Impact of electron-phonon coupling on near-field optical spectra in graphene

The finite momentum transfer $\mathit{q}$ longitudinal optical response ${\sigma }^L(\mathit{q},\omega )$ of graphene has a peak at an energy $\omega =\hslash v_{F}q$. This corresponds directly to a quasiparticle peak in the spectral density at a momentum relative to the Fermi momentum $k_F-q$. Inclusion of coupling to a phonon mode at ${\omega }_E$ results, for $\omega <|{\omega }_E|$, in an electron-phonon renormalization of the bare bands by a mass enhancement factor $(1+\lambda )$, and this is followed by a phonon kink for $\omega$ around ${\omega }_E$ where additional broadening begins. Here we study the corresponding changes in the optical quasiparticle peaks, which we find continue to track directly the renormalized quasiparticle energies until $q$ is large enough that the optical transitions begin to sample the phonon kink region of the dispersion curves, where linearity in momentum and the correspondence to a single-quasi-particle energy are lost. Nevertheless there remain in ${\sigma }^L(\mathit{q},\omega )$ features analogous to the phonon kinks of the dispersion curves which are observable through variation of $q$ and $\omega$.

Figure 1

The real part of the finite momentum longitudinal optical conductivity ${\sigma }^L(q,\omega )/{\sigma }_0$ as a function of $\omega /{\mu }_0$ for a selection of values of $q/k_F$ from $0.0\to 0.9$ (as defined in the legend). These include an electron-phonon interaction with $\lambda \approx 0.18$ for a single optical phonon mode at ${\omega }_E=200$ meV (shown by the vertical dashed line), where ${\mu }_0=1$ eV, $A=0.08$ eV, and $W_c=7.0$ eV, and a residual scattering $\eta =0.005{\mu }_0$. Inset: $-\mathrm{Im}{\Sigma }^{\text{EPI}}\left(\omega \right)/{\mu }_0$ as in Eq. (4) for parameters in the main frame.

Figure 2

Schematic of doped Dirac bands. An electron-phonon interaction renormalizes the Dirac bands, changing the slope through the Dirac point and producing kinks at $\omega =\pm {\omega }_E$, where the Fermi level ($\omega =0$) is illustrated by a change in color.

Figure 3

The peaks in ${\sigma }^L(q,\omega )$ plotted as a function of $k_F-q$ coincide with peaks in the EPI renormalized spectral function $A(k,\omega )$, plotted as a function of $k$. For comparison, the bare dispersion is shown by the dashed line, while the green line with circles is the bare dispersion divided by $(1+\lambda )$, which is the slope of the EPI renormalized spectral function (shown in red) at $k=k_F$. The peaks in ${\sigma }^L(q,\omega )$ sample the renormalized dispersion until near $\omega =-{\omega }_E$. Inset: $A(k,\omega )$ as in the main frame but showing both positive and negative frequencies.

Figure 4

Schematic showing the many optical transitions from occupied states at energy $\epsilon =-q_1$ to unoccupied states at $\epsilon =q_2$, which all contribute to ${\sigma }^L(q,\omega )$ at $\omega =q$ when the dispersion curves are linear. The large degeneracy of transitions gives a peak in the optical conductivity at $\omega =q$.

Figure 5

The real part of the finite momentum optical conductivity ${\sigma }^L(q,\omega )/{\sigma }_0$ as a function of $\omega /{\mu }_0$ for a selection of values of $q/k_F$. The dashed lines include the electron-phonon interaction identically to Fig. 1 for $q/k_F$ displayed in the legend, while the solid lines include only the residual scattering of $\eta =0.005{\mu }_0$ but for shifted $q$ values (marked near the peaks). The solid circles mark the analytical clean limit results, with an adjusted Fermi velocity to obtain the same peak positions.