First-principles resonant Raman spectroscopy for 2D materials
Yue Yu,1 Jun Jiang,1 Liangbo Liang,2 Georgios D. Barmparis,3 Sokrates T. Pantelides,4, 5 and X.-G. Zhang1
1Department of Physics and the Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA
2Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, Heraklion
71003, Greece
4Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
5Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235, USA
We explain Resonant Raman scattering as a two-step transition, in which an electron get excited to an excited state and then jump back to ground state immediately. In this process, a set of phonons are absorbed and emitted. The Raman Intensity is then expressed in the form of the Kramers-Heisenberg-Dirac (KHD) formula. We first rewrite all the energy-dependent terms as the integration of an exponential factor in the time domain, and then we replace the energy factors with phonon Hamiltonian operators. By applying the completeness relation to the vectors in KHD formula, we reduce the integrand to a trace of the product of exponential operators. The operator product and trace are evaluated with the help of Feynman path integrals. Our triple integration method offers a simple and fast numerical approach that could include the contribution of multiple phonons in KHD formula. The ground state and excited state phonon modes are calculated to provide frequencies and equilibrium position of the phonon Hamiltonians. We compared the computed Raman spectroscopy with experimental data and get a good agreement. We also explored the resonance properties of Raman and shows a strong dependence of Raman signal on the energy of incidental laser.