Phonon Effects in Graphene

Electron-Phonon Coupling Mechanism in Two-Dimensional Graphite and Single-Walled Carbon Nanotubes

G.G. Samsonidze, E.B. Barros, R. Saito, J.Jiang, G. and M.S. Dresselhaus

PRB 75, 155420 (2007)

URL: http://link.aps.org/abstract/PRB/v75/e155420

The authors have analyzed the effects of phonons on the fermi energies and wave vectors of graphene. Their analysis is based on the group theory of the wave vector and demonstrates both the power of the technique and my lack of understanding.

They observe two phenomena associated with the phonons. First, there is a Pierls instability, which means the phonon modes open up a frequency-dependent band gap. The second is a Kohn anomaly, which is electron screening of a particular phonon mode.

Section 2 and Appendix A are very useful, as they show the general procedure for introducing phonon effects into the tight-binding model. When the phonon mode breaks the symmetry of the lattice, the unit cell of graphene must be enlarged to become a supercell of six atoms. This leads to a 6 by 6 Hamiltonian instead of the more familiar 2 by 2 version. However, the larger Hamiltonian is what Wigner calls a supermatrix --- a matrix composed of smaller matrices. There are two diagonal 3 by 3 matrices for the on-site terms, H[AA] and H[BB]. The hopping terms are described by two 3 by 3 matrices, H[AB] and H[BA], with every entry equal to t (for k=0). The latter is surprising, as it suggests that every A site is connected to every B site --- a fact that is not obvious from the diagrams provided by the authors.

This Hamiltonian gives a six-band spectrum, with the middle four bands degenerate. The K-point phonon mode breaks the degeneracy from fourfold to twofold and opens a bandgap that depends on the phonon coupling strength and amplitude.

The appendix gives the corresponding Hamiltonian for phonons with wave vectors not at a highly symmetric point of the Brillioun zone. It is not as symmetric. Although the authors do not analyze the energy bands of the general tight-binding Hamiltonian with phonon interactions, I assume that a phonon that does not respect any symmetries of the underlying lattice would lift all the degeneracies. The only exception might be the Kramers degeneracy imposed by time-reversal symmetry.

One aspect of this work I don't understand is why the phonon modes considered by the authors are more important than others. Are they the modes of lowest energy? Do the phonon bands cross the electron bands at the fermi energy? It seems that they are simply the easiest to analyze, but that does not mean their physical effects are the most important.