Controlling Spin-Orbit Interactions in Silicon Quantum Dots Using Magnetic Field Direction

Silicon quantum dots are considered an excellent platform for spin qubits, partly due to their weak spin-orbit interaction. However, the sharp interfaces in the heterostructures induce a small but significant spin-orbit interaction that degrades the performance of the qubits or, when understood and controlled, could be used as a powerful resource. To understand how to control this interaction, we build a detailed profile of the spin-orbit interaction of a silicon metal-oxide-semiconductor double quantum-dot system. We probe the derivative of the Stark shift, $g$-factor and $g$-factor difference for two single-electron quantum-dot qubits as a function of external magnetic field and find that they are dominated by spin-orbit interactions originating from the vector potential, consistent with recent theoretical predictions. Conversely, by populating the double dot with two electrons, we probe the mixing of singlet and spin-polarized triplet states during electron tunneling, which we conclude is dominated by momentum-term spin-orbit interactions that vary from 1.85 MHz up to 27.5 MHz depending on the magnetic field orientation. Finally, we exploit the tunability of the derivative of the Stark shift of one of the dots to reduce its sensitivity to electric noise and observe an 80% increase in $T_2^{*}$. We conclude that the tuning of the spin-orbit interaction will be crucial for scalable quantum computing in silicon and that the optimal setting will depend on the exact mode of qubit operations used.

Popular Summary

Building a working universal quantum computer is one of the big challenges of modern science. Silicon technology that is used for everyday computers can also be used to make a quantum bit based on a subatomic magnetic dipole, also known as the spin of an electron. One of the challenges for these kinds of qubits is the variability of their properties, which arises from atomic-level differences between the qubits that change their spin-orbit interaction. Here, we show how to control this interaction to make qubits that are more uniform and with improved performance.

The spin-orbit interaction depends on the external magnetic field direction with respect to the silicon lattice of the device. Using a silicon metal-oxide-semiconductor double quantum-dot structure, we study the spin-orbit interaction by measuring the $g$ factor, coherence time, and singlet-triplet mixing as functions of the magnetic field direction. We show how to control this interaction and how to change the spin properties of the quantum dots.

Our results confirm many recent theories related to spin-orbit interaction in silicon. They also present a pathway for how to design silicon quantum dots for large quantum computers in order to improve their performance and make them more similar to each other, significantly easing the requirements for the manufacturing of silicon qubits.

Figure 1

Sample $g$-factor measurement. (a) Schematic cross section at the position of the dots (QD1 and QD2) along the dotted line in panel (b). (b) False color SEM image of our sample. The MW antenna used to drive the qubits is on the top, the qubit dots are in the middle, and the SET used to sense the dots is at the bottom. (c) $g$-factor of the electron occupying the G1 dot as a function of external magnetic field angle in plane and at 45° out of plane of the sample in the (1,0) charge configuration. The solid lines correspond to the estimate extracted from the complete $g$-tensor. Inset: Example of ESR frequency as a function of magnetic field with a linear fit that is used to extract the $g$-factor. (d) The derivative of the Stark shift as a function of in-plane magnetic field angle. (e) Out-of-plane $g$-factors measured (colored symbols) with estimates from complete a $g$-tensor (solid lines). (f) Notation of the magnetic field angles with respect to the sample and isosurface of the $g$-factor of the G1 dot based on a single $g$-tensor. We have subtracted 1.9 in order to visualize the anisotropy of the $g$-factor. Blue arrows correspond to laboratory coordinates, and red arrows correspond to the principal axes of the $g$-factor ellipsoid. The data in panels (c) and (e) are taken along the lines shown on the surface.

Figure 2

The $g$-factor difference between the dots. (a) Difference of $g$-factors between the dots under G1 and G2 with occupation (1,0) and (0,1) as a function of in-plane magnetic field angle, together with sinusoidal fit. (b) Difference of $g$-factors between the dots under G1 and G2 as a function of out-of-plane magnetic field angle along [110] (blue) and [$1\overline{1}0$] (magenta). (c) Isosurface representation of the absolute value of $g$-factor difference as a function of magnetic field orientation. The magenta arrows represent crystal lattice orientation. The red, blue, and magenta curves correspond to the fits in panels (a) and (b). (d) Pulse sequences (black and blue arrows) used to measure the $g$-factor difference in panels (a) and (b). C1 and C2 indicate the two different control points in the two different sequences.

Figure 3

The derivative of the Stark shift and $T_2^{*}$ measurement. (a) Stark shift and (b) decoherence time $T_2^{*}$ of the qubit defined by the G1 quantum dot in the (3,0) charge configuration as a function of the external magnetic field in-plane angle ${\phi }_B$.

Figure 4

Singlet-triplet ${\mathrm{T}}^{-}$ mixing as a function of magnetic field direction. (a) Square of the coupling strength of singlet and triplet ${\mathrm{T}}^{-}$ states as a function of external magnetic field in-plane angle. Inset: Pulse sequence used to measure the coupling. (b) Square of the coupling strength of singlet and triplet ${\mathrm{T}}^{-}$ states as a function of external magnetic field out-of-plane angle while at ${\phi }_B=51°$. We show in (a) and (b) the fit based on the SOI spin-flip model and the estimate from the differences in the off-diagonal terms in the $g$-tensors. Inset: Example fit of the triplet probability as a function of ramp rate across the anticrossing.