Lorentz Boost in Graphene

Novel Electric Field Effects on Landau Levels in Graphene

Vinu Lukose, R. Shankar, and G. Baskaran

PRL 98, 116802 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e116802

Pretty slick! The authors solve a problem in graphene by using a Lorentz boost to transform away the electric field.

The authors investigate a graphene system with a magnetic field perpendicular to the surface and an electric field parallel to the plane. The magnetic field leads to the formation of Landau levels.

Since the low-energy physics is described by a Lorentz invariant Hamiltonian with the speed of light replaced by the fermi velocity, the authors use a Lorentz boost to eliminate the electric field. This transformation works as long as the electric field is smaller than the magnetic field. (If B is smaller, can one transform to a frame where there is only an electric field?) The resulting Hamiltonian is a Landau-level Hamiltonian with a rescaled magnetic field.

The authors solve the model, then transform back to the lab frame. The surprising result is that the spacing of the Landau levels decreases with increasing magnetic field, and become degenerate for E = B. This does not happen in a conventional 2DEG, where the level spacing is independent of E. Another difference between graphene and the 2DEG is that the centers of the harmonic oscillator functions in graphene shift with E, and the dependence on position vanishes when E=B!

To show that these results are not a superficial consequence of the low-energy theory, the authors diagonalize a tight-binding model and demonstrate the same phenomena. The shift of the centers of the various Landau levels leads them to predict a new kind of dielectric breakdown for graphene.

I found the idea of using a boost to eliminate the electric field very clever.