Controlling Synthetic Spin-Orbit Coupling in a Silicon Quantum Dot with Magnetic Field

Tunable synthetic spin-orbit coupling (SSOC) is one of the key challenges in various quantum systems, such as ultracold atomic gases, topological superconductors, and semiconductor quantum dots. Here we experimentally demonstrate controlling the SSOC by investigating the anisotropy of spin-valley resonance in a silicon quantum dot. As we rotate the applied magnetic field in plane, we find a striking nonsinusoidal behavior of resonance amplitude that distinguishes SSOC from the intrinsic spin-orbit coupling (ISOC), and associate this behavior with the previously overlooked in-plane transverse magnetic field gradient. Moreover, by theoretically analyzing the experimentally measured SSOC field, we predict the quality factor of the spin qubit could be optimized if the orientation of the in-plane magnetic field is rotated away from the traditional working point.

Figure 1

(a) Schematic of the device layout. The aluminum electrodes and the bar micromagnet used in the experiment are in false colors. Inset: Cartesian coordinate and labels for different magnetic fields with the angle $\varphi$ referring to the in-plane orientation of ${\mathbf{B}}_{\mathrm{ext}}$. (b) Transport current $|I_{SD}|$ as a function of $V_{\mathrm{G}1}$ and $V_{\mathrm{G}2}$ with a bias voltage $V_{SD}=-2\mathrm{mV}$ and an external magnetic field $B_{\mathrm{ext}}=200$ mT along the y axis (i.e., $\varphi =\pi /2$). The PSB results in a current suppression in the bias triangles with a blockade region indicated by an energy gap $E_{\mathrm{ST}}$ between the two dashed lines. Inset: schematic of the energy levels involved in the PSB, where the delocalized states S(1, 1) and T(1, 1) are only weakly split by exchange interaction and the localized states S(0, 2) and T(0, 2) are split by a much larger energy $E_{\mathrm{ST}}$ involving an orbital excitation of the QD under gate G2. (c) Transport current $|I_{SD}|$ as a function of detuning $\epsilon$ and external magnetic field $B_{\mathrm{ext}}$, with the detuning axis highlighted by a white arrow in (b). The blockade region with an energy gap $E_{\mathrm{ST}}$ between the two dashed lines is also denoted. The leakage current due to spin-valley mixing is labeled by line V. Note line V has a slope approximately $5.96\mathrm{ueV/meV}$ of valley splitting with respect to $\epsilon$ (Appendix pp2), which may be caused by the strong dependence of valley splitting on the electric field under gate G1 or G2.

Figure 2

(a) Transport current $|I_{SD}|$ as a function of the external magnetic field $B_{\mathrm{ext}}$ and microwave frequency f. Red dashed lines denote the resonant lines where PSB is lifted by the driven spin-flip transition. Data with high leakage current background are cleared for clarity (blank regions) (Appendix pp3). The bottom diagram shows the calculated energy levels for spin-valley mixing. The spin and valley composition of the hybridized states $|\tilde{2}⟩$ and $|\tilde{3}⟩$ is indicated by the varied color of the corresponding lines near the anticrossing. Two double-headed arrows mark the corresponding spin-valley transitions A and B. Panels (b) and (c) show the transport current $|I_{SD}|$ as a function of the magnetic field strength $B_{\mathrm{ext}}$ and the magnetic field orientation $\varphi$ for the resonance A and B, respectively. Notice the anisotropy magnitude of line A (112 mT) is a little smaller than line B (134 mT), which may be attributed to the incomplete magnetization of the micromagnet under lower applied fields.

Figure 3

(a) The measured peak position of resonance B (blue data points) and different stray field components as a function of the magnetic field direction $\varphi$. The experimental data are fitted using a cosinusoidal function (blue curve). Inset: example of the measured current $|I_{SD}|$ (violet circle) and the fitted Gaussian function (red curve) as a function of the scanning magnetic field strength $B_{\mathrm{ext}}$, with the field direction at $\varphi ={325}^{\circ }$. The nonzero background current of $|I_{SD}|$ in the inset is most likely caused by high microwave power. (b) Plot of both the experimental (blue data points) and simulated (considering different transverse magnetic field gradients) resonance amplitude $I_p$ of resonance B as a function of the magnetic field direction $\varphi$. The scaling factor of $C=1.9$ is used in Eq. (1) for the calculation of all the simulated curves.

Figure 4

(a) Illustration of different magnetic field gradients and their effects on the oscillating electron spin. The transverse magnetic field gradients $d{\mathbf{B}}_{\mathrm{tr}}^{\mathrm{in}}/dy$ and $d{\mathbf{B}}_{\mathrm{tr}}^{\mathrm{out}}/dy$ enable spin flips when the electron is driven by the oscillating microwave fields. The longitudinal field gradients $d{\mathbf{B}}_{\mathrm{long}}/dy$ and $d{\mathbf{B}}_{\mathrm{long}}/dx$ lead to spin dephasing and $d{\mathbf{B}}_{\mathrm{long}}/dx$ also introduces spin addressability in our device. (b) Numerically simulated magnetic field gradients and the calculated quality factor Q as a function of the external magnetic field direction $\varphi$.

Figure 5

(a) Transport current $|I_{SD}|$ as a function of $V_{\mathrm{G}1}$ and $V_{\mathrm{G}2}$ with a bias voltage $V_{SD}=-4\mathrm{mV}$ and an external magnetic field $B_{\mathrm{ext}}=1$ T along the y axis. (b) Stability diagram of the measured DQD with charge sensing.

Figure 6

Illustration of the extraction of the lever arm based on Fig. 1 in the main text.

Figure 7

(a), (b) Full data of Fig. 2, with an additional resonance line I showing the intravalley spin-flip transition. (c) The transport current $|I_{SD}|$ as a function of the magnetic field strength $B_{\mathrm{ext}}$ and the magnetic field orientation $\varphi$ for the intravelly spin resonance I. Notice the anisotropy magnitude of line I (91 mT) is much smaller than line A (112 mT) and line B (134 mT), which should be attributed to the incomplete magnetization of the micromagnet under lower applied fields. (d) Background leakage current (peak baseline) and measured peak width (FWHM) as a function of the in-plane angle $\varphi$.

Figure 8

Comparison of the exact solution [Eq. (D12)] and the approximate solution [Eq. (D13)] of $F_{\mathrm{SV}}$ as a function of the magnetic field strength $B_{\mathrm{ext}}$.

Figure 9

(a),(b) The top view and the side view, respectively, of the micromagnet with the estimated electron position (blue circle). The Cartesian axis is the same as that in the main text and the size parameters used for simulation are denoted in the table inside the figure.

Figure 10

Diagram of different micromagnet designs (a)–(c) and the corresponding magnetic field gradients (mT/nm) $d{\mathbf{B}}_{\mathrm{tr}}^{\mathrm{tot}}/dy$ (d)–(f), $d{\mathbf{B}}_{\mathrm{long}}/dy$ (g)–(i), $d{\mathbf{B}}_{\mathrm{long}}/dx$ (j)–(l) as a function of the in-plane magnetic field direction $\varphi$ (degree). The yellow star shows the position of the electron in the $x-y$ plane.