Diamagnetism in disordered graphene

The orbital magnetism is studied in graphene monolayer within the effective-mass approximation. In models of short-range and long-range disorders, the magnetization is calculated with self-consistent Born approximation. In the zero-field limit, the susceptibility becomes highly diamagnetic around zero energy, while it has a long tail proportional to the inverse of the Fermi energy. We demonstrated how the magnetic oscillation vanishes and converges to the susceptibility, in going from a strong-field regime to a weak-field regime. The behavior at zero energy is shown to be highly singular.

Figure 1

Density of states (above) and the magnetization at $T=0$ (below) in the long-ranged disorder with several strength $W$’s. The plot is against the Fermi energy, and the values are per spin and per valley. Vertical dashed lines show the energies of the Landau level in the clean limit.

Figure 2

Density of states (above) and the magnetization at $T=0$ (below) in the long-ranged disorder with $W=0.02$, in several magnetic fields specified by $n_c={({\epsilon }_c∕\hslash {\omega }_B)}^2$. Dashed curves show the zero-field limit.

Figure 3

Density of states (above) and the magnetization at $T=0$ (below) in the long-ranged disorder with $W=0.02$, in several magnetic fields. The vertical and horizontal axes are renormalized in units of factors $\propto \hslash {\omega }_B$. Dashed vertical lines are the energies of the Landau levels in the ideal limit.

Figure 4

Density of states (above) and the magnetization at $T=0$ (below) in the long-ranged disorder with $W=0.1$, in several magnetic fields. Dashed curves show the zero-field limit.