Saddle point anomaly of Landau levels in graphenelike structures

Abstract

Studying the tight-binding model in an applied rational magnetic field $(H)$ we show that in graphene there are very unusual Landau levels situated in the immediate vicinity of the saddle point ($M$ point) energy ${\ensuremath{\epsilon}}_{M}$. Landau levels around ${\ensuremath{\epsilon}}_{M}$ are broadened into minibands (even in relatively weak magnetic fields, $\ensuremath{\sim}40--53$ T), with the maximal width reaching 0.4--0.5 of the energy separation between two neighboring Landau levels, though at all other energies the width of Landau levels is practically zero. In terms of the semiclassical approach a broad Landau level or magnetic miniband at ${\ensuremath{\epsilon}}_{M}$ is a manifestation of the so-called self-intersecting orbit signifying an abrupt transition from the semiclassical trajectories enclosing the $\mathrm{\ensuremath{\Gamma}}$ point to the trajectories enclosing the $K$ point in the momentum space. The structure of broad magnetic minibands is clearly fractal, demonstrating fast oscillations of the band energy $E(k)$ with the magnetic wave vector $k$. Remarkably, the saddle point virtually does not affect the diamagnetic response of graphene, which is caused mostly by electron states in the vicinity of the Fermi energy ${\ensuremath{\epsilon}}_{F}$. Experimentally, the effect of the broadening of Landau levels can possibly be observed in twisted graphene where two saddle point singularities can be brought close to the Fermi energy. At zero temperature the longitudinal conductivity is expected to be proportional to the calculated magnetic density of states at the Fermi level.