Supporting Information. Davydov Splitting and Excitonic Resonance Effects

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1 Supporting Information Davydov Splitting and Excitonic Resonance Effects in Raman Spectra of Few-Layer MoSe2 Kangwon Kim,,1 Jae-Ung Lee,,1 Dahyun Nam, and Hyeonsik Cheong Department of Physics, Sogang University, Seoul 04107, Korea Contents: Figure S1. Raman spectra of 4 6TL MoSe2 measured with 8 excitation energies. Figure S2. Raman spectra of 7 8TL MoSe2 measured with 7 excitation energies. Figure S3. Low-frequency Raman spectra of MoSe2 measured with 6 excitation energies. Figure S4. Thickness dependence of Raman spectrum of MoSe2 measured with excitation energies 1.58 ev and 1.96 ev. Figure S5. Thickness dependence of Raman spectrum of MoSe2 measured with excitation energies 2.33 ev and 2.41 ev. Figure S6. Thickness dependence of Raman spectrum of MoSe2 measured with excitation energies 2.54 ev and 2.71 ev. Figure S7. Thickness dependence of Raman spectrum of MoSe2 measured with excitation energies 2.81 ev and 3.82 ev. Figure S8. Polarized Raman spectra of 3TL MoSe2 with 1.96 ev excitation energy. Figure S9. Normalized Raman spectra near A2u mode showing Davydov splitting for 4, 7 and 8TL MoSe2 for various excitation energies. 1

2 Figure S10. Normalized Raman spectra near E1g mode showing Davydov splitting for 4, 7, and 8TL MoSe2 for various excitation energies. Figure S11. Normalized Raman spectra near A1g mode showing Davydov splitting for 7 and 8TL MoSe2 for various excitation energies. Calculation of vibrational mode frequencies based on linear chain model Figure S12. Schematic of linear chain model. Figure S13. Peak positions of vibrational modes from experiment and linear chain model. Figure S14. Raman intensity as a function of excitation energies for peak c and A1g/A1 (3) mode. Figure S15. Optical microscope image of MoSe2 sample and photoluminescence (PL) spectra of few-layer MoSe2. References 2

3 Figure S1. Raman spectra of (a) 4TL, (b) 5TL, and (c) 6TL MoSe2 measured with 8 excitation energies indicated. Detailed line shapes of the A1g mode are shown on the right. 3

4 Figure S2. Raman spectra of (a) 7TL and (b) 8TL MoSe2 measured with 7 excitation energies indicated. Detailed line shapes of the A1g mode are shown on the right. 4

5 Figure S3. Low-frequency Raman spectra of MoSe2 measured with 6 excitation energies indicated. A plasma line of the 2.41 ev laser is indicated by ( ), and the Brillouin scattering peak of the Si substrate is indicated by (*). 5

6 Figure S4. Thickness dependence of the Raman spectrum of MoSe2 measured with excitation energies (a) 1.58 ev and (b) 1.96 ev. Detailed line shapes of the A1g mode are shown on the left. 6

7 Figure S5. Thickness dependence of the Raman spectrum of MoSe2 measured with excitation energies (a) 2.33 ev and (b) 2.41 ev. Detailed line shapes of the A1g mode are shown on the left. 7

8 Figure S6. Thickness dependence of the Raman spectrum of MoSe2 measured with excitation energies (a) 2.54 ev and (b) 2.71 ev. Detailed line shapes of the A1g mode are shown on the left. 8

9 Figure S7. Thickness dependence of the Raman spectrum of MoSe2 measured with excitation energies (a) 2.81 ev and (b) 3.82 ev. Detailed line shapes of the A1g mode are shown on the left. 9

10 Figure S8. Polarized Raman spectra of 3TL MoSe2 with the 1.96 ev excitation energy. The E2g 1 mode appears in both polarizations whereas the peak at 291 cm 1 disappears in cross polarization. 10

11 Figure S9. Normalized Raman spectra near the A2u mode showing Davydov splitting in (a) 4TL, (b) 5TL, and (c) 6TL MoSe2 for various excitation energies. 11

12 Figure S10. Normalized Raman spectra near the E1g mode showing Davydov splitting in (a) 4TL, (b) 7TL, and (c) 8TL MoSe2 for various excitation energies. The splitting could not be resolved for 5TL or 6TL. 12

13 Figure S11. Normalized Raman spectra near the A1g mode showing Davydov splitting in (a) 7TL and (b) 8TL MoSe2 for various excitation energies. 13

14 Calculation of vibrational mode frequencies based on linear chain model The vibrational modes can be described by the linear chain model. 1,2 We assume that the elastic response of the crystal is a linear function of the forces. By assuming that each atom is connected with other atoms by springs, one can formulate the equations of motion that describe the vibrational modes. Up to second-nearest-neighbor interactions and surface effects are considered. The schematic of the model for the case of 3TL is shown in Figure S12. The vibrational modes along the in-plane and out-of-plane directions are considered separately. Figure S12. Schematic of linear chain model. 14

15 The equations of motion can be derived by the following procedure. For example, a Mo atom in the first layer (Mo1) is connected by three springs. The force applied to this Mo atom can be described by du F u u u u u u dt 2 Mo1 Mo Mo ( ) ( ) ( ) 2 e Mo 1 Se Mo 1 1 Se Mo 1 1 Se2 Similarly, the force applied to Se atoms in the first layer can be written as follows. 2 du 1 Se1 F 1 Se 2 e( u 1 umo ) e( u u ) Se Se Se1 Se1 dt 2 du 2 Se1 F 2 Se ( 2 e u 2 u 1 ) ( u 2 umo ) ( u 2 u 1 ) ( u 2 u ) Se1 Se1 Se1 Se1 1 Se Mo 1 Se2 Se1 2 dt By considering the forces applied to atoms in each layer, a 9 9 matrix can be formulated for 3TL- MoSe2. The dynamical matrix D can be written as follows. 15

16 D e e e e Se Se Se e e Mo Mo Mo Mo e e Se Se Se Mo Se Se Mo Se Se Se Mo Mo Mo Mo Mo Se Se Se Mo Se e e Se Mo Se Mo Se e e Mo Se Mo Se e e e e Se Mo Se For the normal modes, the equations of motion can be expresses as 2 du 2 DU U, 2 dt U ( u, u, u, u, u, u, u, u, u ). The frequencies of vibrational modes where Se Mo 1 1 Se1 Se Mo 2 2 Se Mo 2 Se3 3 Se3 can be obtained from the eigenvalues of the dynamical matrix D. By comparing with the peak frequencies obtained from Raman measurements, the spring constants for in-plane and out-ofplane direction can be obtained separately. The fitted results and the experimental data are compared in Figure S13. 16

17 Figure S13. Peak positions of vibrational modes from experiment and linear chain model: (a) breathing, (b) shear, (c) A1g, (d) E1g, (e) A2u, and (f) E2g 1 modes. Red symbols indicate Raman active modes whereas cyan symbols correspond to either infrared active modes or Raman active modes that are forbidden in backscattering. In (e) and (f), the green data correspond to the Raman active surface mode. The Davydov splitting is too small to be resolved for A2u and E2g 1. 17

18 Figure S14. Raman intensity as a function of excitation energies for (a) peak c and (b) A1g/A1 (3) mode. 18

19 Figure S15. (a) Optical microscope image of MoSe2 sample. (b) Photoluminescence (PL) spectra of few-layer MoSe2. 19

20 REFERENCES (1) Froehlicher, G.; Lorchat, E.; Fernique, F.; Joshi, C.; Molina-Sánchez, A.; Wirtz, L.; Berciaud, S. Unified Description of the Optical Phonon Modes in N -Layer MoTe2. Nano Lett. 2015, 15, (2) Luo, X.; Zhao, Y.; Zhang, J.; Xiong, Q.; Quek, S. Y. Anomalous Frequency Trends in MoS2 Thin Films Attributed to Surface Effects. Phys. Rev. B 2013, 88,