Hierarchy of gaps and magnetic minibands in graphene in the presence of the Abrikosov vortex lattice

We determine the bands and gaps in graphene subjected to the magnetic field of an Abrikosov lattice of vortices in the underlying superconducting film. The spectrum features one nondispersive magnetic miniband at zero energy, separated by the largest gaps in the miniband spectrum from a pair of minibands resembling a slightly broadened first Landau level in graphene, suggesting the persistence of $\nu =\pm 2$ and $\pm 6$ quantum Hall effect states. Also, we identify an occasional merging point of magnetic minibands with a Dirac-type dispersion at the miniband edges.

Figure 1

Spectrum of Dirac electrons in graphene in a magnetic field of an Abrikosov vortex lattice, with one degenerate band at $E=0$. Energy is scaled as $v\sqrt{2\hslash eB}\equiv \frac{2}{3^{1/4}}v\sqrt{\pi }/a$, and the magnetic field as $4\pi {\lambda }^{2}B/{\varphi }_0$. Inset: Second and third miniband dispersions over the folded magnetic Brillouin minizone near their touching condition.

Figure 2

Distribution function $P\left(s\right)$ of the normalized level spacings in the miniband energies at $\gamma$ and $\mu$ points of the Broillouin minizone. Energy levels were taken from more than 30 states sampled $(E=0$ and the first band excluded) for ten values of $0.5\le \alpha \le 1.5$, with a step of $\delta \alpha =0.1$.

Figure 3

Spectrum of Dirac electrons in the presence of a square vortex lattice. Energy is scaled by $v\sqrt{2\hslash eB}$, and the spectrum includes one degenerate band at $E=0$. Inset: Second and third miniband dispersions over the folded magnetic Brillouin zone in the vicinity of their touching condition.