Stability of the shell structure in two-dimensional quantum dots

We study the effects of external impurities on the shell structure in semiconductor quantum dots by using a fast response-function method for solving the Kohn-Sham equations. We perform statistics of the addition energies up to 20 interacting electrons. The results show that the shell structure is generally preserved even if there are clear signs of disorder in the energetics. The Coulomb interaction and the variation in ground-state spins have a strong effect on the addition-energy distributions, which in the noninteracting single-electron picture correspond to level statistics showing mixtures of Poisson and Wigner forms.

Figure 1

Addition energies for clean and impurity-containing QD’s with the confinement strengths of ${\omega }_0=5\mathrm{meV}$ (a) and $3\mathrm{meV}$ (b).

Figure 2

Noninteracting single-electron energies for a quantum dot containing a single repulsive impurity particle on the dot plane $(d=0)$ and at $R=0\cdots 8a_B^{*}$ from the center.

Figure 3

(Color online) Addition energies (scale from dark blue to dark red) for a quantum dot containing a single repulsive impurity particle at $R=0\cdots 7.5a_B^{*}$ from the dot center. The filled circles denote the changes in the ground-state spin. The solid, dashed, and dotted lines correspond to $S=0$ or $1∕2$, $S=1$ or $3∕2$, and $S=2$ or $5∕2$, respectively.

Figure 4

Electron densities of a 12-electron quantum dot with an impurity particle located at $R=0$, 1, 3, and $6a_B^{*}$ from the dot center, respectively.

Figure 5

Noninteracting addition-energy distributions of 1000 impurity configurations corresponding to $N=4$ (a) and $N=12$ (b). The approximative contributions of the Poisson- and Wigner-type distributions are also shown. The energy bin width is $0.01{\mathrm{Ha}}^{*}$.

Figure 6

Addition-energy distributions of 1000 different impurity configurations in a quantum dot of 1–20 electrons. All the possibilities for the ground-state spins are taken into account.

Figure 7

Addition-energy distributions for $N=6$ (a) and $N=20$ (b) quantum dots in noninteracting (white) and interacting (dark gray) cases. The intersections are plotted in light gray.

Figure 8

Addition-energy distributions for two quantum dots of 4 and 14 interacting electrons, respectively. The results have been calculated both by (i) taking all the possible ground-state spins into account (dark gray) and (ii) neglecting Hund’s rule and thus setting $S=0$ and $1∕2$ for the ground states of $N=4,14$ and $N=3,5,13,15$, respectively (white). The intersections are plotted in light gray.