Imaging Fractal Conductance Fluctuations and Scarred Wave Functions in a Quantum Billiard

We present scanning-probe images and magnetic-field plots which reveal fractal conductance fluctuations in a quantum billiard. The quantum billiard is drawn and tuned using erasable electrostatic lithography, where the scanning probe draws patterns of surface charge in the same environment used for measurements. A periodicity in magnetic field, which is observed in both the images and plots, suggests the presence of classical orbits. Subsequent high-pass filtered high-resolution images resemble the predicted probability density of scarred wave functions, which describe the classical orbits.

Figure 1

Illustration of the device. Erasable electrostatic lithography charge spots are drawn on the device surface to define a 1.4 by $2.9\mu \mathrm{m}$ quantum billiard from the subsurface 2DES. Biased surface electrodes separate the source and drain 2DES regions.

Figure 2

(a) Magnetoconductance of the quantum billiard at 20 mK. (b) Result of box counting algorithm to determine fractal dimension of magnetoconductance data where $D_F=1.44$. (c) Power spectrum of magnetoconductance data from 0 to 20 mT.

Figure 3

(a) Series of 2 by $2\mu \mathrm{m}$ SGM conductance images made by scanning the probe over the top half of the quantum billiards with 0.2 mT increments in the magnetic field. Only the top half is scanned for efficiency. (b) Plot of average image correlation as a function of magnetic period.

Figure 4

(a),(b) High resolution 2 by $3.5\mu \mathrm{m}$ SGM conductance images of the quantum billiard in magnetic fields of 42.0 and 39.1 mT. The approximate billiard boundary is shown. (c) High-pass filtered version of image (b). (d) Power spectrum of magnetoconductance data from 30 to 50 mT. (e) Deviation map of image (c) highlighting nodes. (f) Copy of image (c) with profile plot and annotation to identify features characteristic of scarred wave functions. (g) Result of cube-counting algorithm used to determine the fractal dimension of image (b) where $D_F=2.19$.