Optimal geometry of lateral GaAs and Si/SiGe quantum dots for electrical control of spin qubits

We investigate the effects of the orientation of the magnetic field and the orientation of a quantum dot, with respect to crystallographic coordinates, on the quality of an electrically controlled qubit realized in a gated semiconductor quantum dot. We find that, due to the anisotropy of the spin-orbit interactions, by varying the two orientations it is possible to tune the qubit in the sense of optimizing the ratio of its couplings to phonons and to a control electric field. We find conditions under which such optimal setup can be reached by solely reorienting the magnetic field, and when a specific positioning of the dot is required. We also find that the knowledge of the relative sign of the spin-orbit interaction strengths allows to choose a robust optimal dot geometry, with the dot main axis along [110], or $\left[1\overline{1}0\right]$, where the qubit can be always optimized by reorienting the magnetic field.

Figure 1

A schematic illustration of the considered lateral quantum dots in the two cases of GaAs and Si/SiGe. An electron from the 2DEG at the interface, depicted as a thin red line, is trapped in an anharmonic trap potential with the trapping frequency ${\omega }_{x^{\prime }}\left({\omega }_{y^{\prime }}\right)$ along the major (minor) axis of the dot. The electron spin in the dot is controlled via Rabi oscillations, induced by the oscillating resonant electric field $\mathbf{E}$. The applied magnetic field $\mathbf{B}$ modulates the Rabi frequency and the spin relaxation rate.

Figure 2

(Left) Optimal orientation of the magnetic field ${\beta }_{\mathrm{opt}}\in [-\pi /4,\pi /4]$ as a function of the ratio ${\mathrm{\Gamma }}_0/\gamma$ and spin-orbit mixing angle $\nu$ in a circular dot ($\epsilon =0$). (Right) ${\beta }_{\mathrm{opt}}$ as a function of ${\mathrm{\Gamma }}_0/\gamma$ and ellipticity $\epsilon$ for a dot with $\delta =\pi /4$ and $\nu =0$ (only the Rashba interaction is present).

Figure 3

The tunability $\mathcal{T}$ (left column) and maximal quality $\mathcal{Q}$ (right column). With $\epsilon =0.25$ (top) and spin length ratio $l_r/l_d=5,1/5$ (bottom).

Figure 4

(a) Left figure shows the tunability of the figure of merit, $\mathcal{T}$. Right figure shows the maximal quality $\mathcal{Q}$. Each blue point corresponds to the parameters ($\nu ,{\mathrm{\Gamma }}_0/\gamma$) of the corresponding figure in (b). (b) The normalized figure of merit $\zeta =\zeta (\delta ,\beta )$ is displayed in a $3\times 3$ array of figures. In all figures elliptic quantum dots with $\epsilon =0.75$ are considered in host materials where the Rashba and Dresselhaus spin-orbit couplings have the same sign, i.e., $l_r{l}_d>0$. The red dashed vertical line shows an optimal dot orientation with red points indicating the optimal configuration (${\delta }_{\mathrm{opt}},{\beta }_{\mathrm{opt}}$).