Electronic asymmetry in self-assembled quantum dot molecules made of identical $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots

We show that a diatomic dot molecule made of two identical, vertically stacked, strained $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ self-assembled dots exhibits an asymmetry in its single- and many-particle wave functions. The single particle wave function is asymmetric due to the inhomogeneous strain, while the asymmetry of the many-particle wave functions is caused by the correlation-induced localization: the lowest singlet ${}^1\Sigma _g$ and triplet ${}^3\Sigma$ states show that the two electrons are each localized on different dots within the molecule; for the next singlet states ${}^1\Sigma _u$ both electrons are localized on the same (bottom) dot for interdot separation $d>8\mathrm{nm}$. The singlet-triplet splitting is found to be $\sim 0.1\mathrm{meV}$ at interdot separation $d=9\mathrm{nm}$ and as large as $100\mathrm{meV}$ for $d=4\mathrm{nm}$, orders of magnitude larger than the few meV found in the large $(50–100\mathrm{nm})$ electrostatically confined dots.

Figure 1

(Color online) (a) Contour plot of the hydrostatic strain $\mathrm{Tr}\left(ϵ\right)$ in the two vertically coupled quantum dots. The interdot distance $d$ is measured from one wetting layer to the next. The isostrain values are also marked in the figure. (b) Molecular-orbital energy levels vs interdot distance. The dashed line is the average of ${\sigma }_g$ and ${\sigma }_u$. ${\delta }_{sp}$ is the single dot $s\text{−}p$ energy-level splitting. (c) Sketch of bonding-antibonding splitting. $S_T$, $S_B$ are “$s$” while $P_T$ and $P_B$ are “$p$” single dot orbitals on top and bottom dots. Here $S_T+S_B={\sigma }_g$ and $P_T+P_B={\pi }_u$ are bonding states, while the $S_T-S_B={\sigma }_u$ and $P_T-P_B={\pi }_g$ are antibonding states.

Figure 2

(a) The effective single-particle energy levels $e_{\mathrm{T}}$ and $e_{\mathrm{B}}$ of dot-centered orbitals on the top dot and bottom dot, respectively. (b) Energy of two-electron states calculated from CI using all confined molecular orbitals.

Figure 3

(Color online) Panel (a) gives the weights of configurations ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3$, and ${\Phi }_4$ of the many-particle CI wave functions. Panels (b) and (c) depict the probability of finding the second electron at position $\mathbf{r}$ given that the first electron has been found on the center of the bottom dot (indicated by the arrows) for the interdot distances (b) $d=4\mathrm{nm}$ and (c) $d=7\mathrm{nm}$.

Figure 4

The probability of two electrons occupying the same dot [Eq. (3)] for (a) ${}^1\Sigma _g^{\left(a\right)}$, (b) ${}^1\Sigma _u$, (c) ${}^1\Sigma _g^{\left(b\right)}$ states. $Q_{\mathrm{TT}}$ $\left(Q_{\mathrm{BB}}\right)$ is the probability of both electrons being on the top (bottom) dot, while $Q_{\text{total}}$ gives the total probability.