Ground State of Two-Dimensional Finite Electron Systems in the Quantum Hall Regime

We study electronic structures of quasi-two-dimensional finite electron systems in high magnetic fields. The solutions in the fractional quantum Hall regime are interpreted as quantum liquids of electrons and vortices. The ground states are classified according to the number of vortices inside the electron droplet. The theory predicts observable effects due to vortex formation in the chemical potentials and magnetization of electron droplets. We compare the transitions in the theory to those found in electron transport experiments on a quantum dot device and find significant correspondence.

Figure 1

Mean-field SDFT phase diagrams of the ground state of parabolically confined electron droplets in high magnetic fields. The ground states are classified according to the number of vortices inside the electron droplet (gray scale). The confining potential is $\hslash {\omega }_0=5\mathrm{meV}$ in (a) and $7.67N^{-1/7}\mathrm{meV}$ in (b). In the upper right hand corner of the right diagram the number of vortices is greater than 8.

Figure 2

Current peaks in the electron transport experiments and transitions in the SDFT (red lines). The dashed lines denote the MDD boundaries. The right dashed line and the dotted lines correspond to transitions associated with the emergence of off-electron vortices one by one inside the electron droplet. The experimental data are from Fig. 2b in Ref. 6.

Figure 3

Chemical potential of the 24-electron quantum dot calculated with the SDFT and compared to the experiments from Ref. 6 and VMC and LLL exact diagonalization results in the vicinity of the MDD window. Noise in the experimental data has been reduced by using a Gaussian filter. The numbers between the arrows indicate the total spin. The roman numbers between the triangles indicate the number of vortices inside the electron droplet in the fractional quantum Hall regime. Charge-density-wave solutions in the SDFT (dotted line) have been discarded as unphysical.

Figure 4

(a) Ground state magnetization per electron calculated with the SDFT. The confining potential is $7.67N^{-1/7}\mathrm{meV}$. The beyond-MDD states are characterized by a plateau region in this plot, where magnetization per electron is close to ${\mu }_{\mathrm{B}}^{*}$. The dashed and the dotted lines correspond to the same transitions as in Fig. 2. (b),(c) Detail plots of the oscillations in the magnetization of the 5-electron and 24-electron droplets, respectively. The finite-size precursor of the $\nu =1/3$ quantum Hall state is identified by using conditional wave functions [11]. The confinement strength is 5 meV in (a) and 3.62 meV in (b). The oscillations in plot (a) are smoothed out due to use of finite temperature.