Theory of silicon spin qubit relaxation in a synthetic spin-orbit field

We develop the theory of single-electron silicon spin qubit relaxation in the presence of a magnetic field gradient. Such field gradients are routinely generated by on-chip micromagnets to allow for electrically controlled quantum gates on spin qubits. We build on a valley-dependent envelope function theory that enables the analysis of the electron wave function in a silicon quantum dot with an arbitrary roughness at the interface. We assume the presence of single-layer atomic steps at a Si/SiGe interface and study how the presence of a gradient field modifies the spin-mixing mechanisms. We show that our theoretical modeling can quantitatively reproduce the results of experimental measurements of qubit relaxation in silicon in the presence of a micromagnet. We further study how a field gradient can modify the EDSR Rabi frequency as well as the quality factor of a silicon spin qubit. We show that this strongly depends on the details of the interface roughness. Interestingly, for a quantum dot with an ideally flat interface, adding a micromagnet can give rise to the reduction of the EDSR frequency within some interval of the external magnetic field strength.

Figure 1

(a) Schematic of a disordered quantum dot formed at a Si/SiGe interface comprising two atomic steps. The top gates with applied voltages $\pm V$ are used to trap and confine a single electron in the silicon layer. The atomic steps have the width $a_0/4$ where $a_0=0.543$ nm denotes the lattice constant. (b) Schematic of trapped single-electron in a confinement potential in the presence of a magnetic-field gradient, ${\mathbf{B}}_{\mathrm{pd}}\left(x\right)$. In addition, there is an out-of-plane electric field $F_z$ and a homogeneous magnetic field ${\mathbf{B}}_0$. The angle between ${\mathbf{B}}_0$ and the $\widehat{z}$ axis is denoted ${\theta }_B$. (c) Schematic level diagram of a single-electron silicon spin qubit in the presence of a micromagnet. Here, only spin-valley mixing is considered. The qubit ground (excited) state $|\tilde{g}\rangle$ ($|\tilde{e}\rangle$) is denoted by a green (blue) line. The avoided crossing happens at the spin-valley hotspot above which there is a physical state $|\tilde{d}\rangle$ between the qubit ground and excited state. The small spin splitting observed in the absence of an external in-plane magnetic field $B_{\left|\right|}=0$ is due to the presence of the micromagnet. Dot arrows denote possible decay channels.

Figure 2

Qubit relaxation rate $1/T_1$ as a function of external in-plane magnetic field $B_{\left|\right|}$ in the presence of a micromagnet. Here the step positions are assumed to be at $x_{s\mathrm{L}}=-0.9x_0, x_{s\mathrm{R}}=0.275x_0$. The violet solid curve describes the total calculated spin relaxation rate $1/T_1$, whereas the other curves indicate the contributions due to electron-phonon interaction (yellow dot-dahsed and dotted) and the $1/f$ charge noise (red dashed and dot-dashed). Experimental data points from Ref. [12] are shown as blue circles. Following Ref. [12], we set the azimuthal angle of the magnetic field ${\varphi }_B=\pi /4, T_{\mathrm{el}}=115$ mK, $c_{mm}=1.8$ T/$\mu \mathrm{m}$, and $S_0=3\mu \mathrm{eV}/\sqrt{\mathrm{Hz}}$. We also set $b_x=92$ mT and $b_{0z}=15$ mT. (Inset) Comparison of the relaxation rate in the presence (violet solid curve) and in the absence (red dotted) of the micromagnet.

Figure 3

The qubit relaxation rate $1/T_1$ as a function of the external in-plane magnetic field $B_{\left|\right|}$ for various positions of a single interface step, $x_{s\mathrm{R}}$. The solid lines are obtained in the presence of the magnetic field and the black dotted lines are obtained in the absence of the micromagnet. All the other parameters are the same as given by the caption of Fig. 2.

Figure 4

(a) EDSR Rabi frequency in the presence of the micromagnet relative to the EDSR frequency in the absence of the micromagnet as a function of the external magnetic field for both disordered and ideally flat quantum dot. (b) The SVM-induced contribution to the EDSR frequency relative to the SOM-induced contribution to the EDSR frequency for a disordered quantum dot. The gray area marks the region where SOM is the dominant mixing mechanism. (Inset) The quality factor $Q={\mathrm{\Omega }}_{\mathrm{EDSR}}{T}_1$ for a disordered quantum dot in the presence and in the absence of the micromagnet. Here we assumed $E_{\mathrm{ac}}=10$ kV/m. All quantum dot parameters are the same as given by the caption of Fig. 2.