Large-scale Mesoscopic Transport in Nanostructured Graphene

Through exponential sample-size scaling of conductance, we demonstrate strong electron localization in three sets of nanostructured antidot graphene samples with localization lengths of 1.1, 2, and $3.4\mu \mathrm{m}$. The large-scale mesoscopic transport is manifest as a parallel conduction channel to 2D variable range hopping, with a Coulomb quasigap around the Fermi level. The opening of the correlation quasigap, observable below 25 K through the temperature dependence of conductance, makes possible the exponential suppression of inelastic electron-electron scatterings and thereby leads to an observed dephasing length of $10\mu \mathrm{m}$.

Figure 1

(a) SEM images of two sets of nanostructured graphene devices in square geometry (inset), denoted set $A$, and Hall bar geometry, denoted set $B$. (b) The sheet resistance of the $2\mu \mathrm{m}$ sample from sample set $A$ is plotted as a function of the backgate voltage. Above 20 K, the current used was $I=50\mathrm{nA}$. The 2 K data were measured at $V_{\mathrm{sd}}=20\mathrm{mV}$. Inset: AFM image of the nanostructured graphene with a periodic triangular lattice of antidots. (c) 2D color maps of conductance shown as a function of the bias voltage, $V_{\mathrm{sd}}$, and the backgate voltage, measured at 2 K. A conductance gap is seen.

Figure 2

Logarithm of the sheet conductance of sample set $A$, in units of $G_0$, is plotted as a function of $T^{-1/3}$. The solid lines are measured data while the open symbols are calculated from Eq. (1). The dashed lines are used to highlight the plateau regime. Lower inset: A cartoon picture illustrating the two parallel types of conduction. The upper inset shows the density of states as a function of $x=\epsilon /k_B{T}_c$ used to account for the temperature dependence of the hopping conduction.

Figure 3

(a) Conductance $G_{\square }$ of sample $B1$ is plotted as a function of sample size $L$. Black open circles are for gate voltage $V_g-V_{\mathrm{CNP}}=12\mathrm{V}$ and blue open circles are for a 7 nm discontinuous gold film as a reference sample. Here $G_{\square }$ was measured at 2 K with $V_{\mathrm{sd}}=20\mathrm{mV}$. The red solid line is a linear fit to the data. The dashed lines are guides to the eyes. (b) Conductance $G_{\square }$ from sample set $A$ is plotted as a function of sample size $L$ at different magnetic field. Inset: $1/\xi (B)$ is plotted as a function of the magnetic field, normalized by $B_0$. Here, ${\xi }_0$ is the zero field value. The black line is the theory prediction [12], the blue symbols are for samples in set $A$, and red and green symbols are for samples $B1$ and $B2$, respectively.