Transport Measurements Across a Tunable Potential Barrier in Graphene

The peculiar nature of electron scattering in graphene is among many exciting theoretical predictions for the physical properties of this material. To investigate electron scattering properties in a graphene plane, we have created a gate-tunable potential barrier within a single-layer graphene sheet. We report measurements of electrical transport across this structure as the tunable barrier potential is swept through a range of heights. When the barrier is sufficiently strong to form a bipolar junction ($n\mathrm{\text{−}}p\mathrm{\text{−}}n$ or $p\mathrm{\text{−}}n\mathrm{\text{−}}p$) within the graphene sheet, the resistance across the barrier sharply increases. We compare these results to predictions for both diffusive and ballistic transport, as the barrier rises on a length scale comparable to the mean free path. Finally, we show how a magnetic field modifies transport across the barrier.

Figure 1

(a) Cross-section view of the top gate device. (b) Simplified model for the electrochemical potential $U$ of electrons in graphene along the cross section of (a). The potential is shifted in region 2 by the top gate voltage and shifted in both regions 1 and 2 by the back gate voltage. (c) Optical image of the device. The barely visible graphene is outlined with a dashed line and the PMMA layer appears as a (blue) shadow. A schematic of the four-terminal measurement setup used throughout the Letter is shown.

Figure 2

(a) Resistance across the graphene sample at 4 K as a function of the top gate voltage for several back gate voltages, each denoted by a different color. (b) Two-dimensional gray scale plot of the same resistance as a function of both gate voltages. Traces in (a) are cuts along the correspondingly colored lines.

Figure 3

(a) Odd component of the resistance: the part which depends on the sign of the density $n_2$ in region 2. (b) Gray scale plot of the odd component of the resistance for many values of $n_1$. Colored lines are the cuts shown in (a).

Figure 4

(a) Using the strongly diffusive model described in the text, one can predict the resistance as a function of the top gate voltage $V_t$ for several values of the density $n_1$ (each represented by the same color as in Fig. 3). Here we plot the odd part of this calculated resistance (cf. Fig. 3) for a barrier smoothness $d=40\mathrm{nm}$ and a width $w=1.7\mu \mathrm{m}$. (b) Similar curves for several densities $n_1$ using the ballistic model described in the text. The curves are plotted only for densities $n_2$ not too close to zero: $n_2>{10}^{11}{\mathrm{cm}}^{-2}$ (the model diverges at $n_2=0$) and dashed lines link the curves at opposite sides of $n_2=0$.

Figure 5

Odd part of the resistance as a function of the back gate voltage $V_b$, with the constraint $n_2=-n_1$. Measurements were taken at 4 K for several values of the perpendicular magnetic field $B$.