sinθcos 2 θ)) 2 (6) where θ is the included angle between unique armchair crystal orientation which remains a C 2 axis. Under ε z, we would obtain the same formula shown above with ε a, a 1, b 1, c 1 replaced by ε z, a 2, b 2, c 2 respectively. S12 / 18

13 A series of a i ε, b i ε and c i ε, where i=1 or 2, could be derived by fitting the experimental data of I (C 2v ) (C (ε)= I 2v ) (C (ε)+ I 2v ) (ε) under different strain with the equation above. Furthermore, a set of parameters (Table S1) related to the sensitivity of different tensor elements can be further determined by linear fitting between a i ε, b i ε, c i ε and ε respectively (Figure S6). In more general case when strain is along random direction (ε c ), C 2 axis also disappears, as a result, the crystal symmetry would be reduced from D3h to C1h point group with d matrix following by d 11 d 12 d 13 (C 1h ) = [ d 21 d 22 d d il 0 0 d d 26 ] (7) d 34 d 35 0 New effective tensors need enough experimental data to derive the sensitive parameters in this case. S13 / 18

14 S9. Parameters used to characterize strain-dependent d. Table S1 Parameters used to characterize strain magnitude i i i i = i = S14 / 18

15 S10. Methods. Growth of monolayer and bilayer MoSe2. Monolayer or bilayer MoSe2 were grown on 90nm SiO2/Si or mica substrate by CVD method using MoO3 and Se powder as precursors mg MoO3 powder was placed at the center of a tube furnace and 36.0 mg Se powder at the upstream side 13cm away from the MoO3 powder. Substrates were placed upon the crucible containing MoO3 powder. The CVD process was performed under ambient pressure in controlled ultrahigh-purity Argon gas. The recipe is: keep at 105, 500 sccm gas flow for 1 hour, ramp to 800 with 15 sccm gas flow in 50 minutes, keep at 800 with 20 sccm gas flow for 20 minutes, and then naturally cool down to room temperature. Preparation or strain 2D MoSe2. The wet transfer was performed to transfer the CVD synthesized MoSe2 flakes onto Acrylic substrate. In a typical transfer process, 9% PMMA (polymethyl methacrylate, 950 K) in anisole solution was spin-coated onto SiO2/Si substrates with MoSe2 flakes and baked at 180 for 2 min. Then the sample was immersed into KOH solution (0.1M) at 80 for 5 min. After being lifted off from the original substrate, the PMMA/MoSe2 film was thoroughly washed with deionized (DI) water and then transferred to the target flexible Acrylic substrate. The PMMA/MoSe2/Acrylic was dried naturally for several hours and after that baked at 80 for 10 min to make the interaction between MoSe2 and Acrylic stronger. For 2H bilayer experiment, the materials were directly grown on mica or exfoliated on Acrylic. Characterization of MoSe2. Optical images of monolayer and bilayer MoSe2 on Acrylic were taken by an Olympus microscope (Olympus BX51). PL spectra were probed by home-built confocal microscope system with laser excitation wavelength of 532 nm and average power of 200 μw. The acquisition time of PL was 10 s. S15 / 18

16 SHG measurements. In our experiment, samples was excited by a femtosecond OPO laser centered at 820 nm with average power of 800 μw (Spectra-Physics Inspire ultrafast OPO system with pulse duration of 100 fs and repetition rate of 80 MHz) 3. The excitation photon was normally incident and the linear polarization direction could be controlled by rotating a half-wave plate (Thorlabs SAHWP05M-700) in front of the objective. The incident laser was focus on samples by a Nikon objective (100x, N.A.= 0.80). In reflection geometry, the generated SHG signal was directly collected without analyzer (I) or extracted its parallel (I ) or perpendicular (I ) components through an analyzer. The signal was detected by grating spectrograph (Princeton SP-2500i) with fixed integration time of 3 s. Theoretical Calculations. All calculations were performed within the Vienna ab-initio Simulation Package (VASP) 4 using a projector-augmented wave (PAW) pseudopotential in conjunction with the Perdew Burke Ernzerhof (PBE) 5 function and plane-wave basis set with an energy cutoff of 400 ev. The 1 3 rectangle cell containing two MoSe2 layers with 4 Mo and 8 S atoms was used. The surface Brillouin zone was sampled by a Monkhorst Pack k-mesh. A vacuum region of 15 Å was applied. Because of the absence of strong chemical bonding between layers, van der Waals density functional in the opt88 form 6 was employed for structural optimization. All structures were fully relaxed until the force on each atom was less than 0.01 ev Å 1. First-principles calculations are hard to model the van der Waals interactions for the lattice-mismatched bilayers, since very large supercells are needed. Thus, the non-bonded interaction between MoSe2 interlayers for the mismatched cases are modeled with classical Lennard-Jones potential 7. E = 4ε [( σ r )12 ( σ r )6 ]. (8) The non-bonded coefficients are shown as below. S16 / 18

17 ε Mo Mo = eV, σ Mo Mo = Å; ε Se Se = eV, σ Se Se = Å; ε Mo Se = eV, σ Mo Se = Å. S17 / 18

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