Ising instability of a Holstein phonon mode in graphene

We study the thermal distribution of phonons in a honeycomb lattice. Due to the two electronic bands there are two out-of-plane phonon modes with respect to the two sublattices. One of these modes undergoes an Ising transition by spontaneously breaking the sublattice symmetry. We calculate the critical point, the renormalization of the phonon frequency and the average lattice distortion. This transition might be observable in doped graphene by Raman scattering and transport experiments.

Figure 1

The two out-of-plane modes: the ${\mu }^{\prime }$ mode (left-hand figure) and the $m$ mode (right-hand figure). The arrows indicate the distortions of the carbon atoms according to Eq. (9).

Figure 2

Frequency shift $\delta {\omega }_3/{\alpha }^2$ in units of $1/t$ vs inverse temperature $\beta t$ for $\mu /t=0,3,6$ (from bottom to top) from Eq. (13).

Figure 3

Renormalized frequency ${\omega }_{\phi }$ of the $\phi$ mode (in arbitrary units) vs. Fermi energy $\overline{\mu }$ (in units of $\sqrt{-\Omega /C}$) from Eq. (23).