Electron-Phonon Bound States in Graphene in a Perpendicular Magnetic Field

The spectrum of electron-phonon complexes in monolayer graphene is investigated in the presence of a perpendicular quantizing magnetic field. Despite the small electron-phonon coupling, usual perturbation theory is inapplicable for the calculation of the scattering amplitude near the threshold of optical phonon emission. Our findings, beyond perturbation theory, show that the true spectrum near the phonon-emission threshold is completely governed by new branches, corresponding to bound states of an electron and an optical phonon with a binding energy of the order of $\alpha {\omega }_0$, where $\alpha$ is the electron-phonon coupling and ${\omega }_0$ the phonon energy.

Figure 1

(left) The electron energy spectrum in graphene, exposed to a perpendicular quantizing magnetic field. The dashed line represents the phonon-emission threshold energy ${\epsilon }_{\mu n}^c$. Here, it corresponds to the Landau level ${\epsilon }_{\mu n}$ with $\mu =-1$ and $n=1$. In contrast to the prediction from Ref. 20, no state corresponds exactly to the energy ${\epsilon }_{\mu n}^c$. Instead, the true spectrum includes a sequence of electron-phonon bound states, which coagulates to the threshold ${\epsilon }_{\mu n}^c$. The most distant bound states with the binding energies $W_{1,2}$ of opposite signs are shown in a larger scale by the bold, solid lines below and above ${\epsilon }_{-11}^c$. (right) Schematic diagram of the absorption spectrum in graphene. The cyclotron-phonon resonance is governed by the peaks corresponding to the electron-phonon bound states, which constitute an asymmetric doublet around $\nu ={\omega }_0$.

Figure 2

Equations for the electron-phonon scattering amplitude. (top) The exact scattering amplitude $\Sigma (\epsilon )$ with a “dangerous” intersection with respect to one electron and one phonon line. (bottom) The irreducible four vertex part $\square (\epsilon )$, approximated by the bare amplitude. The solid and dashed lines correspond to the electron and phonon Green functions, the bold dots to the bare electron-phonon vertices. The vertical dotted line shows the “dangerous” intersection.

Figure 3

The binding energies $W_{\mu nr}^0$ for the bound states, which refer to the threshold ${\epsilon }_{\mu n}^c$ with $n=1$ and $\mu =-1$. The solid and dashed curves correspond to the $r=1$ and 2 bound states, which are most distant from the threshold.