Effect of electron-phonon interaction on spectroscopies in graphene
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We calculate the effect of the electron-phonon interaction on the electronic
density of states (DOS), the quasiparticle properties and on the optical
conductivity of graphene. In metals with DOS constant on the scale of phonon
energies, the electron-phonon renormalizations drop out of the dressed DOS,
however, due to the Dirac nature of the electron dynamics in graphene, the band
DOS is linear in energy and phonon structures remain, which can be emphasized
by taking an energy derivative. There is a shift in the chemical potential and
in the position in energy of the Dirac point. Also, the DOS can be changed from
a linear dependence out of value zero at the Dirac point to quadratic out of a
finite value. The optical scattering rate $1/\tau$ sets the energy scale for
the rise of the optical conductivity from its universal DC value $4e^2/\pi h$
(expected in the simplest theory when chemical potential and temperature are
both $\ll 1/2\tau$) to its universal AC background value $(\sigma_0=\pi
e^2/2h)$. As in ordinary metals the DC conductivity remains unrenormalized
while its AC value is changed. The optical spectral weight under the intraband
Drude is reduced by a mass renormalization factor as is the effective
scattering rate. Optical weight is transferred to an Holstein phonon-assisted
side band. Due to Pauli blocking the interband transitions are sharply
suppressed, but also nearly constant, below twice the value of renormalized
chemical potential and also exhibit a phonon-assisted contribution. The
universal background conductivity is reduced below $\sigma_0$ at large
energies.