Micromagnet-free operation of electron spin qubits in Si/Si$_{1-x}$Ge$_x$ vertical double quantum dots

Abstract

We study a vertical double quantum dot (DQD) in a Si/Si$_{1-x}$Ge$_x$/Si double-well heterostructure for full electrical control of electron Loss-DiVincenzo (LD) spin qubits, using realistic device modeling and numerical simulations. Due to the emerging spin-orbit interaction in the DQD, as well as strain from the gate electrodes, small (percentage range) but finite $g$ tensor variations emerge. In addition, we find a large valley splitting, on the order of $E_v{\sim}250\,μ$eV. As a result, multiple avenues for fast electrical single qubit rotations emerge. An ac electric field gives rise to electric dipole spin resonance (EDSR), while electron spin resonance (ESR) in the presence of an ac magnetic field can be electrically controlled by local gates due to varying $g$ factors in DQDs. We also show that shuttling between neighboring dots, in vertical and horizontal direction, results in ultrafast single qubit gates of less than a nanosecond. Remarkably, this DQD architecture completely eliminates the need for micromagnets, significantly facilitating the scalability of LD spin qubits in semiconductor foundries.

Figures (4)

• Figure 1: Realistic device simulation outline. Geometry of the simulated heterostructure with 300 nm regrowth Si$_{0.7}$Ge$_{0.3}$ layer, followed by the Si/Si$_{1-x}$Ge$_x$/Si double well structure, then 30 nm Si$_{0.7}$Ge$_{0.3}$ buffer layer. The tri-layer Al gates (from bottom: layer 1$\rightarrow$initial SL${=}$screening layer, layer 2$\rightarrow$ P${=}$plunger, S${=}$source, D${=}$drain gates, and layer 3$\rightarrow$ B${=}$Barrier gates) are 40 nm thick, while the Al$_2$O$_3$ oxide layers are 7 nm thick.
• Figure 2: The vertical DQD schematic. a) Sketch of the vertical Si/Si$_{1-x}$Ge$_x$/Si DQD. The ground state $z$-wavefunction probability density $|\Psi_{0}(z)|^2$ at plunger field $F_z{=}-2$ MV/m, which resides mostly in the top Si well, with finite extent into the bottom well. Here, $L{=}5$ nm, $a{=}2.5$ nm, and $V_b{=}15$ meV ($x{=}0.033$). The tunnel coupling in the $z$-double well is $t_c{=}1$ meV. The in-plane confinement frequency is $\omega_{ip}{=}\hbar/\!\left(m_tL_{ip}^2\right)$, where $L_{ip}{=}20$ nm. b) The lowest spin, valley, and orbital states of the DQD without shear strain. Here, $\Delta{=}12\,\mu$eV, $E_v{=}250\,\mu$eV, and $e_0{=}1\,$meV denote the Zeeman, valley, and orbital energy splittings, resp. c) Valley splitting $E_v$ as a function of $F_z$ with (purple line) and without (black line) shear strain.
• Figure 3: Electrical control of Si/Si$_{0.967}$Ge$_{0.033}$/Si DQD spin qubit. a) The qubit $g$ factor as a function of the $P_1$ plunger field $F_z$. The variation around $g_0{=}2$ is a result of strong SOI in presence of the strain inhomogeneity. b) EDSR Rabi frequency $f_\pi$ vs. $F_z$ in presence of an ac electric field along $\mathbf{\hat{z}}$. In a) and b), specific scenarios without the shear strain (black solid line) and with shear strain (purple line) are shown. The magnetic field is $\mathbf{B}{=}(100,0,0)$ mT. c) 2D map of $f_\pi$ with the ac electric field in the $x{-}y$ plane and $F_z{=}-1$ MV/m. Here, $f_\pi$ is plotted as a function of the in-plane magnetic field angle $\theta$ ($\mathbf{B}{=}100(\cos\theta,\sin\theta,0)$ mT) and in-plane ac electric field angle $\phi$ ($\mathbf{E}_{\text{ac}}{=}E_{\text{ac}}^0 \cos(\omega t)(\cos\phi,\sin\phi,0)$ with $E_{\text{ac}}^0=10^4$ V/m).
• Figure 4: $g$-factor anisotropy in DQDs in the presence of gate-induced strain. The quantity plotted is $|g(\theta)-2|$, where $g(\theta){=}\frac{\Delta(\theta)}{\mu_BB}$, and the Zeeman splitting $\Delta$ varies with the magnetic field angle, $\mathbf{B}{=}B(\cos\theta,\sin\theta,0)$. a) The polar plot of $|g(\theta)-2|$ for a vertical DQD under plunger gate $P_1$ with $F_z{=}2$ MV/m and $F_z{=}-2$ MV/m. b) Polar plot of $|g(\theta)-2|$ for left (under $P_1$) and right (under $P_2$) dots with rotated plunger gates at $F_z{=}-2$ MV/m. The plunger gates $P_1$ and $P_2$ are rotated by $30^\circ$ with respect to each other.