Dissipative Quantum Hall Effect in Graphene near the Dirac Point

We report on the unusual nature of the $\nu =0$ state in the integer quantum Hall effect (QHE) in graphene and show that electron transport in this regime is dominated by counterpropagating edge states. Such states, intrinsic to massless Dirac quasiparticles, manifest themselves in a large longitudinal resistivity ${\rho }_{xx}\gtrsim h/e^2$, in striking contrast to ${\rho }_{xx}$ behavior in the standard QHE. The $\nu =0$ state in graphene is also predicted to exhibit pronounced fluctuations in ${\rho }_{xy}$ and ${\rho }_{xx}$ and a smeared zero Hall plateau in ${\sigma }_{xy}$, in agreement with experiment. The existence of gapless edge states puts stringent constraints on possible theoretical models of the $\nu =0$ state.

Figure 1

Longitudinal and Hall conductivities ${\sigma }_{xx}$ and ${\sigma }_{xy}$ (a) calculated from ${\rho }_{xx}$ and ${\rho }_{xy}$ measured at 4 K and $B=30\mathrm{T}$ (b). The $\nu =0$ plateau in ${\sigma }_{xy}$ and the double-peak structure in ${\sigma }_{xx}$ arise mostly from strong density dependence of ${\rho }_{xx}$ peak (green trace shows ${\sigma }_{xy}$ for another sample). The upper inset shows one of our devices. The lower insets show temperature and magnetic field dependence of ${\rho }_{xx}$ near $\nu =0$. Note the metallic temperature dependence of ${\rho }_{xx}$.

Figure 2

Excitation dispersion in $\nu =0$ QH state with and without gapless chiral edge modes, (a) and (b) respectively. Case (a) is realized in spin-polarized $\nu =0$ state [4], while case (b) occurs when symmetry is incompatible with gapless modes, for example, in valley-polarized $\nu =0$ state conjectured in Ref. 14. In the latter state a gap opens at branch crossing due to valley mixing at the sample boundary.

Figure 3

(a) Schematic of transport in a Hall bar with voltage probes. Chiral edge states carrying opposite spin, Eqs. (3), are denoted by red and blue. Transport through the bulk is indicated by dotted lines. (b) Voltage probe in a full spin mixing regime [17] measures $V_{\mathrm{probe}}=\frac{h}{2e^2}(I_1+I_2)$. Note finite voltage drop across the probe, Eq. (2).

Figure 4

Transport coefficients ${\rho }_{xx}$, ${\rho }_{xy}$ and $G_{xx}={\rho }_{xy}/({\rho }_{xy}^2+{\rho }_{xx}^2)$, $G_{xy}={\rho }_{xy}/({\rho }_{xy}^2+{\rho }_{xx}^2)$ for a Hall bar (Fig. 3), obtained from the edge model (6) with account of bulk conductivity [Eqs. (10, 8, 9), see text]. The edge transport shunted by the bulk conduction away from $\nu =0$ results in the ${\rho }_{xx}$ peak. Note the smooth behavior of ${\rho }_{xy}$ at $\nu =0$, a tilted plateau in $G_{xy}$, and a double-peak structure in $G_{xx}$.