Nonlinear saddle point spectroscopy and electron-phonon interaction in graphene

This chapter focuses on ultrafast nonlinear spectroscopy at the ultraviolet M- or saddle-point in the electronic bandstructure of graphene where the position and dynamical evolution of its absorption peak are especially sensitive to electron-phonon interactions. Specifically, we explore how these absorption peak changes are caused by optically-induced modifications of the phonon temperature by way of several electron-phonon scattering processes. We present a detailed theoretical model for electron-phonon interactions based on the concept of deformation potentials. We also include a discussion of the phonon dispersion obtained from dynamical matrices. We derive the electronic self-energy to lowest order in the electron-phonon interaction Hamiltonian, then use it to calculate the interband susceptibility and the differential transmission spectrum. Using literature values for deformation potentials, we find good agreement between theory and experiment, indicating that this formalism provides a good understanding of the microscopic electron-phonon coupling processes that renormalize the electronic transitions close to the M-point and produce the observed differential transmission spectra.