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Nonadiabatic exciton-phonon coupling in Raman spectroscopy of layered materials

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Science Advances  07 Aug 2020:
Vol. 6, no. 32, eabb5915
DOI: 10.1126/sciadv.abb5915
  • Fig. 1 Calculated Raman intensities for hBN as a function of excitation frequency.

    Blue lines, Raman intensity on the BSE level calculated with the full nonadiabatic theory (full lines) and in the adiabatic (Ad.) limit (dashed lines). Red lines, same as blue lines, but without excitonic effects, i.e., on the IP level. Shaded areas, imaginary part of the dielectric function on the BSE (blue) and IP (red) levels. Squares, experimental (Exp.) data from (22), normalized such that the last point matches the theoretical curve. a.u., arbitrary units.

  • Fig. 2 hBN: Electronic transition band structure and exciton envelope wave functions in reciprocal space.

    Top: Electronic transition band structure obtained from Kohn-Sham DFT within the LDA. A rigid shift of 2 eV was added to all conduction band energies. The opacity of the lines corresponds to the optical activity of the transition. Bottom: Exciton envelope wave functions in reciprocal space for the first four bright excitons. Each panel shows the band-summed square of the envelope wave function Ak,c,vS2 as a function of the wave vector k integrated over the kz coordinate. coord., coordinate.

  • Fig. 3 hBN: Raman intensity with only inter- or intra-exciton scattering.

    Raman intensity with all contributions (thick lines) and with only inter- or intra-exciton scattering (darker and lighter thin lines, respectively) taken into account. Top: Full, nonadiabatic case. Bottom: Adiabatic limit. The shaded area represents the absorption spectrum. The Raman intensity is given in the same units in both panels and as in Fig. 1.

  • Fig. 4 Calculated Raman intensities for monolayer molybdenum disulfide as a function of excitation frequency.

    Green and blue lines, Raman intensity of the E′-mode (green) and A1-mode (blue) on the BSE level calculated with the full nonadiabatic theory (full lines) and in the adiabatic limit (dashed lines). Blue shaded area, imaginary part of the dielectric function on the BSE level. Green squares and blue triangles, experimental data from (5), normalized such that the A1-mode intensity at the second (“B”)-exciton matches the theoretical curve. Inset: Zoom-in into the region around the first two excitons, highlighting an 80-meV shift between theory and experiment.

  • Fig. 5 MoS2: Electronic transition band structure and exciton envelope wave functions in reciprocal space.

    Top: Electronic transition band structure obtained from Kohn-Sham DFT within the LDA. A rigid shift of 0.925 eV was added to all conduction band energies. The opacity of the lines corresponds to the optical activity of the transition. Bottom: Exciton envelope wave functions in reciprocal space for the first three bright excitons (the A-, B-, and C-excitons). Each panel shows the band-summed square of the envelope wave function Ak,c,vS2 as a function of the wave vector k. In the case of the C-exciton, we show the sum of the squared wave functions of five energetically close bright excitons.

  • Fig. 6 MoS2: Raman intensity with only inter- or intra-exciton scattering.

    Raman intensity with all contributions (thick lines) and with only inter- or intra-exciton scattering (darker and lighter thin lines, respectively) taken into account. Top: E′-mode. Bottom: A1-mode. The shaded area represents the absorption spectrum. The Raman intensity is given in the same units in both panels and as in Fig. 4. Insets: Zoom-ins into the areas around the three dominant excitonic resonances. In the bottom inset, the purple line depicts the Raman intensity of the A1-mode with an artificially tripled phonon frequency (ph. freq.).

Supplementary Materials

  • Supplementary Materials

    Nonadiabatic exciton-phonon coupling in Raman spectroscopy of layered materials

    Sven Reichardt and Ludger Wirtz

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