Statistics of closed quantum dots: Effects of disorder and interactions

We employ a fast response-function algorithm within the spin-density-functional theory to study the effects of disorder and electron-electron interactions on few-electron quantum dots. We focus on the regime between a clear shell structure and complete distortion by applying a realistic impurity model with a tunable impurity density. A drastic change in the addition-energy distribution computed from a thousand impurity configurations is observed in this regime. However, the electron-electron interactions symmetrize the distribution in accordance with the experiments. We show that the spin probabilities are sensitive to the impurities, but in the high-disorder limit the spin states stabilize and become totally independent of the corresponding single-electron level spacings.

Figure 1

Noninteracting level-spacing distributions ${\Delta }_0={ϵ}_{N∕2+1}-{ϵ}_{N∕2}$ up to $N=30$ with different impurity numbers $N_{\mathrm{imp}}$. Dark regions correspond to high probabilities. Increasing impurity density leads to gradual disappearance of the shell structure.

Figure 2

Addition-energy distributions of a 12-electron QD having 5 (a) and 30 (b) impurities. The results for the $e\text{−}e$ interaction strengths $\alpha =1.5$ and $\alpha =2.0$ are shown in dark gray and white bars, respectively (intersection in light gray). Inset: the same ${\Delta }_{\alpha =2.0}$ distribution in comparison with the noninteracting level-spacing distribution ${\Delta }_0\left(12\right)={ϵ}_7-{ϵ}_6$ of the corresponding single-electron case.

Figure 3

Probabilities for different spin states in $N=15$ (a) and $N=16$ (b) quantum dots with $e\text{−}e$ interaction strengths $\alpha =1.5$ (solid lines) and $\alpha =2.0$ (dashed lines) as a function of the impurity number $N_{\mathrm{imp}}$.

Figure 4

(Color online) Total energy for different spin states in a 16-electron quantum dot versus the sum of the noninteracting level splittings on the fourth energy shell. The parameters used are $\alpha =1.5$, $N_{\mathrm{imp}}=5$ (a) and $\alpha =2.0$, $N_{\mathrm{imp}}=20$ (b).