Imaging quantum-dot-confined electron density in transition to fractional quantum Hall regime

We consider mapping the charge density confined in quantum dots by the Coulomb blockade microscopy (CBM) at the transition to the fractional quantum Hall regime. We apply an exact diagonalization method to determine the exact charge density and its reaction to the scanning probe as well as to calculate the energy maps as functions of the position of the probe. From the energy maps—which are the only experimentally accessible quantity in CBM—we evaluate an apparent charge density solving an inverse integral problem given by the perturbation theory. We discuss the exact and apparent charge densities derived from the energy maps. We find that for magnetic fields corresponding to the integer fillings of the lowest Landau level, when the electron system exhibits a liquid-like reaction to the potential of the probe, the confined charge density can be quite accurately mapped by the CBM. For fractional fillings of the lowest Landau level the probe induces nucleation of single-electron islands which in circular quantum dots evade imaging by CBM. We demonstrate that mapping the molecular charge densities is possible for confinement potentials of lower symmetry that is consistent with the geometry of the single-electron islands distribution. (paper)