Spin-Relaxation Anisotropy in a GaAs Quantum Dot

We report that the electron spin-relaxation time $T_1$ in a GaAs quantum dot with a spin-$1/2$ ground state has a 180° periodicity in the orientation of the in-plane magnetic field. This periodicity has been predicted for circular dots as being due to the interplay of Rashba and Dresselhaus spin orbit contributions. Different from this prediction, we find that the extrema in the $T_1$ do not occur when the magnetic field is along the [110] and $[1\overline{1}0]$ crystallographic directions. This deviation is attributed to an elliptical dot confining potential. The $T_1$ varies by more than 1 order of magnitude when rotating a 3 T field, reaching about 80 ms for the optimal angle. We infer from the data that in our device the signs of the Rashba and Dresselhaus constants are opposite.

Figure 1

(a) Scanning electron micrograph of a device similar to the one measured. The black arrows indicate the crystallographic axes. The dotted red circle represents schematically the single QD position. (b) The spin-orbit field ${\mathbf{B}}_{\text{SO}}$ acting on a conduction electron is shown by red and blue arrows (for a constant magnitude of $\mathbf{P}$), arising from the Rashba and Dresselhaus contributions (chosen to be different in modulus and $\alpha <0$, $\beta >0$) and defined, respectively, as ${\mathbf{B}}_{\text{SO}}^R=(\alpha /g{\mu }_B)(-P_y,P_x)$ and ${\mathbf{B}}_{\text{SO}}^D=(\beta /g{\mu }_B)(-P_x,P_y)$, with $g$ the electron $g$ factor and ${\mu }_B$ the Bohr magneton.

Figure 2

Measured spin-down probability (averaged over 5000 cycles) as a function of the waiting time between injection and readout (see Supplemental Material [21], Sec. I) for different angles $\varphi$ of the 3 T in-plane magnetic field. The solid lines are fits to the data of the form $P_{\downarrow }=a\exp (-t/T_1)+b$. The fitted $T_1$’s are indicated for each curve (in ms). Small variations in $P_{\downarrow }(t=0)$ can arise from variations in the readout configuration in the course of the measurements. Measuring longer $T_1$’s requires longer waiting times, with increased pulse distortion from the bias tee (see Supplemental Material [21], Sec. I), and therefore larger error.

Figure 3

Angle dependence of the spin-relaxation time (a) and rate (b), which are separately extracted from exponential fits with either the relaxation rate or time in the exponent. The magnetic field is nearly in-plane, with $|\xi |<5°$, while the in-plane angle $\varphi$ has a systematic error of $\pm 3°$. The red line is a fit to Eq. (7) with free parameters $({\varphi }_{\text{min}},{\xi }^{*},b)$. The shaded region between the two blue curves indicates the 95% confidence interval. The dashed vertical lines show the positions of the extrema of $\mathrm{\Gamma }$ predicted for a circular dot [at $\varphi =45°$ and 135°; see Eq. (3)].

Figure 4

Calculated values of $\mathrm{\Gamma }$ [Eq. (7)] as a function of the angle of the external magnetic field $\varphi$ and the dot major axis $\delta$, for $\xi =0$ and (a) $l_R/l_D=-1$ and (b) $l_R/l_D=-5$. (c),(d) The result of the fit of the data of $\mathrm{\Gamma }(\varphi )$ from Fig. 3 to Eqs. (7) and (8), with $\vartheta$ and ($\kappa l_{\text{SO}}^{-2}$) (the latter in arbitrary units) as fit parameters, as a function of $\delta$. The shaded area between the two red curves indicates the 95% confidence interval (not taking into account the systematic error in $\varphi$). The two black vertical lines indicate $\delta =45°$, 135°; the two red dotted lines and the green and blue horizontal lines are at $\vartheta =120°$, 150°, 135°, and 90°, respectively.