Spin-orbit coupling and anisotropic exchange in two-electron double quantum dots

The influence of the spin-orbit interactions on the energy spectrum of two-electron laterally coupled quantum dots is investigated. The effective Hamiltonian for a spin qubit pair proposed in Baruffa et al. [Phys. Rev. Lett. 104, 126401 (2010)] is confronted with exact numerical results in single and double quantum dots in zero and finite magnetic field. The anisotropic exchange Hamiltonian is found quantitatively reliable in double dots in general. There are two findings of particular practical importance: (i) the model stays valid even for maximal possible interdot coupling (a single dot), due to the absence of a coupling to the nearest excited level, a fact following from the dot symmetry. (ii) In a weak-coupling regime, the Heitler-London approximation gives quantitatively correct anisotropic exchange parameters even in a finite magnetic field, although this method is known to fail for the isotropic exchange. The small discrepancy between the analytical model (which employs the linear Dresselhaus and Bychkov-Rashba spin-orbit terms) and the numerical data for GaAs quantum dots is found to be mostly due to the cubic Dresselhaus term.

Figure 1

Two-electron energy spectrum of a double dot at zero magnetic field as a function of the interdot distance and the tunneling energy. The spatial symmetries of wave functions, 1, $x$, $xy$, and $y$ are denoted as solid, dashed, dotted-dashed, and dotted line, respectively. The two lowest energies are labeled; they are split by the isotropic exchange energy $J$. The energy separation between the lowest states and the higher exited states is denoted by $\Delta$.

Figure 2

Two-electron energy spectrum of a single dot in perpendicular magnetic field. The lowest states are labeled by the total angular momentum $L$. The Zeeman and spin-orbit interactions are neglected. The two regions marked by boxes are magnified in Figs. 3, 4.

Figure 3

(Color online) Magnified region from Fig. 2. Energy spectrum of a single dot for small perpendicular magnetic field. Only the states with the total angular momentum $L=\pm 1$ are plotted. A constant shift is removed from the spectrum. Each state is labeled by the quantum numbers $\left(J_{+},J_{-},T_i\right)$.

Figure 4

(Color online) Lowest energy levels in the anticrossing region marked in Fig. 2. A constant shift was removed from the spectrum. The quantum numbers $\left(J_{+},J_{-},{\Sigma }_i\right)$ label the states. Insets show the anticrossing regions.

Figure 5

Spin-orbit-induced energy shifts at zero magnetic field as a function of the interdot distance. Exact numerics (solid), first-order model $H_{ex}^{\prime }$ (dotted), and first-order model in HL approximation (dashed) are given. (a) Singlet, (b) triplet $T_0$, (c) triplet $T_{+}$, and (d) triplet $T_{-}$. The results of the second-order model (both $H_{ex}^{HL}$ and $H_{ex}$ give the same) are indiscernible from exact numerical data.

Figure 6

Spin-orbit parameters at zero magnetic field as function of the interdot distance. Numerical value and the Heitler-London approximation for the isotropic exchange (solid) and the anisotropic exchange of the first-order model (dashed).

Figure 7

Spin-orbit-induced energy shifts at 1 T perpendicular magnetic field versus the interdot distance. (a) Energy shift of the Singlet $S$ in the exact numerics (solid) is compared to the numerical (dashed) and the Heitler-London approximation (dotted) first-order model. In (b) similar comparison is made for the second-order model. Panels (c) and (d) are analog of (a) and (b) showing the energy shifts of the triplet $T_{+}$.

Figure 8

Spin-orbit parameters at 1 T perpendicular magnetic field versus the interdot distance. (a) Numerical (solid) and Heitler-London approximation (dashed) anisotropic exchange vectors for the first- and second-order models. (b) Isotropic exchange, Zeeman energy, and the spin-orbit-induced effective magnetic field.

Figure 9

Spin-orbit-induced energy shifts of a double-dot system with interdot distance of 55 nm versus the perpendicular magnetic field. (a) singlet $S$, (b) triplet $T_0$, (c) triplet $T_{+}$, and (d) triplet $T_{-}$. Exact numerics (solid) and the numerical second-order model $H_{ex}$ (dashed).

Figure 10

Spin-orbit parameters of the second-order numerical model $H_{ex}$ in a double-dot system with interdot distance of 55 nm versus the magnetic field.

Figure 11

The spin-orbit-induced energy shift as a function of the interdot distance (left panels) and perpendicular magnetic field (right panels). (a) Singlet in zero magnetic field, (c) singlet at 1 T field, (b) and (d) singlet and triplet $T_{+}$ at 55 nm. The numerical second-order model $H_{ex}$ (dashed line), exact numerics (dot-dashed line), and exact numerics without the cubic Dresselhaus term (solid line).