Density of States and Zero Landau Level Probed through Capacitance of Graphene

We report capacitors in which a finite electronic compressibility of graphene dominates the electrostatics, resulting in pronounced changes in capacitance as a function of magnetic field and carrier concentration. The capacitance measurements have allowed us to accurately map the density of states $D$, and compare it against theoretical predictions. Landau oscillations in $D$ are robust and zero Landau level (LL) can easily be seen at room temperature in moderate fields. The broadening of LLs is strongly affected by charge inhomogeneity that leads to zero LL being broader than other levels.

Figure 1

Photograph of one of our devices (left) and its schematics (right). Graphene is connected to the measurement circuit by the contact deposited along the device’s perimeter. The second capacitor plate is an Al electrode (central rectangle in the left photo).

Figure 2

Graphene capacitance in zero $B$. (a) Capacitance (per unit area) of the device in Fig. 1 (solid curve). For consistency, all the plots presented in the report are taken from the same device that exhibited slightly smaller $\delta n$ that the others. The dashed curve is the best fit and the horizontal line shows $C_{\mathrm{ox}}$. Open symbols are the alternative measurements using the periodicity of magnetocapacitance oscillations. For a given $B$, each period in gate voltage $\Delta V_g$ corresponds to the filling of one LL, and this requires an increase in carrier concentration $\Delta n=4eB/h$ where $h$ is the Planck constant. $C$ is then calculated as $e\Delta n/\Delta V_g$. Despite the lower accuracy of this approach (due to oscillations in $C_q$), it provides a valuable crosscheck. (b) $C_q$ as function of $E_F$ (bottom axis) and $n$ (top axis). The right axis plots the density of states, $D=C_q/e^2$. These are the same measurements as in (a), and the flat region near zero energy results from nonlinear stretching of the $x$ axis. The extrapolation lines (dashed) in Fig. 2b should cross at zero, and the small vertical offset can be explained by parasitic capacitances that we deliberately did not include in the analysis to minimize the number of fitting parameters.

Figure 3

Magnetic oscillations in the density of state. (a) Experiment: $C_q=e^{2}D$ as a function of $V_g$ at various $T$. (b) Theory: $C_q$ in 16 T at low $T$. The dashed curve shows the oscillations if all LLs were equally broadened with the half width at half maximum (HWHM), $\Gamma =15\mathrm{meV}$ [20]. The dotted curve takes into account electron-hole puddles and was obtained as a convolution of the above broadening with an added inhomogeneity in $n$ which was modeled by a Gaussian distribution [21] with standard deviation $\delta n=4\times {10}^{11}{\mathrm{cm}}^{-2}$, the value found in Fig. 2. The solid line is a result of a similar convolution but assuming an asymmetric doping, which was modeled as a sum of two Gaussians with height ratios of $7:3$; $\delta n=3.5\times {10}^{11}{\mathrm{cm}}^{-2}$ for both; and the separation of $7\times {10}^{11}{\mathrm{cm}}^{-2}$. Note that the device in Fig. 3a has changed with respect to its state in Fig. 2. We find such changes to occur often after thermal cycling.

Figure 4

Density of states near the NP. (a) The emergence of zero LL with increasing $B$ near room $T$. The curves cover the range from 0 to 16 T in steps of 2 T. Zero LL becomes clearly visible above 10 T. We avoided $T$ above 250 K because the measurements become hysteretic due to an increased mobility of adsorbates, similar to transport experiments [22]. The inset shows $C_q$ and $D$ at the NP as a function of $B$ in units of $\mu \mathrm{F}/{\mathrm{cm}}^2$ and ${10}^{17}{\mathrm{eV}}^{-1}{\mathrm{m}}^{-2}$, respectively. The solid curve is the best fit [19]. (b) $T$ dependence of $D_{\mathrm{NP}}$ at 16 T (symbols). The curves are theory fits. Inset: same as in the main figure but in terms of the width of zero LL.