Fermi energy dependence of first- and second-order Raman spectra in graphene: Kohn anomaly and quantum interference effect

Abstract

Intensities of the first- and the second-order Raman spectra are calculated as a function of the Fermi energy. We show that the Kohn anomaly effect, i.e., phonon frequency renormalization, in the first-order Raman spectra originates from the phonon renormalization by the interband electron-hole excitation, whereas in the second-order Raman spectra, a competition between the interband and intraband electron-hole excitations takes place. By this calculation, we confirm the presence of different dispersive behaviors of the Raman peak frequency as a function of the Fermi energy for the first- and the second-order Raman spectra, as observed in some previous experiments. Moreover, the calculated results of the Raman intensity sensitively depend on the Fermi energy for both the first- and the second-order Raman spectra, indicating the presence of the quantum interference effect. The electron-phonon matrix element plays an important role in the intensity increase (decrease) of the combination (overtone) phonon modes as a function of the Fermi energy.