Phonon spectroscopy through the electronic density of states in graphene

In contrast to the normal state of conventional metals for which phonon renormalizations drop out of the electronic density of states, we demonstrate that they remain in the case of graphene and their signature is large and measurable. Furthermore, the electron-phonon interaction, which is fixed in conventional metals, can be tuned over a significant range in graphene by changing the gate voltage. Indeed, a factor of 2 in magnitude should be easily achievable. These two features allow for a normal-state spectroscopy for examining many-body interactions, such as the electron-phonon interaction. We present a procedure to trace the magnitude of the amplitude of the predicted phonon structures which will increase significantly with increasing doping. We also find that the Dirac point zero in the electronic density of states can be lifted and will become quadratic signifying the presence of many-body renormalizations in graphene.

Figure 1

(Color online) $-d\text{Re}\Sigma \left(\omega \right)/d\omega$ vs $\omega$ for a truncated Lorentzian electron-phonon spectral density peaked around 200 meV. Curves are for ${\mu }_0=0$, 150, 500, and 700 meV.

Figure 2

(Color online) (a) $N\left(\omega \right)/N_0$ (in eV) vs $\omega$ for the bare chemical potential ${\mu }_0=150\text{meV}$. The solid curve gives the phonon renormalized case and the dotted gives the bare band case. The arrows show the bare (long) and renormalized (short) value of $\mu$. (b) Same as for (a) but with ${\mu }_0=500\text{meV}$.

Figure 3

(Color online) (a) $dN\left(\omega \right)/d\omega$ vs $\omega$ (solid curve) for ${\mu }_0=150\text{meV}$. The dotted curve sets a baseline and is the bare band case. The dashed, which is for comparison, is $\left[1-d\text{Re}\Sigma \left(\omega \right)/d\omega \right]\text{sgn}\left(\omega +{\omega }_d\right)$. (b) Same as for (a) but with ${\mu }_0=500\text{meV}$. (c) The absolute height of the phonon peaks at about $\omega ={\omega }_E$ (solid dots) and $-\left({\mu }_0+{\omega }_E\right)$ (open circles) along with the $\lambda$ variation in ${\mu }_0$, the latter indicated as the red solid curve using the right-hand axis (indicated by the arrow).

Figure 4

(Color online) (a) $1-\left|dN\left(\omega \right)/d\omega \right|$ vs $\omega$ for ${\mu }_0=0$ and ${\omega }_E=155\text{meV}$ (solid blue curve). The inset on the lower right compares the phonon region with the input electron-phonon spectral density (red-dashed curve). (b) $1-\sqrt{N\left(\omega \right)/N_0\left(\omega \right)}$ vs $\omega$ (solid blue curve) compared with the result using the procedure of Li et al. (long-dashed red curve).