Giant Anisotropy of Spin Relaxation and Spin-Valley Mixing in a Silicon Quantum Dot

In silicon quantum dots (QDs), at a certain magnetic field commonly referred to as the “hot spot,” the electron spin relaxation rate ($T_1^{-1}$) can be drastically enhanced due to strong spin-valley mixing. Here, we experimentally find that with a valley splitting of $78.2\pm 1.6\mu \mathrm{eV}$, this hot spot in spin relaxation can be suppressed by more than 2 orders of magnitude when the in-plane magnetic field is oriented at an optimal angle, about 9° from the [100] sample plane. This directional anisotropy exhibits a sinusoidal modulation with a 180° periodicity. We explain the magnitude and phase of this modulation using a model that accounts for both spin-valley mixing and intravalley spin-orbit mixing. The generality of this phenomenon is also confirmed by tuning the electric field and the valley splitting up to $268.5\pm 0.7\mu \mathrm{eV}$.

Figure 1

(a) Scanning electron microscope image of a DQD device identical to the one measured. Two circles are used to proportionally denote the position and size of the dots. Inset: the crystallographic directions with respect to the sample. (b) Charge stability diagram of the DQD. The relative voltage magnitude at each step of the pulse sequence for measuring $T_1$ is overlaid on the data. (c) Normalized spin-up fraction as a function of the waiting time $t_{\text{wait}}$ for different angles $\varphi$ of the 0.8 T in-plane magnetic field with $E_{\mathrm{VS}}=78.2\pm 1.6\mu \mathrm{eV}$. The solid lines are exponential fits to the data with the values of $T_1$ (ms) indicated aside.

Figure 2

(a) Relaxation rates as a function of the magnetic field strength with an in-plane angle of $\varphi =117°$. The fittings include contributions from different relaxation channels obtained through the model discussed in the main text. (b) Angular dependence of the relaxation rate measured with different magnetic field strengths. The red and blue solid lines are numerical results based on the spin relaxation model and the parameters from experiment, while the corresponding shaded areas indicate a 95% confidence interval with a sinusoidal fit. Inset: $T_{1,\min }^{-1}(\varphi )$ as a function of the parameter $\theta$ for ${\mathit{B}}_{\mathrm{ext}}=0.8\mathrm{T}$ (red) and ${\mathit{B}}_{\mathrm{ext}}=1.5\mathrm{T}$ (blue). (c) Anisotropy magnitude as a function of the magnetic field strength under real experimental conditions or certain assumptions. The shaded areas indicate the amount of anisotropy suppressed by corresponding mechanism. Inset: numerical simulation of the spin relaxation hot spot as a function of the external magnetic field angle.

Figure 3

(a) Illustration of the intuitive classical picture of the interaction between the effective spin-orbit magnetic field ${\mathit{B}}_{\mathrm{SO}}$ and the external magnetic field ${\mathit{B}}_{\mathrm{ext}}$. The dashed circle shows the rotation of ${\mathit{B}}_{\mathrm{ext}}$. (b) Modified intuitive picture of the interaction between ${\mathit{B}}_{\mathrm{SO}}$ and ${\mathit{B}}_{\mathrm{ext}}$.

Figure 4

(a) Valley splitting $E_{\mathrm{VS}}$ as a function of the gate voltage $V_p$. A linear fit shows a tunability of $0.667\pm 0.020\mathrm{meV}{\mathrm{V}}^{-1}$. The deviation from the linear fit at small $V_p$ perhaps results from an interface localized interaction [36]. (b) Relaxation rates as a function of the external magnetic field along different directions and the calculated anisotropy magnitude with experimental parameters. Solid lines are numerical results based on our spin relaxation model and the parameters from experiment. Inset: numerical simulation of the spin relaxation hot spot as a function of the orientation of the external magnetic field.