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Phonon structure in dispersion curves and density of states of massive Dirac Fermions

Dirac fermions exist in many solid state systems including graphene, silicene and other two dimensional membranes such as are found in group VI dichalcogenides, as well as on the surface of some insulators where such states are protected by topology. Coupling of those fermions to phonons introduces new structures in their dispersion curves and, in the case of massive Dirac fermions, can shift and modify the gap. We show how these changes present in angular-resolved photoemission spectroscopy of the dressed charge carrier dispersion curves and scanning tunneling microscopy measurements of their density of states. In particular we focus on the region around the band gap. In this region the charge carrier spectral density no longer consists of a dominant quasiparticle peak and a smaller incoherent phonon related background. The quasiparticle picture has broken down and this leads to important modification in both dispersion curves and density of states.