Electron localization function for two-dimensional systems

The concept of the electron localization function (ELF) is extended to two-dimensional (2D) electron systems. We show that the topological properties of the ELF in two dimensions are considerably simpler than in molecules studied previously. We compute the ELF and demonstrate its usefulness for various physical 2D systems focusing on semiconductor quantum dots that effectively correspond to a confined 2D electron gas. The ELF visualizes the shell structure of harmonic quantum dots and provides insight into electron bonding in quantum-dot molecules. In external magnetic fields, the ELF is found to be a useful measure of vorticity when analyzing the properties of quantum-Hall droplets. We show that the current-dependent term in the ELF expression is important in magnetic fields.

Figure 1

(Color online) ELF (red solid lines) with the corresponding electron densities (blue dashed lines) for two-dimensional closed-shell quantum dots. The maximum-density values are scaled to 1.

Figure 2

(Color online) Ground-state total densities and ELFs for four-minima quantum-dot molecules having 8, 12, and 16 electrons, respectively. The total spin $S=0$ in all the cases.

Figure 3

(Color online) [(a) and (b)] Electron density and ELF of the maximum-density droplet in a 12-electron quantum dot. [(c) and (d)] Intersections of (a) and (b) together with the results for $N=6$ and $N=20$ quantum dots. [(e) and (f)] Paramagnetic current densities and the kinetic-energy densities of the maximum-density-droplet states, respectively.

Figure 4

(Color online) (a) Electron density and the (b) ELF of a single-vortex solution in a six-electron parabolic quantum dot at $B=11\mathrm{T}$. The lower panel shows middle intersections of the figures in the upper panel.

Figure 5

(Color online) (a) Electron density and (b) the ELF of a two-vortex solution in a six-electron rectangular quantum dot at $B=16\mathrm{T}$. The lower panel shows intersections of the upper figures at $y_1$ and $y_2$, marked in the upper right figure.