Manipulating quantum Hall edge channels in graphene through scanning gate microscopy

We show evidence of the backscattering of quantum Hall edge channels in a narrow graphene Hall bar, induced by the gating effect of the conducting tip of a scanning gate microscope, which we can position with nanometer precision. We show full control over the edge channels and are able, because of the spatial variation of the tip potential, to separate copropagating edge channels in the Hall bar, creating junctions between regions of different charge carrier density, that have not been observed in devices based on top or split gates. The solution of the corresponding quantum scattering problem is presented to substantiate these results, and possible follow-up experiments are discussed.

Figure 1

A schematic representation of the SGM setup. By applying a voltage to the metallic tip, we can locally gate a region of choice.

Figure 2

A scanning electron microscope image of a device of the same design as the one described in this paper. The distance between the side contacts is 6 $\mu \mathrm{m}$, while the width of the Hall bar is about $800\mathrm{n}\mathrm{m}$. The probes used to measure $R_{xx}$ and $R_{xy}$ are indicated, as are the tip positions: left (L), center (C), and right (R).

Figure 3

(a) The conductivity of the device at $T=$ $4.2\mathrm{K}$ and the linear fit that is used to extract the field effect mobility. (b) The four-probe resistance of the device at $T=$ $4.2\mathrm{K}$ and the fit that is used to estimate mobility and residual charge.

Figure 4

The quantum Hall effect as measured in the device, while sweeping the back gate at $T=4.2\mathrm{K}$ and $B=8\mathrm{T}$.

Figure 5

(a) $R_{xy}$ vs $V_{bg}$ with the tip placed at the right (R) side of the device, in between the contacts used to measure $R_{xy}$. The data show a strong dependence on the tip bias $V_{\mathrm{tip}}$ (indicated in the legend). (b) $R_{xy}$ with the tip placed at the center (C) of the Hall bar. $R_{xy}$ is now independent of $V_{\mathrm{tip}}$, demonstrating that the tip effect is well localized.

Figure 6

The longitudinal resistance $R_{xx}$ with the tip positioned at the center of the Hall bar, for various $V_{\mathrm{tip}}$ biases. Several plateaus are indicated, which correspond to the backscattering of one or multiple sets of edge channels.

Figure 7

A 2D map, showing the value of $R_{xx}$ as a function of back-gate voltage $V_{bg}$ and tip voltage $V_{\mathrm{tip}}$. The data were collected by sweeping the back gate from $-30$ to $+30$ V, while increasing the tip bias in steps of 2 V in between sweeps, at $T=4.2$ K and $B=8$ T. The global filling factors ${\nu }_{bg}$ and the filling factors underneath the SGM tip ${\nu }_{\mathrm{tip}}$ are indicated.

Figure 8

A schematic illustration of the system with six leads and four Büttiker probes. The tip is positioned at the center of the ribbon, and the strength of the tip potential is indicated by the blue circle. The tip induces a bipolar junction in the nanoribbon. The equilibration is along the edge between channels $\nu >0$ as marked by green rectangles, and along the $n/p$ junction between channels $\nu >0$ and $\nu <0$, as marked by pink rectangles.

Figure 9

(a) The simulated $R_{xx}$ as a function of back-gate voltage $V_{bg}$ and tip voltage $V_{\mathrm{tip}}$. The dashed white lines show where the global filling factor ${\nu }_{bg}$ changes, and the solid black lines where the filling factor under the tip ${\nu }_{\mathrm{tip}}$ changes. (b) $R_{xx}$ values calculated from Eqs. (6, 7, 8). The factors label the resistance values at the plateaus.

Figure 10

(a) A representative potential map of the system for ${\nu }_{bg}=-6$, ${\nu }_{\mathrm{tip}}=+2$. The dashed circles show where the potential equals the Fermi energy (inner circle) and the energy of the first LL (outer circle). The solid circle indicates the potential at half maximum. (b) Horizontal cross section through the center of the Hall bar in (a), showing how the potential varies with position. (c) An exemplary current density in the system for the potential profile in (a).

Figure 11

(a) A schematic illustration of the backscattering scenario of the lowest Landau level channels only. The tip is positioned at the center of the ribbon, and the strength of the tip potential is indicated by the blue circles. In this scenario the outermost channels ${\nu }_{bg}$ (here ${\nu }_{bg}=-6$) are backscattered and do not equilibrate with the innermost channels $\left({\nu }_{\mathrm{tip}}=+2\right)$. (b) Backscattering of two sets of edge channels. In this scenario the innermost channels $\left(\nu =+6\right)$ do not equilibrate with the outermost channels $\left(\nu =-2\right)$. The equilibration takes place along the edge between channels $\nu =+6$ and $\nu =+2$, marked by green rectangles, and along the $n/p$ junction between channels $\nu =-2$ and $\nu =+2$, as marked by pink rectangles. (c) A schematic illustration of the general backscattering scenario. Here, the channels ${\nu }^{\prime }$ and ${\nu }^{″}$ equilibrate along the $n/p$ junction, whereas the outermost channels ${\nu }_{bg}$ are backscattered and the innermost ones ${\nu }_{\mathrm{tip}}$ undergo an indirect equilibration along the edge.

Figure 12

A schematic drawing of the system for the calculation of longitudinal resistances, with the numeration of the leads. The pink and green rectangles mark the Büttiker probes which absorb all the current flowing through the rectangles.