Broken one-particle symmetry in few-electron coupled quantum dots

Few-electron systems confined in vertically coupled quantum dots are studied by exact methods and through the local spin density approximation (LSDA). Special attention is paid to the recovering of the one-particle symmetry properties in the LSDA. It is shown that—in spite of accurate energy estimates—the LSDA does not reproduce the one-particle parity and the relative probability of finding a definite number of electrons in the different quantum dots. This symmetry breaking appears for certain electron configurations and intermediate thickness of the interdot barrier and is a direct result of electron-electron correlation. We discuss the effect of correlation on the symmetry properties of the few-electron wave function and determine the limits of the applicability of the LSDA for coupled quantum dots.

Figure 1

(Color online) Lowest-energy levels of two electrons in coupled QDs calculated by the (a) $r$-LSDA and (b) $u$-LSDA as a function of interdot barrier thickness $b$. Exact results for states $(M_z,S_z)=(0,0),(0,1)$, and (1,1) are plotted by solid, dashed, and dotted curves, respectively, the LSDA results for the corresponding states $1s^2,1s^{1}2s^1$, and $1s^{1}1p^1$ are shown by full dots, open dots, and crosses.

Figure 2

Energy differences $\mathrm{\Delta }E$ between the $r$-LSDA (upper panel) and $u$-LSDA (lower panel) and exact results for the lowest-energy levels of two electrons in coupled QDs as a function of interdot barrier thickness $b$. Solid, dashed, and dotted curves show the energy differences for states (0,0), (0,1), and (1,1), respectively.

Figure 3

(Color online) (a) Expectation value of the one-electron parity and (b) the relative probability of finding one electron in each QD, $P(1\mid 1)$, and both electrons in one QD, $P(0\mid 2)$, for the two-electron system calculated by the exact method (solid, dashed, and dotted curves) and by the $u$-LSDA (dots and crosses) for the three-lowest energy states, defined in the panel in part (a), as a function of interdot barrier thickness $b$. In part (a) the dotted curve for the state (0,1) is not visible, since it coincides with the abscissa.

Figure 4

(Color online) (a) Lowest-energy levels of three electrons in coupled QDs and (b) relative probabilities of finding one electron in one QD and two electrons in the second QD $\left[P\right(1\mid 2\left)\right]$ and all three electrons in one QD $\left[P\right(0\mid 3\left)\right]$, for the lowest-energy states (0,1/2) and (1,1/2) calculated by the CI method (solid and dashed curves) and $u$-LSDA (dots) as a function of interdot barrier thickness $b$. Solid curve (open dots) corresponds to the state (0,1/2) and dashed curve (full dots) corresponds to the state (1,1/2). Inset in (a), expectation value of the one-electron parity calculated by the CI method (curves) and $u$-LSDA (dots) as a function of barrier thickness $b$.

Figure 5

(Color online) (a) Lowest-energy levels of four electrons in coupled QDs and (b) expectation value of one-electron parity as a function of the interdot barrier thickness $b$. Solid, dashed, and dotted curves display the CI results for (0,0), (0,1), and (1,1) states. Squares, full dots, open dots, and crosses show the $u$-LSDA results for configurations $1s^{2}2s^2,1s^{2}1p^2$ (singlet), $1s^{2}1p^2$ (triplet), and $1s^{2}1p^{1}2s^1$.

Figure 6

(Color online) Relative probabilities of finding one electron in one QD and three electrons in the second QD $\left[P\right(1\mid 3\left)\right]$ and the two electrons in each QD $\left[P\right(2\mid 2\left)\right]$ calculated by the CI method (solid curves) and $u$-LSDA (dashed and dotted curves) as a function of the barrier thickness $b$ for four electrons with the quantum numbers (a) $M_z=0$ and $S_z=0$, (b) $M_z=1$ and $S_z=1$, and (c) $M_z=0$ and $S_z=1$.

Figure 7

Energy difference $\mathrm{\Delta }E$ between the energy of two electrostatically coupled two-electron QDs $\left[P\right(2\mid 2)=1]$ and the energy of four electrons in coupled QDs in a state with $M_z=0$ and $S_z=1$ as a function of barrier thickness $b$.

Figure 8

(Color online) Addition energy $A_N$ calculated by the $u$-LSDA and $r$-LSDA for the $N$-electron coupled QDs separated by a barrier of thickness (a) $b=3\mathrm{nm}$ and (b) $b=8\mathrm{nm}$. For $N=1,2$, and 3 we also show the exact results. The lines are guides to the eye.