Theory of Spin Relaxation in Two-Electron Lateral Coupled Quantum Dots

A global quantitative picture of the phonon-induced two-electron spin relaxation in GaAs double quantum dots is presented using highly accurate numerics. Wide regimes of interdot coupling, magnetic field magnitude and orientation, and detuning are explored in the presence of a nuclear bath. Most important, the giant magnetic anisotropy of the singlet-triplet relaxation can be controlled by detuning switching the principal anisotropy axes: a protected state becomes unprotected upon detuning and vice versa. It is also established that nuclear spins can dominate spin relaxation for unpolarized triplets even at high magnetic fields, contrary to common belief.

Figure 1

Calculated energies of the lowest states for (a) variable interdot coupling (at $B=5\mathrm{T}$) and (b) detuning (at $B=2\mathrm{T}$). Singlet states are given by dashed lines, triplets by solid lines. The blue strokes mark singlet-triplet anticrossings. In (a), the energy of $T_0$ is subtracted, and in (b), the quadratic trend in $E$ is subtracted. The green arrows denote points of exact compensation, and the red oval in (b) shows where nuclear spins dominate the $T_0$ relaxation.

Figure 2

Calculated relaxation rates of (a) the first excited state [$S$ or $T_{+}$, see Fig. 1a] and (b) the triplet $T_0$ as a function of the in-plane magnetic field orientation $\gamma =\arccos (B_x/B)$ (angle) and the interdot distance $2d/l_0$ (radius of the polar plot), for a double dot at $B=5\mathrm{T}$. The $x$ and $y$ axes correspond to crystallographic axes [100] and [010], respectively. The dot orientation $\mathbf{d}||[110]$ is marked by a line. The blue half circles indicate the $S\mathrm{\text{−}}{T}_{+}$ anticrossing, also marked on Fig. 1a. The $x$ axis is converted to the tunneling energy $T$ and the exchange $J$, in addition to $2d/l_0$. The rate is given in inverse seconds by the color scale. The system obeys $C_{2v}$ symmetry, so point reflection would complete the graphs.

Figure 3

Calculated relaxation rates of (a) the first excited state and (b) the triplet $T_0$ as a function of the in-plane magnetic field orientation $\gamma$ (angle) and the magnetic field magnitude (radius of the polar plot) for a double dot with $T=0.1\mathrm{meV}$. The layout with respect to the crystallographic axes is the same as in Fig. 2. The rate is given in inverse seconds by the color scale.

Figure 4

Calculated relaxation rates of (a) the first excited state, (b) $T_0$, and (c) $T_{-}$ as a function of the in-plane magnetic field orientation $\gamma$ (angle) and detuning energy (radius of the polar plot), for a double dot with $2d/l_0=4.35$ ($T=10\mu \mathrm{eV}$), chosen along Ref. 11, and $B=2\mathrm{T}$. The layout with respect to the crystallographic axes is the same as in Fig. 2. The rate is given in inverse seconds by the color scale. The blue lines indicate the singlet-triplet anticrossings, which are in line with the marks in Fig. 1b. The dashed red lines in panel (b) confine the area where hyperfine coupling dominates.