Geometrical approach for the study of G′ band in the Raman spectrum of monolayer graphene, bilayer graphene, and bulk graphite

In this paper, a geometrical approach is presented for the study of the double-resonance process, giving rise to the G(') band in monolayer graphene, bilayer graphene, and bulk graphite. It is shown that there are four discrete peaks present in the G(') band spectrum obtained from the stacking of two graphene layers, and these discrete peaks arise from the quantization of the first Brillouin zone caused by its finite size along the c axis. Our analysis includes the study of the selection rules imposed on the electron-radiation and electron-phonon Hamiltonian interactions involving pi electrons near the K point. We show that the anisotropy in the optical absorption (emission) near the K point in the first Brillouin zone of graphite should be taken into account in order to gain an understanding of the selection rules for optical transitions in bilayer graphene. The validity of considering a linear dispersion for pi electrons along the K-Gamma direction is taken into consideration. We present four numerical equations, giving the dependencies of the frequencies of the four peaks composing the G(') feature in the Raman spectrum of bilayer graphene on the laser excitation energy in the visible range. We also show that the two-peak shape of the G(') band in the Raman spectrum of bulk graphite is in fact caused by the convolution of an infinite number of peaks.