Finite-thickness effects in ground-state transitions of two-electron quantum dots

Using the exactly solvable excitation spectrum of two-electron quantum dots with parabolic potential, we show that the inclusion of the vertical extension of the quantum dot provides a consistent description of the experimental findings of Nishi et al. [Phys. Rev. B 75, 121301(R) (2007)]. We found that the second singlet-triplet transition in the ground state is a vanishing function of the lateral confinement in the three-dimensional case, while it always persists in the two-dimensional case. We show that a slight decrease of the lateral confinement leads to a formation of the Wigner molecule at low magnetic fields.

Figure 1

(Color online) The magnetic dependence of the additional energy $\Delta \mu$ in two-electron QDs with lateral confinements $\hslash {\omega }_0=4.2,3.7,3.5,2.9\mathrm{meV}$ [the first, second, and fourth values are experimental values for samples A, B, and C, respectively (Ref. 8)]. The confinement in the third $\left(z\right)$ direction $\hslash {\omega }_z=8\mathrm{meV}$ is fixed for all samples. The results for $\mid g^{*}\mid =0.3\left(0.44\right)$ are connected by solid (dotted) line for $\hslash {\omega }_0=4.2,3.5,2.9\mathrm{meV}$ and by dashed (dotted) line for $\hslash {\omega }_0=3.7\mathrm{meV}$. The solid gray lines display the experimental spacing $\Delta V_g$ as a function of $B$. The arrows identify the position of experimental ground-state transitions (Ref. 8).

Figure 2

(Color online) Magnetic field dependence of the addition energy $\Delta \mu$ for the 2D model with $\hslash {\omega }_0=2.9,3.5\mathrm{meV}$ and for the 3D model ($\hslash {\omega }_0=2.9\mathrm{meV}$, $\hslash {\omega }_z=8\mathrm{meV}$). Ground states are labeled by $(M,S)$, where $M$ and $S$ are the total orbital momentum and the total spin, respectively. Gray vertical lines indicate the position of the experimental crossings between different ground states.

Figure 3

(Color online) The interval $\Delta B$ in which the singlet state (2,0) survives as a function of the lateral confinement for 2D and 3D calculations. The confinement in the third $\left(z\right)$ direction $\hslash {\omega }_z=8\mathrm{meV}$ is fixed for the 3D calculations.

Figure 4

(Color online) Top: the magnetic dependence of the ground state in 2D and 3D $(\hslash {\omega }_z=8\mathrm{meV})$ approaches for a lateral confinement $\hslash {\omega }_0=2\mathrm{meV}$. The 2D (left) and 3D (right) electron densities are displayed for different ground states ($M$, $S$) at corresponding magnetic fields. The largest 3D density grows from a central small core over a ring to a torus with the increase of the magnetic field.