Imaging Transport Resonances in the Quantum Hall Effect

We use a scanning capacitance probe to image transport in the quantum Hall system. Applying a dc bias voltage to the tip induces a ring-shaped incompressible strip (IS) in the 2D electron system (2DES) that moves with the tip. At certain tip positions, short-range disorder in the 2DES creates a quantum dot island in the IS. These islands enable resonant tunneling across the IS, enhancing its conductance by more than 4 orders of magnitude. The images provide a quantitative measure of disorder and suggest resonant tunneling as the primary mechanism for transport across ISs.

Figure 1

(a) $10\times 10\mu \mathrm{m}$ in-phase charging image taken at ${\nu }_{\mathrm{bulk}}=0.89$ ($B=7.0\mathrm{T}$) with a bias of $+1.5\mathrm{V}$ applied to the tip. (b) Bubble induced by dc tip bias, surrounded by an incompressible ring (IR). Scale bar indicates the size of the IR for the image shown. (c) Intersection of the IR with a hot spot of fixed position in the 2DES creates filamentary arcs in the image that reproduce the shape of the IR inverted through the hot spot.

Figure 2

$3\times 3\mu \mathrm{m}$ charging images. (a), (b) In-phase and lagging-phase electron-bubble images, ${\nu }_{\mathrm{bulk}}=0.83$ (7.5 T), $V_{\mathrm{tip}}=+1.5\mathrm{V}$. Circles drawn in the images show tip positions where the IR resistance varies to result in the in-phase and lagging-phase signals shown by matching arrows in the RC model in (c). (d), (e) In-phase and lagging-phase hole-bubble images for ${\nu }_{\mathrm{bulk}}=1.1$ (5.5 T), $V_{\mathrm{tip}}=-1.5\mathrm{V}$. (f) Density profile (top) and Landau level energy profile (bottom) of a short length-scale density fluctuation creating a quantum dot allowing resonant tunneling across the IR. Dotted, gray, and solid lines (bottom) depict empty, partially filled, and filled Landau levels, respectively. Coulomb blockade of the dot leads to subfilament fringes visible in the images.

Figure 3

(a) Color scale plot of the in-phase charging signal as a function of tip bias voltage and magnetic field with the tip at a fixed position. Lines (i) and (ii) display the threshold for forming a hole and electron bubble, respectively. Dashed (dotted) lines indicate biases and fields where the IR intersects a hot spot, allowing resonant tunneling into a hole (electron) bubble under the tip. (b) $6\times 6\mu \mathrm{m}$ in-phase electron-bubble image at 7.0 T and $+1.5\mathrm{V}$. The blue marker shows the position of the tip during the measurement in (a). Two arcs are identified, and the positions of the hot spots responsible for each arc are indicated. (c) The IR intersects hot spot 1 at biases and fields given by line (iv) in (a). (d) At lower biases the IR is smaller, and intersects hot spot 2 at biases and fields given by line (iii).

Figure 4

(a) $15\times 15\mu \mathrm{m}$ in-phase electron-bubble image, taken at ${\nu }_{\mathrm{bulk}}=0.83$ (7.5 T), $V_{\mathrm{tip}}=+1.75\mathrm{V}$. (b) $15\times 15\mu \mathrm{m}$ in-phase hole-bubble image at the same location at ${\nu }_{\mathrm{bulk}}=1.26$ (5.0 T). A tip bias of $-1.75\mathrm{V}$ is chosen to produce an IR of the same diameter as in (a). (c) Composite image constructed from (a) and (b) with hole-bubble data in blue and electron-bubble data in red. Semitransparent repeating ellipses are overlaid as a guide to the eye. (d), (e) Schematics illustrating local IR width modulation from disorder in (d) electron and (e) hole bubbles. The black marker and arrow indicate the location of a hot spot and direction of a fixed local density gradient. Colored arrows indicate the direction of the density gradient from the tip.