Excitonic gap, phase transition, and quantum Hall effect in graphene

We suggest that physics underlying the recently observed removal of sublattice and spin degeneracies in graphene in a strong magnetic field describes a phase transition connected with the generation of an excitonic gap. The experimental form of the Hall conductivity is reproduced and the main characteristics of the dynamics are described. Predictions of the behavior of the gap as a function of temperature and a gate voltage are made.

Figure 1

(Color online) Schematic illustration of the spectrum and the Hall conductivity in the $n=0$ and $n=1$ Landau levels for four different cases. (a) $\Delta =0$ and no Zeeman term. (b) Nonzero $\Delta$ and no Zeeman term. (c) $\Delta =0$ and the Zeeman term is taken into account. (d) Both $\Delta$ and the Zeeman term are nonzero. Thickness of the lines represents the degeneracy $\times 4$, $\times 2$, and $\times 1$ of the energy states; $L=\sqrt{\hslash v_F^2\mid eB\mid ∕c}$.

Figure 2

(Color online) Upper line with triangles: The experimentally observed transition points between the lowest plateaus in the Hall conductivity from Ref. 8. Lower line: The theoretical phase diagram in the plane of the magnetic field $B$ and the chemical potential $\mu$ for $\gamma =18\mathrm{K}$ and $T=30\mathrm{mK}$. First and second order phase transitions are denoted by solid and dashed lines, respectively.

Figure 3

(Color online) Hall conductivity in the experiment (Ref. 8) (upper panel) and in the theoretical model (lower panel) for magnetic fields $B=9\mathrm{T}$ (circle), $11.5\mathrm{T}$ (pentagon), $17.5\mathrm{T}$ (hexagon), $25\mathrm{T}$ (square), $30\mathrm{T}$ (diamond), $37\mathrm{T}$ (up triangle), $42\mathrm{T}$ (down triangle), and $45\mathrm{T}$ (star). The parameters in the model are $b=0.04$, $\gamma =18\mathrm{K}$, ${\gamma }_{\mathrm{tr}}=6\mathrm{K}$, and temperature $T=30\mathrm{mK}$.

Figure 4

(Color online) The gap $\Delta$ versus $V_g$ for $B=30\mathrm{T}$ and four different values of temperature $T=30\mathrm{mK}$, $5\mathrm{K}$, $10\mathrm{K}$, and $15\mathrm{K}$, from top to bottom.