Zero-Energy State in Graphene in a High Magnetic Field

The fate of the charge-neutral Dirac point in graphene in a high magnetic field $H$ has been investigated at low temperatures ($T\sim 0.3\mathrm{K}$). In samples with small gate-voltage offset $V_0$, the resistance $R_0$ at the Dirac point diverges steeply with $H$, signaling a crossover to a state with a very large $R_0$. The approach to this state is highly unusual. Despite the steep divergence in $R_0$, the profile of $R_0$ vs $T$ in fixed $H$ saturates to a $T$-independent value below 2 K, consistent with gapless charge-carrying excitations.

Figure 1

The resistance $R_{xx}$ (a) and Hall conductivity ${\sigma }_{xy}$ (b) in Sample K7 versus (shifted) gate voltage $V_g^{\prime }=V_g-V_0$ at 0.3 K with $H$ fixed at 8, 11, and 14 T. Peaks of $R_{xx}$ at finite $V_g^{\prime }$ correspond to the filling of the $n=1$ and $n=2$ LLs. At $V_g^{\prime }=0$, the peak in $R_{xx}$ grows to $190\mathrm{k}\Omega$ at 14 T. The inset shows sample K22 in false color (dark red) with Au leads deposited (yellow regions). The bar indicates $5\mu \mathrm{m}$. Panel (b) shows the quantization of ${\sigma }_{xy}$ at the values $(4e^2/h)(n+\frac{1}{2})$. At 0.3 K, ${\sigma }_{xy}=0$ in a a 2-V interval around $V_g^{\prime }=0$.

Figure 2

The resistance $R_{xx}$, conductivity ${\sigma }_{xx}$ and the Hall conductivity ${\sigma }_{xy}$ in K7 vs the shifted gate voltage $V_g^{\prime }$, with $H$ fixed at 14 T (we used [5] ${\sigma }_{xy}=-R_{xy}/[(w{R}_{xx}/L{)}^2+R_{xy}^2]$). As $T$ decreases to 0.3 K, the zero-energy peak in $R_{xx}$ (Panel a) rises steeply to $190\mathrm{k}\Omega$. Panel (b) shows that, as $T$ decreases, double peaks in ${\sigma }_{xx}$ are clearly resolved. Between the peaks, ${\sigma }_{xx}$ falls rapidly but saturates below 2 K. The Hall conductivity at 0.3 K (thin curve) displays a clear plateau ($|{\sigma }_{xy}|<0.02e^2/h$) in the interval $-1V<V_g^{\prime }<1V$. Panel (c) compares $R_{xx}$ (of $n=0$ LL) vs unshifted gate $V_g$ in the samples K5, K7, K8, and K29 at 0.3 K. In each sample, $R_{xx}$ peaks at $V_0$. As $V_0$ increases, $R_0(14)$ rapidly decreases.

Figure 3

The $T$ dependence of ${\sigma }_{xx}^0$ in K7 ($=L/w{R}_0$) and the $H$ dependence of $R_0$ at low temperature. Panel a shows curves of ${\sigma }_{xx}^0$ vs ${log}_{10}T$ with $H$ fixed at 6–14 T. For $H>8\mathrm{T}$, the gap $\Delta$ causes ${\sigma }_{xx}^0$ to decrease markedly until saturation at the residual value ${\sigma }_{\mathrm{res}}$ occurs below 2 K. Panel (b) displays the steep increase in $R_0$ vs $H$ in K7 at selected $T$. At 0.3 K, $R_0$ appears to diverge at a field near 18 T [see Fig. 4b]. Panel (c) compares the $R_0(H)$ profiles in Samples K7, K18, and K22. In sample K18 ($V_0=20\mathrm{V}$), the divergence in $R_0$ becomes apparent only above 14 T, whereas in K22 ($V_0=-0.6\mathrm{V}$) $R_0$ starts to diverge at fields lower than in K7. In K7, we have plotted $R_0$ values measured by sweeping $V_g$ at fixed $H$ (solid symbols) with $R_0$ measured by sweeping $H$ with $V_g^{\prime }$ fixed at 0 (solid curve), to show consistency.

Figure 4

(a) The contour plot of $R_0(T,H)$ (K7) in the $T\mathrm{\text{−}}H$ plane (vertical bar shows values of $R_0$). The contour lines emphasize the unusual approach to the large-$R_0$ state. At low $T$, $R_0$ is unchanged on a horizontal path ($H$ held fixed), but it rises rapidly on a vertical path (increasing $H$ at fixed $T$). Panel (b) displays $\log R_0$ vs $1/\sqrt{1-h}$ in K7 at $T=0.3\mathrm{K}$, with $h=H/H_0$, where $H_0=18\mathrm{T}$. The linear segment at large $R_0$ shows that the divergence is consistent with $R_0(h)\sim \exp [2b/\sqrt{1-h}]$ with $b\sim 0.7$. In Panel (c), the plot of $(d\ln R_0/dH{)}^{-2/3}$ vs $H$ shows a high-field linear segment that extrapolates to zero at $\sim H_0$ (18 T).