Vortex Clusters in Quantum Dots

We study electronic structures of two-dimensional quantum dots in strong magnetic fields using mean-field density-functional theory and exact diagonalization. Our numerically accurate mean-field solutions show a reconstruction of the uniform-density electron droplet when the magnetic field flux quanta enter one by one the dot in stronger fields. These quanta correspond to repelling vortices forming polygonal clusters inside the dot. We find similar structures in the exact treatment of the problem by constructing a conditional operator for the analysis. We discuss important differences and limitations of the methods used.

Figure 1

Angular-momentum $L$ of the six-electron QD from SDFT (open circles). The plateaus are characterized by vortex holes in the electron density. The number next to a plateau gives the number of vortices inside QD. The horizontal lines correspond to ground-state $L$ values from ED.

Figure 2

Upper panel: Electron densities of the six-electron QD calculated with CSDFT for single-vortex (a), three-vortex (b), and six-vortex (c) configurations. Lower panel: Most probable electron positions $\{{\mathbf{r}}_i^{*}{\}}_{i=1}^N$ (crosses) and conditional electron densities (contours) and phases (gray scale) from ED for the cases of the upper panel: $L=21$ (d), $L=30$ (e), and $L=45$ (f). We probe with the rightmost electron. The densities are on a logarithmic scale (values $e^{-i}$, $i=1,3,\dots ,17$), and the phase changes from $\pi$ to $-\pi$ on the lines where shadowing changes from the darkest gray to white.

Figure 3

Occupation of angular-momentum eigenstates for solutions with three (left) and four (right) vortices. Triangles mark the CSDFT and squares the ED results, respectively. The minima at the low $l$ values show the formation of vortices in the QD.

Figure 4

Left: SDFT electron density of a 24-electron QD at 5 T showing 14 vortices. Right: Electron density (gray scale, the white regions have low electron density and correspond to vortices) and currents (arrows) on the edge of the QD.