Three electrons in laterally coupled quantum dots: Tunnel vs electrostatic coupling, ground-state symmetry, and interdot correlations

The phase diagram for the ground-state symmetry of three electrons confined in a pair of laterally coupled dots is determined as function of the interdot distance and the magnetic field. With a few exceptions the ground-state spin and parity symmetry sequence of a circular harmonic quantum dot is conserved. Reentrant behavior of some energy levels as ground states is found as a function of the magnetic field. The disappearance of interdot tunnelling due to a strong magnetic field leads to ground-state degeneracy of the even and odd parity energy levels. It is shown that at a high magnetic field the system can be closely approximated by a two-electron system confined in one dot and a spectator electron localized in the other. Broken-parity eigenstates with a classical charge distribution are constructed and used to discuss the interdot electron-electron correlations.

Figure 1

(Color online) Magnetic field dependence of the three electron spectra for different values of the interdot distance $d$. Energy levels of ${}^{1∕2}\mathrm{O}$, ${}^{3∕2}\mathrm{O}$, ${}^{1∕2}\mathrm{E}$, and ${}^{3∕2}\mathrm{E}$ symmetry states are plotted in blue, black, green, and red, respectively. Numbers in the energy levels labels given in (a) and (b) stand for the absolute value of the angular momentum in $\hslash$ units. Dotted and dashed lines in (e–g) show the singlet and triplet energy levels in a reference two-electron system confined in the potential given by Eq. (6). Crosses and squares in (g) and (h) mark the energy levels of spin-polarized states of even and odd parity, respectively. Numbers 0, 1, $1∕2$, and $3∕2$ in Figs. 1e, 1f, 1g, 1h give the spin quantum number of the plotted energy levels. For clarity, the two-electron spectrum in (g) and (h) was shifted by $+0.1$ and $+0.25\mathrm{meV}$, respectively.

Figure 2

(Color online) Magnetic field-interdot distance phase diagram for the ground-state symmetry. Blue, green, black, and red regions correspond to a ground state with ${}^{1∕2}\mathrm{O}$, ${}^{1∕2}\mathrm{E}$, ${}^{3∕2}\mathrm{O}$, and ${}^{3∕2}\mathrm{E}$ symmetry, respectively. The white region corresponds to a negligible interdot tunnel coupling and a near degeneracy of the ground state with respect to spatial parity symmetry.

Figure 3

Contour plots of the charge density for $d=26.736\mathrm{nm}$ and various magnetic fields. The plots (a–k) correspond to the ground state. Plot (l) at $B=5\mathrm{T}$ corresponds to an excited state of ${}^{3∕2}\mathrm{E}$ symmetry.

Figure 4

Charge density contours for the lowest-energy state of ${}^{3∕2}\mathrm{O}$ symmetry for various values of the magnetic field and $d=26.736\mathrm{nm}$.

Figure 5

Pair correlation functions for $d=26.736\mathrm{nm}$ and various magnetic fields. One of the electrons is fixed at the position $(-28\mathrm{nm},0)$ (indicated by the cross). All the plots (a–k) with the exception of the plot (l) for the state ${}^{3∕2}\mathrm{E}$ at $B=5\mathrm{T}$ correspond to the ground state.

Figure 6

Charge accumulated in the right quantum dot for the broken-parity Hamiltonian eigenstates as function of the phase of the superposition [Eq. (7)] for $S=1∕2$ (dashed line) and $S=3∕2$ (solid line) at $d=26.736\mathrm{nm}$ and for the spin-polarized state at $d=80\mathrm{nm}$ (dotted line).

Figure 7

Charge density (a–d) and pair correlation function (e–h) plots for $d=26.736\mathrm{nm}$ at $B=8.75\mathrm{T}$. Plots (a), (b), (e), (f) correspond to parity eigenstates and (c), (d), (g), (h) to Hamiltonian eigenstates of a broken parity. In (e) and (g) [(f) and (h)] an electron is fixed at $(-28\mathrm{nm},0)$, [($-19\mathrm{nm}$, $19\mathrm{nm}$)]. The fixed electron position in (e–h) is marked by a dot.

Figure 8

Charge density (a) and pair correlation functions (c, d) plots for $d=40\mathrm{nm}$ at $B=8.4\mathrm{T}$. Charge density of the two-electron spin polarized state in a single dot with a point repulsive center at $40\mathrm{nm}$ from the center of the dot plotted in (b). In (c) and (d) an electron is fixed (indicated by the dot) at ($-23\mathrm{nm}$, $16\mathrm{nm}$) and at $(-31\mathrm{nm},0)$, respectively.

Figure 9

Charge density (a) and pair correlation functions (c, d) plots of the spin-polarized broken-parity state for $d=60\mathrm{nm}$ at $B=8.5\mathrm{T}$. Charge density of the two-electron spin-polarized state in a single dot with a point repulsive center at $60\mathrm{nm}$ from the center of the dot plotted in (b). In (c) and (d) an electron is fixed at ($-32$, $15\mathrm{nm}$) and at $(-36\mathrm{nm},0)$, respectively. The electron position is marked by a dot.

Figure 10

Charge density (a) and pair correlation functions (c) plots for the spin-polarized broken-parity state for $d=80\mathrm{nm}$ at $B=8.5\mathrm{T}$. Charge density of the two-electron spin polarized state in a single dot with a point repulsive center at $80\mathrm{nm}$ from the center of the dot plotted in (b). In (c) an electron is fixed at ($-44$, $0\mathrm{nm}$)—marked by a dot.