The high-field state at the Dirac Point in graphene*

The discovery of the quantum Hall Effect in graphene has generated considerable interest in the state at the Dirac Point in a magnetic field $H$. In intense $H$, the 4-fold degeneracy of the $n=0$ Landau Level (LL) is lifted by the enhanced exchange energy. Among the broken symmetry states proposed are the quantum Hall ferromagnet, the quantum Hall insulator state, excitonic condensation, and charge-density-wave formation. A subset of these theories propose counter-propagating edge states that remain conducting in large $H$. We have performed measurements of the resistance $R_{xx}$ and Hall resistance $R_{xy}$ to fields of 33 T at temperatures $T$ from 0.3 to 50 K in $\sim$6 graphene samples. We find that, as $T$ decreases below 10 K, $R_0$ (= $R_{xx}$ at the Dirac Point) undergoes a steep increase with a divergence consistent with a field-driven transition to an insulating high-field state. The divergence in $R_0$ fits well to the Kosterlitz-Thouless (KT) form $\exp(b/\sqrt{h-1})$ with $h=H/H_c$ and $b\sim 1.4$. The critical field $H_c$ is sample dependent (12 T to 33 T ), and correlates with the disorder as measured by the offset gate voltage $V_0$ and the zero-$H$ mobility. The divergence in $R_0$ is strictly confined to the $n=0$ LL (bracketed by the sublevels $\nu = \pm 1$). The peaks with $n=\pm 1$ remain near the values $h/e^2$. Using an ultralow-power (3 fW), voltage-regulated technique, we show that the KT-fit to $R_0$ is valid over 3 decades (40 k$\Omega$ to 40 M$\Omega$). The steepness of the $R_0$ vs. $T$ curves implies a bulk gap $\Delta$ of magnitude 15-20 K that decreases when $H$ falls below $H_c$. We compare our findings with the various proposed models. We will also report thermopower and Nernst measurements taken to fields of 14 T.\\[4pt] Refs. J. G. Checkelsky, L. Li and N. P. Ong, prl {\bf100}, 206801 (2008); ibid. cond-mat arXiv:0808.0906v1.