Theory of spin relaxation in two-electron laterally coupled Si/SiGe quantum dots

Highly accurate numerical results of phonon-induced two-electron spin relaxation in silicon double quantum dots are presented. The relaxation, enabled by spin-orbit coupling and the nuclei of ${}^{29}$Si (natural or purified abundance), is investigated for experimentally relevant parameters, the interdot coupling, the magnetic field magnitude and orientation, and the detuning. We calculate relaxation rates for zero and finite temperatures (100 mK), concluding that our findings for zero temperature remain qualitatively valid also for 100 mK. We confirm the same anisotropic switch of the axis of prolonged spin lifetime with varying detuning as recently predicted in GaAs. Conditions for possibly hyperfine-dominated relaxation are much more stringent in Si than in GaAs. For experimentally relevant regimes, the spin-orbit coupling, although weak, is the dominant contribution, yielding anisotropic relaxation rates of at least two orders of magnitude lower than in GaAs.

Figure 1

Calculated conversion between the single-electron tunneling energy $T$ ($x$ axis), the two-electron exchange coupling $J$ (left $y$ axis), and the interdot distance $2d/l_0$ (right $y$ axis), neglecting nuclear spins. The arrow gives $E_{\text{nuc}}$, Eq. (10), of natural silicon.

Figure 2

Calculated energies of the lowest states, varying (a) the interdot coupling (at $B=3$ T) and (b) the detuning (at $B=0.5$ T, $2d/l_0=2.85$). Singlet states are given by dashed and triplets by solid lines. In (a), the energy of $T_0$ is subtracted, and in (b), the quadratic trend in $E$ is subtracted. The arrow in (a) marks where $J=E_{\text{nuc}}$ (for natural silicon).

Figure 3

Schematic energy spectrum of an unbiased double dot showing the singlet $S$ (dashed line) and the triplet $T_0$ (solid line). For large interdot distances, the exchange coupling $J$ is given by the hyperfine splitting $E_{\text{nuc}}$, Eq. (10), and the eigenstates change to $|\downarrow \uparrow \rangle$ and $|\uparrow \downarrow \rangle$.

Figure 4

Calculated relaxation rates of (a) the singlet, (b) the triplet $T_0$, and (c) the triplet $T_{+}$ as functions of the in-plane magnetic field orientation $\gamma$ (angle) and the interdot distance $2d/l_0$ (radius of the polar plot), for a double dot at $B=3$ T. The $x$ and $y$ axes correspond to crystallographic axes $\left[100\right]$ and $\left[010\right]$, respectively. The dot orientation $\mathbf{d}$ $\left|\right|$ $\left[110\right]$ is marked by a line. 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 5

Calculated channel resolved relaxation rates vs interdot distance for both parallel (top) and perpendicular (bottom) to $\mathbf{d}$ in-plane magnetic field orientation ($B=3$ T). The relaxation channels of $T_{+}$ are black, of $T_0$ are blue, and of $S$ are red.

Figure 6

Calculated relaxation rates of (a) the singlet, (b) the triplet $T_0$, and (c) the triplet $T_{+}$ as functions 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 meV. The layout with respect to the crystallographic axes is the same as in Fig. 4. The rate is given in inverse seconds by the color scale.

Figure 7

Schematic energy spectrum of a biased double dot without magnetic field. The singlets are given in dashed lines, the triplets in solid lines.

Figure 8

Calculated relaxation rates of (a) the first excited state [$S$ or $T_{-}$; see Fig. 2b], (b) $T_0$, and (c) $T_{+}$ as functions of the in-plane magnetic field orientation $\gamma$ (angle) and detuning energy (radius of the polar plot), for a double dot with $2d/l_0=2.85$ ($T$ $=$ 0.1 meV) and $B=$ 0.5 T. The layout with respect to the crystallographic axes is the same as in Figs. 4 and 6. The rate is given in inverse seconds by the color scale.

Figure 9

Calculated relaxation rates of a detuned double dot in an in-plane magnetic field ($B=$ 0.5 T, $\gamma =3\pi /4$) as a function of detuning. The straight lines give the spin-orbit-induced relaxation, the wiggly lines the hyperfine-induced relaxation rates (natural silicon).

Figure 10

Calculated relaxation rates of individual transition channels as a function of in-plane magnetic field for a weakly coupled quantum dot. The magnetic field is oriented along $\left[\overline{1}10\right]$ ($\gamma =3\pi /4$) and the dots along $\left[110\right]$ ($\delta =\pi /4$). The interdot distance is $2d/l_0=$ 2.85, yielding the tunneling energy $T=$ 0.1 meV. The dashed, black line gives the analytical relaxation rate, evaluated with Eq. (A14).