Stability of vortex structures in quantum dots

We study the stability and structure of vortices emerging in two-dimensional quantum dots in high magnetic fields. Our results obtained with exact diagonalization and density-functional calculations show that vortex structures can be found in various confining potentials. In nonsymmetric external potentials we find off-electron vortices that are localized giving rise to charge deficiency or holes in the electron density with rotating currents around them. We discuss the role of quantum fluctuations and show that vortex formation is observable in the energetics of the system. Our findings suggest that vortices can be used to characterize the solutions in high magnetic fields, giving insight into the underlying internal structure of the electronic wave function.

Figure 1

Spin-density-functional-theory (SDFT) electron densities and currents in an elliptic (a–d) and rectangular (e–h) six-electron quantum dot in different magnetic fields. The potential parameters in effective a.u. are $\left(\hslash \omega ,\delta \right)=\left(0.5,2\right)$ and $\left(L_x,L_y\right)=\left(2\sqrt{2\pi },\sqrt{2\pi }\right)$ in elliptic and rectangular dots, respectively. The increasing of the magnetic field leads to a formation of a vortex pattern beyond the maximum density droplet (MDD) solution $\left(B=12\mathrm{T}\right)$ in both geometries.

Figure 2

Chemical potentials for $N$-electron rectangular $\left(\beta =2\right)$ quantum dots as a function of the magnetic field. The dot is expected to be fully spin-polarized. After the MDD window there are oscillations in the chemical potential which matches with the appearing of additional vortices one-by-one into the quantum dot.

Figure 3

Magnetization of a rectangular (solid line) and elliptical (dashed line) six-electron quantum dot as a function of the magnetic field. The ellipse eccentricity $\delta$ and the rectangle side-length ratio $\beta$ are both set to 2. The numbers in the figure denote the correponding number of vortex holes in the electronic structure.

Figure 4

SDFT solutions of six-electron, elliptically $\left(\delta =2\right)$ confined quantum dots at different magnetic fields. The confinement strength is set to $\hslash \omega =0.5{\mathrm{Ha}}^{*}$. Left panel: Conditional single-determinant wave functions of the Kohn-Sham states. The fixed electrons are marked with crosses and the probing electron is the rightmost electron at the top. The contours show the logarithmic electron density of the probe electron and the grey-scale show the phase of the wave function. The phase changes from $\pi$ to $-\pi$ at the lines where shadowing changes from the darkest grey to white. The vortices that cause charge deficiency in the center of the dot are marked with $+$ signs. Right panel: Electron densities show vortex holes and partial localization of the electrons forming six density maxima.

Figure 5

Electron densities (grey scale) and current densities (arrows) of exact diagonalization (ED) solutions for an elliptical quantum dot with $\delta =1.1$. The number of electrons is 6. (a) The ground state with a two-vortex structure. (b) The first excited state with a single vortex at the origin. The right column shows the corresponding electron densities at the longest major axis of the ellipse. The confinement strength is $\hslash \omega =0.5{\mathrm{Ha}}^{*}$ and the magnetic field is $B=13.4\mathrm{T}$.

Figure 6

Electron densities (gray scale) and current densities (arrows) for two-vortex solutions in elliptical QD’s. The number of electrons is 6. (a) Exact diagonalization (ED) result at $\delta =1.1$. (b) ED result at $\delta =1.2$. (c) SDFT result at $\delta =1.2$. The inset shows the corresponding electron densities at the longest major axis of the ellipse. The vortices are localized in the SDFT solution, and the ED solution shows slight delocalization due to quantum fluctuations. The confinement strength is $\hslash \omega =0.5{\mathrm{Ha}}^{*}$ and the magnetic field is $B=17\mathrm{T}$.

Figure 7

Electron densities (grey scale) and current densities (arrows) of ED solutions for elliptical quantum dots with $\delta =2$. The magnetic field is (a) $17\mathrm{T}$, (b) $21\mathrm{T}$, and (c) $22\mathrm{T}$. The inset shows the corresponding electron densities at the longest major axis of the ellipse. The number of electrons is 6 and the confinement strength is $\hslash \omega =0.5{\mathrm{Ha}}^{*}$.