Theory of Anisotropic Exchange in Laterally Coupled Quantum Dots

The effects of spin-orbit coupling on the two-electron spectra in lateral coupled quantum dots are investigated analytically and numerically. It is demonstrated that in the absence of magnetic field, the exchange interaction is practically unaffected by spin-orbit coupling, for any interdot coupling, boosting prospects for spin-based quantum computing. The anisotropic exchange appears at finite magnetic fields. A numerically accurate effective spin Hamiltonian for modeling spin-orbit-induced two-electron spin dynamics in the presence of magnetic field is proposed.

Figure 1

Calculated double dot spectrum as a function of the interdot distance and tunneling energy. Spin is not considered, and the magnetic field is zero. Solid lines show the two-electron energies. The two lowest states are explicitly labeled, split by the isotropic exchange $J$ and displaced from the nearest higher excited state by $\Delta$. For comparison, the two lowest single-electron states are shown (dashed lines), split by twice the tunneling energy $T$. State spatial symmetry is denoted by darker (symmetric) and lighter (antisymmetric) lines.

Figure 2

The spin-orbit induced energy shift as a function of the interdot distance (left) and perpendicular magnetic field (right). (a) Singlet in zero magnetic field, (c) singlet at 1 Tesla field, (b) and (d) singlet and triplet $T_{+}$ at the interdot distance 55 nm corresponding to the zero-field isotropic exchange of $1\mu \mathrm{eV}$. The exchange models $H_{\mathrm{ex}}^{\prime }$ (dashed line) and $H_{\mathrm{ex}}$ (dot-dashed line) are compared with the numerics (solid line).

Figure 3

(a) The isotropic and anisotropic exchange as functions of the interdot distance at zero magnetic field. (b) The isotropic exchange $J$, anisotropic exchange $c/c^{\prime }$, the Zeeman splitting $\mu B$, and its spin-orbit part $\mu B_{\mathrm{so}}$ at perpendicular magnetic field of 1 T. (c) Schematics of the exchange-split four lowest states for the three models, $H_{\mathrm{iso}}$, $H_{\mathrm{ex}}^{\prime }$, and $H_{\mathrm{ex}}$, which include the spin-orbit coupling in no, first, and second order, respectively, at zero magnetic field (top). The latter two models are compared in perpendicular and in-plane magnetic fields as well. The eigenenergies are indicated by the solid lines. The dashed lines show which states are coupled by the spin-orbit coupling. The arrows indicate the redistribution of the couplings as the in-plane field direction changes with respect to the crystallographic axes (see the main text).