Electronic Veselago lenses in graphene: Valley polarization due to trigonal warping

K. J. A. Reijnders, M. I. Katsnelson

Published 2017-03-22Version 1

We study the effect of trigonal warping on the focussing of electrons by n-p junctions in graphene. We find that perfect focussing, which was predicted for massless Dirac fermions, is only preserved for one specific sample orientation. In the general case, trigonal warping leads to a different position of the focus for graphene's two valleys. We derive these positions using the semiclassical approximation and find very good agreement with tight-binding simulations. Considering the transmission as a function of potential strength, we find that trigonal warping splits the single Dirac peak into two distinct peaks, one for each valley. This leads to valley polarization, which increases with sample length. These effects are clearly observed in the transmission curves obtained from both tight-binding simulations and a billiard model that incorporates trigonal warping. Since their origin is classical rather than quantum mechanical, they are predicted to be very robust. The general mechanism that we discuss will also play an important role in other Dirac materials, which generally exhibit stronger band bending than graphene.