Scanning gate imaging in confined geometries

This article reports on tunable electron backscattering investigated with the biased tip of a scanning force microscope. Using a channel defined by a pair of Schottky gates, the branched electron flow of ballistic electrons injected from a quantum point contact is guided by potentials of a tunable height well below the Fermi energy. The transition from injection into an open two-dimensional electron gas to a strongly confined channel exhibits three experimentally distinct regimes: one in which branches spread unrestrictedly, one in which branches are confined but the background conductance is affected very little, and one where the branches have disappeared and the conductance is strongly modified. Classical trajectory-based simulations explain these regimes at the microscopic level. These experiments allow us to understand under which conditions branches observed in scanning gate experiments do or do not reflect the flow of electrons.

Figure 1

Schematic of the gate geometry and measurement setup. Gates are labeled $g_1$ to $g_{12}$. The pink area indicates the movable tip-depleted region.

Figure 2

(a) Conductance $G(x,y)$ of QPC 1 on sample A. Gray solid lines mark the outlines of the gates. Roman numbers label the channel region (II) and the two regions outside the channel (I and III). (b) Derivative $dG(x,y)/dx$ of the data in (a).

Figure 3

Scanning gate images for different channel gate voltages as indicated in the images. (a) Conductance as a function of tip position for the QPC transmitting four spin-degenerate modes and $V_{\mathrm{ch}}=-400$ mV to electrostatically define the channel. (b)–(f) Series of scans with different, nondepleting $V_{\mathrm{ch}}$ showing $\mathrm{\Delta }G=G-4e^2/h$ in color. The QPC transmits two spin-degenerate modes in the absence of the tip.

Figure 4

Trajectories of electrons injected through the QPC (conductance $4e^2/h$) for different situations: (a) only the QPC, no channel ($V_{\mathrm{ch}}=200$ mV), and no tip; (b) QPC and tip, but no channel ($V_{\mathrm{ch}}=200$ mV); (c) QPC and shallow channel potential ($V_{\mathrm{ch}}=0$ mV), but no tip; (d) QPC and shallow channel potential (${\mathrm{V}}_{\mathrm{ch}}=0$ mV) with tip; (e) three distinct backscattered trajectories in QPC, tip, and shallow channel potential ($V_{\mathrm{ch}}=0$ mV); and (f) one backscattered trajectory with depleting QPC, channel ($V_{\mathrm{ch}}=-350$ mV), and tip.

Figure 5

(a) Smoothed conductance cuts along $y$, roughly 80 nm from the channel center of sample B for different channel gate voltages between complete pinchoff ($-350$ mV) and a flat band below channel gates (250 mV). (b) Smoothed conductance cuts along $x$ at $y=5\mu \mathrm{m}$ for a set of channel gate voltages. Smoothing is extended over a range of $1 \mu \mathrm{m}$ [shaded region in (a)]. (c) Symbols denote the measured average conductance of the system with the tip in the center of the channel as a function of $V_{\mathrm{ch}}$. Solid lines show the calculated transmission and reflection probabilities (see the text). (d) Simulated scanning gate images with a QPC conductance of $6e^2/h$ at different channel gate voltages.