Singlet-triplet splitting, correlation, and entanglement of two electrons in quantum dot molecules

Starting with an accurate pseudopotential description of the single-particle states, and following by configuration-interaction treatment of correlated electrons in vertically coupled, self-assembled $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dot molecules, we show how simpler, popularly practiced approximations, depict the basic physical characteristics including the singlet-triplet splitting, degree of entanglement (DOE), and correlation. The mean-field-like single-configuration approaches such as Hartree-Fock and local spin density, lacking correlation, incorrectly identify the ground-state symmetry and give inaccurate values for the singlet-triplet splitting and the DOE. The Hubbard model gives qualitatively correct results for the ground-state symmetry and singlet-triplet splitting, but produces significant errors in the DOE because it ignores the fact that the strain is asymmetric even if the dots within a molecule are identical. Finally, the Heisenberg model gives qualitatively correct ground-state symmetry and singlet-triplet splitting only for rather large interdot separations, but it greatly overestimates the DOE as a consequence of ignoring the electron double occupancy effect.

Figure 1

Possible configurations for two electrons in two vertically coupled quantum dots. (a) Spin configurations in the MO basis. ${\sigma }_g$ and ${\sigma }_u$ indicate the bonding and antibonding states, respectively. (b) Spin configurations in the dot-localized basis. “$T$” and “$B$” indicate the top and bottom dots.

Figure 2

Geometry of the two vertically coupled quantum dot molecule. The interdot distance $d$ is measured from wetting layer to wetting layer.

Figure 3

The bonding-antibonding splitting ${\Delta }_{\sigma }=ϵ\left({\sigma }_g\right)-ϵ\left({\sigma }_u\right)$ (solid line) and singlet-triplet splitting $J_{S-T}=E\left({}^3\Sigma \right)-E\left({}^1\Sigma _g^{\left(a\right)}\right)$ (dashed line) vs interdot distance $d$. We also show the single-dot s, p orbitals splitting ${\delta }_{sp}=e_s-e_p$ and the s orbital Coulomb interaction $J_C$. We define “strong coupling” by ${\Delta }_{\sigma }\sim {\delta }_{sp}(<4\mathrm{nm})$ and “weak coupling,” ${\Delta }_{\sigma }⪡{\delta }_{sp}(>5\mathrm{nm})$.

Figure 4

Selected Coulomb integrals for molecular orbitals. $J_{gg}$ is the self-Coulomb energy of the ${\sigma }_g$ orbital and $J_{gu}$ is the Coulomb energy between ${\sigma }_g$ and ${\sigma }_u$ orbitals, while $K_{gu}$ is the exchange energy between ${\sigma }_g$ and ${\sigma }_u$ orbitals.

Figure 5

(Color online) (a) Two-electron states calculated from CI using all confined MO from LCBB (level 1), including the singlet ${}^1{\sum }_g^{\left(a\right)}$, ${}^1{\sum }_u$, ${}^1{\sum }_g^{\left(b\right)}$ states and the threefold degenerated triplet states ${}^3{\sum }_u$ as well as two threefold degenerated triplet states ${}^3\Pi _u$. (b) Two electron states calculated from the single-configuration approximation (level 3). (c) Comparison of the singlet-triplet splitting calculated from level-1, -3, and -4 theories.

Figure 6

Isospin, defined as the difference in the number of electrons occupying the bonding $\left(N_B\right)$ and antibonding $\left(N_{AB}\right)$ states, of the ${}^1\Sigma _g^{\left(a\right)}$ state in level-1 and level-3 theories.

Figure 7

(a) Effective single-particle energy levels of $s$ orbitals localized on the top $\left(e_T\right)$ and bottom $\left(e_B\right)$ dots. (b) Intradot Coulomb energy $J_{\mathrm{TT}}$, $J_{\mathrm{BB}}$, interdot Coulomb energy $J_{\mathrm{TB}}$ and interdot exchange energy $K_{\mathrm{TB}}$ (magnified by a factor 100). The dashed line gives the single-dot $s$ orbital self-Coulomb energy $J_C$.

Figure 8

(Color online) Comparison of pair-correlation functions calculated from (a) level-1 and (b) level-3 theory for the ${}^1\Sigma _g^{\left(a\right)}$ state and (c) level-1 and (d) level-3 theory for the ${}^1\Sigma _g^{\left(b\right)}$ state at $d\sim 7\mathrm{nm}$. On the left-hand side, the first electron is fixed at the center of the bottom dot, while on the right-hand side, the first electon is fixed at the center of the top dot, as indicated by the arrows.

Figure 9

(Color online) Comparison of the DOE calculated from (a) level-1, (b) level-3, and (c) level-4 theories for two-electron states. In panel (c), both the DOE of the Hubbard model (solid lines) and of the Heisenberg model for ${}^1\Sigma _g^{\left(a\right)}$ state (dashed line) are shown.