Two-electron spectra and the spin transition in ellipsoidal quantum dots

Size and shape effects on two-electron spectra and the spin transition in ellipsoidal quantum dots in magnetic fields are studied. Calculated results show that the level crossing of two electrons in quantum dots is dramatically influenced by the shape and size which can strongly change single-electron levels and Coulomb interaction energies. The spin transition is quite different between prolate and oblate quantum dots. The spectra are almost independent of the total spin as the vertical confinement is much weaker than the lateral one. The quantum behaviors of ellipsoidal quantum dots show some relations with those of vertically coupled quantum dots, and they are well explored.

Figure 1

Energy levels of spin-singlet (solid lines) and -triplet (dashed lines) states normalized by ${\gamma }_{xy}$ as a function of ${\gamma }_{xy}^{-1∕2}$ for ellipsoidal QD’s with ${\gamma }_z∕{\gamma }_{xy}=4$.

Figure 2

Coulomb energies of ellipsoidal QD’s with ${\gamma }_z∕{\gamma }_{xy}=4$ defined by Eq. (13) and obtained by the exact diagonalization (solid lines) and the perturbation calculation (dashed lines) as a function of ${\gamma }_{xy}^{-1∕2}$.

Figure 3

Energy levels of spin-singlet (solid lines) and -triplet (dashed lines) states with ${\gamma }_{xy}=1.0$ as a function of ${\gamma }_z^{-1∕2}$ in (a) prolate QD’s and (b) oblate QD’s where ${\gamma }_z$ is subtracted in the total energy to clearly show the differences between the lower-energy levels.

Figure 4

Coulomb energies of ellipsoidal QD’s with ${\gamma }_{xy}=1.0$ defined by Eq. (13) and obtained by the exact diagonalization (solid lines) and the perturbation calculation (dashed lines) as a function of ${\gamma }_z^{-1∕2}$.

Figure 5

Energy levels of spin-singlet (solid lines) and -triplet (dashed lines) states normalized by ${\gamma }_{xy}$ as a function of ${\gamma }_B$ in (a) a spherical QD with ${\gamma }_{xy}={\gamma }_z=0.5$ and (b) a prolate QD with ${\gamma }_{xy}=0.5$, ${\gamma }_z=0.4$. $c-$, $e-$, $f-$, and$d-$ ($c+$, $e+$, $f+$, and $d+$) represent the corresponding states with negative (positive) azimuthal quantum numbers.

Figure 6

$\Delta E$ obtained by the exact diagonalization (solid lines) and the perturbation calculation (dashed lines) and normalized by ${\gamma }_{xy}$ as a function of ${\lambda }_B$ for ${\lambda }_0=0.8$, 1.0, and 1.2 with (a) ${\gamma }_{xy}=2.0$ and (b) ${\gamma }_{xy}=0.5$.