Electronic Transport and Quantum Hall Effect in Bipolar Graphene $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ Junctions

We have developed a device fabrication process to pattern graphene into nanostructures of arbitrary shape and control their electronic properties using local electrostatic gates. Electronic transport measurements have been used to characterize locally gated bipolar graphene $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ junctions. We observe a series of fractional quantum Hall conductance plateaus at high magnetic fields as the local charge density is varied in the $p$ and $n$ regions. These fractional plateaus, originating from chiral edge states equilibration at the $p\mathrm{\text{−}}n$ interfaces, exhibit sensitivity to interedge backscattering which is found to be strong for some of the plateaus and much weaker for other plateaus. We use this effect to explore the role of backscattering and estimate disorder strength in our graphene devices.

Figure 1

(a) Scanning electron microscopy (SEM) picture showing several complete two-probe devices with local gates. Scale bar is $2\mu \mathrm{m}$. The electrodes contact the widest parts of the structure. The local gates cover the graphene constrictions and part of the central graphene bars. (b) Schematic side view of the devices. Graphene is contacted by source (S) and drain (D) electrodes, and separated from the back gate plane by 300 nm ${\mathrm{SiO}}_2$, and from the local gate (LG) by the top dielectric (20 nm HSQ and 15 nm ${\mathrm{HfO}}_2$). The back gate (with voltage $V_{\mathrm{BG}}$) is coupled to the entire graphene structure via the capacitor $C_{\mathrm{BG}}$. The local gate only couples to part of the structure via $C_{\mathrm{LG}}$.

Figure 2

(a) $G(V_{\mathrm{BG}})$ for a graphene $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ junction, extracted from (b), showing the two conductance minima associated with Dirac valleys in the graphene leads and under the local gate. Inset: false color SEM picture of a patterned graphene bar with contacts and local gate. Scale bar represents $1\mu \mathrm{m}$. (b) Two-dimensional plot of $G(V_{\mathrm{LG}},V_{\mathrm{BG}})$ for the device shown in (a).

Figure 3

(a) Color map of conductance $G(V_{\mathrm{LG}},V_{\mathrm{BG}})$ at magnetic field $B=13\mathrm{T}$, and $T=4.2\mathrm{K}$. The black cross indicates the location of filling factor zero in LGR and GLs. Inset: Conductance at zero $B$ in the same $(V_{\mathrm{LG}},V_{\mathrm{BG}})$ range and the same color scale as main figure (white denotes $G>10.5e^2/h$). (b) $G(V_{\mathrm{LG}})$ extracted from (a), solid line (red online), showing fractional values of the conductance. Numbers on the right indicate expected fractions for the various filling factors [numbers below the trace (red online) indicate the filling factor, ${\nu }^{\prime }$, in LGR]; see also (g). (c) $G(V_{\mathrm{LG}})$ [projection of diagonal trace (orange online) from (a) onto $V_{\mathrm{LG}}$ axis]. Numbers below the trace (orange online) indicate filling factor, $\nu$, in the GLs. (d), (e), and (f) different edge state diagrams representing possible equilibration processes taking place at different charge densities in the GLs and LGR. The center shaded (purple online) region indicates the LGR. Left and right light gray (yellow online) boxes indicate contact electrodes. (g) Simulated color map of the theoretical conductance plateaus expected from the mechanisms shown in (d), (e), and (f) for different filling factors in the GLs and LGR. The numbers in the rhombi indicate the conductance at that plateau. The color scale is identical to that of (a).

Figure 4

Two-terminal conductance of a lateral heterojunction in a QH state (see inset) as a function of the longitudinal conductivity under local gate (LGR) for the states with $\nu =-2$ and ${\nu }^{\prime }=\pm 2$, $\pm 6$, $-10$ [horizontal trace in Fig. 3a and solid line in Fig. 3b, both red online]. Results obtained from the 2D transport model [23, 24, 25] for LGR of size $500\mathrm{nm}\times 700\mathrm{nm}$ are shown. Finite ${\sigma }_{xx}$ has considerable effect on the conductance of the state with ${\nu }^{\prime }$, $\nu =-2$, but very little effect on other states. This explains the difference in roughness of the observed plateaus for ${\nu }^{\prime }\ne -2$ and ${\nu }^{\prime }=-2$.