Correlation hard gap in antidot graphene

We have measured low-temperature variation of resistance and nonlinear current-voltage behavior in antidot graphene in the vicinity of the charge neutrality point. The data are found to be consistent with the manifestations of a variable-range hopping electronic density of states (DOS) with a small hard gap of $\sim 1$ meV around the Fermi level, in conjunction with a parallel tunneling conduction channel that exists at the center of the gap. The hard gap is confirmed by the appearance of a low-conductive plateau at low-bias electric field, whereas the parallel tunneling conduction channel, with temperature-independent conductance, is manifest through the nonlinear electric field variation. Unified good agreement between the temperature and electric field dependencies of conductance, for both channels, is obtained with the predictions of a proposed DOS model. An increase in the gap size with applied magnetic field is observed.

Figure 1

Measurement setup and hard gap in antidot graphene. (a) Experimental setup with a microscope image of antidot graphene sample indicated by the red region. The black scale bar denotes 5 μm. The scanning electron microscopic image of the antidot lattice is shown in the top left corner. The magnetic field was applied perpendicular to the sample. (b) A two-dimensional (2D) map of dI/dV as a function of the gate voltage and the source drain bias voltage $V_{\mathrm{sd}}$. An insulating region (blue color) was observed at $V_{\mathrm{G}}\sim 2.8\mathrm{V}$, which disappears as $V_{\mathrm{G}}$ deviates away from the charge neutrality point (CNP). (c) Arrhenius plots of conductance $G-G_0$ as a function of 1/T. Conductance is measured at CNP under various magnetic fields. The conductance shows good linear relation with different slopes at different magnetic fields, implying a different thermal activation gap. Inset summarizes fitted slopes as a function of the applied magnetic fields. (d) Derived sheet resistance as a function of the gate voltage. Data in (b) and (d) were measured at 100 mK under zero magnetic field.

Figure 2

Proposed density of states (DOS) model and temperature-dependent transport. (a) A schematic figure of the model DOS distribution. A hard gap $2k_{\mathrm{B}}{T}_1$ in the variable-range hopping (VRH) states opens around the Fermi level, where the correlation gap $k_{\mathrm{B}}{T}_1$ is much smaller than the band structure gap Δ. A small DOS bump at the center of the gap [charge neutrality point (CNP)] signifies the parallel conduction channel that arises from the Anderson-localized states. (b) $\ln (G-G_0)$ is plotted as a function of $T^{-1}$. Open circles are measured data, and blue line is the prediction of the proposed DOS model. The top right inset gives the raw conductance data plotted as a function of $T^{-1/3}$. A saturating temperature-independent parallel channel $G_0$ is clearly seen at T <1 K, as indicated by the red dashed curve. The bottom left inset is $\ln \left[d\ln \left(G-G_0\right)/d\ln \left(T\right)\right]$ plotted as a function of $\ln \left(T\right)$, where a crossover from $T^{-1}$ to $T^{-1/3}$ is seen as shown by the fitted red straight lines.

Figure 3

Electric-field-dependent transport under different magnetic fields. (a) The hard gap $T_1$ (black squares) and conductance $G_0$ (red circles) are plotted as a function of the magnetic field. Dotted curves serve as a guide to the eye. (b) Conductance $G$ is plotted as a function of electric field. Different colored symbols represent $G$ measured at different applied magnetic field. Black curves are the theory predictions obtained by the proposed density of states (DOS) model [using Eq. (3)] with a hard gap as shown in Fig. 2. The inset shows the linear relation between the inverse of fitted localization length ξ and the plateau conductance $G_0$ observed at low temperatures. All data were measured at $T=100$ mK.

Figure 4

Tight-binding simulated quasiflat band and wave functions of antidot graphene. (a) Tight-binding simulated band structure of antidot graphene with red curve being the quasiflat bands inside the band structure gap. (b) The wave function for the quasiflat bands at sublattices $A$ and B. (c) Log-log plot of the wave function density ρ as a function of $r-r_0$, as shown by the black squares. The blue line is the linear fitting. Red circles denote the total wave function density within the distance $r$, i.e., N($r$) as a function of $r-r_0$. The red curve serves as a guide to the eye.